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On-board data reduction for the Envisat Advanced Synthetic Aperture Radar : evaluation of the impact… McLeod, Ian 1995

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ON-BOARD DATA REDUCTION FOR THE ENVISAT ADVANCED SYNTHETIC APERTURE RADAR: EVALUATION OF THE IMPACT ON INTERFEROMETRIC AND WAVE MODE APPLICATIONS by Ian McLeod B.Sc. (Electrical Engineering), Queen's University, Kingston, Ontario, Canada, 1993 A THESIS SUBMITTED IN PARTIAL F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F APPLIED S C I E N C E in T H E F A C U L T Y O F G R A D U A T E STUDIES D E P A R T M E N T O F E L E C T R I C A L E N G I N E E R I N G We accept this thesis as conforming to the required standard T H E UNIVERSITY O F BRITISH C O L U M B I A September 1995 © Ian McLeod, 1995 In presenting this thesis in partial .fulfilment of the requirements for an advanced degree at the University of British Columbia, 1 agree that the Library shall make it freely available for reference and -study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be" granted . by the head of my department .or by his or her representatives, It is understood that copying or publication; of this thesis for financial gain shall not be allowed without my written permission. : Department of . E^UE^rAXC^: : G^ki^effl.,^ Q The University of British Columbia Vancouver, Canada Date 1 ^ T Z - ^ M t - -Zl . •Il'jf ' ' DE-6 (2/88) Abstract The ENVISAT remote sensing satellite, to be launched by the European Space Agency in 1998, will carry the Advanced Synthetic Aperture Radar (ASAR) microwave sensor. In order to ease the severity of data storage and downlink restrictions, an on-board SAR data reduction algorithm was designed by MacDonald Dettwiler and Associates (MDA) for E N VIS A T in 1993 [14]. This algorithm, called Flexible Block Adaptive Quantization (FBAQ) was tested extensively during development, to determine the impact of the algorithm on SAR image quality. However, many common uses of SAR data involve further processing of the SAR images. Two very common examples of this are Interferometric SAR (InSAR) where phase differences between two SAR images of a common scene can be used to estimate topography, and Wave Mode SAR, where the power spectrum of SAR images of ocean waves are used to determine wave characteristics such as direction and wavelength. For these applications, various qualities of the SAR data are exploited which may not be directly related to image quality. Therefore, in this study, an extension of the previous work performed at M D A , the evaluation of the F B A Q algorithm was expanded to include the SAR applications of Interferometry and Wave Mode. The goals of the work were first to quantitatively assess the degradation of InSAR and Wave Mode results due to FBAQ, and secondly to determine if the quality of the results were acceptable. To accomplish the above goals, Wave Mode and InSAR processing was performed on ERS-1 SAR data (modified to reflect the properties of ENVISAT data [16]) that had been encoded using ii FB AQ. Error measures taken against results produced using un-encoded data were used to quantify the impact of FBAQ on both InSAR height estimation and Wave Mode ocean parameter estimation. For InSAR, all possible data reduction ratios were used to determine which was acceptable for precision generation of Digital Elevation Models (DEMs). For Wave Mode, only 8-bits/sample to 2-bits/sample FBAQ was used due to storage limitations during acquisitions over oceans. To determine whether the