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Revealing the effects of subsurface structure on the antenna coupling of ground penetrating radar Luzitano, Robert D. 1995

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REVEALING THE EFFECTS OF SUBSURFACE STRUCTURE ON THE ANTENNA COUPLING OF GROUND PENETRATING RADAR by Robert Daniel Luzitano B.Sc, University of Utah, 1986 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Geophysics and Astronomy) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June 1995 © Robert Daniel Luzitano, 1995 In presenting t h i s jthesis in p a r t i a l f u l f i l l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia,--1 agree that j the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head! of my department or by his or her representatives. I t i s understood that copying or publication of th i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of ( ^ ^ p l i y S f c S $ As/ziynom^ The University of B r i t i s h Columbia Vancouver, Canada j Date 0 uyv-e ^7 l^^f Abstract The character (amplitude, phase, frequency, and polarization) of a reflection in a ground penetrating radar (GPR) profile contains a wealth of information about the reflector. However, most of the current research relating reflection strength to material properties, such as soil moisture and the existence of contaminants, involves only single—parallel—component data. This scalar view of a vector phenomenon leaves most of the information contained in the reflected wavelet untapped. Moreover, an amplitude anomaly in the parallel component may be largely, or at least in part, due to a polarization anomaly. This polarization contribution is ignored by the common scalar approach which attributes the anomaly entirely to such properties as water saturation or a suspected contaminant plume. Anomalous polarization degrades the receiver antenna coupling due to a lower polarization match between antenna and wavelet. Usually more obvious, are the coupling changes due to material/structural variations within the near-field which affect antenna properties. Before character analysis can be applied reliably, the effects of variable antenna coupling must be considered. A theoretical review of the effect of ground conditions on antenna radiation patterns, and also of wavelet depolarization, provides the GPR interpreter with the insight to recognize coupling effects in the data. When a survey traverses into material having significantly higher dielectric constant, the antenna centre frequency decreases, and the radiation pattern directivity increases due to a narrower beamwidth and possibly smaller side lobes. The possible effect in the data is lower energy from out of the plane scatterers and a decrease in the maximum dip that can be imaged. An increase in conductivity will decrease the radiated power and the accompanying dispersion will smear the radiation pattern nodes, resulting in a more omnidirectional radiation pattern for pulse antennas. Wavelet depolarization occurs, to some degree, for most cases of reflection and refraction. The severity of depolarization depends on the contrast in electrical properties, incident angle, incident polarization, and ii Ill the orientation of the reflecting area. Generally, wavelet depolarization increases with an increase in reflector asymmetry, such as in scattering geometry, continuity, roughness, and anisotropy. To estimate the power loss in the parallel component due to anomalous polarization, an instantaneous polarization match estimate was developed and applied to field data from two test sites of different structural complexity. In this initial investigation, the TM survey mode was confirmed to suffer a greater degree of depolarization resulting in degraded coupling compared to the TE survey mode. Generally, degraded coupling was also observed at reflector rough spots (a 5 - 20% power loss) and at points of wavefront interference. Although the polarization coupling is probably a second order effect for most cases, at least one situation was documented in 1974 where depolarization was a first order effect causing the parallel component to be extinguished. Additional target types and environments should be investigated for their depolarizing characteristics. iv Table of Contents Abstract ii List of Tables viii List of Figures ix Acknowledgments xiii Chapter 1 INTRODUCTION 1 1.1 The GPR Method 1 1.2 The Motivating Problem 4 1.3 Thesis Overview 6 Chapter 2 A N T E N N A C O U P L I N G A N D W A V E L E T D E P O L A R I Z A T I O N 9 2.1 Review of Pertinent Antenna Fundamentals 9 2.1.1 Electric Dipole Radiation 9 2.1.2 Directive Properties 13 Radiation Patterns 13 Antenna Directivity 14 Antenna Gain and Radiation Efficiency 16 2.2 Radiation of the Pulse GPR Antenna Over Half-space Earths 17 2.2.1 Effects of the Ground on Steady-State Radiation Patterns 21 Effect Due to Dielectric Contrast 21 Effect Due to Antenna Height . 23 Directivity and Gain 23 Effect Due to Conductivity 24 Near-field vs. Far-field; Finite vs. Infinitesimal Dipoles 25 2.2.2 Steady-State Radiation Patterns and Pulse Radar 26 Time Dependence of Radiation Patterns 26 Effect of Antenna Height on Frequency 27 Effect of Velocity Dispersion on Radiation Patterns 28 2.3 Antenna Polarization 29 2.3.1 Definition and Description of Polarization 30 Phase, Sense, Tilt, and Polarization Ratio 31 Degree of Polarization 33 The Complex Polarization Factor 34 V 2.3.2 Review of Fresnel Coefficients and the Polarizing Angle 35 2.3.3 Polarization Patterns and Reciprocity 37 2.3.4 Polarization in GPR Surveys 40 2.4 Antenna Reception 41 2.4.1 The Radar Range Equation 41 2.4.2 Impedance and Polarization Match Factors 44 Impedance Match (mq) 45 Polarization Match (mp) 46 2.5 Wavelet Depolarization 48 2.5.1 Depolarization Due to an Arbitrarily Dipping Reflector 49 Changing Polarization from Vertical Linear to Horizontal Linear .... 51 Reflection from a Smooth Arbitrarily Dipping Infinite Plane 52 2.5.2 The Effect of Scattering from a Finite Plane 58 2.5.3 Other Cases of Depolarization and Characteristic Measurements with Applications 59 2.6 Summary: Amplitude Variation and Antenna Coupling 60 Chapter 3 FIELD TEST SITES AND DATA ACQUISITION 62 3.1 The Abandoned Overpass Ramp 62 3.1.1 Site Description and Relevance 62 3.1.2 Coverage of Single and Two-Component Surveys 65 3.2 Geophysical Test Pit 67 3.2.1 Site Description and Relevance 67 3.2.2 Survey Coverage 67 3.3 Alice Lake 69 3.3.1 Site Description and Objectives 69 3.3.2 Profile and Polarization Experiment 69 3.4 Advice For Acquiring Two-Component Data 71 Chapter 4 "STANDARD" PROCESSING AND SIGNAL CHARACTER 73 4.1 Seismic Reflection Processing Applied to GPR Data 74 4.2 GPR Specific Processing Problems 77 4.2.1 Time-Zero Correction 77 4.2.2 Dewow Filtering and Wow Variation with Near-Surface Structure 79 Origin and Character of Wow 79 Dewow Filtering 81 4.2.3 Gaining Criteria 85 Using AGC and SEC gains 85 An Offset Dependent SEC Gain: EXPGAN.F 87 vi 4.3 A Recommended Standard GPR Processing Stream 89 4.3.1 Processing the Ramp 100 MHz Profile 91 4.3.2 Processing the Test Pit 200 MHz Profile 95 4.4 Processing to Interpret Signal Variation: Instantaneous Attributes 98 Chapter 5 INSTANTANEOUS POLARIZATION M A T C H 102 5.1 Introduction 102 5.2 Generalized Polarization Match 103 5.3 Required Processing Stream 106 5.4 Calibration of the Instantaneous Polarization Match 108 5.4.1 Outline of Assumptions and Procedure 110 5.4.2 Plotting 7R, <5R, and m p as Shaded Traces (Trace Scaling) 113 5.4.3 Estimating Constant Values for 7R and <5R . 113 5.4.4 Features in the m p Calibration Traces 117 5.5 Uncertainties in m p Traces 118 5.5.1 Effect of Timing Errors Between the Two Components 118 5.5.2 Uncertainty Due to Time Varying System Performance 121 Time Test Data 121 Statistics of Time Test Average Power 121 Statistics of m p Calibration and Uncertainty Propagation 124 5.5.3 Unknown Uncertainty For Theoretical Limitations 127 5.6 Color Coding Scheme for Polarization Match Images 129 5.7 Polarization Match Images 131 5.7.1 The Abandoned Overpass Ramp Experiments 132 TE Profile and Polarization Match 132 TM Profile and Polarization Match 136 TE and TM Common Midpoint Soundings 141 Summary of Observations on the Ramp 149 5.7.2 The Test Pit Experiments 150 TE Profile 150 TE Common-Midpoint Soundings 154 CMP Sounding 3: 154 CMP Sounding 6: . . 156 CMP Sounding 10: 158 Summary of the Test Pit Observations 158 5.8 Summary of The Occurrence of Anomalous m p 160 vii Chapter 6 S U M M A R Y A N D CONCLUSIONS 161 C I T E D R E F E R E N C E S 164 Appendix A T H E O R E T I C A L CASES O F D E P O L A R I Z A T I O N 170 A.1 Single Scatterers 170 A . l . l Arbitrary Scattering Direction from a Perfect Conductor 170 A.1.2 Backscatter from an Imperfect Conductor 171 A.1.3 Bodies Having Symmetry 172 A.1.4 Backscatter from Perfect Conductors Having Symmetry 172 A.1.5 Scattering from a Sphere 173 A.1.6 Other Simple Shapes (Circular Cylinders) 175 A.2 Depolarization by Edges of Perfect Conductors 176 A.3 Reflector Roughness 178 A.3.1 Statistical Descriptions of Roughness 178 A.3.2 Perfect Conductors 178 A.3.3 Random Roughness of Finite Conductivity 178 A.3.4 Backscatter from Randomly Rough Surfaces of Finite Conductivity .... 179 A.3.4.1 RMS Slope and Illumination Angle 180 A.3.4.2 Wavelength Dependence 181 A.3.4.3 Dependence on Electrical Properties and Incidence Polarization .... 182 A.4 Anisotropy 182 A.4.1 Definition, Classification, and Occurrence 182 A.4.2 Anisotropy of Reflector Roughness 183 A.4.3 Anisotropy Due to Bedding (Transverse Isotropy) 184 A.4.4 Intrinsic Anisotropy 185 List of Tables Table 4.1 Station spacing required to avoid spatial aliasing of a 45° dip 76 Table 4.2 Material properties for theoretically-based gaining of GPR data 87 Table 5.1 Notable whole trace variations of the time test (Figure 5.11) 124 Table 5.2 Statistics of power variation within selected time windows of Figure 5.11 . 124 Table 5.3 Statistics of match variation within selected time windows of the lake soundings 128 ix List of Figures Figure 1.1 Sketch of a GPR constant offset profiling survey and resulting data. 3 Figure 2.1 A.Field components radiated from an electric dipole. B. Geometry of the near-field problem 10 Figure 2.2 Radiation patterns of 1, 1/2, and 3/2 wavelength dipoles 14 Figure 2.3 Two common representations of the radiation pattern function 15 Figure 2.4 Wavefronts from an electric dipole on the interface between air and a half-space earth 18 Figure 2.5 A CMP sounding illustrating the direct air and ground arrivals, reflected waves, and the critically refracted air wave 20 Figure 2.6 Radiation patterns of a horizontal electric dipole at 3 heights over half-space earths of water and dry sand (after Smith, 1984) 22 Figure 2.7 Directivity and gain as a function of antenna height and dielectric contrast 24 Figure 2.8 Pulse peak to peak radiation patterns 27 Figure 2.9 Propagation of the polarization ellipse 31 Figure 2.10 Polarization chart 32 Figure 2.11 Polarization geometries of TE and TM reflection and transmission 36 Figure 2.12 Decomposition of E$ into its Cartesian components 37 Figure 2.13 Sketch of Cartesian components of the E radiation pattern 39 Figure 2.14 Polarization modes of common GPR survey antenna orientations 40 Figure 2.15 Bistatic radar and target geometry of the radar range problem 42 Figure 2.16 Reflection and refraction of a wave having arbitrary polarization 49 Figure 2.17 Reference and interface coordinate systems as viewed in the wavefront... 50 Figure 2.18 Depolarization of a vertically polarized wave by a perfectly conducting plane with a pure cross-dip 51 X Figure 2.19 Generalized geometry for the general depolarization problem of reflection from a smooth infinite plane 52 Figure 2.20 Geometry of the incidence and wave front planes 53 Figure 2.21 Detail of vectors lying in the incident wave front 54 Figure 3.1 Plan and section view of the abandoned overpass ramp 63 Figure 3.2 Example of CPT —resistivity data at the ramp 64 Figure 3.3 Single-component GPR coverage of the ramp 65 Figure 3.4 Two-component GPR coverage of the ramp 66 Figure 3.5 Photo and cross-section of the test pit 68 Figure 3.6 CMP coverage of the test pit 69 Figure 3.7 Alice Lake location map 70 Figure 3.8 Effect of Jane Rea on a cross component CMP sounding 72 Figure 3.9 The effect of Jane Rea's position between crossed antennas 72 Figure 4.1 Example of raw GPR data—from the test pit 78 Figure 4.2 Definition of wow and its variation with offset 80 Figure 4.3 Wow profile from the ramp 81 Figure 4.4 Wow profile of the test pit 82 Figure 4.5 Comparison of raw and dewowed data from mean- and median-based filters 84 Figure 4.6 The effect of gaining a CMP with EXPGAN.F and the gain function 89 Figure 4.7 A standard processing stream for GPR data 90 Figure 4.8 Raw GPR profile of the ramp 92 Figure 4.9 Dewowed and time drift corrected profile of the ramp 93 xi Figure 4.10 Final processed profile of the ramp 94 Figure 4.11 Dewowed and time drift corrected profile of the test pit 95 Figure 4.12 Test pit profile after EXPGAN.F 96 Figure 4.13 Phase migrated profile of the test pit 97 Figure 5.1 Sketch of the wavelet depolarization and antenna matching problem .... 104 Figure 5.2 Required processing for instantaneous polarization match 107 Figure 5.3 Single quadrant polarization rose from Alice Lake, BC 109 Figure 5.4 Gamma and phase difference traces for the instantaneous polarization match calculation I l l Figure 5.5 Two-component lake data and resulting calibrated polarization match with and without median filtering 112 Figure 5.6 Instantaneous phase of the polarization rose lake bottom reflection 114 Figure 5.7 Testing the sensitivity of polarization match to the phase term 115 Figure 5.8 Comparison of polarization match traces with phase difference set to 0° and 90° 116 Figure 5.9 Sensitivity of match values to timing error between components 120 Figure 5.10 Time test: multiple soundings to test the repeatability of the pulseEKKO™ IV 122 Figure 5.11 Power variation of selected time windows on envelope traces of the time test 123 Figure 5.12 Time window power variation analysis of the five two-component lake data 126 Figure 5.13 Mean amplitude variation of the median filtered match traces from Alice Lake 127 Figure 5.14 The amplitude and polarization match color coding scheme applied to the lake soundings 130 Figure 5.15 Ramp: two-component TE instantaneous amplitude profile 133 xii Figure 5.16 Ramp: TE instantaneous amplitude and polarization match 135 Figure 5.17 Ramp: single-component TM amplitude profile 137 Figure 5.18 Ramp: two-component TM amplitude profile 138 Figure 5.19 Ramp: TM amplitude and polarization match profile 140 Figure 5.20 Ramp: two-component TE amplitude CMP sounding 142 Figure 5.21 Ramp: two-component TM amplitude CMP sounding 144 Figure 5.22 Ramp: dramatic amplitude null on a single-component TM amplitude CMP sounding 145 Figure 5.23 Ramp: polarization match of the TE and TM CMP soundings 148 Figure 5.24 Test pit: two-component TE amplitude profile 151 Figure 5.25 Test pit: amplitude and polarization match profile 153 Figure 5.26 Test pit: two-component amplitude and polarization match of a CMP sounding at the 3.0 m station 155 Figure 5.27 Test pit: two-component amplitude and polarization match of a CMP sounding at the 6.0 m station 157 Figure 5.28 Test pit: two-component amplitude and polarization match of a CMP sounding at the 10.0 m station 159 Acknowledgments In memory of my great-grandfather Herman Pontes, and to my grandparents the Bunches and the Stewarts First I would like to thank my advisor, and friend, Professor Tad Ulrych, for encouraging me to undertake this GPR odyssey. It's been a privilege to have an advisor with such a contagious enthusiasm for science. I've benefitted from his frequent praise and confidence in my work—although I think he exaggerates sometimes (perhaps just a bit). I owe enormous thanks to Guy Cross (now Dr. Cross) for always lending an ear and being a valuable resource through a variety of interesting theoretical discussions and practical advice. Professor Rosemary Knight made this work possible by obtaining funds to purchase and maintain the pulseEKKO™ IV radar. Her enthusiasm and encouragement maintains an exciting and collaborative environment within—and beyond—"The GPR Group." I am grateful to Don Gillespie and Rex Crider of the B.C. Ministry of Transportation and Highways for loaning us their 200 MHz antennas and introducing us to the abandoned overpass ramp—an excellent test site. I am also very grateful to fellow grad. student A l Rempel for the idea of using Alice Lake as a test site. He asked for my requests of polarization experiments and acquired them without my help during record cold weather in the middle of January—what a guy. I would like to thank Professor Doug Oldenburg for being the catalyst to my GPR involvement via the Expo Geophysical Survey and also for informing me of consulting opportunities that crossed his desk. I also appreciate Professor Garry Clarke for being available to explain some of the glaciology group ice radar experiments and for letting me borrow some of his interesting references on ice radar polarization and amplitude analysis. Partial funding was provided by Natural Sciences and Engineering Research Council of Canada (NSERC) Research and Infrastructural grants to Professor Tad Ulrych. Additional fi-nancial support was received through funding to Professor Rosemary Knight from Environment Canada's Environmental Innovation Program, funded by the EIP, Canadian Space Agency, and Energy Mines and Resources. I have benefitted from the friendship and interaction with many fellow students in the department. The many interesting and fruitful theoretical discussions with Mike Perz, Partha Routh, Zhiyi Zhang, Colin Farquharson, Dave Dalton, Mike Knoll, and Jane Rea. The field work was not practical for one person and those who I coerced into volunteering certainly deserve mention: (The Alice Lake Crew: Kevin Gerlitz, Dave Goertz, and Jane Rea), John Rennie, Karl Butler, Colin Farquharson, Brad Isbell and Skagway, Partha Routh, Xingong L i , xiii xiv Ken Matson, Taimi Mulder, and Mike Perz. Also deserving of mention are those who maintained an interesting level of insanity in the terminal room, making the many long nights and sunny weekends more bearable: Dr. Barry C. Zelt, Mike (Mesh-head) Perz, Costas (DoK) Karavas, and Weimin Zhang. Special thanks to Mike Perz for his hours of guidance through the "Insight'TITA processing confusion, and to John Amor and Dave Dalton for their time and patience in dealing with my computer crises—"G.D.F.!" Thanks also to Roger Roberts (Ohio State University) and Greg Turner (Macquarie Uni-versity, Australia) for promptly providing me with a copy of their excellent Ph.D. theses; their modelling results confirmed my understanding of antenna coupling and provided additional insight. No GPR library is complete without these two references. I am indebted to Jane Rea for her prompt review of a draft of this manuscript; this is one debt I look forward to paying. I am also grateful to Professors Rosemary Knight and Matt Yedlin for finding time in their schedules—at an especially busy time—to critically review my thesis. I also owe a big thanks to my sister and brother-in-law, Becky and Rich Whitacre, in Federal Way, Washington, who contributed to the continuation of this work, indirectly, by allowing me to invade their home for weeks at a time for matters financial and occupational. I am also grateful to my dearest friend Becki Woolf for being there for me to vent some of my thesis frustrations, and for the many laughs over the years. Finally, thanks to Mom & Dad for their love, and support through crises financial and otherwise. Chapter 1 INTRODUCTION 1.1 The GPR Method Ground penetrating radar (GPR) is a relatively new geophysical method that has evolved over the last few decades for high resolution imaging of the shallow subsurface. Initially GPR was applied to very low loss geologic materials such as dry salt in salt mining applications (Unterberger, 1978) and glacial ice in the 1950's (Fisher, 1991). In glacier sounding appli-cations the method is known as ice radar or radio echo sounding (RES). In both of these applications, signal penetration of over 2 km is possible. The early 1970's saw many theo-retical and feasibility studies on radar sounding in rock and soil (especially coal and frozen soil) with the late seventies and early eighties producing an increase in various applications (Ulriksen, 1982). Ulriksen (1982) investigated many civil engineering applications of GPR including stratigraphic profiling in soil and rock, estimating the water content of snow pack and locating avalanche victims, locating pavement and road bed problems such as voids and delamination, and mapping the distribution of soil moisture. Since the early eighties GPR has been increasingly successful in an exciting variety of applications, due to improved technology. Current state-of-the-art systems are real-time digital allowing for the stacking of soundings dur-ing acquisition thereby increasing the signal-to-noise and the depth of penetration. The ability to apply basic processing in the field and subsequent advanced processing, can greatly enhance the image and ultimately allow the inference of material properties from reflected amplitudes. Topics presented at the most recent International Conference on GPR (Redman et al., 1994a) included the application of GPR to pavement and nondestructive testing (structural concrete, and bridge decks), permafrost studies (interaction of permafrost with buildings and for construc-tion planning in permafrost terrain), archaeological investigations, borehole studies (mapping fractures and lithology, crosshole tomography, and polarimetric imaging), 3D imaging and pre-l Chapter 1: INTRODUCTION 2 sentation, automated detection and classification of buried targets (pipes, and mine tunnels), geologic and stratigraphic mapping (faulting, depositional environment analysis, hydrogeologic investigations), mining and tunneling (cavity and ore detection within mines and for mine planning), geotechnical problems (railway grade inspection, imaging drilling obstacles, seep-age monitoring, utility detection), and subsurface pollution (detecting thin hydrocarbon layers, waste disposal delineation and characterization, and contaminant mapping and monitoring). Sub-bottom profiling of fresh water lakes and rivers is also accomplished on occasion (Gorin and Haeni, 1988). Another application, recently featured in the news, involved the location of buried bodies during a homicide investigation. In many respects GPR is similar to seismic reflection imaging. A pulse of energy (radar) is beamed into the ground and is reflected or scattered back to the surface by contrasts in subsurface (electrical) properties, as Figure 1.1 illustrates. However, when considering the character of radar reflections the seismic analogy must be restricted to shear waves to include the appropriate polarization related phenomena. A primary limitation of GPR compared to seismic methods is the much shallower limit of investigation due to the higher attenuation rates of radar waves, especially in conductive soils such as wet clay (depths of penetration of about 35 m in dry sand and about 1 m, or less in clay). Large advantages to GPR over seismic methods are quick noninvasive data acquisition, usually minimal or no processing required (depending on the application) and the high resolution of less than a metre in most applications. GPR typically uses frequencies in the 10 - 1000 MHz range with a two octave bandwidth. At these frequencies electromagnetic waves are sensitive to dipolar molecules such as water which has a dielectric constant of 80. In contrast, dry soils and rock have dielectric constants of 3-15 (Davis and Annan, 1989) making GPR very sensitive to changes in volumetric water content and hence porosity. Due to the trade-off between depth of penetration and resolution, 100 MHz is most commonly used for stratigraphic studies, 200 - 900 MHz for Chapter 1: INTRODUCTION 3 AIR Figure 1.1 Sketch of constant offset GPR profiling with the resulting image plotted in wiggle trace format. The image includes reflections from the bedrock interface and an anomalous zone. The finite size of the anomalous zone produces a diffraction hyperbola in the image (from Annan and Cosway, 1991). utility detection, and around 1000 MHz or greater in pavement and bridge deck investigations. Shallow seismic methods and GPR are often complementary. Some soil conditions that inhibit one method are actually preferred for the other, e.g. dry coarse grained soils such as sand are best for propagating radar waves (around 100 MHz) but often strongly attenuate seismic energy. However when soil conditions allow, GPR and seismic imaging are often a good complement Chapter 1: INTRODUCTION 4 for geotechnical and environmental problems, particularly in providing mutual constraints for mapping and characterizing bedrock overburden. For a comprehensive introduction to GPR, two references in addition to Ulriksen (1982) are worth studying. Daniels (1989) introduces applications and theory, and a report by Sensors and Software Inc. (Annan and Cosway, 1991) discusses the considerations of survey design. 1.2 The Motivating Problem Contaminant delineation and some engineering applications regularly pose challenging imaging problems for GPR, often due to site characteristics. In particular, urban sites can be problematic due to both above and below ground clutter (utilities, landfill debris, etc.) which may obscure the target signal. Reflection processing may alleviate some or all imaging problems at these sites. Standard seismic reflection processing is direcdy applicable, involving only a time scale change and a few processing steps specific to GPR (Fisher et al., 1992; Annan, 1993; Gerlitz et al., 1993). However, most standard seismic reflection processing steps do not preserve amplitudes even in their seismic application. Therefore, processing steps must be chosen with greater care when pursuing amplitude analysis. Obviously, physically-based amplitude processing should treat GPR specifically. In some applications the target involves an amplitude anomaly. Reflection strength contains information about the target's physical and scattering properties which may be important in identifying and mapping a particular target. Signal strength may increase or decrease in cases of subsurface contamination depending on the contrast between the host medium and the contaminant (Davis and Annan, 1989; Greenhouse et al., 1993; Lawton et al., 1994; Redman et al., 1994b). Amplitude anomalies are also observed due to variable volumetric water saturation/porosity along thin layers (Duke, 1990; Goertz and Rempel, 1992), variations in thickness of thin layers and cracks (Turner, 1993), and changes in petrology (Hammond and Chapter 1: INTRODUCTION 5 Sprenke, 1991). Radar profiles of ice shelves and adjacent fjords often exhibit zones of weak basal (ice-sea interface) reflections attributed to brackish ice (Narod et al., 1988). In some cases anisotropy was a confirmed cause of such anomalies due to oriented crystallization under the influence of marine currents (Campbell and Orange, 1974; Kovacs and Morey, 1978). In this thesis I address the effects of variable antenna-to-ground coupling on amplitude variation and the identification of significant depolarization effects on amplitudes. In this context, the word depolarization is commonly used to mean the altering of an incident polarization upon reflection or transmission, not a complete depolarization. These two issues must be addressed at some level before material properties can be reliably inferred from reflection strength. Additionally, depolarization can also be a distinguishing characteristic of some targets (Roberts, 1994). When analyzing amplitudes, the interpreter must be aware of lateral changes that alter antenna coupling. These lateral changes can be classified as having near-field or refiected-field origins, the latter usually being more difficult to identify. Often in the near-field case, geologic changes that affect antenna coupling are also obvious visible changes in surface material. However, these material changes can be more subtle, involving a change in saturation degree or type as in a chemical spill (Daniels, 1989). Even more subtle are very shallow heterogeneities—within the reactive near-field of the antennas—which may include targets such as utilities or a buried pavement layer. Since these coupling changes originate within the near-field, most or all of the radar trace is affected. Usually such whole-trace changes are unlikely to be misidentified as a change in reflector properties since the entire trace effect is obvious, however, some reflections at depth may be affected anomalously. Reflections from out of the survey plane or from interfaces having large dip may exhibit dramatic amplitude changes due to the change in radiation pattern directivity. In an extreme case, such reflections could appear or disappear suddenly with a sudden decrease or increase in directivity, respectively. Chapter 1: INTRODUCTION 6 Therefore, an understanding of the ground's affect on antenna properties is very important when analyzing reflection amplitudes, as well as in locating the reflector. The least recognizable form of coupling change—from the refiected-field—originates from individual targets at depth, i.e. from a particular interface or discrete scatterer that alters the wavelet's polarization. Part of the recognition difficulty is that two-component data are required to characterize the polarization. Two-component data is very rarely acquired even in the amplitude studies referenced above. Generally depolarization increases with reflector asymmetry, in the form of geometry, continuity, and roughness, thereby degrading the wavelet's polarization match with the receiving antenna and resulting in a lower received power. Although extreme cases of depolarization, causing reflections to extinguish, due to anisotropy in sea ice was confirmed 20 years ago (Campbell and Orange, 1974; Kovacs and Morey, 1978), little attention has been given to the depolarization effects in the ground. To quantify the depolarization effect on radar reflection strength, I developed an instantaneous at-tribute that estimates polarization match thereby identifying the depolarizing target. To this end I wrote some processing software that alleviates noise and display problems particular to GPR while preserving amplitude information. Recognizing cases where the polarization and coupling may be significantly affected is also of concern; therefore the related theory is discussed. 1.3 Thesis Overview The thesis consists primarily of two parts: a theory-based discussion of antenna coupling and the causes of depolarization which degrades coupling, and an applied section that provides a quantitative assessment of depolarization effects on amplitudes in GPR data. I present the depolarization estimate within the image by calculating the polarization match factor as an instantaneous attribute. After development and testing, I applied this attribute to two data sets from well characterized test sites, each site having a different level of structural complexity. Chapter 1: INTRODUCTION 7 En route I treat some GPR specific processing problems by means that preserve amplitude information. The theory required to understand the fundamentals of antenna coupling is presented in Chapter 2 which discusses antenna radiation and matching, and depolarization as they relate to GPR. This treatment includes reviews of pertinent antenna fundamentals and polarization. The variation of antenna radiation with earth type and antenna position is reviewed from GPR literature. However, most literature on GPR antenna radiation patterns refer to steady-state (continuously radiating) radiation patterns without acknowledging the departures of the transient (GPR) case. By considering the relation between steady-state and pulse radiation, I explain many of the discrepancies noted in the GPR literature between steady-state theoretical patterns, and GPR (pulse) pattern measurements and modelling. Since antenna coupling can be adversely affected by wavelet depolarization, a simple, yet generalized case of depolarization is reviewed by derivation. The depolarization section (Section 2.5) testifies by theory that depolarization upon reflection and transmission is the norm rather than the exception. To anticipate the depolarization potential of field sites, the Appendix provides a quick reference to increasingly complex cases of depolarization. Given that depolarization usually occurs, its significance to amplitude variation is addressed in the instantaneous polarization match part of the thesis. Two test sites are utilized to investigate the polarization matching problem: an abandoned overpass ramp and a test pit containing two concrete blocks. These sites are excellent for testing many processing issues such as resolution, deconvolution, forward modelling, scattering, depolarization, and migration. Much more data were collected than was finally used in this study and others may find these data sets useful. Chapter 3 introduces these sites and explains their relevant qualities. An additional experiment site on a frozen lake is also described, where a polarization rose was measured to quantify the antenna's polarization. Chapter 3 concludes with Chapter I: INTRODUCTION 8 advice on performing two-component GPR surveys. Standard reflection processing for GPR and some amplitude preserving constraints are addressed in Chapter 4. Processing problems particular to GPR that are treated include time zero drift correction, dewowing, and the merits and application of various gains. Wow, a time varying trace bias, is shown to be influenced by heterogeneities within the near-field of the antennas. The test sites are further characterized by presenting processing results in Section 4.3. Although an extra processing step, Section 4.3 also illustrates the success of migration for a test site profile which provides a simple and obvious case for migration. Finally, amplitude analysis and reflection characterization are introduced as applications of the standard instantaneous attributes amplitude (or envelope), phase, and frequency. Chapter 5 introduces instantaneous polarization match to assess the severity with which depolarization affects amplitude variation in GPR data. A generalized match calculation accepts any antenna polarization thereby accounting for imperfect polarization of the antenna pair. The polarization of our 100 MHz pulseEKKO™ IV antennas was measured from the lake data and applied to the match calculations. The two field sites provide tests of the polarization match measurement and an initial investigation of depolarization/amplitude effects. This investigation is only a beginning to illustrate the depolarization problem and the application of a tool for measuring the depolarization effect in GPR data. Further investigation of different environments and target types, such as conductors, is encouraged. The final chapter (Chapter 6) summarizes the causes of variable antenna coupling and the ramifications for amplitude analysis and modelling. Chapter 2 ANTENNA COUPLING AND WAVELET DEPOLARIZATION 2.1 Review of Pertinent Antenna Fundamentals Antennas serve as the connection between a transmission line and the surrounding environ-ment. Typically, the radar transmitter supplies a voltage pulse at the antenna terminals which produces a current pulse down the antenna whereby an electromagnetic wave is radiated into the surrounding environment. An antenna's design—its physical size, shape, and electrical components—determine the radiating properties such as the radiation pattern, the efficiency of energy transfer between the environment and transmission line, and the characteristics of the radiated pulse such as polarization, and the power and phase spectra. Therefore, understanding these properties for a GPR antenna and how these properties are affected by changing ground conditions is essential to accomplishing more than a crude interpretation of the variation of signal character along a.reflection. A number of references provide the fundamental theory of antennas, their design, and definitions of radiation properties (Kraus, 1950; Stutzman and Thiele, 1981; Ulaby et al., 1981; Mott, 1986). For the reader's convenience, in this sec-tion I will briefly review some fundamental concepts and define the common radiation pattern descriptions which are used throughout the thesis. 2.1.1 Electric Dipole Radiation A radiation pattern describes the distribution of power radiated (or received) by an antenna, as a function of radiating angle. The standard radiation pattern is a far-field description and therefore does not depend on range (Kraus, 1950; Ulaby et al., 1981). In the near-field an off-axis observation point receives different magnitude and phase contributions from different segments of the antenna due to geometrical differences, i.e. asymmetry in range and angle over the length of the dipole. Figure 2.1 illustrates the radial (r) and tangential (0,</>) components 9 Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 10 Figure 2.1 A. Radial and tangential components of the electromagnetic field radiated by a vertical dipole in spherical coordinates. B. Geometry of the near-field problem. The affect of varying the near-field range on amplitude and phase contributions from different elements of the antenna becomes negligible in the far-field (after Kraus, 1950). of the fields radiated from a dipole antenna at the origin of a spherical coordinate system. The general solution for an electric dipole of elemental length dz (known as a Hertzian dipole) in free space is E = hle-i(kr-.t)[(2L_^_'\ Air zk 1 z krs) C ° S e ? + ( 7 + ^ - foS" 1 S l n * ® > d Z ( 2 A ) H = .Jo e - . - ( * r - « « ) + J _ \ s i n Q $ d z 47r \r r2 J (2.2) where I 0 is electric current, constant across the length dz, the wavenumber lc = 2TT/\, and r\ — (^) 2 = g is the intrinsic impedance of the medium with magnetic permeability \x and electric permittivity e (Ulaby et al., 1981). For an antenna of length L these field solutions are integrated over the antenna length with the prescribed current level for each Hertzian element. As the observation range is increased, the antenna dimension quickly becomes less significant, resulting in a relatively rapid decay of the radial electric field as predicted by equation (2.1). The 0 component persists in the far-field due to the 1/r radiation term. The far-field limit is defined as the range along the y-axis at which the phase difference from the extreme ends of Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 11 the antenna is < 7r/8. This condition occurs when 21? r > — (2.3) where L is the longest dimensional length of the dipole (Kraus, 1950; Ulaby et al., 1981). Off the y-axis the ranges from the antenna ends are approximated in the far-field as L L r + r — — cos 6 , and r =s r + — cos 6 (2.4) (Kraus, 1950). In the far-field, E may be approximated by E « — — sinfl © dz (2.5) ATT r and H is approximated by H -—- sm0 & dz = ^ . (2.6) in r n Note that the E and H fields contain the spherical wave Green's function e ' ( f c r " ( ) , vary in phase, are mutually orthogonal, and orthogonal to the propagation direction. The wavenumber k is complex in lossy media with the imaginary part describing attenu-ation. The complex wavenumber is written as k = /3-ia , (2.7) where j3 and a are positive and have units 1/length where /3 is angular wavelength 0 = — (2.8) and a is the attenuation term which becomes zero in lossless media. The phase velocity is related to /? by LO V = J- (2-9) Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 12 The inverse of a is the attenuation distance also known as the skin depth 6S = - (2.10) a over which the amplitude decays by a factor of e. The wavenumber is evaluated from the media's electrical properties by a YQ\J^f y V l + tan* / - 1 , (2.11) and fi = Y0]fW VVl+tan2/ +1 , (2.12) where tan / is the dissipation or loss tangent and / is the loss angle defined as tan / = , (2.13) u>e' where A0 = wavelength in free space, a = conductivity, e' = real part of the complex permittivity e = e' + ie" , and er = the complex permittivity, relative to that of free space er = f-. The real part of the relative permittivity e'r is the well known dielectric constant. In low-loss media the loss tangent is small (0 < tan / << 1) and the wavenumber may be approximated by a « nVHE^l = [E^l , ( 2 . i4 ) p* 2-^B. (2.15) A (Lorrain et al., 1988). The low-loss case is nearly always assumed in the GPR literature, where the phase velocity is approximated by LO C C v = ~a w w ~rf . (2-16) where the relative magnetic susceptibility /v is assumed equal to 1 (Lorrain et al., 1988; Davis and Annan, 1989). Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 13 2.1.2 Directive Properties Numerous parameters are available to describe antenna directive properties (Kraus, 1950; Stutzman and Thiele, 1981; Ulaby et al., 1981; Mott, 1986). Only the parameters used in the remainder of the thesis will be defined here. An antenna's radiation description applies to both transmission and reception. This duplexity, known as antenna reciprocity, exists because the directional sensitivity of antennas to radiation, either outgoing or incoming, arises from the antenna's (or an array's) shape and dimensions, and resulting current distribution. Most directive properties of an antenna are defined in terms of the power density patterns given by the time average Poynting vector K a v = | » e ( E xH*) , (2.17) where the * denotes the complex conjugate. The Poynting vector is in the direction of propagation, i.e. radial, and its magnitude is given by Radiation Patterns Radiation pattern functions describe the angular distribution of power density or a com-ponent of field strength and are usually restricted to the far-field. The radiation pattern of a single dipole antenna depends largely on the length of the dipole in terms of wavelength. As illustrated in Figure 2.2 the main lobe tends to narrow with increasing antenna length, and side lobes occur for antenna lengths that are odd fractions of a wavelength. Of course a pulse system radiates a range of frequencies, resulting in a more complex pattern. In terms of centre frequency, our pulseEKKO™ IV GPR uses a A/3 to A/2 antenna. The radiation pattern function is customarily normalized, cos ( ^ L cos (9) — cos (\kh) sin 6 (2.18) Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 14 Figure 2.2 Dipole antenna radiation patterns (in cross section) of Ug for three different length antennas. Rotating these cross sections about the strike of the antenna yields the space pattern. Signal polarity alternates between lobes and is illustrated by the + or - in the less obvious case (c) with more than two lobes (from Kraus, 1950). representing the field variation at a constant far-field radius R (Ulaby et al., 1981). Figure 2.3 is an example of two standards for plotting the radiation pattern function. Definitions for directive properties, such as beam width, directivity, effective area, and gain, follow simply from Figure 2.3. Antenna Directivity Figures 2.2 and 2.3 clearly illustrate that antennas generally radiate in preferred directions. Directivity quantifies this characteristic by comparing the radiation intensity in a given direction to the intensity of the same antenna if it radiated equally in all directions, i.e. compared to an isotropic antenna. Unless specified, directivity is measured in the direction of maximum radiation intensity, sometimes called maximum directivity (Ulaby et al., 1981). Therefore, directivity is the ratio of power radiated in a given direction to the average radiated power. The average radiation level is obtained by integrating F„ over a sphere centred on the antenna divided by the spherical solid angle 4TT. Therefore, directivity is expressed as D ( M ) = A / / t ( ^ ) ^ = D A ( M ) <2'20) Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 15 20 10 0 10 20 (a) Polar Diagram j§ -10 . I " 1 5 "ro 'Ti -20 N -25 "ro * 1-30 P l/2 dB B eamw dth-i —ii A / y \ A • • r nun \ A / -35 -50 -40 -30 -20 -10 0 10 20 30 40 50 Degrees off Axis (b) Rectangular Plot Figure 23 Two common representations of the radiation pattern function. Three different measures of beamwidth are also illustrated in (b): half power width /3lfl, 10 dB width, and null width /? n u l l (from Ulaby et al., 1981). where dfi is the elemental solid angle = sin<9 dd d<j), and D 0 is the maximum directivity. The integrals in the denominator of (2.20) are approximately equal to the product of the half-power beam widths fixz and pyz (in radians) in the xz and yz planes respectively (see /3i/2 labeled in Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 16 Figure 2.3b). Therefore, maximum directivity may be approximated by D ° ~ ~S~eT a 2 1 ) yxzh'yz (Ulaby et al., 1981). Antenna Gain and Radiation Efficiency The antenna gain G(0,<f>) measures the radiated power density Kr(0,</>), compared to that of a hypothetical lossless isotropic antenna Kn, for equal power supply Pt. Therefore, G(M) = ^ # ^ , (2.22) where Kn is given by K r i = T ^ / / K r ( M ) d O (2.23) 47TJ7J J J and rji is the radiation efficiency defined as the fraction of the power supplied (Pt) that is actually radiated (P 0) from the lossy antenna Vl = ^ (2-24) (Ulaby et al., 1981). The power lost (P/ = Pt - P 0) is the ohmic loss in the antenna in the form of heat. The Sensors & Software pulseEKKO™ IV antennas have an efficiency of about 10% (Dr. Peter Annan, personal communication, 1991). The power radiated, P0, is simply the radiated power density Kr(t9,</>) integrated over a surrounding spherical surface, P 0 = r 2 J J K r(0, <f>) dO . (2.25) 27T 27T An ideal antenna experiences no power loss, therefore, P t = P o i = 47rr 3K r i (2.26) Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 17 Finally, gain can be written solely in terms of the subject antenna G(0, <f>) = 47r?7/Kr 47rr 2 r)iKt 4 7 r r 2 K r (2.27) J / K r d O P 0 Pt and is simply related to directivity through antenna efficiency G(M) = »WD(M) (2.28) (Ulaby et al., 1981). Therefore, antenna gain is a percentage of directivity, scaled by the ohmic losses in the antenna. Note that the pattern descriptions primarily describe the main lobe or its significance relative to the remaining radiation pattern. The pattern descriptors defined in this section pertain to the radiated power, the most commonly discussed radiation pattern. However, patterns also involve phase and polarization. Usually the phase is assumed to be reasonably constant across the main lobe with little concern for the side lobes, although it is understood that the phase varies at least from lobe to lobe by 180° as indicated by the + and - signs in Figure 2.2. The polarization of the signal also varies with radiation angle (Kraus, 1950; Stutzman and Thiele, 1981; Roberts, 1994), but again this variation is usually neglected (this variation is illustrated in Section 2.3.3). 2.2 Radiation of the Pulse GPR Antenna Over Half-space Earths The ground surface within the near-field range dramatically affects the radiating properties and impedance of an antenna. A discussion of the effect on impedance is reserved for Section 2.4.2, whereas this section discusses the effects on radiation patterns. The resulting radiation pattern is related to the division of the wavefield into the two half-spaces when the antenna is on or near the interface. Figure 2.4 illustrates the two wavefields where the radar wave velocity in air is much greater than in the subsurface. Waves A and B are spherical waves propagating at their respective phase velocities of the two mediums. Wave C is an evanescent wave which satisfies the boundary conditions with wave B. As Figure 2.4 illustrates, this wave Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 18 decays exponentially with height. Wave D is commonly known in the radio literature as a lateral wave, analogous to the seismic head wave. The lateral wave satisfies the boundary conditions as energy critically refracts from wave A thereby imposing a 1/r2 amplitude decay on wave A at the interface (Brekhovskikh, 1980). Figure 2.4 Wave fronts from an electric dipole on an interface, air over a half-space earth. Waves A and B are spherical waves propagating at their respective velocities. Wave C is the evanescent "surface" wave propagating along the surface at the earth velocity having an amplitude that decays exponentially with height. Wave D is critically refracted from the air wave (after Annan, 1973). A two layer earth introduces interference between the half-space wave fields of Figure 2.4 and reflections from the subsurface interface. Wave B and D reflect off the subsurface interface (at the earth-air critical angle), returns to the surface and critically refracts into the air to travel, once again for wave D, along the surface as the critically refracted air wave. The distance at which the critically refracted air wave is first observed is the critical distance r c, is defined as r c = 2<itan0 c , (2.29) where d is the depth to the reflector and the critical angle 0C = s i n - 1 (j^J. These waves are commonly observed in GPR "shot" or common midpoint (CMP) soundings as illustrated in Figure 2.5. Within the near-range, r < rc, the wave field primarily consists of the interaction between the two spherical waves, as in the half-space. At ranges about rc a complicated Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 19 interference occurs between the near-range wavefield and reflected D of comparable amplitude. In the far-range, r > rc, multiple reflections are important and the normal modes become dominant (Rossiter et al., 1973). Additionally, the amplitude along the wave fronts varies due to the antenna radiation pattern. Fortunately for GPR, antenna radiation is preferentially directed into the medium of higher dielectric constant that is positioned within the near-field. This power division between the two mediums varies as approximately e 3 / 2 for planar antennas on a dielectric surface (Brewitt-Taylor et al., 1981). Bow-tie antennas, which are planar electric dipoles, are commonly used by some GPR manufactures such as Geophysical Survey Systems Inc. (GSSI). The concern in this section is the effect of varying ground electrical conditions on the radiation pattern and the region of illumination. Although insightful, Smith's (1984) numerically generated radiation patterns are often referenced in GPR literature without their limitations to GPR noted. Smith generated these patterns using a geometrical optics solution for an infinitesimal antenna radiating continuously at a given frequency at different heights over two very different lossless half-space earths. The geometrical optics solution is restricted to the far-field and excludes the fields near the interface, i.e. the direct waves and the associated lateral and evanescent waves. Also a few qualifiers must be realized when adopting steady-state (continuously radiating) radiation patterns for pulse radiation. In a later paper, Smith and Scott (1989) present their physical modelling of a pulse antenna radiation pattern. Additionally, GPR targets are often within the near-field. In fact, modelling by Turner (1993) and water tank measurements of pulse radiation patterns by Wensink et al. (1990) infer that the far-field limit defined by equation (2.3) is not always a reliable guide. In fact, the Wensink et al. results infer that the far-field condition was not fully developed at a distance of 15 wavelengths. Numerical modelling by Turner (1993) and Roberts (1994) produce radiation patterns that are contoured from the antenna into the "far-Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 20 Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 21 field." These two theses each provide a collection of radiation patterns covering a typical range of GPR environments including conductivity up to 20 mS/m in Turner's modelling and up to 100 mS/m in Roberts's modelling. Despite their shortcomings, Smith's (1984) radiation patterns concisely illustrate the effects due to a dielectric half-space and therefore are presented in Figure 2.6. After discussing Smith's (1984) results, their deficiencies to GPR will be addressed in light of the more comprehensive work just referenced. 2.2.1 Effects of the Ground on Steady-State Radiation Patterns The half-space field patterns are explained in terms of refraction and reflection about the .. interface (Smith, 1984). Figure 2.6 illustrates Smith's (1984) half-space patterns for the two cases of air over dry earth and air over water for three different antenna heights. Note that for all cases, in the plane along strike of the antenna (the E-plane), the pattern within region 2 has a null at the critical angle of refraction. This null becomes the maximum pattern amplitude, sharply peaked, in the plane perpendicular to the antenna (the H-plane). This bounding of the main lobe within the critical angle is due to the transmitted angle converging to the critical angle as the incidence angle converges to 90° grazing incidence. Transmission within region 2 beyond the critical angle, the primary sidelobes (9C < 0 < TT/2), arises from energy contributed by the second order lateral waves (Figure 2.4) radiating into region 2 from the interface. Effect Due to Dielectric Contrast Figure 2.6 clearly illustrates that the ground having the higher dielectric constant will result in the narrower main lobe. Water is an extreme example (Figure 2.6) having a dielectric constant of 80, whereas typical earth materials have values in the range of 3 - 40 (Davis and Annan, 1989). The dry earth in Figure 2.6 has a dielectric constant equal to 4.0. The obvious advantage to the narrower beam is a decrease in side-scattered energy with the focusing of energy in the direction of interest. However, this "advantage" can be a disadvantage in problems such as utility detection where prominent diffraction hyperbolae are a distinguishing characteristic. In Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 22 Air / Dry Sand Air / Water E-Plane H-Plane E-Plane H-Plane 180° , . . 180° Figure 2.6 Radiation patterns of a horizontal electric dipole at three heights h over half-space earths of water and dry sand (after Smith, 1984). structural problems a narrow beam can be a disadvantage due to the lower maximum dip that can be imaged. Note in Figure 2.6, however, that these amplitude effects would primarily be noticeable when the directivity is optimized by elevating the antennas above the ground. For the typical case of the antennas on the ground these amplitude effects would probably not occur due to the existence of large sidelobes. Instead, with the change in beamwidth of the main lobe, out of the plane or steep dipping events might exhibit a polarity change—a much more subtle effect. Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION Effect Due to Antenna Height 23 Another important feature of the field patterns is the contrast in magnitudes above and below the interface as a function of antenna height. The amplitude of the back lobe is a minimum for « 0. As the antenna is raised, more of the incident field is reflected, therefore, contributing to the radiation field in the air. Also note that the field patterns within the earth are broad at h = 0 and have significant amplitude for both pre- and post- critical angles. Amplitudes for 0 > 6C are at their maximum for h = 0 since the energy put into the lateral/evanescent waves is not required to propagate any distance before reaching the surface. Since the evanescent waves are damped exponentially, they become insignificant before ^- — 0.35 (Figure 2.6) resulting in transmitted patterns that are only significant within the cone 0 < 9 < 0C, i.e. no side lobes for large h/Ai (Smith, 1984). Directivity and Gain A direct consequence of the field distribution about the interface is the effect on directivity. Recall that directivity measures the concentration of power in the front main lobe (into the earth) compared to the side lobes and back lobe. Also recall the relation between directivity and gain, (2.28) G = 77/D. In the case of a lossless antenna /// = 1, and therefore G = D. As shown by the field pattern plots, directivity is increased with increasing £2, being much greater in the case of water (£2 = 80). This is due to a decrease in the width of the main lobe since 0C decreases with increasing £2. Directivity also depends on antenna height (Figures 2.6, and 2.7). As antenna height increases, the two competing effects of decreasing side lobes and an increasing back lobe results in maximum directivity at h« 0.1 Ai. This height of maximum directivity is also noted in calculations by Burke et al. (1983), Turner (1993), and measured by Wensink et al. (1990). However, note in Figure 2.7, for the dry earth example, £2i = 4, that directivity maintains a relatively low value regardless of antenna height. The dependence Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 24 of directivity on antenna height greatly increases as £21 increases, such as with increasing soil saturation. For the extreme case, of surveys performed over water, directivity becomes especially sensitive to antenna height. 100 1 1 1 1 1 : 1 1 Figure 2.7 Directivity (lossless antenna) and gain (lossy antenna) as a function of antenna height h, for different values of e2i where e 2 i = ff- Solid line: lossless media (directivity), dashed line: region 2 low-loss (gain) with tan/ = • = 1.0 (after Smith, 1984). Effect Due to Conductivity Another interesting feature of Figure 2.7 is the significant, yet reasonably small difference between lossless and low-loss directivities. The overall shape of the directivity curve is affected little by the lossless assumption. Turner (1993) also notes that the radiation pattern for low conductivity is similar to the lossless case. The difference appears as decreased sharpness of the nulls and peaks at the critical angle. However, the numerical modelling by both Turner, and Roberts, are for the transient case, i.e. pulse radiation, which complicates the comparison of conductivity effects with Smith's results which are for a given frequency. The consequences of this difference are discussed in Section 2.2.2. Numerical modelling results by both Turner, and Roberts, exhibit a decrease in the sharpness of radiation lobe boundaries with increasing conductivity, becoming nearly omnidirectional. However, this is likely primarily due to dispersion which accompanies conductivity. Dispersion is required in an attenuating medium in order to maintain causality and wavelet asymmetry (Aki and Richards, 1980). Section Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 25 discusses the effect of dispersion on pulse radiation patterns. For the monochromatic case, the small difference between lossless and lossy directivities is due to the increase in losses of the antenna via the impedance load where the lossy directivities in Figure 2.7 are actually gains (Smith, 1984). As in the lossless case, the radiation pattern in the ground broadens along the interface when the antenna is elevated, however this observation appears more obvious in the lossy results of Roberts. This observation probably only appears more obvious, being accentuated by the shallow depth of penetration. While the subsurface radiation decreases and becomes more omnidirectional with increasing conductivity, the above ground main lobe radiation maintains both strength and shape. Energy reflected into the air becomes greater due to an increase in reflection coefficient. This results in an increased relative—or absolute if radiated through the sidelobes—significance in reflection amplitudes from above ground objects in GPR surveys (Turner, 1993; Roberts, 1994). Near-field vs. Far-field; Finite vs. Infinitesimal Dipoles Near-field patterns tend to exhibit sharper nodes and peaks, particularly in the E-plane, however unlike the far-field, these near-field cusps appear to be more related to the antenna than the critical angle (Roberts, 1994). Roberts notes that increasing antenna height does not focus the near-field as much as the far-field. Although most of the radiation is constrained within the critical angle, each elemental length of antenna radiates into the subsurface within its own critical angle cone. Therefore, a near-field location observes radiation through a larger angular portion of ground surface than does a far-field location. Closely related to the near-field versus far-field comparison is the finite versus infinitesimal dipole comparison. In the far-field, the radiation pattern of a bow-tie antenna is very similar to that of an infinitesimal dipole. However, noticeable differences do occur in the very near-field, especially in the E-plane. The near-field pattern is broader from a bow-tie antenna than Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 26 from the point dipole. Generally, however, they appear impressively similar in Roberts (1994) modeled comparison. 2.2.2 Steady-State Radiation Patterns and Pulse Radar Upon considering the relation between steady-state radiation patterns and pulse radiation I have realized some important consequences that arise in adopting the steady-state half-space radiation patterns for pulse radiation. These consequences will be discussed after Section highlights the transient nature of radiation patterns and the fundamental difference between the pulse and steady-state cases. Two primary consequences involve the frequency dependence of radiation patterns since the pulse case involves a broader frequency band than the steady-state nearly monochromatic case. Therefore, Section discusses the effect of antenna height on frequency and Section discusses the effect of velocity dispersion on radiation patterns. Time Dependence of Radiation Patterns When a continuously radiating antenna resides above the interface the steady-state pattern builds over a short time interval from the instant that the antenna begins to radiate (usually tens or hundreds of ns). This pattern evolution is due to the above ground steady-state pattern being the superposition of direct and reflected wave fields and therefore requires the difference in travel times of the two wave fields to build up the power level. Pulse radar surveys measure entirely within this transient time by their very nature. To realize the power level predicted by the steady-state field pattern the cumulative power over the entire pulse wave train or time trace must be measured. However, simply measuring the maximum peak to peak amplitude is a more convenient method of constructing pulse radiation patterns that remains meaningful as illustrated by Smith and Scott (1989) in Figure 2.8. Note, however, that this convenient measure is unlikely to exhibit the same changes to the above ground patterns that occur with changing antenna height as in the steady-state case since these changes are primarily due to the reflected wave field. The patterns by both Turner (1993) and Roberts (1994) are whole trace pulse Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 27 + Figure 2.8 Time domain peak to peak field values as a function of radiating angle for an antenna at three different heights above an air/emulsion interface. The model consisting of an emulsion of mineral oil and saline scales the electrical properties of a red clay. The wavelet is measured directly and adjusted to a constant radius. Finally the wavelet is plotted as a function of take-off angle (after Smith and Scott, 1989). radiation patterns, i.e. a superposition of radiation patterns—such as Smith's (1984)—with the radiated power level scaled by the wavelet's power spectrum. Effect of Antenna Height on Frequency Continuously radiating antennas are assumed monochromatic, whereas the pulse wavelet, by definition, contains a range of frequencies. Figure 2.6 clearly illustrates that at a given Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 28 height, h, different frequencies undergo different radiation patterns. Therefore, a finite height effectively acts as a "directivity filter" applied to the wavelet with the filter's peak response at the frequency fp - > wi m a spectrum shaped like the curves in Figure 2.7. Note that for h = 0 all frequencies are radiated with equal directivity. Since in the special case of h = 0 the radiation pattern does not require any time to evolve, the steady-state solution may be adopted directly as the radiation pattern of the pulse wavelet. However, note that the main refracted lobe's shape, relative to the peak radiation, may be adopted directly regardless of height or frequency (for nondispersive media). Although the actual magnitude will decrease with increasing h due to spherical spreading and attenuation, the main lobes's relative magnitude remains constant (Figure 2.6). Essentially, the changes of the radiation patterns and directivity, due to antenna height and frequency, involve an exchange of energy between the side and back lobes. A consequence of different frequencies experiencing different radiation patterns is a de-creased sharpness of lobe boundaries. Although this decreased sharpness has been noted in loss-less pulse patterns when compared to Smith's (1994) results (Smith and Scott, 1989; Winsink et al., 1990, Turner, 1993; Roberts, 1994), the explanation of frequency dependent radiation patterns has not been offered. Lobe sharpness should also decrease with increasing antenna height since increasingly lower frequencies become affected. This variation with antenna height is noticeable in both Turner's and Roberts's results but was not acknowledged. This blurring of radiation lobes is also enhanced with increasing conductivity due to dispersion. Effect of Velocity Dispersion on Radiation Patterns Although velocity has been shown to be nearly constant over much of the GPR frequency band (Turner, 1993; Turner and Siggins, 1994) at least a small variation is required, even in nearly lossless media. Attenuating media requires velocity dispersion for a causal and asymmetric wavelet (Aki and Richards, 1980). An increase in velocity dispersion with Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 29 increasing attenuation rate seems to be a reasonable assumption (Kalinin et al., 1967) and therefore dispersion will increase with conductivity. Dispersion complicates the direct adoption of the steady-state patterns since different frequencies yield different critical angles. Therefore, dispersion causes the critical node in the E-plane or the peak in the H-plane to smear over a range of radiation angles, further blurring lobe boundaries. Smith and Scott (1989) illustrate this by direct measurement of the wavelet radiation pattern (Figure 2.8). Notice that the normalized main lobe broadens with decreasing antenna height in Figure 2.8b, whereas it remains constant in the monochromatic nondispersive case (Figure 2.6). At zero height, the lobe in Figure 2.8b is broader and flatter than expected. This is attributed to the lab measurements being made in the near-field when the antennas are positioned at the surface (Prof. Smith, personal communication, 1992). To reiterate a point made in Section; note that many of the frequency dependencies of radiation patterns do not occur when the antennas are on the surface. The primary exception is the smearing effect due to dispersion which is only minimized in this case. This section has shown that changes in near-field electrical properties can significantly alter the radiation pattern in a number of ways and possibly cause some reflections to extinguish or appear due to increasing or decreasing directivity. Additionally, the traditional single (parallel) component GPR survey may exhibit significant amplitude anomalies which are actually polarization anomalies. Therefore, understanding the relation between wavelet and antenna polarization and their coupling is also important. 2.3 Antenna Polarization Antenna polarization is defined by the polarization of the electromagnetic wave radiated within the main lobe of the radiation pattern. The polarization characteristics of an antenna are generally constant over the main lobe, however, side lobe polarizations may differ considerably Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 30 from the main lobe (Stutzman and Thiele, 1981). An antenna may be designed to radiate any desired planar polarization from linear to the general case of elliptical. Elliptical and circular polarizations are also classified into right-handed and left-handed polarization depending on whether the instantaneous electric field vector rotates counterclockwise or clockwise, respectively, with the wave propagating towards the observer (Figure 2.9). This section proceeds with a very brief review of polarization descriptions referenced in the thesis followed by a reminder of the connection between polarization and reflection / transmission, i.e. the Fresnel equations. Finally, polarization topics more specific to GPR are covered in Section 2.3.3 on antenna polarization patterns and reciprocity, and Section 2.3.4 on polarization due to survey geometry. 2.3.1 Definition and Description of Polarization The polarization of an electromagnetic wave is usually described in terms of the orientation of the electric field vector. Linear polarization is the most simple form with the electric field strength oscillating in a single direction. Remaining forms of polarization, circular and elliptical, involve a rotation of the electric vector. The rotation is constrained in two dimensions, i.e. planar, for waves generated by antennas. Any planar polarization may be generated from two orthogonal (component) linear polarizations having different peak magnitudes and phases. The component polarizations correspond to Ei and E 2 in Figure 2.9 where the resulting ellipse is traced by the tip of the instantaneous electric field vector S. Circular polarization occurs in the special case where the components have equal magnitude but oscillate 90° out of phase. Alternatively, two orthogonal circular polarizations are sometimes used as the component polarizations. Orthogonal circular polarizations rotate in opposite directions. Elliptical polarization is the general form from which many polarization descriptors are defined. Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 31 Figure 2.9 A. In rotational polarizations the instantaneous electric vector traces a helical path with propagation. When this motion is viewed within the plane of rotation, as shown, the elliptical nature becomes clear. Part Billustrates the ellipse in detail and its various descriptors discussed in Section The ellipse is viewed with the wave approaching. Phase, Sense, Tilt, and Polarization Ratio Parameters that numerically describe the fundamental characteristics of the polarization ellipse are commonly used, particularly in the V L F geophysical prospecting method. These parameters include phase, sense or direction of rotation, tilt, and polarization—or axial—ratio. All of these ellipse descriptors are defined easily from Figure 2.9 where the wave is propagating towards the reader (+x direction). Note that the polarization components Ei and E2 are the Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION maximum coordinate amplitudes in the z and y direction respectively, vector can be written as 32 The instantaneous electric —• £ = £x5t + £yy - E i cos tot x + E 2 cos (ut+6) y (2,30) where 6 is the phase by which the y-component leads the ^-component (Stutzman and Thiele, 1981). This phase difference or phase of polarization determines the rotational sense and to a large extent the tilt and ellipticity or axial ratio of the ellipse as illustrated in the polarization chart of Figure 2:10. Negative phase produces counter-clockwise or right-handed rotation and positive phase produces clockwise or left-handed rotation. When the phase is either 0° or 180° linear polarization results with a tilted orientation determined by the relative amplitudes of the components. Tilt is simply the angle r between the semi-major axis and the reference x-axis (Figure 2.9) and can be calculated from t a n 2 r = 2 E i E 2 cos 6 EJ-El (2.31) 90° 63° 45° ^2 2 \ ! 1 26.6° 0.5 0° 0 s D a % s 0 § s Counter- Clock-clockwise wise -180 -135° -90° -45° 0° 5 +45° +90° +135° +180° Figure 2.10 Chart of polarization ellipses as a function of the ratio |a (or 7) and the phase difference 6 (wave approaching) (after Kraus, 1950). 1 Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 33 (Beckmann, 1968). The axial ratio IARI is simply the ratio of the major axis to the minor axis. The axial ratio can also be described by the angle x given by (Stutzman and Thiele, 1981). Similar to IARI but a measurement that follows more directly from radar recordings is the angle 7 that describes the relative magnitudes of Ei and E2 from (Stutzman and Thiele, 1981). Note that 7 is only equal to the tilt for the case of linear polarization (<5 = 0° or 180°, or 7 = 0° or 90°) (Figures 2.9 and 2.10). As Figure 2.10 illustrates, the polarization of the wave is fully described by the angle pair 7 and <5. An alternative pair of angles that specifies the polarization state is the tilt r and x - These parameters describe the type of complete polarization. However, electromagnetic waves can also be partially polarized, or completely unpolarized. Therefore, another measurement quantifies the degree of polarization. Degree of Polarization Electromagnetic radiation from physical objects, or radio waves influenced by inhomoge-neous media or irregular terrain, usually produce an interference of many different polarizations (Ulaby et al., 1981). Therefore, the measured components can be completely uncorrected for waves that are incoherent or unpolarized. The components of completely polarized waves, however, are well correlated. Varying degrees of polarization also exist between these two ex-tremes. A natural measure of the degree of polarization or degree of coherence is the magnitude of the normalized cross-correlation between the components X = c o t - 1 ( A R ) 1 < | A R | < 00 - 45° < x < 45° (2.32) , _ i ^2 7 = tan — 0° < 7 < 90° (2.33) P12 -( E i E g ) (2.34) 1/2 Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 34 where < > indicates an average operator which is required when the amplitudes or phases vary with time. The average operator is given by T { ' " ) = T ^ o \ f / "' d t ( Z 3 5 ) 0 (Ulaby et al., 1981). For a completely polarized wave the components' amplitudes and phase difference do not vary with time, therefore, averaging is not necessary and p\2 = 1.0. For completely unpolarized waves, p\2 = 0.0. Partially polarized waves have degrees of polarization between 0.0 and 1.0. Note that this measurement gives no indication of the type of polarization, but only measures how well a wave is polarized. A very useful parameter that concisely describes polarization is the complex polarization factor. The Complex Polarization Factor The complex polarization factor is defined by the ratio of the vertical to horizontal electric field strength components, or more specifically the ratio of plus- and minus-polarized waves E+ P = E _ 7 (2.36) (Beckmann, 1968). This factor is also known as the polarization ratio (Mott, 1986). The plus and minus notation is adopted from the reflection coefficients for a perfectly conducting plane, being +1 and-1 for vertical (TM) and horizontal (TE) polarizations, respectively. This notation avoids confusion in geometries where the labels horizontal and vertical are not very relevant. If the wave is considered by its individual frequency components, i.e. the monochromatic case with frequency to, then E + and E~ can be written as complex phasors E ~ = | E ~ | e , ( w < - k ' r ) , E + = | E + | e i ( a ' ' ~ k ' r + * ) , (2.37) resulting in a complex valued polarization factor eiS = | p | e ! a r &P (2.38) P = E+ E " Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 35 (Beckmann, 1968). Since the complex polarization factor contains both relative magnitude and phase difference between the two-components, the polarization is uniquely described. The complex plane is utilized as a polarization chart with the following correlations to polarization: Im P = 0 = linear P = 0 = horizontal P = oo = vertical Im P > 0 — right elliptical Im P < 0 = left elliptical P = i right circular P = -i = left circular. Note that the polarization factor does not include information on amplitude unlike the Stokes parameters which are another common polarization description involving four terms, only three of which are independent. Amplitude is independent of polarization since polarization is strictly the direction of vibration regardless of the amplitude. The conciseness of the polarization factor makes it very convenient for depolarization studies and is used extensively in the discussion on depolarization in Section 2.5. 