quality of the FBAQ Wave Mode results were acceptable, they were compared to 2-bits/sample linearly truncated Wave Mode data, which represents one of the current methods used to compress Wave mode data aboard ERS-1 and ERS-2. The results of the study were as follows. For InSAR, only 4-bit FBAQ, with an average RMS height estimation uncertainty increase of less than 3 %, was found to produce topographical estimation acceptable for precision DEM generation. The 4-bit FBAQ also presented no problems for image registration and phase unwrapping, two important processing steps in InSAR processing. It was verified, however, that InSAR processing was also possible for 3-bit and 2-bit FBAQ encoding levels, which may find use in lower precision mapping. As well, important insights were gained into the characterization and spatial distribution of FBAQ encoding noise and coherence magnitude degradation, important factors in interferometric quality. For Wave Mode, the 2-bit FBAQ was found to produce results of better quality than those currently available using the 2-bit linear truncation option of ERS-1 and ERS-2. Location of spectral peaks in the ESA Wave Product were on par with 8-bit data, while the spectral peak magnitudes were degraded by about 10 %. This compares favorably, however, to the current 2-bit Wave Mode data which was found to experience degradation as high as 20%. As well, insight was gained into the spectral error distribution of FBAQ data, the impact of ships within the wave scene, the importance of normalization procedures in wave mode analysis, and the impact of data saturation on FBAQ data quality. Overall, the FBAQ algorithm was found to perform acceptably well for InSAR and Wave Mode applications. Table of Contents Abstract ii Table of Contents v List of Tables ix List of Figures ix List of Acronyms xvi Acknowledgment xvn 1 I N T R O D U C T I O N 1 1.1 Objectives of the Research 1 1.2 Outline of Report 3 2 PRINCIPLES O F S Y N T H E T I C A P E R T U R E R A D A R 5 2.1 Basics of Synthetic Aperture Radar 5 2.2 SAR Signal Data and Data Reduction 8 2.2.1 Characteristics of SAR Signal Data 9 3 D A T A E N C O D I N G T H E O R E T I C A L D E V E L O P M E N T 11 3.1 Minimum Mean Square Error Quantization 11 3.1.1 Properties of the Minimum Mean Square Error Quantizer 13 3.2 Minimum Mean Squared Error Quantization for SAR signal Data: Block Adaptive Quantization 14 3.3 FBAQ 16 Section I: SAR Interferometry Study 4 R E P E A T - P A S S SAR I N T E R F E R O M E T R Y 19 4.1 InSAR Equations 20 4.2 Interferometric Processing Steps 24 4.2.1 Obtain Two Single Look Complex Images of a Scene 24 4.2.2 Register the Images 24 4.2.3 Upsample the Images .25 v 4.2.4 Form an Interferogram 26 4.2.5 Rat Earth Compensation 26 4.2.6 Smooth the Interferogram 27 4.2.7 Unwrap the Interferogram 27 4.2.8 Perform Height Calculations (Calibration) 27 4.2.9 Re-sample to Ground Range Co-ordinates 28 5 THEORETICAL DEVELOPMENT OF INSAR ANALYSIS 29 5.1 Interferogram Quality 29 5.1.1 Sources of Phase Noise 30 5.2 Phase Noise Measurement 31 5.2.1 Error Distribution 34 5.3 Coherence Magnitude 37 5.3.1 Maximum Likelihood Coherence 38 5.3.2 Other Uses of Coherence Magnitude 43 5.4 Summary 44 6 INSAR EXPERIMENTAL METHODOLOGY 46 6.1 Data Selection 48 6.2 Data Pre-conditioning 48 6.3 Data Encoding/Decoding 49 6.4 SAR Processing 49 6.5 Upsampling and Interferogram Formation 53 6.6 Raw Interferogram Analysis 53 6.7 Flat Earth Compensation 53 6.8 Smoothed Interferogram Analysis 53 6.9 Coherence Analysis 54 6.10 D E M Error Analysis 55 6.11 Phase Unwrapping Error 56 7 