2.3.2 Review of Fresnel Coefficients and the Polarizing Angle This section is included here to remind the reader that the Fresnel reflection and transmission coefficients are defined for the end member (orthogonal) linear polarizations, transverse electric (TE) and transverse magnetic (TM), i.e. E~ and E + respectively. For an in depth review consult texts such as Lorrain et al. (1988). The T E and T M polarizations are referenced from the plane of incidence where the electric field vector for TE polarization is oriented perpendicular (transverse) to the incidence plane (Figure 2.11). A T M polarized wave has the magnetic field vector perpendicular to the incidence plane, therefore the electric vector lies within the incidence plane. For the remainder of this thesis Beckmann's notation (1968) is adopted which Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 36 TE polarization TM polarization 'y 'y Figure 2.11 Polarization geometries of TE and TM reflection and transmission (after Lorrain et al., 1988). is in terms of the electric field vector: E~ for T E , or horizontal, polarization and E + for T M , or vertical, polarization. The Fresnel coefficients are stated here for later reference and to recall their relation to the E~ and E + polarizations. For horizontal polarization the reflection and transmission coefficients are cos$i — \/ep — sin Oi cos Oi + v ep — sin Oi 2 cos Oi cos Oi + y/ejj. — s in 2 Oi T " = where ejj, £2/^ 2 (2.39) (2.40) (2.41) and e is the electric permittivity and \i is the magnetic permeability of the indicated mediums For vertical polarization the Fresnel coefficients are £//cost9 1 - \fefi — s i n 2 01 R+ = T+ = ejx cos 0X + y/efi — s in 2 0X 2\/Tp, cos 0± e//cos(9a + yjep — s i n 2 ^ (2.42) (2.43) (Beckmann, 1968). Recall that the polarizing angle, or Brewster angle, is the angle at which the reflected wave has only E~ polarization since all of the E + component is transmitted. The Brewster Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 37 angle can be utilized for measuring the relative permittivity of the reflecting surface (Lorrain et al., 1988) since tan#g = (2.44) For angles less than 0R> the reflected vertical component has negative polarity compared to the incident vertical component E^", becoming positive polarity for angles greater than #B-For conductive media E+ does not go to zero but does acquire a minimum at the quasi- or pseudo- Brewster angle (Beckmann, 1968). 2.3.3 Polarization Patterns and Reciprocity Most GPR antennas are essentially electric dipoles and therefore are linearly polarized. However, actual antennas do emit a small cross-polarized component due to stray currents on the antenna housing or support structures. Additionally, theory predicts a small cross-polarized component from electric dipoles. The radiation pattern of the cross-component becomes apparent when the electric field is written in Cartesian coordinates whereas in spherical coordinates the total field is a single-component (E$) as described by equation (2.5) and illustrated by Figure 2.2a. Figure 2.12 reveals that the cross-component arises from the wave front segments that are not parallel to the dipole which occurs between the H-plane (y-axis) X Figure 2.12 Decomposition of (tangent to the wavefront) into its Cartesian coordinates. This view illustrates the wavefront in the ;cy-plane (after Roberts, 1994). Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 38 and the E-plane (x-axis). Written in Cartesian coordinates, the electric field becomes three components E x = E0sin0 , (2.45) Ey = Eg cos 9 cos <[> , (2.46) Ez = Eg cos 9 sin<£ , (2.47) (Roberts, 1994) and are sketched in Figure 2.13. Recall from equation (2.5) that EQ contains a sin0 term. Therefore, although EX and EQ appear similar, EX is shaped as sin20 whereas EQ is shaped as sin0. The resulting cross-component radiation patterns have four lobes with maximums at 45° from the antenna strike. Although significant, the cross-component maximums are about 12 dB down from the peak EX component (Roberts, 1994). Cross-component nodes occur at the x- and y- axes where the radiation changes polarity. The primary importance of these cross-component patterns is in illustrating the potential for a side scatterer to contribute to noise in cross-polarization measurements and possibly confuse interpretation. In Section 2.1.2 the concept of antenna reciprocity was introduced where the directive properties of an antenna apply to both transmission and reception. However, antenna reciprocity is not sufficient for two antennas at fixed positions to receive the same amplitude when their transmitting and receiving roles are interchanged, particularly when the polarization is altered along the ray path. When simply interchanging transmission and reception roles of a bistatic pair of identical antennas, then, for either direction, the radiated polarization pi is altered to p2 during propagation to the receiver. This form of reciprocity is known as location reciprocity. However, transmitting backwards, starting with p2 usually does not result in the polarization pi; instead p2 will usually depolarize even more to P3. An example is an unpolarized wave incident at the Brewster angle resulting in a purely E" polarized reflection. However, if the E~ wave is propagated back on the same ray path, the reflection will remain E~~ polarized, instead of returning to the original unpolarized wave. For p2 to change back to pi upon reverse Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 39 Z A ) Far-field pattern of E x component z A B) Far-field pattern of E „ component z A C ) Far-field pattern of E z component Figure 2.13 Sketch of the Cartesian components of the E radiation pattern. The + and - denote signal polarity in the multi-lobed cross components which are 12dB down from the E* component (after Roberts, 1994). reflection is a very restricted form of reciprocity known as polarization reciprocity (Beckmann, 1968). Polarization reciprocity holds only for special cases, such as E~ at the Brewster angle (in both forward and reverse directions polarization remains as E~). General conditions for polarization reciprocity are derived by Beckmann (1968) to be Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 40 where E * are the + and - polarizations after reflection. Generally, these conditions do hold for good conductors with large radii (Section 2.5 and Appendix). The type of polarization for Ei is in part determined by survey geometry, particularly the relative orientation of the transmit dipole to the survey line. 2.3.4 Polarization in GPR Surveys In GPR surveys the orientation of the dipole antennas, relative to each other and the survey line, determines the mode of linear polarization, i.e. T E or T M mode as illustrated in Figure 2.14. Usually in ground surveys (as opposed to glacial) the antennas are oriented parallel and broadside to each other and perpendicular to the survey line, yielding a T E wavelet, i.e. polarized perpendicular to the survey "plane of incidence." Antennas oriented end-to-end (endfire) and parallel to the survey line (common in glacier surveys) result in a T M wavelet. There are two problems with the T M mode: 1. due to the radiation pattern more energy is radiated out of the survey plane, resulting in stronger out of the plane reflections, and 2. severe depolarization of the reflected wavelet is more likely due to scattering at the Brewster angle. This antenna configuration is commonly used in glacier surveys because of the low frequencies used (5 - 25 MHz, typically) requiring antennas that are about 4 to 15 m long. The most TE (common for ground surveys) TE Cross Tx Rx Tx Survey Line Rx TM (common for glacier surveys) Tx Rx TM Cross Tx •Survey Line Rx Figure 2.14 Plan view of common polarization modes of GPR surveys and the antenna orientation for measuring the parallel and cross polarization components. Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 41 practical means to deploy antennas of this size is by towing them in line. Usually the survey objective in glacier studies is only to map the bed, and possibly internal layers, with little interest in anomalous reflection character. As already suggested above, wavelet polarization may change during propagation for various reasons (Section 2.5) which can dramatically affect the received power (Section 2.4). Assessment of wavelet depolarization (Chapter 5) requires 2-component data which includes the cross component which is measured as shown in Figure 2.14. 2.4 Antenna Reception The reception of a signal incident on an antenna depends on a number of factors, some of which are fairly obvious, such as the relative orientation between the antenna and signal. This mutual orientation is important in two ways: 1. reception varies with incidence angle due to the antenna's radiation pattern, and 2. the matching of the signal's polarization with the antenna polarization. The second is probably less obvious and will be defined in Section Other factors concern the signal propagation path and system design. Propagation related factors include spherical spreading, intrinsic attenuation of the medium, scattering losses, target scattering cross-section, and total range. System related factors include the antenna's effective area to intercept the signal, the radiation pattern, antenna efficiency, and impedance match at the terminals. All of these factors are considered by the radar range equation which measures the power accepted at the receiver antenna terminals (PR) relative to the power supplied at the transmitter antenna terminals (PT). This radar range problem is illustrated in Figure 2.15. 2.4.1 The Radar Range Equation The usual form of the radar range equation assumes that both transmit and receive antennas are oriented with their pattern maxima toward the target. Additionally, the incident wavelet is assumed to have a polarization that is optimally aligned with the receive antenna polarization and the load impedance matches the antenna input impedance. Often a lossless medium is also Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 42 Gain G f - 4 Spreading Loss 1/4jcr Scatferinq Patrern eceiver Effective Area A e R Spreading Loss 1/471 r,? Scattering Cross-Section £ Figure 2.15 Bistatic radar and target geometry for the radar range problem. A hypothetical scattering pattern (similar to a radiation pattern) is sketched around the target. The target scattering cross-section is also sketched as the uniform shape around the target. assumed such as air. These additional complications are easily incorporated, however, as will be shown. A common form of the radar rage equation is 1R) (2.49) p R = p T GT(flT , < M A eR(flR,^RL ( 47 r r 1 r 2 ) 2 (Mott, 1986) where subscript T designates a transmitter term and subscript R designates a receiver term, G T - transmitter antenna gain, rj = range from transmitter to reflection point (target), r2 = range from target to receiver AER = receiver effective area, and C = the scattering cross-section of the target. The effective area (units of square wavelength) measures how effectively the receiving antenna intercepts and converts incident power KR„ into received power PR at the antenna terminals PR = K R N A E R . Antenna area is related to gain and directivity by A 2 A 2 A e R = T,G = 4 ^ ° (2.50) (2.51) Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 43 where r)a is the aperture efficiency, equivalent to radiation efficiency, but for receiving instead of radiating. This distinction between r]a and rji is made for the general case of transmit and receive antennas of different efficiencies. The scattering cross-section, with units of area, is the equivalent target area as if the target reradiated the incident power isotropically. Although the incident power is not really reradiated isotropically, this parameter only considers the apparent target size for the power scattered in the direction of the receiver as if the scattering were isotropic. This parameter actually varies with scattering geometry due to the scatterer's general shape and roughness properties, as well as reflectivity. For a complex target, such as an aircraft, a change in scattering angle of only one degree can change the scattering cross-section such that the received power changes by many decibels (Mott, 1986). The scattering cross-section assumes no knowledge of the target's scattering characteristics by assuming, instead, that the measured power level would be the same at any observation point. The power density incident at the target is t n T — — ' ( } which is partially intercepted and scattered (or reradiated). The portion of the scattered field directed to the receiver is described by the scattering cross-section (. The scattering cross-section can be calculated as the spherical solid angle 4ir times the ratio of the scattered wave intensity directed toward the receiver r|l<Rn to the power density incident at the target C = 4 T T ^ ^ . (2.53) Therefore, the power radiated by the transmitter antenna and intercepted by the target is ( K ^ ^ ^ ' t . (2.54) Since ( assumes isotropic scattering, the wavelet power continues to decay by spherical spreading once more from target to observation site. Therefore, the power incident at the Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 44 receiving antenna is CKtn = P T G T C 4 T T ^ ( 4 7 r r 1 r 2 ) 2 K R n = = 3 ( 2 - 5 5 ) (Mott, 1986). Including the receiver area gives the standard radar equation (2.49). Intrinsic attenuation is easily incorporated by including the attenuation parameter a in an exponential term PR ^ P i G t ^ t ' ^ A e R ( g R ' ( c-2«(ri+r2) ( 2.56) (47rr 1r 2) 2 (Duke, 1990) where a is the imaginary part of the complex wavenumber in lossy media (Section 2.1.1). Additional scattering losses from "point" scatterers such as boulders and cobbles may be incorporated into the intrinsic attenuation parameter a. The range equation in the form of (2.56) includes all propagation losses for a homogeneous host material and the effect of the radiation patterns. However, this equation still involves the common assumptions of a perfect match between the incident wavelet polarization and antenna polarization and a perfect match between the load and input impedances at the antenna terminals. For G P R neither of these assumptions are valid much of the time and these matches can vary significantly over a survey area. 2.4.2 Impedance and Polarization Match Factors As just noted, the radar range equation assumes that the receiver antenna is optimally oriented to receive the incident wavelet (in terms of polarization) and that the impedance across the antenna terminals is perfectly matched. These two matchings are primary links concerning the environment's effect on antenna radiation and reception. The receiver antenna effective area A C R (2.51) is modified by the polarization and impedance match factors m p and m q , respectively. Note that the matching at both the transmitting and receiving end are combined into single match factors applied to the receiver antenna. The terminology for the modified and unmodified antenna areas is sometimes inconsistent. I refer to the modified antenna area Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 45 as the mismatched antenna effective area AeM. where A eM = m p m q A e R = m p m q G(9,4>) (2.57) (Stutzman and Thiele, 1981). The match factors simply rescale the antenna area by varying be-tween 0.0 and 1.0. This section will define these match factors and discuss their environmental link. Impedance Match (mq) The impedance match factor m q is the fraction of power transmitted across the antenna terminal - transmission line junction, i.e. the power transmission coefficient. The mathematical definition is where V = voltage reflection coefficient, Z L = load impedance across the antenna terminals, and Z j n p = antenna input (transmission line) impedance. Since m q ranges from 0 to 1, a perfect match (mq •= 1) indicates that Z j n p = Z L (Stutzman and Thiele, 1981). Note that m q pertains to the transmission of power from the antenna to the transmission line, by quantifying the impedance contrast of the connection. The immediate environment has a significant effect on antenna behavior, in addition to the effects on the radiation pattern discussed in Section 2.2. In GPR surveys the impedance match changes due to variations along the survey line within the reactive near-field of the antennas. Moving a pulseEKKO™ GPR antenna from an isolated position in the air to the ground changes the impedance load resulting in an increase in the antenna's effective length, typically on the order of 20% (Dr. Annan, personal communication, 1992). This effect is observed in the data as a decrease in the centre frequency by a similar percentage. A shift m, l q = i _ in2 = i - [ZL - Z j n p |2 |ZL + Z i n p | 2 (2.58) Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 46 in centre frequency closer to 40% is commonly observed in our data sets, particularly for relatively conductive ground. Clearly the wavelet shape is affected by the antenna (load) impedance by controlling the current in the antenna. Turner's (1993) numerical modelling results illustrate the effect of antenna height on the impedance of a dipole as a function of frequency. These results indicate that when the dipole height is greater than one-tenth the free-space resonant wavelength the impedance is no longer affected by height or ground conditions. This relation makes an argument for elevating the GPR antennas to slightly more than 0.1 Ad above the surface when amplitude information is important, particularly at a site that may have significant heterogeneities within the near-field. Polarization Match (mp) The polarization match factor m p scales the receiver antenna area by the degree of match between the receive antenna polarization and incident wavelet polarization. A match of 1.0 correlates to a precise match, or copolarization. Orthogonal (or cross) polarization, eg. a horizontal linear wavelet upon a vertical linear antenna, has a match of 0.0. The power received is proportional to IV 0 CI 2, where V o c is the open circuit voltage at the antenna terminals due to an electric field incident on the antenna of effective length h. This relation is described by V o c = h * - E . (2.59) The complex conjugate effective length is required since h is defined in a transmitting reference frame whereas the reception case is considered here. The complex conjugate reverses the reference direction so that receive antenna polarization components and wave polarization have the same reference axes. The match factor is the power received normalized by the maximum possible received power. Therefore, Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION Al where b.R and e are the complex unit vectors for the effective length of the receive antenna and the incident wavelet, respectively. These unit vectors also represent the polarization states. Note that e = hx if the wavelet polarization state is not altered along the propagation path. As indicated by (2.60), the problem essentially involves the projection of the incident wavelet polarization onto the receiver antenna polarization. General expressions for hji and e may be written as e = C 0 S 7 X + el6 sin7y (2.61) and h R = cos 7R x + el6R sin 7R y (2.62) where 7 = t a n ~ 1 | ^ , (2.63) the same parameter as defined in equation (2.32) but with E= being the amplitude component measured with the receiver antenna parallel to the transmitter and E j_ being the cross component measured with the receiver antenna perpendicular to the transmitter antenna. The parameter 8 measures the phase difference by which E= leads Ej_ (Section The general expression for the match factor is derived in Chapter 5 where m p is developed as an instantaneous attribute. Obviously any alteration of the wavelet polarization will have some effect on the coupling with the receiver antenna. To avoid confusion, note the distinction between the polarization match factor and the degree of polarization that was introduced in Section Recall that the degree of polarization, or degree of coherence pxy is a cross-correlation measurement quantifying the degree to which a wave is polarized. Two similarities between these measurements provide a possible source of confusion: 1. neither parameter describes the type of polarization and 2. both parameters vary between 0.0 and 1.0. Recall that a completely unpolarized or incoherent Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 48 wave has pxy = 0.0, whereas pxy = 1.0 indicates a completely polarized wave. A value between 0.0 and 1.0 indicates a partially polarized wave. However, the match factor indicates the degree of match between two polarizations, i.e. between the receiver antenna and an incident wavelet. A perfectly polarized signal (pSig = 1.0) and a perfectly polarized antenna (pmt = 1.0) may be cross polarized to one another, resulting in m p = 0. 2.5 Wavelet Depolarization Depolarization is a well known problem in many disciplines that measure electromagnetic waves such as in satellite communication or remote sensing. The word depolarization is commonly used in this sense to mean the repolarization of the wave, usually to a more general state, such as from linear to elliptical. The choice of receiver antenna polarization can be crucial for many applications such as satellite communication and meteorological imaging. A linearly polarized signal transmitted from a satellite to the earth becomes depolarized due to Faraday rotation in the ionosphere. This depolarization varies with time, due to spacecraft motion and variations in the ionosphere, resulting in power fluctuations when received by a linearly polarized antenna. Faraday rotation rotates the polarization vector with the polarization remaining nearly linear. Although using a circularly polarized receiver antenna results in a 3 dB signal loss, the power fluctuation is avoided (Stutzman and Thiele, 1981). In another application, antenna polarization may be chosen to either enhance or diminish radar echoes from precipitation in order to either track a storm or improve radio communication (Beckmann, 1968). Generally, asymmetry between the incident wavelet and a surface or anisotropy will result in depolarization. In this section I will illustrate, by two examples, how depolarization upon reflection is the rule rather than the exception. The discussion then proceeds to outline the effects of more realistic (complex) cases of depolarization with a quick reference provided in the Appendix. Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 49 2.5.1 Depolarization Due to an Arbitrarily Dipping Reflector The treatment of reflection and refraction of electromagnetic waves usually found in standard textbooks, such as Lorrain et al. (1988), is limited to cases in a privileged coordinate system. In this privileged system the incidence plane is identical to one of the polarization planes. From this geometry the Fresnel coefficients are defined. Recall from Section 2.3.2 that reflection and refraction each involves a pair of Fresnel coefficients. A pair of coefficients treats the two end-member linear polarizations, transverse electric (TE) (coefficients R~ and T~) and transverse magnetic (TM) (coefficients R + and T + ) which are defined relative to the incidence plane. Figure 2.16 illustrates this privileged geometry for arbitrary polarization with the T M and T E electric field components labeled E + and E~, respectively. The standard limited treatment often results in an simplified perception of the general case involving arbitrary reflector geometry. Figure 2.16 Reflection and refraction of the electric field vector components of an incident wave having arbitrary polarization (after Beckmann, 1968) To illustrate the prevalence of depolarization due to reflection and refraction, the remainder of this section summarizes a primary derivation from Beckmann (1968). First the complex geometry of the problem will be introduced by a simple example. Both cases describe the polarization reflected from a smooth infinite plane; the most innocent of surfaces. Depolar-3 Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 50 ization becomes greater for rough finite surfaces and random media. The problem is to find the reflected polarization parameter (introduced in Section p2 when pi, the incident and reflected geometry, and the scatterer geometry and electrical properties are known. Of course, theoretically, this problem could be reversed to obtain material properties. The main complication due to arbitrary reflector dip arises from the Fresnel coefficients being defined in the interface coordinate system which differs from the reference, or incidence propagation, coordinate system. Figure 2.17 illustrates these two different coordinate systems. The ref-erence coordinate system is based on the propagation vector with its horizontal and vertical components in the x and z direction, respectively. The incident polarization is defined in the reference coordinate system by the vertical, E + , and horizontal, E", components. However, the T M Fresnel coefficients operate on the field component in the qj direction (Figure 2.17) defined by the intersection of the incidence and wave front planes while the T E Fresnel coefficients operate on the component tangent to the interface, t. Therefore, as Figure 2.17 illustrates, linear polarization defined by a single vertical component is actually reflected as two components that undergo different amplitude and phase changes. The resulting reflected polarization will, in general, be elliptical. In the reference coordinate system, this simple case results in the Figure 2.17 The reference coordinate system and the interface coordinate system as viewed in the wave front (wave approaching the reader). Polarization is defined in the reference coordinate system. In this case Hx is simply vertical linear polarization Ef. However, the Fresnel coefficients operate on the polarization components in the interface coordinate system, i.e. Et and E qi . Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 51 reflected field having two components, one parallel to the single-component incident field and a cross component in the E~, horizontal, direction. Although the incident polarization could be defined in the interface system, such a definition is not practical when more than one interface is involved. The following two examples from Beckmann (1968) should greatly clarify this complicated physical explanation. Changing Polarization from Vertical Linear to Horizontal Linear In this example, a perfectly conducting, smooth, infinite plane has a pure cross-dip orientation, i.e. dipping perpendicular to the propagation direction as shown in Figure 2.18. The incident wave, E i , is vertically polarized and propagates towards the reader, impinging the tilted plane near grazing incidence. Since the plane is a perfect conductor, the reflected tangential component is equal and opposite to the incident tangential component. The qj component is, in this case, normal to the plane with the reflected component equal and parallel to the incident field. Therefore, the resulting reflected field, E2, has purely horizontal linear polarization. z Figure 2.18 A perfectly conducting plane oriented with a pure cross-dip of 45° depolarizes vertical polarization to horizontal polarization (after Beckmann, 1968). Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION Reflection from a Smooth Arbitrarily Dipping Infinite Plane 52 This example will illustrate how the E " and E + components become depolarized, each reflecting with their own + and - components, i.e. parallel and cross. Ultimately a general solution for the polarization of the reflected wave will be derived in terms of the incident polarization, pi and the Fresnel coefficients. Figure 2.19 illustrates the complex geometry of the general problem. To reiterate, the reference coordinate system is defined by the horizontal and vertical components of the incident propagation vector with x in the horizontal direction and z in the vertical direction. Therefore, the propagation vector K i lies in the xz-plane. The incident electric field E i has reference components E^" and E j with e i - = y and e{+ in the xz-plane. The incident ray makes an angle &\ with the z-axis and strikes the arbitrarily dipping reflector at the reference origin. Reflector orientation is described by its unit normal, n, with Figure 2.19 also showing the k Figure 2.19 Reflector and field geometries for the general depolarization problem of reflection from a smooth infinite plane with arbitrary orientation. The ghost xyz coordinate axes on the incident wave are provided to clarify the Ei vector orientations (after Beckmann, 1968). Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 53 geologic descriptors, strike and dip. Note that the angle 0,- describes the "local" incidence angle which differs from B\. Figure 2.20 details the incidence geometry near the origin, or reflection point, and illustrates the wave front and polarization planes. Figure 2.20 also clarifies the vector directions qi and t, described earlier, upon which the Fresnel coefficients operate. Recall that q i is the unit vector along the intersection of the wave front and incidence planes and t is the unit vector tangent to the reflector, together forming an orthogonal pair. The intersection of the reflected wave front and the reflector remains along t, however the angle /?2, concluding t and e% , will be different from j3\. Figure 2.21 details the E j vector components within the wave front, i.e. a similar view as depicted in Figure 2.17. Note that the cross-dip is described by f3\ which is concluded by t and e i - = y. Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 54 Figure 2.21 Detail of vectors lying in the incident wave front (after Beckmann, 1968). As already described, the two incident field, components, and E ^ , each produce two reflected components, resulting in depolarization of each incident component. In other words, each of the + and - incident components acquires a cross polarized component upon reflection. Writing the reflected fields in terms of their parallel and cross components yields E+ = E+ + E+ (2.64) and E 2 p + E 2 c • (2.65) From these cross and parallel components, four generalized reflection coefficients are defined (2.66) E+ r++ _ 2P E+ E r r ~ = E 2p Ei^ (2.67) Rewriting equations (2.64 and 2.65) in terms of the generalized reflection coefficients and the complex polarization factor, p i = Ef /Er , yields E+ = E r ( P i r + + + r + - ) (2.68) Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 55 and E 2- = E7(T— + pir-+) . (2.69) From equations (2.68 and 2.69) the reflected polarization, p2, can be written in terms of the incident polarization and the generalized reflection coefficients P1r++ + r+-P2 = r _ _ , • (2.70) r + piT + Equation (2.70) is not the final result of this problem since the generalized reflection coefficients remain to be solved from equations (2.66 and 2.67). These solutions require the reflected cross and parallel components to be solved from the incident t and q~i components through the Fresnel reflection coefficients, R~ and R + (equations 2.39 and 2.42). We proceed by rewriting E i and E2 in terms of components upon which the Fresnel coefficients may be applied. Therefore, from Figure 2.21 Ex = E u t + E l q c u (2.71) and E 2 = E 2 t t + E 2 q c £ . (2.72) First we consider only the reflection of ET by temporarily setting E+ to 0 and (2.71) becomes Ex = ET = ETtt + E^ qT (2.73) where ETt = ET cos Pi , and Ef q = E f sin fix . (2.74) Applying the Fresnel coefficients yields E J t = R~ETt = R-Efcos /? i (2.75) and ETq = R+ETq = R + Ers in /? i . (2.76) Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 56 A solution for T~ ~ can now be acquired by considering the reflected energy that is parallel to E ^ . Since E j = 0, the reflected parallel component is given by E j p = E2 = E 2 • e V . (2.77) Applying (2.72, 2.75, and 2.76) leads to E2p = Ej-^R-cos^it-e^" + R +sin ft q£ • e V ) • (2.78) The definition of fa provides the relations t • e 2 = cos fa , and q 2 • e2 = sin fa (2.79) which, together with (2.78) and the definition of T~ ~ (equation 2.67), yields r ~ = R~ cos fa cos fa + R+ sin fix sin fa . (2.80) Similarly, with r + _ _ E 2 • e 2 + Er yields (2.81) t-.e^+ = sin fa and q 2 • e2+ = cos fa (2.82) T + = -R cos fa sin fa + R+ sin fa cos fa . (2.83) The remaining two generalized reflection coefficients r + + and V + are obtained by considering the reflection of E i = E ^ . The resulting coefficients are T + + = R~ sin fa sin fa + R + cos fa cos fa , (2.84) and r _ + = -R~ sin fa cos fa + R+ cos fa sin fa . (2.85) Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 57 Finally, substituting into (2.70) yields the general solution of depolarization upon reflection _ pi ( R - tan fa tan p2 + R + ) + R + tan Pi - R " tan p2 P 2 = R+ tan px tan p2 + R" + pi(R+ tan p2 - R ~ tan Pi) A similar solution also exists for refraction, where the transmitted wave will also generally depolarize. Insight may be gained by looking at some simplifying cases of equation (2.86). If the plane is tilted only about the y—axis, then the reflected ray remains in the incidence plane with the tilt simply adding to the incidence angle. Therefore, 03 =0 and P\= Pi = 0, simplifying (2.86) to R+ P2 = j p P i • (2-87) Note that this orientation is the privileged orientation in which the Fresnel coefficients are defined. As this equation illustrates, only the two end-member polarizations, horizontal (pi = 0) and vertical (pi = oo) maintain their polarization. These two polarizations are the characteristic polarizations. For any scatterer there is always at least one and possibly two polarizations that will not depolarize, known as the characteristic polarizations (Beckmann, 1968). For a perfect conductor (a - oo) the Fresnel reflection coefficients are R~ = -1 and R + = 1, thus simplifying (2.86) even further to P2= - P i • (2.88) This result indicates that the sense of polarization is reversed, i.e. right-handed rotation becomes left-handed. There are two incidence geometries where even noncharacteristic polarizations are not depolarized by a plane. One occurrence involves a finitely conducting plane tilted only about a single axis. Near grazing incidence on such a plane R~ = R + resulting in p2 = pi. This case is only likely to occur in GPR surveys for very steep dips since offsets are usually small. The second case occurs at "vertical" incidence when n = — K i , i.e. the wavelet is reflected back to the source. However, this case only holds for an infinite plane. Finite or curved surfaces will depolarize at vertical incidence. Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 58 In general, the GPR case involves near vertical incidence, increasingly so with depth, and therefore should experience less depolarization, particularly for primary reflections. According to recent modelling results of polarization in GPR surveys (Roberts, 1994), depolarization should decrease with depth for specular incidence less than 45°. Also the reflection point shown in Figure 2.19 will occur up dip with the reflected ray contained in the incidence plane. However, the depolarization due to targets of small radius of curvature, reflector roughness and (see Appendix) can be very significant for GPR. 2.5.2 The Effect of Scattering from a Finite Plane If the plane from the last section is finite, nonspecular scattering must be included. The geometrical optics approximation will fail when the dimensions of the plane are sufficiently small. When geometrical optics fails, physical optics usually provides a sufficient approxi-mation. Geometrical optics considers only specular reflection, whereas physical optics also includes nonspecular scattering described by a scattering diagram. The scattering diagram is essentially the radiation pattern for a particular scatterer, complete with sidelobes and nulls. The main lobe direction coincides with the specular reflection. For ease of description, assume that the finite plane is square with sides parallel to the x and y coordinate axes. For a finite horizontal plane with dimensions X and Y, the scattering diagram side lobes crowd towards the specular direction as the dimensions (X/A, Y/A) increase. As the dimensions approach infinity, or A —• 0, only the main lobe remains, having zero width, i.e. a ray as in the geometrical optics approximation. Since the polarization factor p is independent of amplitude, it remains valid even at the scattering diagram nulls where the amplitude is zero. The shape of the sheet only affects the diffraction pattern, i.e. the spatial distribution of scattered amplitudes and therefore does not affect depolarization from a sheet. A scatterer may be considered in terms of specular and nonspecular contributions. Geometrical optics is only valid when the nonspec-ular contributions are negligible. An often significant case of depolarization not handled by Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 59 geometrical optics is backscatter at non-vertical incidence. Backscatter from a plane sheet will be depolarized except for normal incidence and the two characteristic polarizations, vertical and horizontal (Beckmann, 1968). 2.5.3 Other Cases of Depolarization and Characteristic Measurements with Applications Sections 2.5.1 and 2.5.2 illustrated that for the most simplified case even geometrical optics predicts that depolarization is the rule rather than the exception. Backscatter from a plane sheet (not treated by geometrical optics) will be depolarized in most cases. A number of other depolarizers with relevance to GPR are worth noting: single finite scatterers, edges, rough surfaces, and possibly anisotropy. Cases concerning these depolarizers are described in quick reference form in the Appendix. The descriptions in the Appendix are results compiled mostly from Beckmann (1968) which serve at least two purposes. One, the prevalence of cases and their limiting conditions will become more apparent. Two, the importance of depolarization from a particular target or material might be anticipated for GPR surveys. A particularly fascinating topic is the possibility of measurable anisotropy in GPR surveys which is discussed in Section A.4.4. Once a potential occurrence is identified, the instantaneous polarization match, that I develop and apply in Chapter 5, can estimate how severely depolarization is affecting amplitudes. This instantaneous attribute also identifies the responsible depolarizing structure. Measures of depolarization can also help identify some distinguishing properties of targets such as pipes. Recent modelling by Roberts (1994) shows distinct differences in depolarization between slow velocity horizontal cylinders (metal and/or water filled pipes) and fast velocity horizontal cylinders (air filled nonmetal pipes), the latter being a better depolarizer. More specifically, the characteristic polarizations or similarly the optimal polarizations are directly related to scatterer properties. The term characteristic polarization was defined in Section (second to last paragraph). Optimal polarizations result in the minimum or maximum scattered power for a given scattering geometry (Cho, 1990). A simple example of optimal Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 60 polarization involves backscattering from a long thin pipe. Maximum backscattered power occurs when the transmitting dipole is parallel to the pipe and minimum power results for the perpendicular orientation. However, not all scatterers have optimal polarizations, which can be predicted mathematically through the scattering matrix. The scattering matrix transforms the polarization from incident to scattered, based on the scatterers properties. Clearly transverse waves carry a wealth of information in their polarization. For seismic exploration, Crampin (1985) estimates that shear waves contain nearly 4 times the information available in P-waves. This section has concentrated on the general occurrence of depolarization to identify possible effects on amplitude variation and refers to only a few applications of polarization measurements. To further illustrate the value of polarization measurements, additional applications are referenced here. Giuli (1986) provides a comprehensive review of polarization techniques for signal enhancement, improved target discrimination and resolution (particularly spectral resolution using maximum entropy methods), and target classification and identification. Gjessing (1986) considers the theoretical design of radar systems to adapt themselves to imaging particular target characteristics by optimizing the illumination and detection based on a priori information. Hopefully, more of these applications find use in GPR. 2.6 Summary: Amplitude Variation and Antenna Coupling This chapter has illustrated the potential for antenna coupling to change due to variations in the very near surface electrical properties, and due to variations in the depolarization characteristics of a target or the host medium. Changes in antenna coupling directly affect detection and resolution through antenna directivity or gain, and the impedance and polarization matches, thereby changing the maximum dip that may be imaged and the strength of subsurface reflectors. Near-surface variations could include a target within the reactive near-field of the antennas. In addition to a change in electrical contrast across a boundary, the textural or Chapter 2: ANTENNA COUPLING AND WAVELET DEPOLARIZATION 61 depolarizing characteristics of the boundary are also important. Changes in depolarization along an interface could produce a noticeable amplitude anomaly. A subtle case is feasible where the amplitude anomaly does not significantly involve the typical parallel component, i.e. the cross-component amplitude changes while the parallel component amplitude appears constant. In such a case, the anomaly would be missed in the usual single-component survey. Chapter 3 FIELD TEST SITES AND DATA ACQUISITION In order to unambiguously evaluate the success of imaging techniques, data should be acquired at well characterized sites. Such evaluations also provide insight into the imaging technique whether the technique is a geophysical method such as GPR or seismic exploration, or a processing algorithm such as deconvolution or migration. These sites should be characterized well enough that the answer to the imaging problem is, essentially, already known. Such sites may range from specifically designed targets emplaced in a simple or homogeneous host material, to geologic reality in the case of a near vertical earth cut (road cut or a gravel pit wall), ideally with material property measurements around target interfaces. To investigate the effect of depolarization on amplitudes I acquired data at two field test sites: an abandoned overpass ramp having simple structure, and a geophysics test pit having complex structure. Field experiments were also conducted on a frozen lake which allowed me to measure the polarization of our 100 MHz pulseEKKO™ IV antennas. The data sets also allowed intermediate processing steps to be evaluated. All GPR data in this thesis were acquired with the department pulseEKKO™ IV manufactured by Sensors & Software Inc. (SSI) in Mississauga, Ontario. This chapter describes the three field sites and their relevance for testing various processing steps, particularly the instantaneous polarization match. 3.1 The Abandoned Overpass Ramp 3.1.1 Site Description and Relevance The abandoned overpass ramp is located at the northwest corner of the intersection of Miller and Cesena Roads on Sea Island in Richmond, B.C. This location is at the entrance to the Vancouver International Airport. The overpass ramp was abandoned prior to being paved when traffic route plans were changed. As Figure 3.1 illustrates, the ramp road surface is 130 m 62 Chapter 3: FIELD TEST SITES AND DATA ACQUISITION 63 Plan View 1 = Soil sample and CPT 2= CPT 157.5 m Figure 3.1 Plan and cross-sectional view of the abandoned overpass ramp. Two cone penetrometer sites and two soil sample sites are located in plan view. The cone penetrometer was also instrumented with an electrical resitivity collar (after Reilly, 1992). long with a 10 m elevation gain; a dip of 4.4°. The ramp was constructed in one to two foot horizontal layers, or lifts, of fill which were compacted before dumping the next lift. Sieve analysis of a soil sample indicates that the fill is a sand and gravel with a trace of silt. A sample of the natural sediment, which underlies the ramp, was classified as a low plastic silt with a trace of fine sand (Reilly, 1992). Additionally, two cone penetrometer (CPT) sites on the ramp (located in Figure 3.1) provided in-situ electrical resistivity logs courtesy of Matt Kokan from the University of British Columbia (UBC) Department of Civil Engineering. The CPT data shown in Figure 3.2 indicate that the lifts have both a cone bearing and electrical resistivity signature of a cyclic nature. Due to the coarse fill material, the ramp drains quickly and is unlikely to ever be saturated. The water table occurs within the first metre of the level natural surface as indicated by the water level in nearby drainage ditches. Due to the fine grain Chapter 3: FIELD TEST SITES AND DATA ACQUISITION 64 U B C I N S I -r U T~ EZ S T I N G Site Location. KOKAN CPT Date B: m i l r e o 1 . r o d On Site Loc MILLER ROAD Cone Used. UBC HDC3 - RES Common t Q RCPT1 PP BT CONE BEARING RESISTIVITY RESISTIVITY Ot (bar) (oha-«> (oh»-«> Depth Increment . . 025 m Max Depth i 23. 50 ni Figure 3.2 An example of the cone penetrometer data (site 2). Ten lifts of fill are clearly identifiable within the upper 6.5 m of the cone bearing profile and have a definite influence on the electrical resistivity profile. size of the silt, however, the water table is unlikely to be a distinct boundary: being diffused, instead, due to a capillary fringe. The ramp is well suited for investigating many GPR issues. The large contrast in material properties between the fill and silt and the precisely known boundaries of the ramp make it easy to delineate in GPR images. The repetition of parallel planar lift interfaces and the pinchout of lifts at their intersection with the surface make the ramp ideal for investigating temporal resolution and testing deconvolution algorithms. With the availability of in-situ electrical Chapter 3: FIELD TEST SITES AND DATA ACQUISITION 65 properties, the reflection sequence could be accurately modeled. The planar nature of the interfaces also means that out of the plane reflections are not a problem, i.e. the structure is essentially two dimensional. The wedge nature of the ramp edge is well suited for investigating the effect of surface material thickness on antenna radiation and wave propagation. For my depolarization investigation, the ramp provides somewhat of a control case since the smooth interfaces (at 100 MHz) should have a minimal effect on wavelet polarization. Decreased polarization match was anticipated to occur at the ramp edge but the character of this anomaly and the influence of the pinchout on the anomaly was unknown. The planar 2D structure is also favorable for investigating the dependence of polarization match on offset. However, variable offset studies are limited since amplitudes should be corrected for the unknown radiation pattern. The polarization match images of the ramp were anticipated to provide insight for interpreting polarization match anomalies at sites having more complex structure. 3.1.2 Coverage of Single and Two-Component Surveys The ramp was profiled extensively with single-component surveys (Figure 3.3), whereas the two-component surveys, both T E and T M modes, concentrated on the lower ramp and ramp edge (Figure 3.4). I acquired the single-component data set before I had clearly defined a thesis Plan View 1 = Soil sample and CPT v I 2 = CPT 3 j 3 = Soil sample _L———v.- -. in 3 j_Mjin.SyiisyJ.liipi02 4- Stifles::!:;. 3 o 1 r i i 3 10 m 130 m 17.5 m Figure 33 Plan view of the ramp showing the single-component GPR coverage. Chapter 3: FIELD TEST SITES AND DATA ACQUISITION 66 2 1 3 in 3 10 m 130 m 17.5 m Figure 3.4 Plan view of the ramp indicating the two-component GPR coverage. topic and the data set is not really utilized in this thesis. However, the centre strike profile (down the middle of the road surface) is displayed in Chapter 4 as intermediate processing examples and adds to the characterization of this site. The single-component data set is described here for anyone interested in using it. Both 100 and 200 MHz data were acquired with station 0.0 m located at the top of the ramp. The data set also includes a CMP sounding located on the centre line at the 22.0 m station (Figure 3.3). Acquisition of the two-component data set was guided by 6 primary objectives: 1. characterize the polarization match of internal ramp interfaces, 2. characterize the polarization match of the natural sediment off the end of the ramp, 3. identify any structure responsible for degrading the polarization match, 4. investigate what affect a boundary such as the ramp edge has on polarization match, 5. determine if T M data is significantly more susceptible to depolarization than T E data even in such simple structure, and 6. investigate polarization match versus offset. The two-component profiles run 50 m up the ramp to include enough interface reflections in the image to easily distinguish any anomalous polarization match from their background. These profiles also include a 10 m section of the natural silt off the end of the ramp (Figure 3.4). In both the single and two-component data sets a 2.0 m antenna offset was used to avoid clipping of any amplitudes with soundings every Chapter 3: FIELD TEST SUES AND DATA ACQUISITION 67 0.4 m. Two-component CMP soundings are centred 80 m up the ramp to provide a thicker sequence of reflectors and includes one of the CPT borings. 3.2 Geophysical Test Pit 3.2.1 Site Description and Relevance Guy Cross designed and built the test pit to re-evaluate the application of shallow seismic reflection to archaeological prospecting (Cross, 1995). The pit site is located next to TRIUMF on the UBC campus. The pit was excavated to two levels (1 and 2 m depths) in a hard impermeable clay and contains a concrete block on each level (Figure 3.5). Block dimensions were chosen to resemble buried walls. The clay is a glacial till member of the Vashon Drift formation which outcrops throughout most of Vancouver (Armstrong, 1956). The pit was homogeneously backfilled with a coarse grained sand. The angular interfaces and homogeneous fill make the test pit ideal for investigating scattering effects and for testing the processing of diffraction dominated images. These discontinuous interfaces together with the dipping pit walls provide an excellent test for migration algorithms which attempt to reposition scattered energy into a focused image. The discontinuous parallel surfaces also test the temporal and spatial resolution of both antenna frequency and processing steps. Of particular concern is the resolution of instantaneous attributes such as the polarization match developed in Chapter 5. Diffractions were anticipated to exhibit depolarization and match anomalies that possibly vary with scattering geometry. A much more complex level of structure is provided by the pit in contrast to the ramp for a second test of the instantaneous polarization match. 3.2.2 Survey Coverage A two-component data set consisting of a profile and three CMP soundings was acquired in one day with constant ground conditions. On this day the pit was fully saturated as evidenced Chapter 3: FIELD TEST SITES AND DATA ACQUISITION 68 NW 0.0 2.0 CROSS-SECTION OF TEST PIT Line Distance (m) 4.0 i 6.0 _ j 8.0 10.0 SE 11.0 scale 1:1 Figure 3.5 Photograph of the concrete blocks just prior to backfilling the pit and a cross-section of the test pit (after Cross, 1995). by seepage through small puddles at the slightly lower S E end. Past experience has shown that these conditions yield the best images. The lower radar velocity of the saturated fill provides optimum positioning of reflections, particularly from the top of the shallow block (Gerlitz, 1993). The profile traverses the middle of the test pit coincident with the cross-section of Figure 3.5. For the profile an antenna offset of 0.6 m and station spacing of 0.05 m was adopted from Gerlitz (1993). The C M P soundings are centred at the 3, 6, and 10 m stations with maximum offsets of 4.2, 4.0, and 3.0 m, respectively (Figure 3.6). A l l three C M P soundings have a 0.4 m Chapter 3: FIELD TEST SITES AND DATA ACQUISITION 69 initial offset with offset increasing by 0.1 m steps. CMP locations were chosen to investigate the effects of various structural elements on reflections with varying offset. cement Figure 3.6 Coverage of the three CMP soundings along the profile line. 3.3 Alice Lake 3.3.1 Site Description and Objectives A number of experiments were performed on Alice Lake when the lake was covered with 12 cm of ice during an exceptionally cold spell in January, 1993. This small lake is located about 11 km north of Squamish, B.C. in Alice Lake Provincial Park (Figure 3.7). Spot depths were measured with a weight on a fishing line for depth calibration at 25 m intervals along a line running 125 m out, perpendicular, from the north shore near the docks. At 25 m from shore the lake depth is 5.3 m and gradually deepens to 10.8 m at the 125 m mark. A 165 m profile indicates that the lake bottom is smoothly varying beyond 35 m from the shore and dips only 1.8° beyond the 90 m mark. In general, lakes provide excellent data for analyzing target related amplitude variation since the near-field antenna coupling is essentially constant and the intervening water layer is usually very homogeneous. The primary experiment at the lake was the recording of a single quadrant polarization rose. 3.3.2 Profile and Polarization Experiment The profile was acquired primarily to characterize the lake bottom and choose a suitable location for the rose sounding. A location was chosen where the lake bottom is relatively Chapter 3: FIELD TEST SITES AND DATA ACQUISITION 70 smooth and dipping only 1.8° therefore minimizing scattering and hence any change in wavelet polarization at the lake bottom. The T E polarization rose was acquired by rotating the receiver antenna in 5° increments from a parallel to a perpendicular position relative to the transmitting antenna (Figure 2.14). Five soundings were recorded at each 5° interval. The lake bottom reflection is expected to extinguish for perfect dipole antennas oriented in the cross position. Squamish 7 km Figure 3.7 Alice Lake is a small provincial park located about 11 km north of Squamish on B.C. Highway 99. Chapter 3: FIELD TEST SITES AND DATA ACQUISITION 71 This polarization rose was primarily acquired to quantify any deviation from the assumed linear polarization of our pulseEKKO™ 100 MHz antennas. The cross and parallel polarization measurements were later utilized as a calibration for the instantaneous polarization match. 3.4 Advice For Acquiring Two-Component Data There are a few helpful points worth mentioning regarding two-component GPR surveys. Ideally the data will be acquired with a polarimetric receiver antenna which consists of an orthogonal pair of dipoles on separate channels of the receiver. I know of one polarimetric GPR which is built by Ohio State University (Roberts, 1994). Lacking a polarimetric receiver the survey must be conducted for each component. A time period must be chosen over which the soil conditions will not change during acquisition of all components. Although the next point is fairly obvious it is worth emphasizing. Polarization reciprocity (Sections 2.3.3 and 2.3.4) requires that different cross components must be acquired for T E and T M surveys, i.e. TE cross and T M cross are different components (see Figure 2.14). Finally, when acquiring a CMP sounding nobody should be standing between or around the antennas, or at least their position should be the same relative to the antennas for each recording. As I discovered by accident, the cross component is very sensitive to the location of people around the antennas. Figure 3.8 illustrates the effect of my assistant, Jane Rea, standing alternately next to the transmitter and receiver while acquiring a CMP sounding at the ramp. This sounding prompted us to perform another experiment where the antennas were offset by 5.4 m and Jane stepped from the transmitter to the receiver in 0.2 m increments. The effect of Jane's position between and around the antennas is illustrated in Figure 3.9. Finally, there are two processing considerations for two-component data: time correlation between components, and all processing steps must be applied to both components equally. These processing issues are discussed in detail in Chapter 5. Chapter 3: FIELD TEST SITES AND DATA ACQUISITION 72 Trace Number 0ol4 - j | Figure 3.8 Cross component data is very sensitive to the position of objects around the antennas. The alternating character of traces 4 through 16 was caused by my field assistant, Jane Rea, standing next to the receiver during one sounding then next to the transmitter during the following sounding. During soundings 17 through 26 Jane stood next to the same antenna for a few consecutive soundings to confirm the source of the character change. Trace Number 5 10 15 20 25 30 35 40 45 50 55 GO G5 70 75 80 Figure 3.9 The effect of Jane Rea's position on the received cross signal. While the antennas were offset 5.4 m Jane moved from the transmitter to the receiver in 0.2 m steps starting 0.2 m from the transmitter. This experiment was repeated but with Jane continuing around the crossed receiver as shown for traces 57 through 65. Traces 66 - 84 were acquired without Jane's presence. Chapter 4 "STANDARD" PROCESSING AND SIGNAL CHARACTER The cost/benefit to many industry GPR surveys usually dictates very minimal processing of the data. However, applications such as stratigraphic and contaminant imaging often require advanced processing techniques in order to satisfy the survey objectives. Advanced techniques are applied to enhance certain data components or to obtain quantitative measurements such as velocity depth functions or the distribution of signal attributes (Annan, 1993). Fortunately, kinematic similarities between the seismic reflection and GPR methods allow the direct application of most seismic reflection processing techniques to GPR data. The term kinematic implies only the geometrical occurrence and propagation of reflected, or scattered, energy and thus the similarity breaks down for details of amplitude and phase variation. Since the late 1970's seismic reflection processing techniques have been successfully applied to GPR (Fisher, 1991). Limits of this applicability will be discussed in Section 4.1 including a review of spatial aliasing and its prevalence in typical GPR surveys—a limit to the success of some processing applications. Although the spatial sampling problem is well understood in the seismic exploration community it often receives little consideration or is even misunderstood by some of the GPR community. The fidelity of some processing steps is also addressed as a consideration for amplitude (or character) analysis. After demonstrating solutions to data problems specific to GPR in Section 4.2, a standard processing stream is recommended and applied, in Section 4.3, to profiles from the ramp and test pit. Additionally, instantaneous attributes are introduced in Section 4.4 as an interpretational tool for relating reflection character to the subsurface and as a basis for the instantaneous polarization match analysis of the following chapter. 73 Chapter 4: "STANDARD" PROCESSING AND SIGNAL CHARACTER 74 4.1 Seismic Reflection Processing Applied to GPR Data Probably the most comprehensive demonstration to date of the applicability of seismic reflection processing to GPR data was given by Fisher et al. (1992). Fisher acquired a GPR profile in the standard seismic shot gather format, i.e. a single shot recorded at multiple offsets, upon which a standard seismic processing sequence was applied: filtering, static corrections, CMP sorting, velocity analysis, normal- and dip- moveout corrections, stacking and depth migration. Note, however, that Fisher's processing stream does not include deconvolution since the direct application of seismic deconvolution algorithms to GPR data usually yields minimal to insignificant improvement. The limited success has primarily been in removing multiples (LaFle'che et al., 1991). The failure is due to the GPR wavelet being mixed phase and much more nonstationary than the seismic wavelet. Solutions to the GPR deconvolution problem must address these wavelet traits as does the approach of Turner's propagation deconvolution (Turner, 1993; Turner, 1994) which indeed appears to be a significant improvement over the direct seismic adoption. Although a multi-channel GPR profile (Fisher et al., 1992) offers improved depth of penetration, lower noise, and continuous velocity control, such acquisition with present systems is rarely cost effective over single-channel common offset profiling with CMP soundings only at suspected lateral changes in velocity. To convert the time section into a depth section a number of velocity depth functions might be required along the profile, the distribution of velocity control depending on the suspected heterogeneity. Reasonably accurate velocity coverage is also required to successfully apply migration. Migration is applied to GPR data to unclutter the image from diffraction hyperbolas and to generate a more accurate representation of subsurface structure by collapsing diffraction hyperbolas and repositioning dipping reflections to their true location (steepening dips and migrating energy up dip). Since most migration algorithms assume zero offset profiles, the data should be corrected for the constant offset, i.e. normal moveout (NMO) corrected. However, for Chapter 4: "STANDARD" PROCESSING AND SIGNAL CHARACTER 75 the typical offsets used in GPR profiling (about 1 m or less) NMO correction only significantly affects arrivals from depths of less than a metre (Gerlitz, 1993). Usually a more serious migration requirement is spatial aliasing, which does not seem to be sufficiently appreciated by many GPR practitioners. Spatial aliasing affects dipping wavefronts such as from diffractions or dipping structure. Aliasing causes dips to change sign and possibly decreases the dip depending on the severity of aliasing (Yilmaz, 1987). For an interpretation of nonmigrated data, aliasing is not a problem unless the structure becomes visually incoherent. However, dipping events that have a visual degree of spatial aliasing, yet remain visually coherent, have a significant upper portion of their frequency spectrum affected by spatial aliasing. Migration will reposition this affected energy in the wrong direction, thereby blurring the image (Yilmaz, 1987). Migration artifacts are not the only hazard of spatial aliasing as other processing methods are also susceptible such as FK filtering. Two of the best solutions to spatial aliasing are interpolating traces, and decreasing the station spacing in acquisition. A sufficient station spacing is easily calculated which is worth demonstrating here to illustrate the degree of aliasing in most discretely sounded GPR data sets. GPR profiles often exhibit zones of severe spatial aliasing which appear as incoherent arrivals. However, even dipping features that are clearly identifiable visually may still be spatially aliased at the higher frequencies. The degree of spatial aliasing can be determined by either viewing the data in the FK domain or by a simple calculation for survey planning. Viewing the data in the F K domain clearly reveals the frequency and wavenumber beyond which spatial aliasing occurs, and therefore this assessment is recommended whenever considering processing steps that are sensitive to spatial aliasing (Yilmaz, 1987). When deciding on a station spacing for discrete soundings the spatial resolution requirements of the survey must be considered. Although many practitioners seem to favor (somewhat arbitrarily) 1 m station spacing (at 100 MHz), I typically use a spacing of 0.5 m or less. I sometimes use a coarser Chapter 4: "STANDARD" PROCESSING AND SIGNAL CHARACTER 76 spacing when the resolution of steeper wavefronts is less important than coverage, particularly in reconnaissance. A 0.5 m station spacing at least provides visual coherency for most structure. A station spacing that avoids spatial aliasing even at the higher frequencies may be determined from A X = At V • 0 W 4/max Sin 0 by anticipating a suitable velocity v, maximum frequency fmax, and dip of interest 0 (Yilmaz, 1987). Table 4.1 indicates the maximum allowable station spacing in order to avoid spatial aliasing at the wavelet's highest frequency for a dip of 45° or 8.84 ns/trace assuming a velocity of 0.08 m/ns. Note that a decrease in velocity requires a decrease in station spacing since the time dip (ns/trc) will be steeper. Table 4.1 clearly illustrates that most discretely sounded GPR surveys alias even shallower dips, and dips of 45° are aliased throughout a significant upper portion of the spectrum. "Continuously" sounded surveys often yield a trace about every 0.2 m or less, depending on the number of soundings stacked into a single trace and the rate at which the antenna is towed. ANTENNA (MHz) MAXIMUM FREQUENCY STATION SPACING (m) 50 75 0.38 100 150 0.19 200 300 0.094 300 450 0.063 Table 4.1 Station spacing required to avoid aliasing a 45° dip (8.84 ns/trc at a velocity of 0.08 m/ns) up to the expected maximum frequency for the given antenna. F m a x is assumed to be 1.5 times the antenna centre frequency for the pulseEKKO™ IV (Annan and Cosway, 1991). When pursuing amplitude analysis, processing methods must be considered more carefully since many of the seismic processing steps do not preserve amplitudes even in their seismic reflection application. A couple of notable examples are gaining and migration. Obviously the Chapter 4: "STANDARD" PROCESSING AND SIGNAL CHARACTER 77 chosen gain function should increase smoothly and steadily with time with a consistent and predictable application across the traces. Since A G C gain (Automatic Gain Control) destroys relative amplitude variation it should never be applied to data before amplitude analysis. Some recommended gains will be discussed in Section 4.2.3. A less obvious fidelity problem concerns migration. The primary amplitude problem with migration is that the theory requires infinite aperture in both time and space whereas the images are finite and therefore provide incomplete amplitude information. Generally, migration is used only to provide a focused image of structural geometry and amplitude analysis is restricted to nonmigrated data. A poor choice of processing parameters can produce artifacts from most processing steps, especially when applying advanced methods such as migration. The text by Yilmaz (1987) provides an excellent overview of both the theoretical and practical sides of seismic reflection processing. 4.2 GPR Specific Processing Problems A couple of GPR data problems that routinely require processing are relatively uncommon or unknown in seismic reflection data. These problems are clearly visible in the raw GPR profile of Figure 4.1: time 0 drift, and a time decaying trace bias known as wow which appears throughout the trace as grey shading. Often the pulseEKKO™ IV traces also have a dc shift. This section demonstrates high fidelity solutions to these problems, thereby allowing subsequent character analysis. Additionally a theoretically-based gaining criteria for GPR will be discussed. 4.2.1 Time-Zero Correction Raw pulseEKKO™ data always exhibits an upward drift in time-zero, the first arrival of transmitted energy (Figure 4.1). To avoid a large logarithmic drift during the survey, the pulseEKKO™ IV manual recommends waiting 20 minutes for the control unit to stabilize before acquiring data. My experience indicates that a 30 minute waiting period is necessary to avoid having the time-zero drift off the top of the screen. Even after this stabilization Chapter 4: "STANDARD" PROCESSING AND SIGNAL CHARACTER 78 Distance (m) 0 1 2 3 4 5 6 7 8 9 1 0 1 1 0 c O O 0 • 0 2 0 a 0 4 0 c 0 6 co 0 • 0 8 CD EE j— 0 u 1 0 $ 0 a 1 2 Two- 0 a 1 4 0 a 1 6 0 • 1 8 0 = 2 0 0 • 2 2 Figure 4.1 Raw pulseEKKO™ GPR profiles exhibit a drift in time zero and prominent dc shift. Raw GPR data from any system also exhibits some amount of wow. A gain must be chosen to reveal reflections beyond a travel time of only 70 ns, in this profile. period a small upward drift in time-zero still occurs. In fact, to avoid drifting off the plot, the p u l s e E K K O ™ positions time-zero, within a certain percentage of the full time scale, down from the top of the plot. The time that the pulse is actually transmitted (transmit time-zero) is found by subtracting the air wave travel time (assuming an air wave velocity of 300 m///s) from the first break time on the plot. The time correlating to zero depth is obtained by adding the ground wave travel time to the transmit time-zero. First arrivals could be leveled to actual time (transmit time-zero equal to zero on the plot), or to some arbitrary time which preserves the record of the pre-arrival noise level. A number of simple methods exist to level the time drift. A n obvious quick choice is a semblance based static shift. A major problem with this approach is that the first pulse does not Chapter 4: "STANDARD" PROCESSING AND SIGNAL CHARACTER 79 have a constant character along the profile, i.e. amplitude, phase and frequency may change. This variable character is primarily due to the direct ground wave interfering with the direct air wave. This interference varies due to surface velocity and also due to coupling changes along the survey line. Since a semblance approach tries to maximize coherency between traces, the peak of the first lobe will tend to align causing the first breaks to drift up or down depending on interference character. The best method is to pick the first break and shift it to some desired time. This solution is of course time consuming. Fortunately the time drift is slowly varying (logarithmically) upward and becomes almost linear as the survey proceeds. Therefore, first break picks are only required at the end of straight line segments of the nearly linear drift curve. In the best case, only the first and last traces require picking. Time corrections are then simply interpolated between the pick times. This method is especially quick for data acquired after the 30 minute stabilizing time of the control unit. 4.2.2 Dewow Filtering and Wow Variation with Near-Surface Structure Origin and Character of Wow The time decaying trace bias observed in all raw GPR (and RES) data (Figure 4.1), known as wow, is not completely understood, but is known to vary in magnitude and in decay rate depending on near-field structure. As illustrated in Figure 4.2 wow strength also decreases with increasing offset between transmitter and receiver. The wow component of the trace begins abruptly within the very large amplitude interference, of the direct air and ground arrivals then slowly decays with time appearing similar to the decay curve of an induced polarization measurement. Dr. Barry Narod (UBC Dept. of Geophysics & Astronomy) claims that the wow is explained entirely (including the variation in magnitude with near-surface structure) as the saturation of the receiver electronics (personal communication, 1992). Dr. Peter Annan, of Sensors & Software Inc., believes, however, that the wow also consists of a telluric response and that the receiver saturation is perhaps the smaller contribution (personal communication, Chapter 4: "STANDARD" PROCESSING AND SIGNAL CHARACTER 80 0.4 Offset (m) 2 3 0.5 Figure 43, A raw CMP sounding clearly exhibits the decrease of wow with offset due to a decrease in amplitude of the large direct arrivals that saturate the receiver electronics. The wow component is illustrated as the dashed curve on the enlarged trace. Note the sudden onset of the wow. (after Gerlitz et al., 1993). 1992). Both arguments predict a larger wow in more conductive ground. Indeed, I have observed that wow magnitude and decay rate tend to be greater in more conductive ground and that this signature is affected by objects in the near-field as illustrated by wow profiles from the ramp (Figure 4.3) and the test pit (Figure 4.4). At the ramp there is such a large contrast in wow magnitude between the lossy silts and resistive ramp fill that the wow in the ramp is often barely visible at the scale of Figure 4.3. In the test pit, the wow is clearly affected by near-surface structure. However a consistent correlation between wow character and structure is not obvious (Figure 4.4). A noticeably larger wow occurs over the southeast edge of the shallow block next to the deep part of the pit and also at the seepage puddles and where the clay wall shallows to the surface at the southeast end. However, this correlation is not always consistent since the north edge of the pit exhibits only slightly elevated wow amplitude. Wow variation in both profiles is very similar to that in the same profiles acquired in previous Chapter 4: "STANDARD" PROCESSING AND SIGNAL CHARACTER 81 Distance (m) 138 132 126 120 114 108 102 96 90 84 Figure 43 Wow profile traversing the edge of the abandoned overpass ramp. The dynamic range of this plot is insufficient to see both the large wow in the silt and the relatively small wow in the ramp fill. years. This repeatability proves that these wow variations are not simply due to a change in instrument performance with time but rather due to a change in near-surface material. Dewow Filtering Sensors & Software Inc. (SSI) plotting routines automatically apply a running residual mean filter (invisible to the user) to remove the wow, however this filter is far from being the best solution. Although an obvious treatment, this filter introduces extra lobes throughout the trace and destroys amplitude variation. These extra lobes are especially undesirable when studying the relations between subsurface structure and the GPR waveform. A filter of high fidelity is required for amplitude and polarization studies. A high pass filter might seem worth considering but is doomed by the high frequency onset of the wow signature and also produces artifacts throughout the trace (Gerlitz et al., 1993). Fortunately the median filter fulfils our requirements by extracting blocky shaped components of the signal, i.e. components having Chapter 4: "STANDARD" PROCESSING AND SIGNAL CHARACTER 82 Distance (m) 4 5 6 7 10 1 I Figure 4.4 Wow profile of the test pit. The pattern of wow strength and duration appears somewhat correlated to near-surface structure. Compare to the pit cross-section of Figure 3.5. The stepped appearance of the amplitude decay is due to a lack of plotting resolution. The decay is actually smooth. sharp onset (positive or negative) but otherwise slowly varying. The utility of the median filter for dewowing radar profiles was, to my knowledge, first recognized by Guy Cross (1987; Clarke et al., 1989). The scale of the slowly varying signal that is extracted by the running median depends on the filter length which is defined as 2n+l samples (Leany and Ulrych, 1992). With proper selection of filter length, the median trace is a reasonable approximation of the wow. Therefore, to dewow G P R data, the median trace is simply subtracted from the input trace, i.e. a running residual median filter (Clarke et al., 1989). Choosing the filter length requires some trial runs: too short a filter length results in a median trace containing too much signal, while too long a filter does not remove enough wow. Fortunately the apparent overabundance of signal in the Chapter 4: "STANDARD" PROCESSING AND SIGNAL CHARACTER 83 median trace can also be alleviated by median filtering the median trace. The result of a few iterations of median filtering with the same window length converges to a trace consisting of blocks, or series of equal values, of at least n+1 samples duration known as the root signal (Leaney and Ulrych, 1992). The root signal is not affected by additional iterations and is devoid of blocks having lengths less than n+1 samples. Although we do not want the root signal for our filter, a second iteration does decrease the risk of obtaining too much signal by removing small scale features in the initial median trace. A small amount of signal-like features may remain after the second iteration which could be due to the possibility of the receiver electronics becoming resaturated by a sufficiently large reflection that arrives before the initial wow decays significantly. Although the difference between the first and second iteration is usually small, I believe that the two iteration median is a safer choice. The superior fidelity of the residual median filter is illustrated in Figure 4.5 which compares raw data and dewow results from mean (SSI) and median-based filters. Note that the median-based filter also removes the commonly occurring dc noise; the longest duration block. The only negative aspect of this filter is the extra effort required to choose an optimum filter length, i.e. a few trial runs are necessary. My experience indicates that the optimum filter length for 100 and 200 MHz data typically varies from about 2n+l = 25 to 51 points with the shorter window being required for areas of greater attenuation. In some cases, however, even 15 points may be necessary. An empirical criterion based on the dominant low frequency signature of the wow in the amplitude spectrum (Gerlitz et al, 1993) has proven inadequate at various sites. The lack of success of this criterion is not entirely surprising since the wow also contains some energy at high frequencies due to the sharp onset. Filter length is best chosen from a few trial runs, however current limited experience seems to indicate that these trials can be concentrated to a narrower range of potential lengths based on ground type. Higher attenuating ground appears to produce wow of greater magnitude but with a more rapid initial decay rate, therefore requiring a shorter filter Chapter 4: "STANDARD" PROCESSING AND SIGNAL CHARACTER 84 (STJ) awn Aeyw - OMI Chapter 4: "STANDARD" PROCESSING AND SIGNAL CHARACTER 85 length. Sites that have significant lateral changes in attenuation rates may require different filter lengths for different sets of traces. Fortunately, this added effort rarely seems necessary since an average filter length usually provides adequate results overall. An example of such a case is the edge of the abandoned overpass ramp (Figure 4.3). Hopefully, further investigation will yield a less subjective criterion. 