INSAR EXPERIMENTAL RESULTS 57 7.1 Data Selection 57 7.2 Data Pre-conditioning 59 7.3 Data Encoding/Decoding 59 7.4 SAR Processing 61 7.5 Image Registration 65 7.6 Interferogram Formation and Analysis 65 7.7 Rat Earth Compensation 67 7.8 Smoothed Interferogram Analysis 67 vi 7.9 Coherence Magnitude Analysis 69 7.9.1 Average Coherence Magnitude Analysis 69 7.9.2 Average Phase Analysis 77 7.9.3 Phase Noise and Coherence Change vs. Original Coherence 83 7.9.4 Bin Analysis 87 7.10 Impact on D E M Generation 93 7.10.1 Addition of Variances 93 7.10.2 Change in Coherence Magnitude Measurement 97 7.10.2.1 Coherence Magnitude Bias Error 97 7.10.3 Overall Impact on D E M Height Uncertainty 102 7.10.4 The Impact of Calibration Error 103 8 R E G I S T R A T I O N A C C U R A C Y 106 9 P H A S E U N W R A P P I N G 110 9.1 Phase Residues I l l 10 I N S A R C O N C L U S I O N A N D R E C O M M E N D A T I O N S 116 Section II: SAR Wave Mode Study 11 W A V E M O D E S Y N T H E T I C A P E R T U R E R A D A R 121 11.1 ESA Wave Mode Algorithm 122 12 W A V E M O D E E X P E R I M E N T A L M E T H O D O L O G Y 126 12.1 Evaluation Criteria 127 12.1.1 Detailed Analysis of Wave Spectra 129 12.1.2 ESA Wave Product Analysis 130 12.2 Linear Quantization Comparison 131 13 W A V E M O D E E X P E R I M E N T A L R E S U L T S 132 13.1 Processed Images and Wave Spectra 133 13.2 Raw Data Comparison 140 13.3 Detailed Analysis of Wave Spectra 141 13.3.1 Normalized Spectrum Statistics 141 13.3.2 Interpolated Spectral Peak Analysis 141 13.3.3 Integrated Wave Energy Analysis 144 13.3.4 Clutter Analysis 145 13.4 ESA Wave Product Analysis 145 13.5 Discussion of Results 147 vii 13.5.1 Saturation Level 147 13.5.2 Under-Estimation of Absolute Spectrum Magnitudes 152 14 CONCLUSIONS A N D R E C O M M E N D A T I O N S O F W A V E M O D E S T U D Y . . . 155 15 S U M M A R Y A N D CONCLUSIONS 157 15.1 SAR Interferometry Study 157 15.2 SAR Wave Mode Study 158 B I B L I O G R A P H Y 159 A APPENDIX A : Data Pre-conditioning 163 A . l Rationale and Methodology 163 A . 2 Test Results 165 B APPENDIX B: The Impact of Ships on the E S A Wave Mode Algorithm 167 B. l Theory 167 B.2 Experiment 168 C APPENDIX B: E S A Wave Product at Other F B A Q Reduction Levels 173 D APPENDIX D : A S Q N R and Saturation 174 viii List of Tables Table 5-1 Average Phase Error Estimates for Images and Raw Interferograms (before phase smoothing) 35 Table 7-1 ERS-1 Parameters for scenes used in the study 59 Table 7-2 Signal Data Statistics for Sardinia orbit 327 and Toolik Lake orbit 943 60 Table 7-3 Signal Data Statistics for Sardinia orbit 241 and Toolik Lake orbit 1029 . . . 60 Table 7-4 SLC Image Statistics for Sardinia orbit 327 and Toolik Lake orbit 943 64 Table 7-5 S L C Image Statistics for Sardinia orbit 241 and Toolik Lake orbit 1029 64 Table 7-6 Average Phase Error Analysis for Raw Interferogram 66 Table 7-7 Average coherence magnitude results 70 Table 7-8 Average phase error statistics for smoothed interferogram (2x10 smoothing) 77 Table 7-9 Summary of bin analysis for Sardinia 89 Table 7-10 Summary of bin analysis for Toolik Lake 90 Table 7-11 Estimates of total RMS height uncertainty for Sardinia (normal baseline = 126 m) 95 ix Table 7-12 Estimates of total RMS height uncertainty for Toolik Lake (normal baseline = 40.4m) 95 Table 7-13 Change in coherence magnitude measurements for Toolik Lake measured using 4 by 20 coherence magnitude estimator 99 Table 7-14 Estimates of total RMS height uncertainty for Toolik Lake using change in coherence magnitude measurements (normal baseline = 40.4m) 