4.2.3 Ga in ing Cr i t e r i a Signal strength rapidly decays in GPR data and therefore a gain is required to view the image. The choice of gain depends on the objectives of subsequent analysis. As already mentioned near the end of Section 4.1 an A G C gain is unsuited for amplitude studies since A G C destroys amplitude variation. Depending on the objective, any predictably smooth gain function that is applied consistently should suffice for amplitude studies. One such gain is simply ty, the power y usually being about 2.0 or greater. For most seismic data t2 works well (Claerbout, 1985) whereas the higher attenuation rate of the GPR signal usually requires a power of at least 2 but less than 3. A theoretically-based gain, such as SSI's spreading and exponential compensation (SEC), compensates for spherical divergence and may also include an average intrinsic attenuation term. Using A G C and SEC gains One of the most popular gain types is instantaneous A G C which is applied merely to reveal any presence of signal by boosting amplitudes to some common level. This gain is useful during acquisition (applied to the display, only) since the limit of signal penetration is clearly observed regardless of changes in material properties along the survey line. A G C commonly involves measuring the rms amplitude within a window of a specified length then determining a multiplier that gains the rms value to a specified amplitude level which is then applied to the sample at the middle of the window. The instantaneous A G C operates with a Chapter 4: "STANDARD" PROCESSING AND SIGNAL CHARACTER 86 sliding window. Variations of AGC also exist: measuring the window amplitude level as the mean or median value, and applying the multiplier to the nth sample instead of the window midpoint. Another type of AGC calculates and applies the multiplier within fixed time gates instead of a sliding window (Yilmaz, 1987). The user must exercise caution in choosing the window length. Too short a window, a fast AGC, results in all arrivals having amplitude levels that are too similar, giving the data the appearance of white noise. Too long a window produces undergained shadows about large amplitude events since the mean amplitude is biased by large events within the window. For seismic data, the AGC window often contains 10 to 30 cycles. However, an entire GPR trace is often only about 10 pulse widths or less. For GPR data, an AGC window of only 2 or 3 pulse widths works surprisingly well. For various reasons one may desire to apply a theoretically-based gain instead of a more simple ty function. One such case is in preparing data for deconvolution where a geometrical spreading compensation and possibly an exponential gain is applied (Yilmaz, 1987). Theoretically-based gains for GPR are based on the propagation terms in the radar range equation; the gain function having the general form g(t) = te+at (4.2) thereby including both geometrical spreading and exponential attenuation, where a is the attenuation parameter. In practice a reference time must be used since (4.2) will diminish amplitudes for t < 1. The SEC gain in SSI's processing package uses the pulse width TW as a reference time in (1 + T/Tw) eAT r > 0 g(*) = { ( « ) 1 T < 0 where r = (t - (Tw + t0)) , to = time zero, A = a v/8.69 (units = dB/m) and 8.69 is the conversion factor to decibels (8.69 = 20 log 1 0 e), v = estimated radar wave velocity, and a has the units 1/m. The required Chapter 4: "STANDARD" PROCESSING.AND SIGNAL CHARACTER 87 input parameters v and A can be estimated from the list of material properties in Table 4.2. To keep the gain function from becoming extremely large there are two methods of imposing a ceiling value: 1. a maximum gain value determined automatically from the pre time-zero noise level, or 2. a constant maximum gain value provided by the user. The automatically determined maximum is calculated as the value that will increase the average pre-arrival noise level to a specified fraction of the full plotting width. The specified fraction is usually between 0.01 and 0.1. Since the automatic option results in a variable maximum gain that may produce artificial lateral variation in amplitudes, it is not a good feature for amplitude studies. MATERIAL DIELECTRIC CONSTANT CONDUCTIVITY (mS/m) VELOCITY (m/ns) A (dB/m) Air 1 0 0.3 0 Distilled Water 81 0.01 0.033 0.002 Fresh Water 81 0.5 0.033 0.1 Sea Water 81 30,000 0.01 1,000 Dry Sand 3-5 0.01 0.15 0.01 Sat. Sand 20-30 0.1-1.0 0.6 0.03-0.3 Limestone 4-8 0.5-2 0.12 0.4-1 Shale 5-15 1-100 0.9 1-100 Silts 5-30 1-100 0.7 1-100 Clays 5-40 2-1,000 0.6 1-300 Granite 4-6 0.01-1 0.13 0.01-1 Dry Salt 5-6 0.01-1 0.13 0.01-1 Ice 3-4 0.01 0.16 0.01 Table 42 Some material properties relevant to GPR providing a guide for estimating velocity and attenuation values for the exponential gains SEC and EXPGAN.F (after Davis and Annan, 1989). An Offset Dependent SEC Gain: EXPGAN.F The SEC gain is limited to a constant offset and therefore cannot be used to gain CMP soundings. Since I was interested in at least a crude first order observation of amplitude and polarization variation with offset I programed an offset dependent version of SEC called Chapter 4: "STANDARD" PROCESSING AND SIGNAL CHARACTER 88 EXPGAN.F. In addition to allowing variable offset, EXPGAN.F can also use a velocity-depth function (actually v m s (m//is) and time in fis). Instead of using the pulsewidth as a reference time, EXPGAN.F uses the far-field limit of the given antenna for both a reference offset and time (depth). I favored this reference criterion because the range equation is only valid for the far-field. However this criterion is not always an improvement over using the pulsewidth. Since the direct air wave is a head wave with an amplitude decay of 1/r2 (Section 2.2) a different gain criterion is used for air arrival times. Usually we are not interested in the direct air wave, particularly in profiles. Moreover there is sometimes an advantage to purposely undergaining or diminishing the large air wave amplitude thereby reseating the plot to better reveal reflections. Therefore, EXPGAN.F provides three options for gaining the air wave: multiplying by r 2 , r, or a constant chosen by the user. Finally, the maximum gain level is also optional to eliminate the risk of introducing gain artifacts in the data. This unrestricted gain does not become too large for soundings of shallow penetration. If the maximum gain option is chosen and the limit is reached, EXPGAN.F notifies the user and writes to a file the trace number, noise level, sample number, gain limit, and peak gain. Figure 4.6 illustrates a CMP sounding before and after applying EXPGAN.F and the gain function calculated with velocity decreasing with depth. One disadvantage of theoretically-based gains is they often require trial runs of different input parameters. Table 4.2 is only a guide and an effective attenuation value is often higher than one might first assume since the effective attenuation also involves scattering losses. Through experience with different data sets, I have noticed that often once an effective attenuation value is chosen, the latest times seem to be overgained with the appearance of the gain function increasing too rapidly. This overgaining has also been observed by Prof. Garry Clarke (UBC Dept. of Geophysics & Astronomy) in glacier sounding data where the geometrical spreading was estimated empirically to be only r~°A instead of r - 1 0 . Some possible causes of this apparent decrease in spherical spreading include geometrical focusing by geologic structure (Prof. G. Chapter 4: "STANDARD" PROCESSING AND SIGNAL CHARACTER 89 NO GAIN Offset (m) 6 12 18 24 GAINED Offset (m) G 12 18 24 EXPGAN FUNCTION Offset (m) 6 12 18 24 Figure 4.6 The gaining of a CMP sounding from the ramp with EXPGAN.F. Note how the time at which the gain traces overlap decreases with increasing offset due to using a velocity function that decrease with depth. A constant attenuation of 1 dB/m was used. Air arrivals are gained simply as offset squared. Note that the stepped appearance of the gain function is only a plotting artifact due to a lack of resolution: the function is indeed smooth. K. Clarke, personal communication, 1995) and possibly focusing of the radiation pattern from the near-field into the far-field (Section 4.3 A Recommended Standard GPR Processing Stream Although, for many cases, the current SSI processing package provides a sufficient assort-ment of software, I prefer the greater control and flexibility provided by my own software combined with a complete seismic reflection processing package. Particularly, one major prob-lem with the SSI package is the residual mean dewowing filter which is automatically applied to the plots. The artifacts from the SSI dewowing filter (Section make it unaccept-able for stratigraphic and amplitude studies. For my investigations the department's seismic reflection processing package, Insight™ by Landmark-ITA (Inverse Theory & Applications), Chapter 4: "STANDARD" PROCESSING AND SIGNAL CHARACTER 90 combined with the algorithms discussed in the last section (Section 4.2), provided the neces-sary control and flexibility over processing problems. This section will illustrate the effect of various processing steps on data from the ramp and test pit. A standard GPR processing stream is illustrated schematically in Figure 4.7. Dewowing is recommended before the time drift correction since large dc shifts can sometimes complicate first break picking, depending on the picking software. If a theoretically-based gain is used, then a velocity analysis done beforehand will obviously provide better velocities than Table 4.2 and the measured velocities can guide the choice of an attenuation value from Table 4.2. Gained sections usually exhibit significant high frequency noise, especially near the end of the trace. Therefore a low pass filter is applied after gaining which also makes the effects of the filter (including any artifacts) more obvious than if the filter was applied before the noise is gained. Since elevation corrections on GPR profiles are often on the order of the depth of penetration, migration should be performed (if necessary) before applying elevation corrections. Quality Control Deleting redundant traces, splice broken profiles, etc. T Dewow Residual Median Filter I Time Drift Correction Gain AGC CMP Sounding Velocity Analysis on AGC gained CMP • at least surface velocity Filtering Low Pass ? I Migration if necessary I Elevation Correction if necessary Figure 4.7 A standard processing stream for GPR data. Note that many steps are optional. Chapter 4: "STANDARD" PROCESSING AND SIGNAL CHARACTER 91 Many of these steps are optional and in a consulting situation may not be worth the cost in time. Indeed some of these steps are excluded from the processing of the ramp, and test pit. 4.3.1 Processing the Ramp 100 MHz Profile The profile discussed here traverses the length of the ramp down the centre line and consists of excellent quality data. The ramp profile appears as dipping reflections since the profile is not corrected for elevation change which would rotate the reflection to their proper horizontal position. Time zero drift is clearly observed from right to left in the raw profile of Figure 4.8 as the profile was acquired beginning at the top of the ramp (station 0.0 m). Wow is not clearly evident at this plotting scale, but does exist throughout the profile, being much more a problem in the silt (stations 128.8 —140.0). The upper third of the ramp (stations < 48 m) exhibits an apparent elevated attenuation which in part is likely due to a negative dc shift. Most of this elevated attenuation does indeed remain after dewow filtering (Figure 4.9). The profile in Figure 4.9 has also been time drift corrected. Additionally, a much larger trace overlap was used to better illustrate the effect of gaining (in the next step) on faint reflections. If this trace overlap was used for the raw data plot, many features would be difficult to see since the profile would be mostly black due to the wow. The optimum median filter length for the ramp dewow is 2n+l = 41 points. Finally EXPGAN.F gain was applied for Figure 4.10. The offset dependent exponential gain used the surface velocity of 103 m/fis (measured from a CMP sounding) for a constant velocity depth function, an effective attenuation equal to 0.05 dB/m, and a constant antenna offset of 2.0 m. Velocity measurements on internal reflections indicate that Vjms is fairly constant in the ramp. The cyclic nature exhibited by the cone penetrometer data (Figure 3.2) suggests that velocity may also be cyclic within the lifts, decreasing from top to bottom with a discontinuous increase across lift boundaries back to the higher velocity. This cyclic nature would not be revealed by v ^ measurements since they provide an average velocity down to the reflector which, in this case, is fairly constant. The smooth varying gain Chapter 4: "STANDARD" PROCESSING AND SIGNAL CHARACTER C D -CSJ o . CD ro C\J co o . co 8 c S CQ Q to CD CSI. CO co o CO CD CO o • CO o -o CSJ • CD CSJ • CM CO -CO ro-o CM CD CO o CS) CD CO o O o O o o o a D • D o D a a o O o o o o o o o o (sri) eiuiiAeM-oMi Chapter 4: "STANDARD" PROCESSING AND SIGNAL CHARACTER 93 (STI) eiwLABM-OMI Chapter 4: "STANDARD" PROCESSING AND SIGNAL CHARACTER Chapter 4: "STANDARD" PROCESSING AND SIGNAL CHARACTER 95 confirms the suspected attenuating zone within the ramp and also reveals two relatively large amplitude basal reflections (Figure 4.10). Very faint subhorizontal reflections are evident within the silt beneath the lower ramp where the signal fades rapidly due to the substantially greater attenuation of the silt. Although the internal structure of the ramp is primarily planar, the final profile does reveal regions of apparent excessive settlement. 4.3.2 Processing the Test Pit 200 M H z Profile Incremental improvements to the image due to processing are more obvious for the test pit profile than the ramp profile probably due to the higher attenuation rate and relatively larger wow exhibited by the raw test pit profile (Figure 4.1). Wow is clearly evident in the raw data as the grey shaded trace bias with wow duration increasing across the pit from left to right. The profile of Figure 4.11 was dewowed with a 31 point filter and was also time drift corrected. As explained in Section 4.3.1 on the ramD. a larger trace overlap was also used Distance (m) Figure 4.11 The profile of Figure 4.1 dewowed with a 31 point median filter and time drift corrected. Chapter 4: "STANDARD" PROCESSING AND SIGNAL CHARACTER 96 in plotting Figure 4.11. At this stage a number of features are clearly identifiable, especially when compared with the cross-section of Figure 3.5. The northwest wall reflection emerges through the strong interference of the direct ground and air waves at about 1.5 m and 0.03 us and slopes to the pit shelf reflection at 0.05 p,s. A reflection from the top of the shallow block emerges through the strong interference at 0.03 us between about 3.25 and 5 metres. A reflection from the top of the lower block occurs at 0.06 J^S between 6 and 7.5 metres. The sloping southeast wall appears to intersect the lower block due to a diffraction. Other diffractions appear faintly at all corners within the pit. Gaining with EXPGAN.F reveals the reflection from the bottom of the lower block at about 0.08 fis (Figure 4.12). Diffractions are now prevalent with 2.5 m offsets from the lower block. Very high attenuation of the host clay dissipates the signal almost immediately. Lack of any signal after 0.14 us is confirmed by the 1 CD Distance (m) 4 5 6 7 8 10 11 Figure 4.12 The profile of Figure 4.11 after exponential gaining using a velocity of 63 m/ps, an attenuation of 2.0 dB/m for an antenna offset of 0.6 m. Chapter 4: "STANDARD" PROCESSING AND SIGNAL CHARACTER 97 dominance of random noise revealed through sufficient gaining. The gain used a velocity of 60 m//j,s, an attenuation of 2.0 dB/m with an antenna offset of 0.6 m. Finally, Figure 4.13 illustrates the result of phase migration (from an Insight™ module) to demonstrate the worth of migration on this obvious case. Migration was performed very simply with only a single velocity for the entire image: the velocity of the pit fill equal to 60 m/fis. An obvious improvement is the collapse of diffraction hyperbolas to their scattering source, thereby uncluttering the image. The dipping pit walls have also been steepened. Since the south wall reflection and diffraction no longer interfere with the lower block these structures are now more easily identified. A few very localized features around the walls and below the lower block appear to be possibly overmigrated thus indicating their location within the clay which has a significantly lower velocity. Upward crossing hyperbolas from the lower half of the plot demonstrate the effect Distance (m) 4 5 6 7 8 Figure 4.13 The phase migrated result of migrating Figure 4.12 at a single velocity of 60 mips: the velocity of the pit fill. Since the gained profile was not low pass filtered prior to migrating, upward hyperbolic streaks are prevalent in the lower half of the image. Chapter 4: "STANDARD" PROCESSING AND SIGNAL CHARACTER 98 of high frequency random noise on migration; similar to the impulse response of migration. Clearly, data should be low pass filtered before migration to minimize this streaking which could obscure weak reflections. Although the migrated image cannot be used for amplitude analysis (end of Section 4.1) the true structure is much more evident. 4.4 Processing to Interpret Signal Variation: Instantaneous Attributes As discussed in various sections thus far, the character of a reflection, and especially the variation of its character in profile, contains a wealth of information about the reflecting surface and, to some extent, the intervening medium. Before analyzing this variation, however, the data must be processed in a manner that preserves amplitude and phase as discussed in previous sections of this chapter. Usually signal character refers to instantaneous amplitude, phase, and perhaps frequency, collectively the most commonly considered of instantaneous attributes. However, signal character also includes a fourth attribute which receives relatively litde attention: polarization. Much of the analysis of reflection variation within radar profiles, however, has focused on the power reflection coefficient (PRC) usually measured distinctly at basal and internal reflections within glaciers (Narod et al., 1988; Clarke et al., 1989). The PRC is the ratio of reflected power to incident power measured in dB. However, before making these distinct measurements, reflections of interest must be identified, usually an easier task in glacier profiles than in ground profiles which consist of many more reflections. In fact, it is often the instantaneous attributes that help identify reflections of interest. Additionally, instantaneous attributes, when color-coded, allow the interpreter to view their spatial relationships and changes within subsurface structures. Of course the PRC could also be calculated as an instantaneous attribute as illustrated by Prager (1983). However, instantaneous measures of reflection strength are more useful when combined with other attributes such as phase and frequency. It is the combination of amplitude and phase that defines the primary character of a signal (Lendzionowski et al., 1990; White, 1990). An advantage of instantaneous amplitude over the Chapter 4: "STANDARD" PROCESSING AND SIGNAL CHARACTER 99 PRC is that the amplitude is paired with phase and frequency through simple calculations from the analytic, or complex, trace (Taner et al., 1979). Since a primary aim of this thesis is to identify effects of anomalous polarization on reflection strength, I have employed instantaneous amplitude and phase in generating the instantaneous polarization match: the topic of the next chapter. Therefore, the remainder of this section briefly discusses the calculation and physical significance of instantaneous attributes. In expressing a wiggle trace (as in seismic or GPR data) as an analytic signal the recorded data is the real component and the imaginary, or quadrature, component is simply the data trace phase shifted by 90°. This quadrature trace is generated by the Hilbert transform of the real data trace (Bracewell, 1965; Taner et al., 1979; Yilmaz, 1987) with the resulting analytic signal given by a(t) = d(t) + iH[d(t)} = d(t) + d(t) L 7T (4.4) where * denotes convolution. The instantaneous amplitude and phase follow direcdy from the analytic signal as expressed in polar form a(t) = i?( t )e i $ W , (4.5) where the amplitude is given by R(t) = ^Jd\t) + q\t) , (4.6) and the phase is $(*) = t a n - 1 (4.7) where q(t) is the quadrature trace. Frequency is obtained by taking the time derivative of the instantaneous phase. Apparent polarity is the sign of d(t) at the local maximum of R(t) (Taner et al., 1979). Chapter 4: "STANDARD" PROCESSING AND SIGNAL CHARACTER 100 From the complex trace, amplitude and phase are effectively separated, i.e. the two attributes are independent of each other. Frequency and apparent polarity, however, are both derived from amplitude and phase. Instantaneous amplitude describes the amplitude of the whole wavelet, not the rise and fall in amplitude of the individual lobes. When a wavelet is phase rotated the envelope remains unchanged while the wavelet lobe peaks move along the envelope. Therefore, the peaks of the real wiggle trace often do not coincide with the local maximum of the envelope. To illustrate, a sinusoid, by definition, has a constant instantaneous amplitude or envelope. Indeed, the very location of the envelope maximum within a reflection (the corresponding phase value) provides a description of character (Taner et al., 1979). Instantaneous phase describes the wavelet's rotation at an instant in time thereby aiding the trace-to-trace correlation of peaks, troughs or apparent first arrivals. Since phase is independent of amplitude, phase sections also reveal the coherency of weak events and clarify discontinuities such as faults, pinchouts, and unconformities. Since frequency is the derivative of phase, frequency sections may exhibit a large degree of variation, being very susceptible to noise and interference effects. Frequency is especially unstable at small amplitudes, due to the dominance of noise, but is fairly stable when the amplitude is large. The more erratic frequency values, which correlate to envelope minimums, are often unrealistically large or even negative, emphasizing the loose relation to actual frequency (White, 1991). A smoothed frequency, such as weighted by amplitude or energy, helps to stabilize the measurement (White, 1991). Fortunately additional smoothing, and an emphasis of lateral continuity, is effected by the discrete nature of color coded attribute values. Similarly, apparent polarity also often lacks a clear physical correlation (to the reflection coefficient) due to interference and noise with the added assumption of a zero-phase wavelet (Taner et al., 1979). When measured at the envelope maximum of an isolated reflection, the amplitude provides the reflection's Chapter 4: "STANDARD" PROCESSING AND SIGNAL CHARACTER 101 magnitude, the phase describes its polarity and symmetry, and the frequency approximates the centre frequency (White, 1991). Despite their pitfalls, instantaneous attributes are useful interpretive tools that may be directly related to physical properties when noise and interference are not too dominant or at least when the interference effect is recognizable. For interference dominated reflections, even frequency can at least help identify lateral changes in interference due, for example, to changes in the thickness of thin beds or to pinchouts (Taner et al., 1979). Instantaneous attributes are most useful when combined (at envelope maxima) such as in mapping reflection character which has demonstrated success with seismic data in delineating gas reservoirs (Lendzionowski et al., 1990; White, 1991). Character mapping should also prove useful in GPR applications such as in mapping contaminant plumes. Although my investigation focuses on amplitude variation (using phase only in calculating polarization match), clearly various applications of the other attributes should prove useful in GPR applications. Chapter 5 INSTANTANEOUS POLARIZATION MATCH 5.1 Introduction The potential for amplitude variation in GPR data to yield information about material properties via modelling or inversion is widely known. However, commonly considered material properties, such as water saturation, are not always directly responsible for a significant amplitude change. Since radar waves are transverse waves, the usual analogy of GPR and seismic reflection should be made more specifically with shear waves, thus including the appropriate polarization related phenomena. Significant amplitude variations may arise due to a change in a wavelet's ability to couple with the receiver. Before one can reliably infer material properties from amplitudes, the magnitude of variations in coupling and the radar system must be considered. This chapter characterizes two such factors, polarization match factor (mp) and transmitter output. The primary objective of this chapter is to investigate the severity with which depolarization affects amplitudes in the standard single (parallel) component survey. Recall from Section 2.5 that in much of the polarization/scattering literature the use of the word depolarization implies the altering of the initial polarization — in this case the linearly polarized state. Instantaneous polarization match will identify zones of received power deficiencies that result from anomalous wavelet polarization. This information is portrayed by color-coding the radar image as the match between wavelet and receiver polarizations. A general equation is derived for the polarization match which includes antenna terms allowing one to make fewer assumptions regarding antenna polarization. The resulting equation is calibrated to our pulseEKKO™ IV 100 MHz antennas by utilizing two-component recordings of a lake bottom reflection. Polarization estimates are usually covariance based (Kanasewich, 1981; Macbeth and Crampin, 1991; Perelberg and Hornbostel, 1994). However, I chose to base these first GPR polarization match estimates on 102 Chapter 5: INSTANTANEOUS POLARIZATION MATCH 103 complex trace analysis: an efficient and less computationally expensive alternative (Rene' et al., 1986). Since the calculation of instantaneous m p uses single samples from two-component data (transmit and receive antennas parallel and crossed) the results can be very susceptible to noise. This problem is alleviated by minimizing noise in the input data with low pass filtering and by median filtering the resulting m p traces. I investigated three potential sources of error: the affect of timing errors between the two components, the stability of the pulseEKKO™ IV system, and random noise. The merits of a chosen color-coding scheme for instantaneous m p sections will be discussed with regard to the uncertainty and magnitude histograms. Finally, color m p profiles and CMP soundings, and their interpretations, from the ramp and test pit are discussed. In closing, the implications of these results on more traditional (scalar) amplitude modelling is considered and some possible corrections are proposed. 5.2 Generalized Polarization Match Polarization match is defined simply (equation 2.60, restated here as 5.1) from the dot product of the antenna polarization unit vector (IIR) with the incident wavelet polarization unit where the * denotes the complex conjugate (Stutzman and Thiele, 1981). Most GPR antennas are assumed to be linear dipoles, where the wavelet radiated within the main lobe of the radiation pattern is linearly polarized as in Figure 5.1. If the transmitter and receiver antennas are directed at each other, maximum power transfer will occur when the antennas are parallel, since their polarizations are perfectly matched in this geometry. This maximum transfer assumes identical antennas and that the wavelet polarization is not altered during propagation. In this case, the wavelet arrives at the receiver antenna with its E vector parallel to the antenna, a perfect match (mp = 1). If the antennas are oriented orthogonally, then no signal will be vector (e) 2 (5.1) Chapter 5: INSTANTANEOUS POLARIZATION MATCH 104 received since the wavelet and receiver polarizations are mutually crossed, i.e. m p = 0. If, however, the wavelet polarization is altered along the raypath, then the received power will increase in this orthogonal component while decreasing in the parallel component. Random scattering is the dominant mechanism that depolarizes the wavelet. However, as was shown in Section 2.5, depolarization also occurs for most cases of reflection and refraction—to some degree—even for a smooth dipping infinite planar interface. Figure 5.1 illustrates this general matching problem and the polarization parameters used in calculating polarization match. Tx Radiated Wavelet Polarization • Generalized Altered Wavelet Polarization polarization altered along raypath % = tilt of ellipse 5 = phase by which y-comp. (E J leads the x-comp. (EJ Y = t a r i ^ E J E J e = instantaneous E vector Wg - semi-major E component E/ = semi-minor E component Figure 5.1 Diagramatical relationship between general antenna and wavelet polarizations. The polarization of the antenna pair is described by the receiver as hR. Wavelet polarization is described by e. A linear dipole transmitter antenna (Tx) radiates a linearly polarized wavelet. Scattering along the raypath may alter the wavelet polarization, shown here as elliptical: the most generalized case. Imperfect polarization of the antenna pair may be addressed by also describing the receiver antenna polarization—generally—as elliptical. The detail of the polarization ellipse illustrates the relation between the two-component data (the parallel and perpendicular components) and the polarization parameters r and 7 , and the major and minor components of the ellipse. Note that 7 is an angle describing the relative maximum amplitudes of the measured components and equals the tilt r only for linear polarization. Note, also, that E= is defined along the y-axis and Ej_ is along the x-axis of the survey reference frame. Chapter 5: INSTANTANEOUS POLARIZATION MATCH 105 To maintain generality and minimize assumptions about the antenna polarization, the general equations of wavelet and antenna polarizations (2.61, and 2.62) rewritten here, e = cos 7 x + s in 7 elS y = cos 7 x + (s in 7 cos 8 + i s in 7 s in 8)y , (5.2) and — - * ^ hR, = C O S 7 R X + (s in7R.cosc% - i s i n 7 R s in c%)y , (5.3) will be applied to equation (5.1). For two-component data, the parameter 8 is the phase by which the parallel component E= leads the perpendicular component Ej_. From equation (2.63) 7 is an angle describing the relative strengths of the orthogonal electric field vectors, 7 = t a n - 1 f ^ . (5.4) Recall that 7 is only equal to the tilt (T) for the special case of linear polarization (8 = 0° or 180°, or when 7 = 0° or 90°) (Figures 5.1, and 2.10—where E 2 = E= and E i = E ± ) . Applying the general definitions given by (5.2) and (5.3) to (5.1) yields m p = I cos 7 R cos 7 + s in 7 R COS 6R s in 7 cos 8 + s in 7 R s in 6 R s i n 7 s in 8 + i [ s k i 7 R C o s < ! > R s i n 7 s i n 6 — s in 7 R s in <5R s in 7 cos 8] | 2 = I cos 7 R cos 7 + s in 7 R s in 7 (cos (8 — 8\\f) + i\sm 7 R s in 7 (s in (8 — £ R ) ) ] | 2 which becomes m p = cos 7 R C O S 7 + 2 cos 7 R cos 7 s i n 7 R s in7 ( cos (8 — 8R)) + . 0 . 0 sin 7# sin 7 . By using the square of function-product trigonometric relations, (5.6) simplifies to 1 1 m p = - cos 2 (TR - 7) + - cos 2 ( 7 R + 7) + ( 2 s i n 7 R C O S 7 R ) ( s i n 7 c o s 7 ) cos (8 — 8R) which is further reduced to the final result by using the double-angle relations for sin m p = \ [ c o s 2 (7B. - 7) + c ° s 2 ( 7 R + 7) + s i n (2TR) s in (27) cos (8 - <5R)] (5.5) (5.6) (5.7) (5.8) Chapter 5: INSTANTANEOUS POLARIZATION MATCH 106 where 7R and <SR will be utilized as calibration terms for the pulseEKKO™ IV antennas. Figure 2.10 illustrates how 7 and 8 describe polarization. In the case of perfect linear dipoles Ej_ = 0, resulting in 7 ^ = 90°. If our antennas are assumed to be perfect linear dipoles, then (5.8) simplifies to m p = sin2 7 . (5.9) 5.3 Required Processing Stream Throughout the data preparation stream, the most important requirement is to process all traces equally. Much of the processing stream is commonly applied to single-component GPR data (Chapter 4) with some restrictions and a couple of additional steps (Figure 5.2). First one must ensure that the trace pairs from the two components correlate to the same station. The next two steps are to dewow the traces and correct any timing errors. Since the m p calculations depend entirely on the relative amplitude and phase between components, a de wo wing filter that does not distort the signal is essential. Therefore, dewowing requires a residual median filter, whereas the more commonly applied residual average or bandpass filters should be avoided (Gerlitz, et al., 1993; Section When flattening the time zero drift, preserving a reasonable pre-arrival segment is suggested for future noise level assessment. The timing of the first subsurface event must be matched between components. In most cases, matching the first breaks is unreliable due to the orthogonal antenna being in the near-field of the transmitting antenna resulting in arrival time variation across the orthogonal antenna. Events between components may be reliably correlated by viewing the data as envelope and instantaneous phase traces as well as in wiggle trace format, after applying an A G C gain (AGC applied only for this correlation exercise). The envelope trace is often the most helpful since confusing phase changes are avoided. After aligning the zero times between the components, gaining should be accomplished with a smooth varying gain that operates on both components equally. This Chapter 5: INSTANTANEOUS POLARIZATION MATCH 107 QUALITY CONTROL: check trace correspondence between components DEWOW: Residual Median Filter STATIC CORRECTIONS: a) flatten time drift b) match timing between components , V , GAIN: spreading exponential compensation (SEC) OR time to some power DO NOT USE AGC j FILTERING: low pass r , FROM THE ANALYTIC SIGNAL: Instantaneous phase and amplitude (envelope) PMCH.FOR: polarization match program • MEDIAN FILTER: (2n+1 = pulse/4) .-. 100MHz => 2n+1=9,200MHz =>2n+1=5 r DISPLAY: color contour pmch w/ wiggle trace overlay of the parallel component Figure 5.2 Processing stream for instantaneous polarization match. criteria disqualifies AGC. Some suggested gains are a geometrical spreading exponential gain (SSI's SEC, or my EXPGAN.F), or a ty gain. Ideally deconvolution should follow the gaining, as the improved temporal resolution and multiple suppression would improve the resolution of the resulting m p traces. If deconvolution is performed then the preceding gain should probably be geometrical spreading only, or geometrical spreading and frequency dependent gain. This Chapter 5: INSTANTANEOUS POLARIZATION MATCH 108 pre-deconvolution gaining criteria is a common philosophy in seismic processing that addresses the nonstationarity of the wavelet (Yilmaz, 1987). Any additional gain would be applied after deconvolution. Unfortunately deconvolution of GPR data is to date somewhat experimental as mentioned in Section 4.1. Therefore, I excluded deconvolution from the processing stream. After the final gain is applied, low pass filtering is often necessary to reduce the high frequency noise enhanced by gaining and deconvolution. This filter should be designed to minimize artifacts. Calculating m p requires the instantaneous maximum amplitude (envelope) and phase (Section 4.4). The envelope traces also aid interpretation of the m p traces. My programs (PMCH_CAL.F and PMCH.F) apply equation (5.8) to instantaneous amplitude and phase trace files generated by the Insight™ modules IAMP and IPHS. As will be discussed in Section 5.4.1, the resulting m p traces should be median filtered with a filter length (2n+l samples) equal to about 1/4 the wavelet duration. 