100 Table 7-15 Average D E M RMS height estimation uncertainty 103 Table 7-16 Example of the impact of Calibration Error (for the "Ideal" E N VIS A T case 105 Table 7-1 Change in coherence due to use of shift parameters obtained from encoded data 107 Table 7-2 Coherence Magnitude of Sardinia Original (un-encoded) data 108 Table 9-1 Total number of residues in the smoothed interferogram 112 Table 13-1 Raw Data Statistics 140 Table 13-2 Detailed Analysis of Wave Spectra Results 142 Table 13-3 ESA Wave Product Analysis Results (bin magnitude values range from 0-255) 146 Table 13-4 Performance of F B A Q vs. L Q for Scene 1 with reduced saturation 149 Table C-1 ESA Wave Product Bin Magnitude Error O i^n magnitudes values from 0 - 255) 172 x List of Figures Figure 2-1 Simplified SAR imaging geometry 6 Figure 4-1 Satellite across-track InSAR geometry 21 Figure 5-1 Height error as a function of baseline length and interferogram phase error 33 Figure 5-2 Relation of F B A Q encoding phase error to magnitude 36 Figure 5-3 Probability density function of 10 look interferogram phase for coherence magnitudes 0.2,0.4,0.6, and 0.8 41 Figure 5-4 Coherence magnitude vs. interferogram phase standard deviation for 1,8, and 20 looks 41 Figure 6-2 Flow chart of experimental methodology 47 Figure 7-1 Single look Complex image of Sardinia and corresponding phase error due to 3-bit encoding scaled to show detail (orbit 241, slant range vertical, azimuth horizontal) 62 Figure 7-2 Phase Error and Real Error due to 3-bit F B A Q encoding vs. magnitude 62 Figure 7-3 Single look Complex image of Toolik and corresponding phase error due to 3-bit encoding scaled to show detail (orbit 943, slant range vertical, azimuth horizontal) 63 Figure 7-4 Phase Error and Real Error due to 3-bit F B A Q encoding vs. magnitude 63 xi Figure 7-5 Histograms of raw interferometric phase error for 4-bit, 3-bit, and 2-bit encoding for the Sardinia scene. 66 Figure 7-6 Histograms of raw interferometric phase error for 4-bit, 3-bit, and 2-bit encoding for the Toolik Lake scene 67 Figure 7-7 Smoothed original interferograms: magnitude and phase 68 Figure 7-8 Sardinia coherence magnitude images for the original (unencoded), 4-bit, 3- bit, and 2-bit encoding levels 71 Figure 7-9 Toolik Lake coherence magnitude images for the original (unencoded), 4- bit, 3-bit, and 2-bit encoding levels 72 Figure 7-10 Histograms of reference coherence magnitude and 4-bit, 3-bit, and 2 bit change in coherence magnitude due to F B A Q for Sardinia 73 Figure 7-11 Histograms of reference coherence magnitude and 4-bit, 3-bit, and 2 bit change in coherence magnitude due to F B A Q for Toolik Lake 74 Figure 7-12 Absolute coherence magnitude difference images (scaled by a factor of 1000) for Sardinia 75 Figure 7-13 Absolute coherence magnitude difference images (scaled by a factor of 1000) for Toolik Lake 76 Figure 7-14 Sardinia interferogram phase images for reference, 4-bit, 3-bit, and 2 bit F B A Q 79 Figure 7-15 Sardinia phase error due to F B A Q encoding (multiplied by a factor of 10) . . 80 Figure 7-16 Toolik Lake interferogram phase images for reference, 4-bit, 3-bit, and 2 bit F B A Q 81 Figure 7-17 Toolik Lake phase error due to F B A Q encoding (multiplied by a factor of 10) 82 xii Figure 7-18 Histograms of smoothed interferogram phase error due to F B A Q encoding/decoding for 4-bit, 3-bit and 2-bit encoding levels for Sardinia . . . 