5.4 Calibration of the Instantaneous Polarization Match Realistically there is no reason to believe that our GPR antennas are perfect linear dipoles. Since the goal is to identify changes in the wavelet's transmitted polarization, an attempt was made to compensate for the antenna's deviation from an assumed perfect polarization. Such a calibration is readily permitted with the general equation (5.8) that allows for arbitrary polarization of both antennas and the incident wavelet. Therefore, the calibration problem is really a matter of estimating the polarization of the transmitter-receiver antenna pair as 7R and £R (Figure 5.1). Estimating the antenna polarization requires a calibration experiment to record a two-component reflection from a simple target (nearly planar and flat) in a homogeneous nonscattering medium such as air or water. Figure 5.3 displays the single quadrant polarization rose, acquired on Alice Lake, that is utilized for the calibration. The location of the rose sounding was chosen where the lake bottom is relatively smooth and flat to minimize scattering and hence any change in wavelet polarization at the lake bottom. Chapter 5: INSTANTANEOUS POLARIZATION MATCH 109 Jx Rx Receive Antenna Orientation (degrees) | \7 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 o Q o-i—||||i||||||||||||i|||m 0 • 1 - I H44rf H^TTTTI-K "I- -Figure S3 A single quadrant polarization rose (100 MHz) from Alice Lake, BC. These data have been dewowed, time zero drift corrected, and gained by EXPGAN.F. The receiver antenna was rotated from 0° (parallel) at the left side of the section to 90° at the right in 5° intervals, with 5 soundings at each orientation. The horizontal axis gives the antenna orientation in degrees from parallel. The Alice Lake polarization rose (Figure 5.3) exhibits a number of features worth noting which yield insightful, although predictable, results in the m p traces. Note that the first arrivals were arbitrarily positioned at 0.15 fis. Encouragingly, the lake bottom reflection, at 0.8 p,s two-way travel time, diminishes to noise level at the crossed orientation. Another important observation is that the number of events occurring immediately after the direct arrivals (0.18-0.40 us) increases as the receiver is rotated to 90°. These early events were probably multiply scattered within the 12 cm surface layer of ice, with the receiver becoming more sensitive to side scattered arrivals as it approaches the orthogonal orientation. Similar events (0.69-0.76 /is) occur just above the lake bottom reflection with marginally larger amplitudes at the crossed orientation. These deeper events are probably side scattered arrivals from lake bottom objects. The m p values of all of these features will be discussed in Section 5.4.4. Of course the calibration is only concerned with the lake bottom reflection (0.80-0.83 us). Chapter 5: INSTANTANEOUS POLARIZATION MATCH 110 5.4.1 Outline of Assumptions and Procedure Primarily five assumptions are made in using the Alice Lake data to estimate the polarization of our GPR. 1. Antenna polarization does not change with time. 2. The surface ice, water, and lake bottom do not impose any polarization change on the wavelet reflected from the lake bottom, i.e. 7 = 90°, and 6 = 0°. Therefore, antennas that are perfectly polarized should result in a lake bottom reflection with m p = 1.0. Since imperfect antennas will have some measurable signal in the cross component of the lake bottom reflection, the associated m p value will be less than 1.0. 3. Reciprocity holds and the transmitter and receiver antennas are identical. Therefore, any imperfections in the antennas may be combined into one pair of polarization terms, 7R and <5R, for the receiver (as implied in defining mp). 4. Assumption 3 allows for the assumption that the original wavelet polarization is perfectly linear. 5. Side scattering is not significant. These data are perhaps less than ideal for this calibration, since assumption 5 and parts of 2 may be violated to some extent. I am reasonably confident that the surface ice does not noticeably affect the returned lake bottom pulse, however, a repeat experiment when the lake is ice free would confirm this claim. Another uncertainty is the origin of the arrivals 0.1 //s before the primary lake bottom reflection. The most reasonable interpretation is out of the plane scatterers beginning about 1.5 meters above the lake bottom. Perhaps the greatest problem is the unknown conditions of the lake bottom which return significant energy after 0.7 fis through Chapter 5: INSTANTANEOUS POLARIZATION MATCH 111 the lake bottom time in the cross component. Despite these shortcomings, a nearly perfect polarization match occurs for the lake bottom reflection even for noncalibrated m p . The calibration proceeds by using the 5 traces at 0 and 90 degrees rotation from the polarization rose to generate 5 instantaneous m p traces. My program PMCH_CAL.F also generates files containing intermediate calculated traces of 7 and 6 including an average trace for each of the three parameters which are displayed in Figure 5.4. The antenna polarization values are estimated within the lake bottom reflection as the mean value of the 7 and S average traces. These estimates, and other trial values, were tested as calibration constants 7R and CSR in the program PMCH.F. Unfortunately, the resulting 7R, SR (and mp) traces are contaminated with spiky noise, though with obvious trends down the trace (Figure 5.5). This spiky nature is due to making Relative Magnitude: y Trace Pai r * 1 .2 3 4 5 Ave. Phase Difference: 6 Trace Pair # 1 2 3 4 5 JL Trace width scaled to 90° Trace width scaled to 180° Ave. X Figure 5.4 Median filtered traces (for display only) of the intermediate calculations 7 and 6 and their average traces. Recall that 7 = 90° and 6 = 0° indicates perfect linear polarization as would be transmitted if the transmitter antenna was perfect. The lake bottom reflection occurs between 0.8 and 0.83 fis. Chapter 5: INSTANTANEOUS POLARIZATION MATCH 112 o "ca •o CD CD as CD o "cO CO M _oi5 o Q_ "co Q CO a cz o CO £= o _ Q "cO O a> c o o_ E o o $ o O ca . a . CD o ca cu c o CL E o O J L 1.11 . X. L.JlL , ^ - ^ J - V - J V ^ ^~^ v^^ v MMwV'w'^ MYrV'Vll ^ v^Vtoto* ^ w /^w ^v f^tu ilmftrvN^ Wfr^ fVnFri f l i i J L I L iliiliJ. Alliduj illJlill hh4 o CM CO i n CO CO o a o o • • D a D • • • O O O o o o O o o o -1—1 (sri ) euijiABM-OMl I tl a, B 3 Cu ui .9 8 a I •s a * ca CA O 8 •g > o 5 3. eo d p § a I I a J= I J •a « I -JC o ^ 00 a a 1 8 a I 12 •» d vi Cti CA 3 ^ U 2 a ta a 3 >/•> s d >> 3 8 1 I .9 ^  § a o a .2 ^  2 a £ « •a ca a a 8 •a M H .a T3 p I WD « > a « d . •i -i I * D , a * u a <s « a 3 1 Chapter 5: INSTANTANEOUS POLARIZATION MATCH 113 single point calculations which is very sensitive to noise. A median filter effectively reveals the amplitude trends by removing spikes within a particular scale length (as explained in Section After testing various filter lengths with one to three iterations, the best results were obtained with a 9 point filter (0.0072 pis) with two iterations. This filter length is roughly 1/4 the duration of the typical pulseEKKO™ 100 MHz wavelet and removes spikes with durations less than 0.004 us (5 samples, or about 2/15 a wavelet) (Figure 5.5). Both median filtered and unfiltered traces were analyzed for estimating the mean polarization values within the lake bottom reflection, with the calibration constants taken from the filtered traces. 5.4.2 Plotting 7R , 6R, and m p as Shaded Traces (Trace Scaling) Polarization match plotted as shaded traces (+ side shaded) should have a trace width of 1.0 so that a match of 1 touches the zero line of the adjacent trace to the right as in Figure 5.5. To ensure the 0 to 1 trace width, the first sample of each trace is set to 1. Similarly, PMCH_CAL.FOR outputs intermediate calculations of 7 and 6 traces with their first samples set to 90° and 180°, respectively. After median filtering, the scaling sample must be reset. This practice restricts adjacent large match valued features from interfering across adjacent traces and also allows the values to be read, approximately, by eye. However, shaded trace plots can only be reasonably interpreted for extremely simple structure, such as lake bottom topography, and discerning a change in match of 0.10 remains difficult. Interpreting the simplest m p structure with shaded traces is usually impractical. As with any other instantaneous attribute the preferred plotting method is color coding. A suggested color coding scheme is discussed in Section 5.7 after the uncertainty analysis. Despite the drawbacks, trace shading is easier for generating plots quickly and was effective for assessing calibration values. 5.4.3 Estimating Constant Values for 7R and 6R A major problem in estimating 7R and 6R is that the first assumption is violated, i.e. 7R is mostly constant, but S\\ varies throughout the reflection time interval (Figure 5.4). This Chapter 5: INSTANTANEOUS POLARIZATION MATCH 114 problem is mostly explained by the fact that we are comparing the phase of a signal at noise level to the phase of a strong signal. This situation does not allow for a constant SR since the phase of noise is rapidly varying and the phase of no signal should be taken as zero. The instantaneous phase of the polarization rose (Figure 5.6) reveals that a phase jump emerges half way through the lake bottom signal at about the 60° antenna angle^  The magnitude of this phase jump increases to approximately 90° for receiver angles of 80° - 90°. However, the phase during the first half of the orthogonal reflection roughly tracks the phase for the parallel recording, i.e. SR « 0°. For angles greater than 90° the phase difference jumps to near 180°. Since obtoning a constant value for SR is fraught with problems and fairly meaningless, I decided to test the sensitivity of m p traces to this parameter. This test was accomplished by setting S - SR to 0° and 90° (shown in Figure 5.7), thereby eliminating the phase dependence of equation (5.8). The resulting m p traces are very similar to the calibrated and noncalibrated traces though with match values slightly increased (compare Figure 5.7 with the calibrated and noncalibrated results in Figure 5.8). Based upon the observations on phase noted in Figure 5.6, Receive Antenna Orientation (degrees) | 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 I I I I I I l l l I I I i i i i Figure 5.6 Instantaneous phase of the Alice Lake polarization rose. The lake bottom reflection is clearly visible at 0.8-0.82 ps. Beyond an antenna angle of 60° the lake bottom phase begins to split. At 90° the phase initially tracks the more parallel soundings, however, noise is also significant. There are five traces for each antenna orientation at increments of 5°. Chapter 5: INSTANTANEOUS POLARIZATION MATCH 115 one could argue the choice of SR = 90°. Match traces resulting from SR = 0° and SR = 90° are very similar as Figure 5.8 illustrates. Close inspection reveals that SR = 0° traces maintain a slightly more constant match value through the lake bottom time interval. Therefore, SR = 0° is favored over SR = 90°. As Figures 5.7 and 5.8 illustrate, the sensitivity of m p to S is minimal and therefore the choice of SR is not critical. The calibration analysis demonstrates that the pulseEKKO™ IV antennas are fairly close to being perfect linear dipoles with their polarization being best compensated when 7R = 85° and SR = 0°. This preferred value for 7 R yields the highest and most constant match values within the lake bottom reflection interval. However, this is a minor improvement over the noncalibrated case which assumes perfect linear polarization where 7 R = 90°. co CD ca COS(5-5r) = 0.0 Trace Pair # 1 2 3 4 5 Ave. COS(8-Sr) = 1.0 Trace Pair # 1 2 3 4 5 Ave. O • 0 O • 2 5 0 . 3 Trace width scaled to 1.0 Figure 5.7 Median filtered polarization match calculated with the phase term cos(<5 - <5R) = 0° and 90°. The calculation also included the preferred value for 7R = 85°. These traces are only marginally different from the calibrated traces with 5R = 0° shown in Figure 5.8. Chapter 5: INSTANTANEOUS POLARIZATION MATCH 116 (STt) 811111 Chapter 5: INSTANTANEOUS POLARIZATION MATCH 117 5.4.4 Features in the m p Calibration Traces The Alice Lake data proved especially useful since the features described in Section 5.4 illustrate various causes of match values and some potential ambiguities. Match values associated with strong signal in the parallel component persist on all 5 traces (Figure 5.5). The primary reflection from the lake bottom results in a match that is prominently very nearly 1.0 throughout the reflection time interval (0.80-0.83 us). Within the time interval 0.18-0.40 us, the greater energy in the cross component, due to multiple scattering in the surface ice layer, results in match values on the order of 0.10. Since this anomaly does not originate within the water column where it appears, it illustrates the importance of referring to the amplitude variation in the input data when interpreting the match anomalies. A successful application of predictive deconvolution will remove these multiples, thereby avoiding this pitfall in the m p traces. Although no signal exists from 0.4—0.68 us, match values vary about 0.5 due to both components having comparable amplitudes, at noise level. Values in this interval may vary considerably since they are dominated by random noise (examine the variance trace in Figure 5.5). Since the deeper scattered events (0.69-0.76 us) have marginally larger amplitudes in the cross component, some m p traces (particularly traces 2 and 3) exhibit a noticeable decrease in match, but remain near 0.5. These two occurrences of values in the 0.5 neighborhood illustrate an ambiguity in ascribing a source to this value range. This ambiguity was a major consideration in devising a color contouring scheme. A weak reflection at 0.87 p,s produces match values on the order of 0.75 which, unfortunately, is not always distinguishable from some noise events, particularly on trace 4. The average, or stacked, trace enhances these features with the 0.75 match event being easily recognizable and the interval containing no signal (0.4-0.68 us) is quickly converging to 0.5 Chapter 5: INSTANTANEOUS POLARIZATION MATCH 118 5.5 Uncertainties in m p Traces Three primary sources of uncertainty in the polarization match estimate are timing errors, variable system performance, and random noise. Timing errors between the two components could produce significant errors in the m p traces, however, this problem is avoidable with careful time drift corrections and inspection of timing between components. The variation in system performance was investigated with multiple soundings at a single station. These variations are measurable, but usually not visible in the plots of multiple soundings. Fortunately for GPR, random noise is usually much less a problem than in seismic data, but can be significant* -for weak signal as the lake data illustrate. Noise is mostly a problem in the cross component since the signal is usually weak. Random noise and timing errors are the largest contributors to uncertainty in polarization match estimates. 5.5.1 Effect of Timing Errors Between the Two Components The magnitude of timing error necessary to noticeably alter the match trace was investigated by applying increasingly larger static shifts to the perpendicular component of a lake sounding. Figure 5.9 shows the difference in resulting m p values of the delayed component cases from the zero delay case with negative differences colored in blues and positive differences in reds. Differences within + 0.05 match values are colored white. Notice that the events having more extreme m p values are mostly immune to this error (coded white), however significant artifacts may occur immediately adjacent (before and after) to these events. Three such examples of immunity occur at 0.16-0.18 p,s (large match), 0.25-0.4 us (very low match), and fortunately the lake bottom reflection 0.80-0.83 us. The large errors adjacent to these events may be greater than ±0.4, however their duration is less than 0.01 us for timing errors of 10 samples (0.008 us) increasing to very significant durations of 0.05 /J,S for timing errors of 50 samples (0.04 us). Some errors exhibit a periodicity probably related to cycle skipping, with error magnitude becoming largest at intervals of 10 to 15 delay samples (0.008-0.012 us). Figure 5.9 indicates Chapter 5: INSTANTANEOUS POLARIZATION MATCH 119 that the m p traces are not, for the most part, adversely affected if the timing error is less than the order of 0.004 p,s (5 samples or about 13% of the wavelet length) for 100 MHz data. Although a timing error of 0.004 /J,S is usually obvious, errors twice this size can escape scrutiny due to cycle skipping and an uncertain time zero in the cross component. As mentioned in Section 5.3, the first break time of the cross component is not necessarily a reliable reference for correlating time with the parallel component. Particularly at near offsets, the physical geometry of the antenna pair results in a near-field effect where the first arrival of energy in the cross component arrives earlier than for the parallel component. This timing problem arises because the recording geometry places one end of the orthogonal antenna significandy closer to the parallel antenna than the distance between their midpoints. Therefore, the first arrival time varies across the orthogonal antenna. A near-surface reflection is usually a reliable reference for the time correlation. Correlating events between components is often more reliable using the envelope traces as recommended in Section 5.3. Chapter 5: INSTANTANEOUS POLARIZATION MATCH 120 Sensitivity of Match Values to Timing Error Polarization Match Difference (from 0 Delay) - 0 o 4 0 - 0 n 3 0 - 0 G 2 0 - O o 10 - 0 o 0 5 0 . 0 5 0 a 10 0 D 2 0 0 D 3 0 0 Q 4 0 0 a 5 0 Delay of Cross Component (# of samples) 0 10 2 0 3 0 4 0 5 0 Figure 5.9 The error in match value as a function of the time shift applied to the cross component is colored. Note that the sampling interval is 0.0008 fis. The parallel component wiggle trace is repeated to serve as a reference. Blues indicate underestimates of mp (negative errors) and reds indicate positive errors, whereas white indicates errors of < ±0.05 match values. Note that the 0 delay trace is white as well as events that result in strongly matched or mismatched values. These errors were calculated as the difference of "delayed" mp traces from the zero delay m p trace. The mp traces were median filtered before the subtraction. Chapter 5: INSTANTANEOUS POLARIZATION MATCH 121 5.5.2 Uncertainty Due to Time Varying System Performance To investigate the reliability of the signal strength recorded by the pulseEKKO IV, a statistical analysis was performed on multiple soundings. This analysis was also applied to the Alice Lake calibration on both the input and output traces to investigate the propagation of uncertainties. Time Test Data To assess the amplitude consistency of the pulseEKKO™ IV, two time tests were acquired at the ramp and one at the test pit. Time tests are multiple soundings at a single station. The ramp time tests are of different time scales and antenna orientations. The test with the antennas parallel contains 132 traces acquired over 44 minutes yielding 20 seconds between recordings. The second test, with the antennas perpendicular, contains 42 traces with a sounding every 4 seconds yielding a test duration of 164 seconds. A trace every 4 seconds is typical of our acquisition rate. The test pit experiment consists of 71 traces sounded every 3.5 seconds, with the antennas parallel, yielding a test duration of 245 seconds (Figure 5.10). Since all three tests yield similar results, only the pit will be considered here. Statistics of Time Test Average Power To specifically measure system output variation and estimate the resulting uncertainty in reflection power, the average power within chosen time windows was calculated for each envelope trace (Figure 5.11). The median, mean, variance, standard deviation, sample variance, sample standard deviation, and the coefficient of variation of the average power were then calculated for each time window. Plots of the average power variation reveal occasional correlations between time horizons and make a valuable aid for interpreting the window statistics. Chapter 5: INSTANTANEOUS POLARIZATION MATCH 122 0 35 7 0 Sounding (s) 105 140 175 2 1 0 2 4 5 O c O O 0 o 0 2 ^ CO 3, 0 D 0 4 - f E 0 o 0 6 i— JF 0 n 0 8 i % O a l O 0 o l 2 0a 14 Figure 5.10 Time test consisting of multiple soundings at the test pit. A sounding was made every 3.5 seconds. The horizontal axis is labeled as sounding time in seconds. These data were dewowed, time drift corrected, and gained with EXPGAN.F. Note that each trace is a stack of 64 soundings during acquisition. Plots of the variation of average power help discriminate between changes due to system output and changes due to environment noise. The effect of signal to noise ratio on variability is also clearly observable. Mean power was chosen over mean amplitude since the variation is magnified in the power. As expected, Figure 5.11 shows a general increase in variation with increasing travel time due to the generally decreasing signal to noise ratio. System related variations are expected to occur across all time windows, whereas random noise bursts should be more temporally isolated. Figure 5.11 illustrates that medium to large variations tend to be limited to a couple of adjacent time windows with the largest variations occurring primarily in single windows. Table 5.1 lists some notable whole trace variations observed in Figure 5.11. Whole trace variations are probably due to a sudden change in system output, and are usually much smaller than the short duration noise bursts. Statistics within the time windows are given in Table 5.2. Including the noise bursts, power variation seems to typically remain Chapter 5: INSTANTANEOUS POLARIZATION MATCH 123 within about ±13% (or about ± 7% for amplitude variation), with the variation due to the radar system being probably less than half this amount. Sounding (s) 105 140 0 Q 00 o D 02 ih 1 0 „ 0 4 1 CD E 0 D 06 l— J2 0a08 § 0• 10 - § 0 n 1 2 -1 0 Q 14 * > Sh O CL, d a CD a o o to Multi-Sounding Power Variation 8 "i i i i i i i r T ~i i i i i i i i i i \ i i r j l i i i i l i i i \ l i i i i l i i i i l i i i L 2.49 10.23 7.34 9.77 12.25 50 100 150 200 250 Sounding (s) Figure 5.11 Envelope (instantaneous amplitude) traces of the test pit time test (from Figure 5.10). Within the five labeled time windows, statistics were calculated for the average envelope power of each trace. The numbers labeled on the power variation curves correspond to the time window number along the right time axis of the soundings. The coefficients of variation (C = standard dev./mean (%)) are labeled along the right axis of the power variation plot. Chapter 5: INSTANTANEOUS POLARIZATION MATCH 124 Table 5.1 Notable whole trace variations observed in Figure 5.11 Sounding (s) Window #s Comments Regarding Windows 20 - 40 1-4 5 exhibits some correlation 40-45 1, 3,4 2 & 5 overprinted w/ noise ? 45-60 1 - 5 125 - 145 1? - 5 1 too small to observe. 5 not as correlated after 135 s — due to noise? 160 - 170 1 - 5 180 - 193 1 - 5 Statistics of m p Calibration and Uncertainty Propagation To investigate the propagation of uncertainty in calculating m p , the power variation analysis of the previous section is performed on the calibration traces. This analysis is applied to input traces, and intermediate traces 7 and 8, while a similar, amplitude, analysis is applied to both Table 5.2 Window statistics from Figure 5.11 The input f i l e was: timxp2.amp The chosen amplitude measurement i s : MEAN POWER WINDOW # 1 Time= 0.0200 to 0.0300 Range= 0.6181E+10 0.5516E+10 Mdn= 0.5850E+10 Mn= 0.5839E+10 Var= 0.2113E+17 StDv= 0.1453E+09 s**2{Mn}= 0.2975E+15 s{Mn}= 0.1725E+08 ** C = 2.489 % ** WINDOW # 2 Time= 0.0350 to 0.0450 Range= 0.2466E+10 0.1595E+10 Mdn= 0.1813E+10 Mn= 0.1867E+10 Var= 0.3646E+17 StDv= 0.1910E+09 s**2{Mn}= 0.5136E+15 s{Mn}= 0.2266E+08 ** C = 10.23 % ** WINDOW # 3 Time= 0.0540 to 0.0640 Range= 0.5593E+09 0.3752E+09 Mdn= 0.4377E+09 Mn= 0.4420E+09 Var= 0.1052E+16 StDv= 0.3243E+08 s**2{Mn}= 0.1481E+14 s{Mn}= 0.3848E+07 ** C = 7.337 % ** WINDOW # 4 Time= 0.0700 to 0.0800 Range= 0.6582E+08 0.3168E+08 Mdn= 0.5546E+08 Mn= 0.5488E+08 Var= 0.2875E+14 StDv= 0.5362E+07 s**2{Mn}= 0.4049E+12 s{Mn}= 0.6363E+06 ** C = 9.769 % ** WINDOW # 5 Time= 0.0800 to 0.0900 Range= 0.2162E+08 0.1203E+08 Mdn= 0.1662E+08 Mn= 0.1689E+08 Var= 0.4279E+13 StDv= 0.2068E+07 s**2{Mn}= 0.6026E+11 s{Mn}= 0.2455E+06 ** C = 12.25 % ** Chapter 5: INSTANTANEOUS POLARIZATION MATCH 125 the raw and median filtered m p traces. For the input and intermediate traces, mean power is a more revealing measurement, at least graphically, than mean amplitude. However, since the match factor is already in terms of power, match amplitude should be compared to signal power. Generally, the uncertainty in median filtered match varies with the combined noise level df the two components: lowest uncertainty (±0.05 for 95.4% confidence) for high match, largest uncertainty (±0.14) for mid range match, and a medium uncertainty (±0.09) for low match values. Figure 5.12 shows five windows chosen to examine uncertainty propagation for different event types. The window statistics were performed on both the parallel and perpendicular input traces. For average signal strength, the standard deviation of the mean power is about 9% of the mean for the parallel component and about 15% for the perpendicular component of the same events. Assuming a normal distribution, 95.4% of all power values should occur within ±18% and ±30% of the means for parallel and perpendicular components respectively. Some trends in error propagation are observed by comparing the coefficients of variation for 7 and the input amplitudes. For good signal to noise ratio, the standard deviation of 7 is about 4.4% of the mean power, which is about half the coefficient of variation for the stronger amplitude component. For weak signal, the standard variation in 7 is about the same percentage as that of the component with the larger uncertainty. These trends for the propagation of uncertainty in strong and weak signals are based only on the five calibration traces; therefore, these observations may not survive similar experiments at different field sites. The small sample size (only 5 traces) is addressed in the tables of window statistics (Tables 5.2, and 5.3) by sample variances and sample standard deviations. Finally, variation in the m p traces increases for the lower values probably due to the noisier perpendicular component becoming dominant and the parallel component also containing a more significant noise level (Figure 5.13). The coefficient of variation for the nonfiltered match is about 3.5% for high match values (approx. > 0.85), roughly 12% for mid values, and roughly 50% for low values Chapter 5: INSTANTANEOUS POLARIZATION MATCH 126 CD E i— ca TE Component Trace Pair # 1 2 3 4 5 .Tx iRx Ave. Cross Component Trace Pair # 1 2 3 4 5 Ave. • Tx Rx Power Variation 10 CM CD o Q_ CD CD O C' 0 / -111111111111111111111 4 9.17 9.33 r " 1 • 31.43 69.87 13.44 I I I I 1 I!1 11 I I II I I I 1 1 1 I 9.0 8.5 8.0 7.5 \ 2 3 4 5 Trace Pair # 6.5 Power Variation Cf%) 111111111111111111111 1 13.86 19.27 26.03 2 8.04 • 19.26 111111111111111111111 1 2 3 4 5 Trace Pair # Figure 5.12 Time windows chosen for the power variation analysis are marked on the envelope trace section of the 5 trace pairs. The time windows include the strong air-ice interference amplitudes, the weak ringing from the ice layer, the interval of no signal, the strong lake bottom reflection, and some medium strength scattered sub-bottom arrivals. The power variation curves are numbered to correlate with the window number labeled on the sounding time axis. Chapter 5: INSTANTANEOUS POLARIZATION MATCH 127 Variability in Polarization Match Trace Pair # 1 2 3 4 5 Norm. Ave. Var. CO Mean Match Variation 1.0.— - Q - i ! m % 0-6 8 0.4 1 2 3 4 5 Trace Pair # Figure 5.13 Mean amplitude variation of the median filtered mp traces. Time windows are defined in Figure 5.12. The window statistics are summarized in Table 5.3. The variance trace is normalized by 0.09755, the maximum variance. (approx. < 0.25). Assuming a normal distribution, the standard deviations indicate that with a confidence of 95.4% high m p values have an uncertainty of about ±0.06 (±0.05 for median filtered mp), middle values ±0.10 (±0.14 filtered), and for low values ±0.13 (±0.09 filtered). Note that the normalized variance trace in Figure 5.13 exhibits a low variance for extreme match values, particularly for the primary lake bottom reflection. 5.5.3 Unknown Uncertainty For Theoretical Limitations Although the current lack of knowledge about polarization within side lobe radiation and the near-field of our antennas results in an unknown level of uncertainty for signals of these origins, their m p calculations remain somewhat interpretable. Direct arrivals are transmitted through the side lobes of the radiation pattern, within which the polarization is currently unknown. Therefore, polarization match can not strictly be calculated within the interference interval. Also, the radiation pattern is not known for the near-field and the standard radiation pattern Chapter 5: INSTANTANEOUS POLARIZATION MATCH 128 Table 5.3 Window statistics from Figure 5.13 The input f i l e was: pol5bnd2mch85.mdn The chosen amplitude measurement i s : MEAN AMPLITUDE WINDOW # 1 Time= 0.1600 to 0.1800 Range= 0.9161 0.8237 Mdn= 0.8871 Mn= 0.8803 Var= 0.1339E-02 StDv= 0.3659E-01 s**2{Mn}= 0.2678E-03 s{Mn}= 0.1636E-01 ** C = 4.157 % ** WINDOW # 2 Time= 0.2500 to 0.3500 Range= 0.1975 0.4773E-01 Mdn= 0.6780E-01 Mn= 0.1047 Var= 0.4164E-02 StDv= 0.6453E-01 s**2{Mn}= 0.8327E-03 s{Mn}= 0.2886E-01 ** C = 61.61 % ** WINDOW # 3 Time= 0.4500 to 0.5500 Range= 0.4880 0.3188 Mdn= 0.3488 Mn= 0.3756 Var= 0.4639E-02 StDv= 0.6811E-01 s**2{Mn}= 0.9279E-03 s{Mn}= 0.3046E-01 ** C = 18.14 % ** WINDOW # 4 Time= 0.8000 to 0.8200 Range= 0.9732 0.9372 Mdn= 0.9546 Mn= 0.9521 Var= 0.2075E-03 StDv= 0.1440E-01 s**2{Mn}= 0.4149E-04 s{Mn}= 0.6441E-02 ** C = 1.513 % ** WINDOW # 5 Time= 0.8300 to 0.8500 Range= 0.6291 0.4440 Mdn= 0.5040 Mn= 0.5177 Var= 0.4899E-02 StDv= 0.6999E-01 s**2{Mn}= 0.9798E-03 s{Mn}= 0.3130E-01 ** C = 13.52 % ** is by definition a far-field description. Despite these theoretical limitations, match calculations for these signals in this study do have some practical merit. Although the values are more uncertain, systematic variations in match are often observed within the interference interval which can be related to structure very near, or at, the surface. Also, although the estimated near-field limit occurs after the interference interval at both test sites, the match values do not change in any notable manner that might be attributed to the near-field. Therefore, in practice, the near-field limitations do not appear to pose a problem for the match calculations, at least for times later than the interference interval and a limited interpretation also remains possible within the interference interval. Finally, the interpreter must be aware of possible out of the plane events which might be more apparent on the cross component. If these events have a distinct polarization then they could possibly be attenuated with a polarization filter as has been accomplished for seismic data (Perelberg and Hornbostel, 1994). Chapter 5: INSTANTANEOUS POLARIZATION MATCH 129 5.6 Color Coding Scheme for Polarization Match Images My recommended color scheme for instantaneous m p is governed by the relation between m p and E= amplitudes, and the ambiguous origin of values around 0.5 (as explained in Section 5.4.3). Despite being significantly lower than 1.0, values about 0.5 are often relatively unimportant, likely being transitional or due to a lack of signal entirely. Therefore, a color scheme should alert the interpreter of these values, but in a subtie manner. Additionally, values less than 0.5 indicate that energy is predominantly in the cross component; therefore signifying a corresponding deficiency of amplitude in the more conventional parallel component. A color scheme that makes the distinction obvious between values greater than and less than 0.5 would be helpful. A two tone scheme that grades to white at 0.5 honors all these interpretational concerns while remaining very simple. White is distinct, yet neutral amongst the color. An obvious color pair is blue for low values (deficient amplitudes) and red for high values (well coupled amplitudes). An additional concern is the ability to discern contour values in the image. This ability is aided by having distinct steps in color grade between adjacent colors and limiting the number of color bins to about 20 or less. Having too many colors makes the image busy and obscures the primary details. Contour intervals should exhibit the desired resolution, constrained by uncertainties. Con-sidering the uncertainty for mid range match values, 0.1 is a reasonable contour interval through-out the low and mid range. Finer intervals of 0.05 are necessary for the higher values (>0.85). The lowest interval is also 0.05 to distinguish between matches that are nearly 0.0 from those near 0.1. Due to an abundance of values above 0.95, a 0.025 interval is necessary to re-solve minimal match decreases and remains reasonable with the lower uncertainty. Figure 5.14 demonstrates this color coding scheme on the familiar lake soundings. Since polarization match is a power measurement, Figure 5.14 also lists the equivalent amplitude scale as ^/m^. Although the occurrence of significant match anomalies are often predictable by studying the Chapter 5: INSTANTANEOUS POLARIZATION MATCH 130 Calibration Soundings: TE Amplitude and Polarization Match TE Component ,Tx iRx Cross Component Jx Rx Envelope Amplitude (u.V) 4 9 2 . 8 0 6 7 0 . 0 0 8 8 5 a 4 0 1 1 8 6 . 0 0 1 5 3 6 . 0 0 1 9 4 9 . 0 0 2 5 6 5 . 0 0 3 4 2 9 o 0 0 4 8 8 4 . 0 0 7 2 4 1 D 0 0 1 0 1 5 0 . 0 0 1 3 5 7 0 . 0 0 2 0 4 0 0 . 0 0 Trace Pair # 1 2 3 4 5 1 2 3 4 5 Ampl. Scaling Vmp 0 . 2 2 0 . 3 9 0 . 5 0 0 . 5 9 0 . 6 7 0 . 7 4 0 . 8 1 0 . 8 7 0 . 9 2 0 . 9 5 0 . 9 7 5 0 . 9 9 0 . 9 9 5 &c_aung polarization Match 0 . 0 5 0 . 1 5 0 . 2 5 0 . 3 5 0 . 4 5 0 . 5 5 0 . 6 5 0 . 7 5 0 . 8 5 0 . 9 0 0 . 9 5 0 . 9 7 5 0 . 9 9 1 2 3 4 5 Ave.Trc. _| Figure 5.14 The amplitude and polarization match color coding schemes applied to the familiar lake soundings. Chapter 5: INSTANTANEOUS POLARIZATION MATCH 131 two-component amplitude data, the match images are more thorough with the advantage of a numerical estimate of power loss. 5.7 Polarization Match Images The two-component data sets from both the ramp and test pit together address two degrees of structural complexity (Chapter 3). The ramp data set was acquired at 100 MHz in order to maximize penetration while maintaining necessary resolution. The greater structural complexity and shallow depth extent of the test pit required the highest frequency pulseEKKO™ IV antennas: 200 MHz. Note, that all images in this chapter are not corrected for offset and are plotted with an arbitrary zero time in order to display the pre-arrival noise level. Also note that some signals in the figures of input data appear clipped, however, this is only a plotting artifact due to scaling and is not so evident in the larger plots of polarization match. Some theoretical limitations should be remembered when considering the following inter-pretations. As discussed in Section 5.5.3, the match values do not appear to be affected by the near-field limit which is estimated to occur at 0.06 J^S in the ramp images and at 0.05 p,s in the test pit images (equation 2.3). Our lack of knowledge about polarization within the pulseEKKO™'s side lobes, in which the direct arrivals occur, may be partly responsible for the consistently low match within the interference interval (the time interval of direct air and ground wave interference). Despite these unknowns, a limited interpretation is possible for the systematic match variations within the interference interval, some of which appear to be the largest match anomalies. Unfortunately, the accuracy of these anomaly magnitudes is un-known. Also, all CMP interpretations are inherently preliminary since the antennas' radiation pattern has not been characterized with a proper calibration. Since the angle of radiation varies with offset, interpretations of amplitude variation in CMP soundings remain speculative with-out a more comprehensive estimate of the radiation pattern for both power and polarization. Chapter 5: INSTANTANEOUS POLARIZATION MATCH 132 Studying the more simple overpass ramp data first will hopefully yield insight for the more complicated test pit site. 5.7.1 The Abandoned Overpass Ramp Experiments Imaging the lower 50 m of the ramp and continuing 10 m on the natural silts allowed me to characterize the depolarization effects of the ramp's distinct boundary and internal layering. At the, uppermost, 80 m station the ramp thickness should be about 3.8 m consisting of three or four lifts. The T E and T M CMP soundings have initial offsets (measured from antenna midpoint) of 0.8 m and 1.2 m, respectively. For both CMP soundings, offset was increased in 0.4 m steps to 14 m maximum offset. The standard processing, such as gaining (EXPGAN) and filtering (Figure 5.2), used the parameter values chosen in Section 4.3.1. The profiles were not corrected for elevation changes, therefore, the horizontal internal layering of the ramp appears as dipping reflections. T E Profile and Polarization Match Signal strength drops dramatically from the ramp fill to the natural silt, regardless of the component. Figure 5.15 illustrates the relative amplitude strength between the components of the T E pair. The basal reflection (from the base of the ramp) is clearly identifiable as the deepest high amplitude (dark orange) reflection in the T E component. Some subtle undulating structure is noticeable on the otherwise linear reflections. The undulations are probably due to irregularities in the interfaces. The most prominent irregularities correlate with the three zones of relatively large amplitude (light green to orange) in the cross component. The amplitude anomaly on the cross basal reflection (105 - 115 m stations) correlates with an apparent flattening of the reflector, likely responsible for a depression on the surface. Below the basal reflection, weaker T E reflections within the Fraser silt exhibit an apparent shallow dip. These weak events are not discernible as interfaces in the cross component. Primarily the only apparent structure observed off the ramp is a long period medium amplitude T E event (light 000 Chapter 5: INSTANTANEOUS POLARIZATION MATCH 133 Two-Way Time (jis) o o • o "3-o • o CO o CM CD D o O CO 03 M _ § O Q_ LUI 51 81 o O O O O O O O O O O O O — 1 c o o o o o o o o o o o o o CD o CD > LU U O O O C O O O C M C D O ^ C D C O L O L n C O C M ^ 00 CO o o CM CO ^ J " DO CD in N o o o o o o CD CO CM CO CD CM CD CM CO CO O CO —' CM CD CO K CJ n o T3 V It 9 2 T J o u B E 0 E 9 Bu cd a, g g -5 B -O 3 I 1 an s f S.J? S TJ U SI _ i- a a o oe — — c-b CB Chapter 5: INSTANTANEOUS POLARIZATION MATCH 134 green) at about 0.05 /xs. However, calculating the direct transit time with the silt velocity (50 m//is determined from a CMP sounding) reveals this event to be the direct ground arrival. A water table reflection might also occur about this time but is not expected to be a distinct event since the water table is probably a diffuse boundary due to the capillary fringe in these fine grained sediments. Amplitudes within the interference interval are abruptly weaker for both components in the silt with the air wave cross component having a more complicated appearance and the direct ground wave cross component apparently absent. Although theory predicts some depolarization from dipping reflectors, Figure 5.16 reveals that reflections within the ramp are very well matched; the degree of depolarization is mostly negligible. This is expected since the dips are quite low, only about 4.5°, and the interfaces within the ramp dip parallel to the survey line. A few exceptions occur at the irregularities noted above, with matches typically about 0.95 and the lowest values (about 0.6 match or 0.77 for amplitude) occurring below the zone of subsidence noted on the surface. Despite their weak amplitude, the sub-basal reflections are also well matched (often > 0.975), but obviously more variable than the fill reflections, with values down to about 0.85. The long period direct ground arrival within the silt is also extremely well matched. As anticipated, the most obvious match anomaly occurs at the edge of the ramp with a minimum of about 0.15 (an amplitude scaling of only 0.4). Coincident with the edge anomaly is a horizontal zone of relatively low match (averaging about 0.8 and down to about 0.5) which correlates with the short dead interval between the direct pulses in the T E component within the silt. In general the interference interval is associated with lower match values with its greatest match between stations 105 and about 123 m (predominately above 0.9) being below the average within the fill. All primary TE reflections at this site are very well matched, with minor degradation at reflector rough spots. 0 0 0 Chapter 5: INSTANTANEOUS POLARIZATION MATCH 135 Envelope Amplitude (uV) 845.30 1368.00 2099.00 3050.00 4150.00 5482.00 7526.00 10640.00 14440.00 19660.00 28290.00 60620.00 98280.01 Distance (m) 100 120 140 0.00 - i - Oo 16 Figure 5.16 The TE polarization match profile of the lower 50 metres of the ramp . Every second trace is plotted. Chapter 5: INSTANTANEOUS POLARIZATION MATCH T M Profile and Polarization Match 136 Direct comparison of the two-component T M and T E data is hindered by their different dates of acquisition, the soil being significantly drier during the T M survey. However, a single-component T M profile (shown in Figure 5.17) was acquired on the same day as the two-component T E data and is included in this discussion to aid comparison. The single-component T M data exhibits only about half the amplitude of the T E component as well as a lower frequency direct ground pulse within the ramp. Note that the amplitude color bar was rescaled for each profile. Interestingly, despite this general difference in amplitude the long period direct silt pulse has roughly the same signal strength as in the T E component and therefore appears with higher amplitude colors in both T M plots (including the two-component T M profile in Figure 5.18). Figure 5.17 clearly shows the direct ground pulse arriving at progressively earlier times with ramp fill thickness increasing from 0.0 to 0.8 m (station at 105.6 m). This direct T M wave also appears to arrive about 0.01 us ahead of the T E component! This apparent anisotropy is likely due to the relative antenna geometry/near-field effect (described in Section 5.3 regarding intercomponent time correlation). The near-field effect also explains why the time difference exists only for the direct arrival and not the reflections, although two very different soils are also involved. Amplitude and phase changes also occur across the ramp boundary in the direct air wave. The two-component T M profile (Figure 5.18) exhibits only about 60% of the signal strength of the single-component survey. This observation agrees with past experience at the ramp which suggests that higher saturations result in stronger signal, probably due, at least in part, to an optimum range of saturation that enhances the contrast due to porosity at the lift boundaries. Additionally, the higher saturation case also has a lower velocity which should result in a more directive radiation pattern; this directivity effect is probably minor compared to the porosity/saturation differences. Another advantage to the slower velocity is a greater separation between reflection times, thereby decreasing their interference. Other differences 000 Chapter 5: INSTANTANEOUS POLARIZATION MATCH 137 Two-Way Time ( j is) CSJ _CD CD X O Q. £ o O _CD CD an CD O £= B Q I I I I II | I M l I I I I I [ I I I I i I I I I | I II 1 i I I 1 I I I CO CD CD jo CD > L U o o o o o o O O o O O o .—i o o o o o o O o o o o o o • • • • • • D • • o • • D C x i —* C M C O o o o o o C O .—i r> co in C O 00 C M m 00 . — i L D C O o o xT in L D C O C x i •xr cn O J C x i C O «sr L P C D C O * — J C D in C M C D C D - 5 — o SO s a 1 E •t-i g » as — IS <u •a B -o Cd . V3 -o -Q •s s .5 * o o H g «»-s -o s s o o a £ OJ « Its O BJ GO O C IM ft) OA 5 3 E T -3 • B 9! 3 . B 0 ft £ o 0 1 o St o « ! " M o -a — u <° i o * J Z •s i 3 CB - E a 3 -o <g B 0 3 P . 2 oo S ' H B H • B -B a 0 * -a. a B a 0 " 3 CJ 1 OJ oe H B co jj r~ C J •>l • * -s 9) T 3 fc. B 3 « 9 0 — ti, <u 5 B tS •a E a »-Cu > w 1 * co a -O -H O IT! 3 3 EH 6 * OS a i B "O cj OS co -H "S II O _ H CU O 5 . an; B •  B sj - 3 > <o 3 •r^ oo O U co B3 B _ "° .2 S § S u 000 Chapter 5: INSTANTANEOUS POLARIZATION MATCH 138 c o a. E o Two - Way Time (|ns) o o o CO o CSJ C O o O O cd CO N _§ o CL 3 o CL CD _o CD > C LU o O o o o o o O o o o o o in ro o o o o O o o o o o o • o o • a c • • • o • • • in —-t in C O o C O m 00 o o o —< C x i in cr> t > cn 00 o C O C O *=» C O C O — • C O CD in L P C O —1 C O C O — r> — 1 — 1 C M C O m C O • — 1 C O C\J c c 3^ C = o 5 c a X! •O a US 0 01 . S i 5 B 5 g .2 o o tu 6 S ? is H 3 S a ae W Chapter 5: INSTANTANEOUS POLARIZATION MATCH 139 between the single and two-component surveys are fairly minor, although the character of the basal reflection appears different. This difference in character appears to be primarily due to a "trailing lobe" on the basal reflection (particularly between stations 90-110) which maintains roughly the same amplitude in both profiles while the amplitude of the preceding lobe has decreased to about half in the two-component survey. This relative change in amplitude causes the basal reflection in the two-component profile to appear as a more complicated "doublet." The basal reflection remains to be one of the largest amplitude events in both T M profiles and better exhibits the flattening (in Figure 5.17) at stations 100 - 110 m coincident with the subsidence observed on the surface. Internal layering is not so evident in the T M profiles partly due to the longer duration large amplitude ground pulse. However, closer comparison reveals that all of the internal reflections in the TE component are at least detected in the T M component, although to a lesser degree in the two-component survey. The ringing observed in the T M component of Figure 5.18 is likely instrument related since it is not effected by overlying structure and did not appear in a failed T M profile acquired only hours earlier. However, the cause of the ringing is not known. T M sub-basal reflections are not very recognizable as interfaces. Generally the cross component has peak amplitudes that are about half the signal strength as the parallel component. While the direct air wave is the strongest event, the direct ground wave is nearly nonexistent in the cross component. Larger amplitudes in the cross basal reflection often correspond to lower amplitude in the T M component. As in the T E component, the largest cross basal amplitude occurs in the zone of subsidence. Overall the polarization match section in Figure 5.19 exhibits an abundance of values greater than 0.975, however, most of these occurrences are the multiples in the T M component. Generally, the T M reflections are not as well matched as the T E reflections. Match of the basal reflection is quite variable, ranging from about 0.6 (scaling amplitudes by about 0.78) to above 0.99, typically changing every couple of meters laterally. As with the T E case, the basal T M 000 Chapter 5: INSTANTANEOUS POLARIZATION MATCH 140 TM Amplitude and Polarization Match Envelope Amplitude (pY) °® Distance (m) 1 0 0 1 2 0 1 4 0 4 1 5 . 5 0 7 2 1 . 3 0 1 1 5 5 . 0 0 1 8 9 7 . 0 0 2 6 7 8 . 0 0 3 5 9 1 . 0 0 4 5 8 0 . 0 0 5 8 0 3 . 0 0 7 1 8 5 . 0 0 8 9 9 9 . 0 0 1 1 8 1 0 . 0 0 1 7 4 8 0 . 0 0 2 9 7 9 0 . 0 0 0 . 0 0 + 0 . 0 4 0 . 0 8 -+- 0 . 1 2 -+- 0 . 1 6 3 CD co Figure 5.19 TM amplitude and polarization match profile of the ramp. Every second trace is plotted. Chapter 5: INSTANTANEOUS POLARIZATION MATCH 141 reflection within the subsidence zone (stations 100 - 120) exhibits a lower match, at about 0.87. The weaker internal reflection has lower match than the basal reflection, with a common occurrence of values around 0.8. The lowest match for a reflection (down to about 0.4, or a 0.63 scaling of amplitude) occurs along the sub-basal reflection which is poorly defined in the T M component and even less discernible in the T M cross component. Polarizations of the two direct arrivals are distinctly different, with the direct ground phase being mostly well matched, while the direct air phase has a significantly lower match of about 0.8. The lowest match (about 0.1, or 0.19 amplitude scaling) occurs where the direct ground phase begins to interfere with the basal reflection at station 125, with very low matches (0.3 - 0.5) dominating throughout the interference of these two phases. This generally lower polarization match for T M surveys was anticipated, since a random occurrence of incidence at the Brewster angle due to scattering and reflector roughness should cause significant polarization changes. In a T M survey the wavelet predominately has T M polarization, whereas a T E survey wavelet will generally have only a minor T M component. As in the T E survey, the observed rough spots cause relatively significant depolarization. T E and T M Common Midpoint Soundings The common midpoint soundings exhibit four clearly identifiable reflections. The earliest reflection is relatively difficult to track, being interfered with the direct ground arrival. In the T E component (Figure 5.20) the three distinct reflections have minimum offset times of: 1 « 0.066 fxs, 2 « 0.086 /.is, and 3 (the basal reflector) « 0.10 ^s. Throughout the discussion of the CMP soundings these reflections will be referred to by number. A couple of sub-basal T E reflections are discernible with their amplitudes peaking between offsets of 3 - 6 meters. The T E surface ground velocity was estimated manually as 103 ±25 m/^s. The uncertainty estimate on this velocity was obtained by averaging the error of two air wave velocity estimates. All components clearly show the direct air wave out to the farthest offset. The cross component in 000 Chapters: INSTANTANEOUS POLARIZATION MATCH 142 c o C i Two-Way Time (jis) o OJ -3" C D CC o Cxi x r CO CO o o o O O o Cvi o a D D D o • • G o a o O O O o o o o o o o CJ > as o CO CO N O Q _ iiiiiiiii niiniii iniiiiii iiiiiiiii iiniiHi niiiinimiiiiiii iiiiiiiii <x> < CD Q _ _o CD > LLi o o o o o o o o o o o o o C O C O C O O O O O O O O O O O co CM CD 00 IO -d" CM co CM o co cn —< CM CO CO CD CO UO CM CM OO in xT CM CD •sr CM •—> CO CD CO o C M CM 'd-cs N "3 c 3 3" P. s c c = o E o c u S i c -5 c 3 o O c o p. £ o u • 0 ci W H P. ca «l —' £< 3 cj fa is Chapter 5: INSTANTANEOUS POLARIZATION MATCH 143 Figure 5.21 exhibits two or three distinct reflections abruptly constrained by offset. Although the direct ground signal is usually difficult to discern in CMP soundings, it seems to be nonexistent in the cross component. This lack of direct ground signal agrees with the profiles where the direct ground phase is unrecognizable or also nonexistent in the cross component. Features in the two-component T M CMP sounding (Figure 5.21) are not exhibited as clearly as in a single-component T M sounding acquired at the earlier date (Figure 5.22). The two-component T M surveys were conducted in April, 1994, with dry soil, the single-component T M CMP sounding was acquired in January, 1994, with wet soil, and the two-component T E sounding was acquired in November, 1993, with fairly dry conditions. Fortunately, the soil saturations on the two days of two-component acquisition were probably more similar. Although both the two-component and single-component T M soundings exhibit the same events, the single-component T M sounding is included in this discussion since it better exhibits some important features. Due to the marginally lower velocity (85 ±25 m///s), reflections in the single-component sounding (Figure 5.22) are more distinct with a tremendous amplitude drop at 6 meters offset on reflector 1. The single-component CMP sounding also has about twice the signal strength of the two-component sounding. Amplitude drops also occur in the two-component sounding but are not nearly as distinct and are complicated by interference of adjacent reflections. The weak sub-basal reflections are barely noticeable in the T M soundings. Critically refracted air waves are noticeable in both T M soundings (Figures 5.21 and 5.22) but not in the cross or T E components (Figures 5.20 and 5.21 ). The most prominent refracted air wave originates from reflection 1. Generally, intermittent occurrences of amplitude highs in the T M cross component sounding usually correlate with amplitude lows in the T M parallel component (Figure 5.21). Compared to the T E cross component, the T M cross component contains larger amplitudes (relative to the parallel component) and the T M cross reflections Chapter 5: INSTANTANEOUS POLARIZATION MATCH 144 Two-Way Time (\is) o C M -xt- C O C O o O J C D C O o o O o o O C M D • c a • D • o • a • o o o Q o o o o o o o o CO CO N O Q _ o l 1 CD cr o Q . E o O C/3 o i— O 1 CM _ & 1 o _ H — 1 c= c o -CD , . £Z CD O co co — C L E o o O — ^ 1— CM — CD "CJ E CD CD > CZ o o o o o o o O o o o o o m OJ CO — 1 o o o o o o o o • • • • • D o • • a • p • cn — 1 cn CM CO CO cn o o o OJ CM CM cn —c co 03 K CD —i OJ CD 'Xi CM CD CM o CM CO CO r> —i O J CO CD — 1 —> CM t-t ON O w 2 * c °» § " O E g a £ * « •s g .5 -o CO 9 •a a cu •a * 3 2 S3 OJ a> -a i • S H o JT cu ... •> S ~Q £ > o - 5 cu cu a * £ -2 5 OA N B 0> 6* S3 M § s •s m 5 cu « -s -S 2 » S •5 S o S 3 * M a B -a 1 5 s I B 2 " S -o 2 B 5 C ° H B O a aa 83 J> to 2 CC fc. B o <uO r— >g- ° w 03 = c g u O £ o ~ o .5 g - , S « B g..a -o 1 I 3 5 6 0 g § * S is * 000 Chapter 5: INSTANTANEOUS POLARIZATION MATCH 145 3 tN O 3 06 Two - Way Time (|is) o cr o C L E o O o CM •xi- CD CO o CM xt- CD CC o CM o o o O o CM CM • • • • a • • D D • - • o o o o o o o o o o o o IIIIIIIII IIIIIIIII IIIIIIIII IIIIIIIII IIIIIIIII IIH1IIII IIIIIIIII IIIIIIIII IIIIIIIII CD > L U CO O CM —< CO CD CM LO CO LO o o • r- o "d- —• CO CD I D f x CD CM CM CO LO CD CO O O O O o CO xT CM O •xt M -5 B 3 O •s •a B O s c •S E o E 2? ea B o E W 5 2 E £ .2 » 3 <-> m « H oo o 5 ° 1 4> #—s B ° i * -a . — 2 -O ci] u | s 5 ** 2 B U Si c E s -i If f J < -5 a •u E a -a M cr es o v> £ 3 • « ca fa o Chapter 5: INSTANTANEOUS POLARIZATION MATCH 146 are not as constrained by offset. Therefore, the T M data have once again experienced greater depolarization than the T E data. A few possibilities explain the dramatic amplitude decrease (nearly down to noise level) in T M reflection 1 at about 5 - 7 m offset in the single-component (Figure 5.22). Much smaller amplitude decreases also occur in reflections 2 and 3 within offsets of 4 - 5 meters. These smaller amplitude drops are similar to the drops observed in the two-component T M sounding (Figure 5.21). Some explanations include destructive interference, the Brewster angle, and a radiation pattern null. The interference possibility might include two other events, the direct ground pulse and the critically refracted air wave. Although in the single-component sounding (Figure 5.22) at 6 m offset the direct ground wave arrives only 0.018 p,s ahead of reflection 1, interference appears to be minimal, possibly involving only the weak coda of the direct ground wave. More significant interference probably occurs with the critically refracted air wave which is roughly tangent to the reflection hyperbola at this offset. However, the timing indicates that the interference should be, at most, within only the first half of the reflection duration. Interference does not seem to fully account for the extinction of this reflection. An exciting possibility is the Brewster, angle, where the T M component of an incident wave is totally refracted resulting in no reflection. Recall that at the Brewster (or polarizing) angle, reflection occurs only within the T E component. Unfortunately, the velocity variation within the ramp occurs primarily within each layer remaining fairly constant from layer to layer. Since a CMP based velocity analysis yields rms velocities, the resulting velocity depth function for the ramp is nearly constant (as discussed in Section 4.3.1). Therefore a CMP sounding will not provide a meaningful estimate of the velocity contrast across the boundary making an estimate of the Brewster angle unattainable. The radiation pattern null possibility is easily checked by a rough estimate of the ray path geometry against the critical angle. Recall that Smith's (1984) radiation patterns for a half-wave dipole antenna predicts a null at the air-ground critical angle Chapter 5: INSTANTANEOUS POLARIZATION MATCH 147 for the T M mode and a relatively small peak for the T E mode (Figure 2.6). The ray path angle from the antenna to the amplitude extinction was measured from the vertical to be 19° on a 1:1 sketch, and the critical angle is calculated to be 19.5°. Straight ray paths and flat reflectors were assumed in the sketch, which is unlikely to produce a significant deviation from reality for the ramp (the ramp dips only about 4.5°). Although I did not obtain an estimate for the Brewster angle, the estimated radiation pattern null is a strong possibility for causing the dramatic extinction in reflection 1. The Brewster angle remains a plausible explanation, especially for some of the amplitude drops in the two-component T M sounding since they correlate with an amplitude peak in the cross component (Figure 5.21). As with the profiles, the CMP domain exhibits substantially better polarization match in the T E mode than in the T M mode (Figure 5.23). The T E CMP match image illustrates that reflections are generally well matched with subtle decreases (down to 0.95) occurring in reflections 1 and 2 at 2-4 m, and 6-8 m offset respectively. The direct T E air wave maintains a high match for most offsets, dropping intermittently to 0.9 beyond offsets of 10 metres. Reflections beyond this offset are interfered, and their matches also drop to about 0.9. The sub-basal reflections have significantly lower matches of about 0.85 and are quite variable. The lower match in the T M CMP sounding (averaging around 0.85) is also quite variable, typically ranging from about 0.6 to greater than 0.975. The largest amplitude "null" results in a match of only about 0.1, indicating that the Brewster angle remains a very plausible explanation. Smaller amplitude drops in T M reflections 2 and 3 also exhibit decreases in match, although much smaller. Reflection 2, at 0.10 /J,S, consistently has the best match, often greater than 0.975. A similarly matched T M event is the critically refracted air wave from the 0.06 s^ reflection. The direct T M air wave appears to have excellent match at offsets greater than 6 000 Chapter 5: INSTANTANEOUS POLARIZATION MATCH 148 Two - Way Time ( J I S ) o CM CD oo o CM cc 0 0 o o o o O o O J • • 0 c • • D • D E o o O o o o o o o o o N J L n u o L n u o u o i j o L o u o i j o o L n ^ c o »g o — ' w n x j i D i D N C o m c n t r j c n Z Q O D D O D D D D O D n D a Q_ o o o o o o o o o o o o o Chapter 5: INSTANTANEOUS POLARIZATION MATCH 149 metres, a contradictory occurrence to that observed in the T E sounding. This variation may be a near to far-field effect. Figure 5.23 illustrates that for T M surveys at this site the best offset is about 2 metres for maximum overall polarization match. Summary of Observations on the Ramp The ramp proved to be a good control site for investigating the amplitude effect of depolarization due to reflection from relatively smooth planar interfaces. Depolarization at a simple lateral boundary of large contrast is also demonstrated. Probably the most valuable observation confirms that T M wavelets become significantly more depolarized than T E wavelets, probably due to the occurrence of the Brewster angle in the T M component. The "dipping" reflectors impose only a small degree of depolarization in the T E component resulting in a negligible effect on amplitudes. However, the ramp dips only 4.5° and theory predicts that much steeper dips should have a significantly greater affect on the polarization match. Relatively significant decreases in match for these reflections occur at rough spots in both T E and T M profiles, emphasizing the significance of scattering in depolarization. These experiments also indicate a difference in polarization of the direct phases. The direct ground wave is well matched in both T E and T M surveys while the air wave is well matched only in T E surveys, the T M air wave match being significantly degraded (about 0.8). The T M CMP sounding illustrated that the air wave actually becomes very well matched for offsets greater than 5 m suggesting that near-field effects may be partly responsible for the poor airwave match at profiling offsets. In both profiles, the largest polarization match anomaly (on the order of 0.15) occurs at the ramp edge with the interference of the direct ground wave and basal reflection, suggesting that large polarization match anomalies could be common at sites with more complex structure. Chapter 5: INSTANTANEOUS POLARIZATION MATCH 150 5.7.2 The Test Pit Experiments T E Profile The two-component T E profile of the test pit (Figure 5.24) exhibits gross similarities between the components: the largest amplitudes occur in the interference interval and around the lower block. At about 0.03 /is the reflection from the top of the shallow block (green to orange T E amplitudes) weakly breaks through the large amplitude interference of the direct air and ground pulses. The reflection from the top of the deeper block occurs at 0.06 us (orange T E event). Some of the largest amplitudes in the orthogonal component occur at the top and bottom of the lower block. Most of the pit walls are also clearly visible in the parallel component and reasonably clear in the cross component. Most of the diffractions, identifiable primarily in the parallel component, are relatively weak beyond about 0.75 m offset from their source. At greater offsets the hyperbolae often exhibit sudden amplitude drops probably due to a combination of complex raypaths through intervening structure and the antenna directivity. The polarization match profile (Figure 5.25) illustrates that most primary reflections are well matched, including most dipping events, while some of the lowest matches (< 0.45) occur at zones of interference. About 40% of the large amplitude events have m p values of only 0.55-0.85 with some small zones (0.5 m lateral extent) as low as 0.45. The remaining large amplitude events are mostly well matched (> 0.95) but also exhibit lower matched zones (0.45-0.9) of about 0.5 m in lateral extent. Some weak amplitude events are also well matched (> 0.95) such as diffraction tails from corners of the pit floors. Weakest amplitudes (time intervals between events where amplitudes drop below 4000 uW) typically have very low match (< 0.1) such as between 4-5 m and 0.065-0.07 us. Most of the low match/large amplitude events occur within the interference between the direct ground and air phases. Interpreting within this time interval is complicated due to the depths being within the near-field, and the direct arrivals originating from the side lobes of the 000 Chapter 5: INSTANTANEOUS POLARIZATION MATCH 151 Two -WayTime ([is) o CSJ -d- (X' co o CSJ \ r CD oo o o o o • O • o • • o o o o o o o o o o — O E eu a 2 CD £= O CL E o O co CO o CD C o Q -E o O LJJ H1TH11J! UTTfllftl rrTrTfrrrl IrrTtrWIIIfWITfl CD Q _ j O CD > CO CO co •sf CD —< o o * CD CO c o r - — ' C D C D C D L n ^ r C D C S J O C O C O C O O O C O —• CSJ co co in r>-o o LO in cn co oo —• o o CD CO CD 3 T S S c 5 3 iu -B = 14 ^3 c o 3C «; 5 w o c « g w » - E s «•> c <=> E >> ? I 6 u c 2 "a 2 5 E 5 Chapter 5: INSTANTANEOUS POLARIZATION MATCH 152 radiation pattern. Although the direct arrival anomaly magnitudes are not reliable, the relative variation of anomalies within the interference interval appears to be related to shallow structure. The reflection from the top of the shallow block emerging through the direct interference phases (stations 3-5m and about 0.03 p,s) results in slightly lower match values (0.45-0.85). Sharp lateral boundaries often cause isolated large amplitudes in the cross component, resulting in low matches (0.45-0.75) at 2, 4.8, and 9.3 m stations. The interference interval over the deep part of the pit maintains a relatively high match (0.75-0.975). Relatively low match within the station range 6.8-9 m (generally about 0.75) correlates with pools of water on the surface where the pit overflows. Two remaining low match/large amplitude events are from the dipping pit wall at 1-3 m, and from the bottom of the lower block (5.8-7 m, 0.08 us). Explaining the difference between the predominant match of the NW wall reflection (0.75-0.9) and the SE wall reflection (> 0.975) is difficult. Both walls have similar roughness, however the SE wall dips about 10 - 15° more than the NW wall. Perhaps energy scattered from the nearby shallow block, which is not easily identifiable near the interference interval, interferes with the shallow wall reflection to sufficiently alter polarization at the wall. The remaining occurrence of decreased match values appears to be related to interference or complicated raypaths. The low match for the reflection from the bottom of the lower block (minimum « 0.25) appears to be primarily due to interference with the diffraction from the SE wall and various adjacent corner diffractions. The two remaining horizontal primary reflections, from the bottom of the upper block and the top of the lower block, are well matched (> 0.975). The only diffraction hyperbola that maintains a consistently high match (> 0.975) is the relatively unobscured corner of the shallow pit floor at 5.7 m and 0.05 p.s, however, this diffraction is only observable to about 1 m offset. Remaining diffraction hyperbolas exhibit significantly lower, but quite variable, matches. Complicated raypaths through intervening structure probably contribute to this variability. One of these diffractions, off the NW corner 0 0 0 Chapter 5: INSTANTANEOUS POLARIZATION MATCH 153 T_E Amplitude and Polarization Match Envelope Amplitude (jiV) 9 Distance (m) 4 5 B 7 6 9 3 . 8 0 1 1 4 3 D 0 0 1 6 6 4 . O Q 2 2 7 6 . 0 0 3 0 1 3 . 0 0 3 9 6 7 . 0 0 5 3 6 7 o 0 0 7 3 6 1 • 0 G 9 8 5 0 . 0 0 1 3 3 4 0 . 0 0 1 8 7 5 0 . 0 0 3 1 4 5 0 . 0 1 6 1 6 0 0 . 0 0 3 CD if CO 0 . 18 Polarization Match 0 . 0 5 Oo 15 0 . 2 5 0 . 3 5 0 . 4 5 0 . 5 5 0 . 6 5 0 . 7 5 0 . 8 5 0 . 9 0 0 . 9 5 0 . 9 7 5 0 . 9 9 10 11 Figure 5.25 See text for discussion of the 200 M H z polarization match profile. Chapter 5: INSTANTANEOUS POLARIZATION MATCH 154 of the lower block, appears to undergo a phase change resulting in a minimum match of 0.5 at 5 m and 0.075 p,s. Despite the complicated raypaths, long offset / weak amplitude diffraction tails maintain an impressive match (predominately greater than 0.85). T E Common-Midpoint Soundings The T E CMP soundings were centred at stations 3.0, 6.0, and 10.0 to investigate the affects of discontinuous structure on CMP soundings and their polarization matches. Offset between the antenna midpoints was initially 0.4 m and increases 0.1 m between soundings. Generally the two direct phases and reflections are clearly identifiable in all of the CMP soundings. However, as anticipated at this structurally discontinuous site, the reflection hyperbolas are complicated by disruptions and truncations, and exhibit dramatic amplitude/polarization anomalies. Note that all the CMP soundings use the same amplitude contours as the profile, allowing direct amplitude comparisons. Times referenced in the following discussion are minimum offset two-way times unless specified otherwise. CMP Sounding 3: Studying the two-component CMP sounding amplitudes in Figure 5.26 reveals that major amplitude reductions in the T E component often correlate to relatively strong signal in the cross component suggesting significant depolarization. The most noteworthy amplitude anomalies along reflection hyperbolas occur at the 0.05, 0.056, 0.065, and 0.085 us reflections. The first two of these events correlate in the profile (Figure 5.24) with the pit wall and the diffraction from the pit wall - floor corner, respectively. A relatively large amplitude T E diffraction off the top NW corner of the shallow block arrives at or before 0.04 us but is interfered with the direct ground phase at zero offset. The latest notable cross event is temporally isolated at 0.085 /<s and observable out to 3 m offset compared to only 2 m offset in the T E component. Figure 5.26 illustrates that the direct ground wave is the only well matched event (> 0.975) in CMP Sounding 3. The polarization match in Figure 5.26 reveals 000 Chapter 5: INSTANTANEOUS POLARIZATION MATCH 155 CMP3: TE Amplitude and Polarization Match TE Component Offset (m) ,Tx iRx Envelope Amplitude (tu.V) 6 9 3 . 8 0 1 1 4 3 » 0 0 1 6 6 4 . 0 0 2 2 7 6 . 0 0 3 0 1 3 . 0 0 3 9 6 7 . 0 0 5 3 6 7 . 0 0 7 3 6 1 „ 0 0 9 8 5 0 . 0 0 1 3 3 4 0 . 0 0 1 8 7 5 0 . 0 0 3 1 4 5 0 . 0 1 6 1 6 0 0 . 0 0 Tx Cross Component j — Offset (m) 2 3 4 ,^ 1 n , | 0 . 0 0 Offset (m) Polarization Match 0 . 0 5 -0 . 1 5 -0 . 2 5 0 . 3 5 0 . 4 5 0 . 5 5 0 . 6 5 0 . 7 5 0 . 8 5 0 . 9 0 0 . 9 5 0 . 9 7 5 0 . 9 9 Figure 5.26 Input two-component 200 MHz CMP sounding and corresponding polarization match near the NW edge of the shallow block, centered at the 3.0 m station. Every second wiggle trace is plotted. Chapter 5: INSTANTANEOUS POLARIZATION MATCH 156 that the 0.05, 0.056, and 0.065 p.s amplitude anomalies result in the largest mismatch (down to about 0.1; a greater than 3 fold decrease in amplitude). The second largest match anomaly occurs at 0.085 p,s (predominately about 0.3), but is poorly contrasted with the noise that immediately follows. The direct air wave, in all test pit CMP soundings, is relatively poorly matched (averaging 0.80); significantly lower than at the ramp. A critically refracted air wave from the pit wall - floor corner emerges with a good match (> 0.90) at 3 m offset and 0.06 p,s. CMP Sounding 6: Centred just within the NW corner of the lower block, CMP sounding 6 exhibits a major amplitude/polarization anomaly from 0.075 to 0.10 p,s (Figure 5.27). The amplitude plots illustrate the correlation between T E amplitude decreases and the only significant reflection in the cross component. Five reflection events are identifiable in the T E component and are mostly nonexistent in the cross component. These events are identified from the profile in Figure 5.24: 1. a diffraction from the upper pit floor corner at 0.048 /u,s, 2. reflection from the top of the lower block at 0.06 p,s, 3. reflection from the bottom of the lower block at 0.08 p.s, 4. a diffraction from the lower block's bottom corner at 0.10 ^s, 5. probably a ringing multiple of the lower block bottom corner diffraction at 0.12 ^s. Event 2 is the only event that does not exhibit a very noticeable phase change or discontinuity with offset. Figure 5.27 shows that most of these events are very well matched (> 0.975), particularly the simple event number 2. A large match anomaly correlates with the only reflection event in the cross component. This match anomaly encompasses a large range of amplitudes in the T E component, resulting in match values ranging from 0.05 to just over 0.95. The last large TE amplitude event (light orange), beyond 2 m offset, predominately has a match of only 0.9 -0.95 (low for an event of such large amplitude). This match correlates to an amplitude scaling of 0.95 - 0.99. The two weakly scattered events arriving after 0.10 /is exhibit a fair, but variable, match (predominately > 0.90). 000 Chapter 5: INSTANTANEOUS POLARIZATION MATCH 157 CMP6: TE Amplitude and Polarization Match Envelope Amplitude (uV) 6 9 3 D 8 0 1 1 4 3 . 0 0 1 6 6 4 . 0 0 2 2 7 6 . 0 0 3 0 1 3 . 0 0 3 9 6 7 . 0 0 5 3 6 7 . 0 0 7361 . 0 0 9 8 5 0 . 0 0 1 3 3 4 0 . 0 0 1 8 7 5 0 . 0 0 3 1 4 5 0 . 0 1 6 1 6 0 0 . 0 0 TE Component Offset (m) 1 2 3 •Tx .Rx Cross Component Offset (m) 1 2 3 4 .Tx Rx Polarization Match 0 . 0 5 0 . 1 5 0 . 2 5 0 . 3 5 0 . 4 5 0 . 5 5 0 . 6 5 0 . 7 5 0 . 8 5 0 . 9 0 0 . 9 5 0 .975 0 . 9 9 Offset (m) 1 2 3 o J3 5- 0 . 0 8 CD IF 0 . 1 6 o CD *»= co Figure 5.27 The input two-component 200 MHz CMP sounding and corresponding polarization match at the middle (station = 6.0) of the test pit. Every second wiggle trace is plotted in each panel. Chapter 5: INSTANTANEOUS POLARIZATION MATCH 158 CMP Sounding 10: The final CMP sounding, centred at station 10, exhibits little to discuss. Figure 5.28 illustrates that reflections at this location are mostly truncated at 1 m offset due to the receiving antenna being outside the pit; the primary reflector being the pit wall. At the pit boundary the match of the direct ground phase is degraded to about 0.9. The latest two events at 0.08 and 0.10 /ts are probably diffractions from the top and bottom of the lower block. These diffractions are disrupted when the receiver crosses the pit boundary, but continue very weakly at greater offset. Although weak, these diffractions are well matched (predominately > 0.95). Summary of the Test Pit Observations Most of the events in the test pit data set are reasonably well polarized, however very significant depolarization does occur locally. Interference, best observed in profile (Figure 5.25), appears to be the most significant cause of depolarization at this site, in some cases resulting in a match of only 0.1, a 70% decrease in amplitude. Near the bottom of the lower block is the best example of this interference effect due to the many nearby corners. Although dipping reflectors, in general, should depolarize the wavelet, in this data set the alteration usually only degrades the match to about 0.95 (a 3% amplitude decrease). An explanation for the NW wall having a lower match than the SE wall remains very speculative. Although reflector roughness could provide a greater contribution to depolarization than reflector dip alone, the NW wall is not significantly rougher. Although the incident angle remains nearly vertical in constant offset profiles, diffraction hyperbolas could exhibit a systematic variation in match due to a dependence of depolarization on incident angle. However, in this profile only one such example is identifiable where a phase change occurs in the diffraction off the upper NW corner of the lower block and this occurrence should have complicated raypaths. The CMP soundings clearly exhibit match variation with offset and added complications due to intervening structure. Although the majority of events in these data exhibit relatively minor 000 Chapter 5: INSTANTANEOUS POLARIZATION MATCH 159 CMP10: TE Amplitude and Polarization Match TE Component ,Tx |Rx Envelope Amplitude (u.V) 6 9 3 . 8 0 1 1 4 3 o 0 0 1 6 6 4 . 0 0 2 2 7 6 . 0 0 3 0 1 3 . 0 0 3 9 6 7 . 0 0 5 3 6 7 . 0 0 7 3 6 1 . 0 0 9 8 5 0 . 0 0 1 3 3 4 0 . 0 0 1 8 7 5 0 . 0 0 3 1 4 5 0 . 0 1 6 1 6 0 0 . 0 0 Offset (m) 1 2 ,Tx Cross Component | — Offset (m) 1 2 3 CD co Polarization Match Offset (m) 1 2 3 0 . 0 0 0 . 9 5 0 . 9 8 0 . 9 9 Figure 5.28 The input two-component 200 MHz CMP sounding and corresponding match near the SE edge of the pit (midpoint - 10 m station). Every second wiggle trace is plotted. Chapter 5: INSTANTANEOUS POLARIZATION MATCH 160 polarization anomalies, significant depolarization does occur in a number of zones resulting in significant amplitude deficiencies in the parallel component. 5.8 Summary of The Occurrence of Anomalous m p The two data sets revealed a few consistent occurrences of anomalous polarization match. Field experiments at the ramp investigated the decrease in polarization match for the simple case of smooth shallow dipping interfaces and a lateral change in material. Both profiles and CMP soundings demonstrate that the T M wavelet is often matched 0.10 values, or more, lower than the T E wavelet. This difference is—at least in part—explained by the occurrence of the Brewster angle in the T M mode. Depolarization due to smooth shallow dipping reflectors is usually not large enough to significantly affect the polarization match. The CMP soundings confirm an expected incidence angle dependence of depolarization and therefore polarization match, but this "confirmation" is limited by not really knowing the radiation pattern. Although actual match values are unknown for the direct phases, a significant difference in polarization match is recognized between the air and ground waves. The air waves at the test pit, also appear to have significantly different polarization resulting in relatively low T E matches. At both sites the most significant mechanisms for depolarization appear to be scattering from rough spots and interference. In general, degraded polarization matching is probably a second order amplitude effect. However, theory predicts that in some cases this second order problem could become sig-nificant (Beckmann, 1968; Roberts, 1994) or even a first order effect as documented for sea ice (Campbell and Orange, 1974; Kovacs and Morey, 1978). Additional work is required in characterizing the occurrence of significant depolarization, and in incorporating these effects into amplitude modelling. More subsurface environments and target types should be investi-gated for their depolarization characteristics, such as a systematic study of both dielectric and conductive targets of various simple shapes and scattering geometries. Chapter 6 SUMMARY AND CONCLUSIONS Nearly all analyses of reflection amplitudes in GPR data, to date, have involved only the single parallel component in attempting to delineate targets of anomalous amplitude, such as soil saturation and contaminants—a scalar view of a vector phenomena. Before proceeding with amplitude analysis, however, the effects of variable antenna coupling, both near-field and reflected-field, must be considered to some degree. In some cases amplitude targets may be falsely identified or missed entirely due to a change in near-field conditions, or due to depolarizing structure in the reflected-field. In fact, the target might be better identified by its anomalous polarization and perhaps an amplitude anomaly is primarily evident only in the cross component. The possible change in near-field coupling along a survey line is usually sufficiently addressed simply by noting any change in surface conditions, or suspected changes just beneath the surface. With an understanding of how these changes affect antenna properties, their resulting effect on the data can be qualitatively identified or at least correlated from the field notes. Suspected near-surface changes might be ground-truthed with a pick and shovel. Changes in antenna, properties with ground conditions primarily include a shift in centre frequency and changes in radiation pattern shape. Traversing into material having significantly higher dielectric constant will decrease the centre frequency and the beamwidth thereby increasing antenna directivity. Directivity can be maximized by elevating the antennas to one-tenth the dominant wavelength in air. The primary advantage of a more directive radiation pattern is the decreased energy from out of the plane scatterers. However, in some applications increased directivity is a disadvantage by decreasing the energy incident on the steepest dipping reflectors and by making the diffraction hyperbolae of finite targets less evident. An increase in conductivity will decrease the radiated power and smear the radiation pattern nodes making the pattern more omnidirectional for pulse antennas. 161 Chapter 6: SUMMARY AND CONCLUSIONS 162 In most cases, even for a dipping infinite plane, a linearly polarized wave will become elliptically polarized upon reflection and refraction. The magnitude of depolarization depends on the contrast in electrical properties, incident angle, incident polarization, and the orientation of the reflecting area. Frequency is another variable in dispersive media and in the relevant scaling of reflector asperities. Generally, wavelet depolarization increases with an increase in reflector asymmetry in the form of the reflector geometry, continuity, roughness, and anisotropy, thereby degrading the polarization match. The primary question I have investigated is, to what extent, in magnitude and occurrence, does depolarization degrade coupling and thereby affect amplitude variation as viewed in standard (single-component) surveys. Least obvious of the coupling effects is the polarization match between the reflected wavelet and the receiving antenna. Assessment of this effect requires multicomponent data and polarization processing. I developed an instantaneous polarization match estimate and applied it to two test sites having different levels of structural complexity. At the ramp, the T M survey mode was confirmed to suffer a larger degree of depolarization than the T E survey mode. At both sites, very localized match anomalies were identified and correlated to reflector rough spots and wavefront interference. The interference occurrence might be partially due to a degradation of the input envelope trace. Although more computationally involved (Rene' et al., 1986), a covariance based match estimate might prove more robust and should also be investigated. These two sites provide only an initial investigation and other traverse orientations, environments, and target types (such as metal pipes) should be investigated, some of which are likely better depolarizers than the targets I investigated. To reiterate a potential pitfall, the interpreter must be aware of out of the plane events which could produce an apparently prominent match anomaly. In such cases additional lines or a 3-D survey may be required to properly characterize these scattered events. If these events prove to have a distinct polarization, they could be attenuated by polarization filtering (Perelberg and Hornbostel, 1994). Inherent Chapter 6: SUMMARY AND CONCLUSIONS 163 in all signal character analyses is the necessity to apply only processing steps which preserve signal character. The processing stream suggested in Chapter 4, including some GPR specific steps, was applied with high fidelity. An underlying point of this work is that transverse waves contain a wealth of information which remains mostly untapped in GPR applications. Polarization analysis distills most of this information into two parameters such as the real and imaginary part of the polarization parameter. An obvious extension to instantaneous polarization match is to describe the polarization instantaneously (Rene' et al., 1986; Perelberg and Hornbostel, 1994). Polarization properties of a target can be an additional distinguishing characteristic (Beckmann, 1969; Cho, 1990; Roberts, 1994). Polarization, by its strict definition, is actually independent of amplitude (Section, affecting amplitude through coupling with the sensor. Scalar amplitude analysis can remain useful, particularly if combined with polarization information which should also be used to adjust or redefine a single-component amplitude. One possible amplitude "correction" is to rotate the data into optimum alignment, i.e. orient the parallel component along the major axis of polarization. However, if the polarization deviates significantly from linear then the rotation, indeed, only optimizes the amplitudes with significant energy remaining in the perpendicular component. Perhaps the simplest solution that honors this complexity and allows amplitude analysis within a single-component, is to model the resultant reflectivity, i.e. ER(0 = y E^(t) + E\(t) involving the envelope traces. The resultant reflectivity approach acknowledges a salient, fundamental, point from this investigation: transverse wave reflectivity is a multicomponent (vector) parameter. CITED REFERENCES Aki, K., and Richards, P.G., 1980. Quantitative Seismology Theory and Methods, 1,W.H. Freeman and Co., New York, NY, 557 p. Annan, A.P., 1973. Radio interferometry depth sounding: part 1 - theoretical discussion. Geophysics, 38, pp. 557-580. Annan, A.P. and Cosway, S.W., 1991. Ground Penetrating Radar Survey Design, Sensors and Software Inc., Mississauga, Ontario, Canada, 28 p. Annan, A.P., 1993. Practical Processing of GPR Data, Sensors and Software Inc., PEMD#91, Mississauga, Ontario, Canada, 26 p. Armstrong, J.E., 1956. 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Measured underwater near-field E patterns of a pulsed, horizontal dipole antenna in air: comparison with the theory of the continuous wave, infinitesimal electric dipole, Geophysical Prospecting, 38, pp. 805-830. White, R.E., 1991. Properties of instantaneous seismic attributes, The Leading Edge of Exploration, 10, pp 26-32. Yilmaz, O., 1987. Seismic Data Processing, Society of Exploration Geophysics Press, Tulsa, OK, 526 p. Appendix: THEORETICAL CASES OF DEPOLARIZATION This appendix is a compilation of depolarization cases, derived and discussed—mosdy—in Beckmann (1968), with some comments on the limits of particular cases. In a few cases other work is also referenced to better complete, or update, some cases where Beckmann provides a more limiting treatment. A trend noticeable from these many cases is that generally an increase in any form of asymmetry between the scatterer and the incident wavelet will enhance depolarization. Although not complete, this compilation should be useful as a quick reference on cases of depolarization that might be encountered in GPR surveys. Cases involving backscatter particularly concern common offset GPR profiles. The instantaneous polarization match of Chapter 5 measures how severely depolarization affects amplitudes and identifies the depolarizing structure. The following compilation includes cases of single scatterers, backscatter, edges and corners (conductors only), rough surfaces, and anisotropic media. A.1 Single Scatterers The general solution for this case, involving an arbitrary surface, arbitrary electrical properties, and arbitrary scattering geometry, is a complicated expression which does not yield insight without computer modelling. However, three simplifications do provide insight readily. These special cases include: 1. arbitrary scattering direction from a perfect conductor, 2. backscatter from an imperfect conductor, 3. bodies with symmetry (Beckmann, 1968). A. l . l Arbitrary Scattering Direction from a Perfect Conductor As noted in Section a perfect conductor has Fresnel reflection coefficients of R + = 1, and R~ = - 1, regardless of the local angle of incidence. This case assumes that the radii of curvature is large compared to the incident wavelength, and that edge effects are negligible. The first assumption is easily honored by the body being a perfect conductor which makes the body impenetrable to electromagnetic waves. The second assumption is coupled to the 170 Appendix: THEORETICAL CASES OF DEPOLARIZATION 171 first since edge effects become important when the curvature restriction no longer holds. Edge effects are considered in Section A.2. As in the case of a finite plane (Section 2.5.2) the shape of the scatterer determines only the spatial distribution of amplitude and does not affect depolarization. However, unlike the finite plane, a perfectly conducting body with large radii of curvature will not depolarize backscatter. Restrictions (in addition to perfect conductivity): 1. Radii of curvature large compared to A m c . . 2. Negligible edge effects (refer to Section A.2). Depolarization occurrences: 1. Shape of the scatterer determines only amplitude and does not affect the depolar-ization. 2. Backscatter not depolarized. A.1.2 Backscatter from an Imperfect Conductor Once again, this case assumes a large radii of curvature to avoid the contributions of waves returned to the scaterer's surface from within the body such as internal multiples. With this restriction in place, the occurrence of depolarization is the same as that for a finite plane. In general the backscatter is depolarized. Restrictions: 1. Radii of curvature large compared to Ai n c - . 2. Negligible edge effects. Depolarization Occurrences (same as for finite plane): 1. Backscatter is generally depolarized Exceptions (a surface element will not depolarize if one of the following hold) 1. 6 = | (locally grazing incidence) since backscatter does not occur. Appendix: THEORETICAL CASES OF DEPOLARIZATION 172 2. 3i = 0 , or § , i.e. the incident polarization plane is either perpendicular or equal to the local plane of incidence. 3. R + = R". In addition to a perfect conductor, this result holds for a mirroring element ("vertical" incidence) where ii = —K. A.1.3 Bodies Having Symmetry Electromagnetic waves incident on a surface exhibiting symmetry will not depolarize if two additional requirements hold. The reflected plane must coincide with the incident plane, i.e. K i and K2 are co-planar. Also, the incident polarization must be linear with the polarization plane either identical or perpendicular to the plane of symmetry. This second requirement has the effect of the cross components on either side of the symmetry plane canceling. These two polarizations are the characteristic polarizations, meaning that any other incident polarization will become depolarized. Therefore, a sphere—having an infinite number of symmetry planes—will depolarize most incident waves with the exceptions meeting the two additional requirements. Restrictions (for no depolarization): 1. Illuminated surface must exhibit symmetry. 2. K i and K2 are co-planar. 3. Incident polarization must be linear, either coincident with or perpendicular to the symmetry plane. A.1.4 Backscatter from Perfect Conductors Having Symmetry For the general case of backscatter, Beckmann (1968) summarizes work by Chytil (1960, 1961a,b) which considers backscatter from scatterers having one or more symmetry axes. This treatment was restricted to an incident wave that is linearly polarized and incident normally to a symmetry axis. Therefore, this study particularly addresses cases that may be encountered in common (near) offset GPR profiles. In Beckmann's summary of Chytil's work, depolarization Appendix: THEORETICAL CASES OF DEPOLARIZATION 173 is measured by the depolarization ratio, D: the ratio of cross to parallel backscattered powers. A special extreme-valued case of D is the cross-polarization ratio. Some pertinent points of Beckmann's (1968) summary are paraphrased or quoted below. Restrictions: 1. Perfect conductors only. 2. Scatterer dimension > A. 3. Shapes considered: strip, elliptical (including circular) cylinder, parabolic cylinder, and a spheroid. Depolarization Occurrences: 1. Maximum depolarization ratio usually occurs when the polarization plane concludes a 45° angle with the symmetry axis. Roberts (1994) has confirmed this occurrence of maximum depolarization for pipes by modelling. 2. Depolarization ratio, D, decreases as far", where k = 2-K/X, and a is a characteristic dimension of the scatterer (width of a strip, radius of a circular cylinder, etc.) with 1 < n < 4. A smooth curve approximating D(faz) by 'averaging' the oscillations of the latter is approximately proportional to far1 for two-dimensional scatterers (strip, cylinder slab) and far4 for three-dimensional scatterers (prism, spheroid, elliptical disc). 3. In all considered cases, D < 9(ka)~2; for all but the elliptical cylinder at normal incidence parallel to the major axis, D < 0.64(faj)~r, if ka > 1. 4. For surfaces having no edges, the cross polarization ratio is less, in most cases much less, than 10~2 for ka > 10 or a > 1.6A (Beckmann, 1968). A.1.5 Scattering from a Sphere The case of depolarization from spherical scatterers likely has some implications for GPR imaging of sediments such as rounded sands and gravels or boulders. Beckmann (1968) refers to a number of researchers who have derived the rigorous solution for scattering by a sphere of arbitrary electrical properties. Some limiting cases allow for only qualitative results concerning depolarization. For discussion, consider the following general situation. A sphere of radius a centred at the origin of a spherical coordinate system with radius coordinate r, angle 6 measured Appendix: THEORETICAL CASES OF DEPOLARIZATION 174 down from the vertical, and azimuth <f>. For reference, the incident propagation vector ki is in the positive, upward, direction (9 = 0), impinging the sphere normally at 9 = TT. The incident polarization is assumed linear with Ei in the <f> = 0 direction which is also the direction of the e i - unit vector. For reference, an e + component would be in the <j> = 4p direction. Medium properties: Homogeneous sphere in a homogeneous medium. Backscatter: Depolarization involves only a sign change (for most other scatter directions a linear incidence polarization will depolarize to elliptical). Two privileged scattering directions: The scattering directions <j> = 0, and <f> = - | are in the directions of the two coordinate polarizations and are also perpendicular to the incidence direction. If the incidence polarization is linear horizontal (pi = 0), or linear vertical (pi = oo) the scattered wavelet will maintain polarization in the <f> = j direction and become completely cross-polarized in the <f> = 0 direction regardless of the sphere's electrical properties. Generally, the dependence of depolarization on the sphere's electrical properties, size (a/A), and the scattering angle 9 is very complicated. However, limiting cases concerning size can be discussed. Large scatters (a » A) have essentially been considered by some of the previous sections in this appendix as those bodies whose radii of curvature is large compared to a wavelength. The other extreme (a « A) is a case known as Rayleigh scattering. a « A - Rayleigh Scattering: 1. Depolarization is independent of the sphere's electrical properties. 2. For scattering perpendicular to the incidence direction (9 = £ ) , p 2 = 0, i.e. horizontal linear polarization in the cj> = 0 direction. This means that very small spheres actually act as a polarizing filter for scattering directions perpendicular to incidence. Appendix: THEORETICAL CASES OF DEPOLARIZATION 175 3. The degree of polarization (/>) is a function of size (a/A) and scattering direction 9. a. In the scattering direction 9 = | , as particle size increases the degree of polarization decreases from completely polarized (/> = 1) to a mere maximum. b. For spheres of size increased to a ~ — Maximum polarization occurs at larger scattering angles (0max > f) f ° r dielectric spheres. — 0max < f for absorbing spheres. c. For spheres with radii on the order of a wavelength and greater, the occurrence of polarization maxima and minima becomes irregular until the scatterer size converges to the case handled by physical optics (a » A). d. Perfect conductors: The maximum degree of polarization occurs at the Thomson angle (9 = 60°). As the radius increases this angle also increases. A.1.6 Other Simple Shapes (Circular Cylinders) Since a sphere has the most symmetry of any simple shape, solutions for scattering and depolarization from the following simple scatterers are bound to be more complicated. Beckmann (1968) only makes numerous references to treatments and their variations. Some of these shapes include an infinite cylinder with circular cross-section, an infinite cylinder with elliptical cross-section, ellipsoids, a semi-infinite cone, a conducting wedge, a thin conducting circular disk, and a strip or slit. The interested reader can refer to pages 126 and 127 in Beckmann (1968) for these references. However, results from recent modelling by Roberts (1994) on the depolarization from horizontal circular cylinders is worth mention due to their common occurrence in GPR surveys as utility pipes. Roberts mentions in his thesis (1994) that the ability of metallic pipes to strongly depolarize wavelets near normal incidence has been exploited for their detection for years, particularly Appendix: THEORETICAL CASES OF DEPOLARIZATION 176 by polarimetric (two channel cross-dipole) antenna systems. Roberts (1994) modeled the T E and T M fields scattered from horizontal pipes having various azimuthal orientations, depths, diameters, and electrical properties. The modelling results indicate that backscattering from a water filled nonmetallic pipe is similar to that of an empty metallic pipe for small diameters (a/A < 0.1). In this case, the T M backscattered field is significantly greater than the T E backscattered field. When a/A ~ 0.1 the backscattering of T M and T E components becomes extremely frequency dependent. Note that these two types of cylinders are both low velocity cylinders. A fast velocity cylinder such as an air filled nonmetallic pipe produces a backscattered field opposite that of the above two cases, i.e. T E > T M and is the strongest depolarizer of these three cases at common antenna offsets. Depolarization from pipes: —The scattering characteristics of dielectric cylinders depends strongly on the dielectric contrast with the host material. — The degree of depolarization may be utilized to determine whether a pipe is metallic or dielectric and whether it is filled with water or air. A.2 Depolarization by Edges of Perfect Conductors A readily assessable formulation that fully describes diffractions from edges and corners is not available, except for some special cases of conductors (Beckmann, 1968, James, 1986, Roberts, 1994). Most of these formulations concern perfect conductors, however James (1986) also provides evaluations for diffraction by an impedance wedge for the special cases having vertex angles of 0° (half plane), 90°, 180° (flat plane), and 270°. The solutions for diffraction, solely, can be added to the uniform solution, i.e. superposition of solutions for the "large" finite plane (the uniform solution) with the edge effects (diffraction solution). Diffraction coefficients for conductors are approximated from the rigorous solution for an analogous Appendix: THEORETICAL CASES OF DEPOLARIZATION 177 canonical shaped body. For straight edges the diffraction coefficient is approximated from the rigorous solution for a wedge. The resulting diffraction coefficient is dependent on the incident polarization, i.e. different incident polarizations will result in different diffraction patterns. Generally, depolarization will occur upon scattering from a perfectly conducting edge. For non-normal incidence, edge diffracted waves will be depolarized even for horizontal and vertical polarization. However, two characteristic polarizations do exist, linear parallel to the edge and linear perpendicular to the edge, which of course do not depolarize. Note that the relative geometry between these characteristic polarizations and the edge requires normal incidence. For non-normal incidence, linear polarization may rotate to a new orientation, but will not be converted to a rotational polarization, whereas circular polarization will usually become elliptical. These rotational restrictions do not hold for an imperfect conductor. Backscatter will occur only for incidence which is normal to the edge. Restrictions: 1. Perfect conductors only. 2. Scatterer dimension > A. Depolarization Occurrences (same as for finite plane): 1. Diffraction pattern determined by incident polarization. 2. Generally, depolarization occurs, even for vertical and horizontal polarization. 3. Linear polarization will change orientation, but will not become rotational (nonper-fect conductors can cause linear to become rotational). 4. Circular polarization will usually become elliptical. Exceptions, i.e. Characteristic Polarizations 1. Linear polarization - parallel to the diffracting edge. 2. Linear polarization - perpendicular to the diffracting edge. ** Note: These characteristic polarizations imply that incidence must be normal. Appendix: THEORETICAL CASES OF DEPOLARIZATION 178 A.3 Reflector Roughness A.3.1 Statistical Descriptions of Roughness Studies on the affect of reflector roughness on scattering and the resulting depolarization are couched in the description of the statistical characteristics of the roughness (Krishen, et al., 1966, Beckmann, 1968, Phu, et al., 1994). This description is often in terms of the rms slope for an otherwise flat surface, i.e. an increase in roughness means an increase in rms slope. Other related statistical characteristics include the distribution of surface slope (in physical modelling a Gaussian distribution is often used), whether the roughness is deterministic or random, isotropic or anisotropic. For the anisotropically rough case, the effective correlation length is another statistical variable. A deterministic surface is defined as having definite specular points for a given incidence and scattering direction. For random surfaces, however, the number of specular points is a random variable. Not surprisingly, the geometrical optics approach fails when a surface has a low probability of specular points existing for a particular scattering geometry. A.3.2 Perfect Conductors The result from Section A . l . l concerning perfect conductors also applies to perfect con-ductors having random surfaces. This result states that the scattered polarization depends on the incident polarization, direction of scattering, but not the scatterer's shape, if it is a perfect conductor. A.3.3 Random Roughness of Finite Conductivity For imperfect conductors, the scattered polarization does depend on the surface slope, except in the limiting case of A = 0. Unfortunately, the mathematical solution is too involved to yield insight. Krishen, et al. (1966) demonstrated by physical modelling that (nonconductive) targets of the same dielectric constant but different surface statistics can be distinguished by their scattered cross-polarization field. They also concluded that targets of the same surface statistics Appendix: THEORETICAL CASES OF DEPOLARIZATION 179 but different dielectric constants should be likewise distinguishable. Phu, et al. (1994) also observed scattered polarization to vary with the surface statistics for a "very rough" conductive surface (a plastic surface painted with several layers of nickel paint). A.3.4 Backscatter from Randomly Rough Surfaces of Finite Conductivity Insight may be gained from the mathematical solutions if they are simplified for the case of backscatter (Beckmann, 1968). The mathematical simplification requires that the illumination angle not be too large (<5 ~ 55°), and the surface should be very rough, i.e. oJX » 1 where <7S is the standard deviation of the rms slope. Note that the experiments by Krishen et al. and Phu et al. used surfaces with "very rough" surface elements on the order of a wavelength. The same order of roughness in GPR surveys is 1 m at 100 MHz. Beckmann also gives only very limited consideration to the contribution of multiple scattering. Since the surface is random, both the parallel and cross polarization components (and therefore the depolarization) are also random. A useful measurement for discussing the polarization of a randomly scattered field is the cross-polarization ratio which is the ratio of mean cross polarization power to mean parallel polarization power This parameter is an easier description of random polarization, to both measure and calculate, than using probability distributions (Beckmann, 1968). A helpful conceptualization of depolarization from rough surfaces is to consider the scattering in terms of quasi-specular and diffuse components of backscatter (Beckmann, 1968). To simplify the explanation of this concept, momentarily do not consider multiple scattering. The quasi-specular component is backscattered by mirroring surface elements, i.e. the elements whose normals point in the direction of the observer, which is also coincident (or very nearly so) with the source. The diffuse component is scattered by the non-mirroring elements. Q = < ECE* > < EPE; > (* =3- complex conj.) . (A.1) Appendix: THEORETICAL CASES OF DEPOLARIZATION 180 Neighboring quasi-specular elements scatter waves that reinforce each other coherently because they are nearly in phase. Elements contributing to the diffuse component result in significantly different phases between waves scattered from the individual elements. These diffusely scattered waves add incoherently. As noted in Section A. 1.2, mirroring backscattering elements do not depolarize. Therefore, only the diffuse component of the backscattered field is depolarized. The relative number of quasi-specular and diffuse scattering elements is, in part, affected by the rms slope. This concept also extends to multiple scattering similar to the primary bistatic reflection described in Section 2.5.2 A.3.4.1 RMS Slope and Illumination Angle The rms slope controls the relative occurrence of quasi-specular and diffuse scattering elements. As the rms slope decreases, the local angle of incidence 6 has values closer to the illumination angle 9 resulting in a more prominent quasi-specular component. The illumination angle is measured from the vertical of the horizontal reference plane. However, the general dependence of depolarization on illumination angle must also be maintained; where depolarization is a minimum for 9 = 0 (vertical incidence) and a maximum at 9 = %| (grazing incidence). Additionally, as the rms slope increases, 6 will increasingly differ from 9 causing a larger diffuse component thereby increasing depolarization, for the case of single bounce backscatter (Beckmann, 1968). • Generally, depolarization is a minimum at vertical illumination angle for mirroring backscatter and a maximum at grazing illumination angle. • Generally (for single bounce backscatter), an increase in roughness results in an increase in depolarization. Unfortunately the theoretical insights outlined by Beckmann are too incomplete, with their required simplifications, to reconcile with physical modelling, particularly for multiple Appendix: THEORETICAL CASES OF DEPOLARIZATION 181 scattering, Physical modelling results of Krishen et al. (1966) and of Phu et al. (1994) involve significant contributions from multiple scattering. In some cases, the physical modelling results appear to conflict with Beckmann's theoretical predictions, perhaps due to multiple scattering. An additional complication in making this comparison is that Krishen et al. did not measure to the mirroring backscatter angle, the case in which Beckmann concentrates. Unfortunately, the modelling of three dielectric surfaces by Krishen et al. does not provide a clear measure of depolarization as a function of roughness. Results from the more recent modelling of conductive surfaces by Phu et al. exhibit a more diffuse T E co-polarized backscatter enhancement, however, the peak values are similar between the two roughnesses. The cross components appear similar in shape and magnitude indicating a small increase in depolarization for the rougher case, particularly at larger scattering angles. The rougher surface also produces a significantly greater enhancement at the specular angle. A.3.4.2 Wavelength Dependence For depolarization of single bounce backscatter, the dependence on wavelengths should be negligible, except for vanishing wavelengths (Beckmann, 1968). Physical modelling results by Phu, et al. (1994) exhibit a strong frequency dependence in the cross-polarized field. This potential discrepancy may in part indicate that their "very rough" surfaces are rougher (or, conversely, the signal higher frequency) than allowed by the approximations made in Beckmann's treatment. However, the significant multiple scattering is at least mostly responsible as Phu, et al. note in the following explanation for the decrease in backscattered cross-polarization with increasing frequency. The cross-polarization of backscatter depends very much on multiple scattering which depends on the distance between adjacent peaks of the rough surface, relative to the wavelength. The relative increase in this distance, due to decreasing wavelength, reduces the probability of multiple scattering. Therefore, the cross-polarized field decreases at sufficiendy increased frequencies. Appendix: THEORETICAL CASES OF DEPOLARIZATION A.3.4.3 Dependence on Electrical Properties and Incidence Polarization 182 From computer generated plots of cross-polarization ratio versus incidence angle for both single and double bounce backscatter, Beckmann (1968) illustrates that depolarization decreases with increasing dielectric constant. Although not plotted, depolarization is also expected to decrease with increasing conductivity. This theoretical trend at least agrees with Section A. 1.1 in the limit of infinite conductivity. However, the Krishen et al. (1966) results indicate that depolarization is greater for larger values of dielectric constant. Unfortunately, dielectric constant, or conductivity, were not varied in the Phu et al. study and this observed paradox remains unexplained. The trend concerning the dependence on incidence polarization maintains the very general trend noted in the introduction of this appendix: depolarization increases with increased incidence asymmetry between the polarization vector and the reflecting surface. At least for single bounce backscatter, depolarization is a maximum when the incidence polarization plane is 45° to the local surface (angle /?) and a minimum for {3 = 0° or 90° (Beckmann, 1968). This general relation was actually considered in the design of the Krishen et al. surfaces. However, this trend was not actually measured in either of the two experiments referenced in this section. A.4 Anisotropy A.4.1 Definition, Classification, and Occurrence Simply defined, an anisotropic material has one or more physical properties that vary with the direction in which they are measured. Any medium exhibiting oriented fabric at a particular scale (most geologic material, ionized gas permeated by a magnetic field) is anisotropic at that scale. Anisotropy depolarizes waves and affects the propagation velocity. Mechanistically, anisotropy depolarizes electromagnetic waves by the oriented fabric (in terms of electrical properties) imposing preferred directions on the induced current flow which then Appendix: THEORETICAL CASES OF DEPOLARIZATION 183 radiates secondary waves with the preferred polarization directions (Beckmann, 1968). A classic example in optics is the calcite crystal which is birefringent, producing two refracted images due to two privileged polarizations propagating at different velocities (Jenkins and White, 1957). Birefringence is observed in seismic exploration as shear wave splitting (Crampin, 1985). The ionosphere is anisotropic since it is permeated by the geomagnetic field and depolarizes radio waves by Faraday rotation (introduction of Section 2.5). In geologic materials two types of anisotropy are distinguished in the seismic literature both of which may pertain to GPR: transverse isotropy and azimuthal anisotropy. Transverse isotropy exhibits isotropy within a plane (such as a bedding plane or the internal layering of shales), i.e. no azimuthal variation, but is anisotropic with respect to the perpendicular direction (across the layering) (Sheriff andGeldart, 1982; Crampin, 1985). Azimuthal anisotropy (or just anisotropy) is the more general case exhibiting variation azimuthally as well as in the perpendicular direction. Generally, these two classes of anisotropy involve different scales where azimuthal anisotropy is often intrinsic, whereas lateral isotropy is classified by Crampin et al. (1984) as long-wavelength anisotropy since thin beds are involved. Intrinsic anisotropy consists of oriented fundamental grains that make up the material such as crystals or sediment clasts that become oriented during various environments of crystallization or deposition (Campbell and Orange, 1974; Kovacs and Morey, 1978; Crampin et al., 1984). The following sections address the possibility of anisotropy being observed in GPR data. A.4.2 Anisotropy of Reflector Roughness Erosion and deposition (especially of course grained material such as glacial till and sand and gravel) might both produce anisotropic roughness of sufficient scale to affect GPR reflections. Phu et al. (1994) illustrated obvious differences between the fields scattered from isotropically and anisotropically rough surfaces. Their very rough surface (rms height ~ A) was anisotropic in terms of the correlation length in the y direction being twice that in the x Appendix: THEORETICAL CASES OF DEPOLARIZATION 184 direction with lx ~ 2A and ly ~ 4A having corresponding rms slopes of px = 0.707 and py = 0.354. The Phu et al. experiment illustrates that the shape of the scattering pattern depends on the orientation of the scattering plane with respect to the anisotropy. In the direction of shorter correlation length (resulting in higher rms slope) a higher percentage of scattered energy is backscattered with a maximum at the mirroring angle. The longer correlation direction produces a much higher percentage of scattered energy in the specular direction with the maximum shifted from the mirroring angle to the specular angle. This scattering distribution also occurs for the cross component but with a much more subtle variation with scattering angle since the maximum in the parallel component becomes very much larger than the cross component. This relative distribution of energy between components indicates that, generally, depolarization should decrease as the scattering angle approaches the angle with maximum energy in the parallel component. However, the peak in scattered energy is probably much more diffuse for scattering planes having orientations nearly midway between the fabric coordinate system (x, y). The question remains: what is the threshold scale (in terms of dominant wavelength), i.e. at what scale does the anisotropic roughness become a significant influence. A.4.3 Anisotropy Due to Bedding (Transverse Isotropy) Anisotropy due to a sequence of thin beds having "large" velocity contrasts is well known in seismic methods (Levin, 1984). Currently I know of no examples in GPR, however this occurrence is rarely, if ever, investigated. Although such material that is also not prohibitively attenuative to GPR might be rare, its existence seems possible; perhaps a sufficiently large sequence of dry sand and gravel with relatively wet interlayered silt. An obvious question is at what scale and volume of interbedding is required for an observable anomaly in velocity and polarization? For seismic methods, Levin (1984) states a gross guide to interval thickness as the restriction kl < 1.0, where k = the wavenumber and / = the interval length. Levin made Appendix: THEORETICAL CASES OF DEPOLARIZATION 185 the safe but arbitrary choice of / ~ — A (A.2) 10 5 ' v ' which at 100 MHz and a material velocity of 0.1 m/ns yields 0.1 - 0.2 m. This interval length is the thickness that the wavelet averages layers into one transversely isotropic layer. Levin estimated that an upper limit of seismic anisotropy due to bedding might approach 30% and estimated 10% anisotropy for a number of intervals in drill cores. These measures of anisotropy are given by (vn - V J _ ) / V J _ . Blindly accepting these magnitudes for the radar case, an interbedded sequence 0.8 - 2.8 m thick would be required to produce a difference of 5 ns in two-way travel time between the two polarizations. These thicknesses are not geologically unreasonable, however the possible limit of anisotropy due to bedding for the radar case should be investigated as Levin did for the seismic case. A.4.4 Intrinsic Anisotropy Intrinsically anisotropic material of sufficient volume to produce an anomaly in GPR data is probably rare. A few geologic materials worth investigating for sufficient anisotropy are sed-iments having paleomagnetically oriented grains, clays, and ice. In fact, when profiling sea ice thickness with a GSSI (Geophysical Survey Systems, Inc.) GPR Campbell and Orange (1974) documented extreme anisotropy in first-year sea ice due to ice crystal alignment. Anisotropy appeared as the fading or extinction of the ice/water reflection when the transmit/receive an-tenna was rotated azimuthally with maximum and minimum amplitude occurring 90° apart. This phenomenon was observed in areas of uniformly thick sea ice of thicknesses ranging from 0.25 - 2 m. However, anisotropy was never observed on freshwater ice and multiyear sea ice usually appeared isotropic. An oriented ice core revealed that subparallel intercrystal platelets were oriented parallel to the tidal current. The maximum ice bottom signal occurred when the wavelet polarization was perpendicular to the platelets and the minimum signal correlated Appendix: THEORETICAL CASES OF DEPOLARIZATION 186 with polarization parallel to the platelets. These observations were confirmed in a later survey by Kovacs and Morey (1978). Unfortunately, Campbell and Orange were unable to measure the cross component which should exhibit a large amplitude increase with the extinction of the parallel component. 


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