83 Figure 7-19 Histograms of smoothed interferogram phase error due to F B A Q encoding/decoding for 4-bit, 3-bit and 2-bit encoding levels for Toolik 83 Figure 7-20 Scatter plots of interferogram phase error due to F B A Q encoding for Sardinia 84 Figure 7-21 Scatter plots of interferogram phase error due to F B A Q encoding for Toolik Lake 85 Figure 7-22 Scatter plot of change in coherence magnitude due to F B A Q encoding vs. initial coherence magnitude for Sardinia 86 Figure 7-23 Scatter plot of change in coherence magnitude due to F B A Q encoding vs. initial coherence magnitude for Toolik Lake 87 Figure 7-24 Histograms of phase error and coherence magnitude change due to encoding for 3 bit F B A Q . Bin 3 is initial coherence magnitude values 0.2 to 0.3, bin 8 is initial coherence magnitude values 0.7 to 0.8 88 Figure 7-25 (a) RMS encoding phase error, (b) mean change in coherence, and (c) RMS coherence magnitude change vs. original coherence magnitude 91 Figure 7-26 Total rms height uncertainty vs. original coherence magnitude for Sardinia 96 Figure 7-27 Total rms height uncertainty vs. original coherence magnitude for Toolik Lake 96 Figure 7-26 Total rms height uncertainty vs. original coherence magnitude for Toolik Lake derived using measured change in coherence magnitude 101 Figure 8-1 Hypothetical effect of adding more residues to an area of low residue concentration 109 Figure 9-1 Phase residue maps for Sardinia 114 Figure 9-2 Phase residue maps for Toolik Lake 115 Figure 11-1 ESA Wave Mode Product Layout 124 Figure 12-1 Flow chart of experimental methodology 128 Figure 13-1 Scene 1 off coast of Portugal A . Original 8-bits/sample. B . Decoded 2-bits/sample F B A Q 134 Figure 13-2 Scene 1 wave spectra: (a) Wave spectrum of 8-bit Original, (b) Decoded 2-bit F B A Q wave spectrum, Absolute Difference Spectrum, (d) Histogram of pre-conditioned (Original) data 135 Figure 13-3 Scene 2 off coast of Portugal A . Original 8-bits/sample. B . Decoded 2-bits/sample F B A Q 136 Figure 13-4 Scene 2 wave spectra: (a) Wave spectrum of 8-bit Original, (b) Decoded 2-bit F B A Q wave spectrum, Absolute Difference Spectrum, (d) Histogram of pre-conditioned (Original) data 137 Figure 13-5 Scene 3 off coast of Portugal A . Original 8-bits/sample. B . Decoded 2-bits/sample F B A Q 138 Figure 13-6 Scene 3 wave spectra: (a) Wave spectrum of 8-bit Original, (b) Decoded 2-bit F B A Q wave spectrum, Absolute Difference Spectrum, (d) Histogram of pre-conditioned (Original) data 139 Figure 13-7 Histogram of Scene 1 original raw data with reduced saturation levels . . . . . 149 Figure 13-8 ASQNR performance measured with respect to the continuous input data for varying levels of histogram standard deviation 151 Figure 13-9 ASQNR performance measured with respect to the 8-bit input data for varying levels of 8-bit histogram standard deviation (8-bit curve still measured with respect to continuous input) 152 xiv Figure B - l The Effect of Ship Location: (a) ship located near edge of vignette, (b) wave spectrum of (a); (c) ship located near center of vignette, (d) wave spectrum of (c) 170 Figure B-2 Collapsed Views of the Spectra of Figure B - l 171 Figure D-1 Summary of ASQNR Experiment demonstrating the effect of saturation on ASQNR measurements taken with respect to 8-bit data 174 xv List of Acronyms A S A R Advanced Synthetic Aperture Radar A S Q N R Average Signal to Quantization Noise Ratio B A Q Block Adaptive Quantization D E M Digital Elevation Model DETS Data Encoding Techniques Study F B A Q Flexible Block Adaptive Quantization ERS-1 Earth Resource Satellite 1 ERS-2 Earth Resource Satellite 2 E S A European Space Agency ESTEC European Space Research and Technology Center InSAR Interferometric Synthetic Aperture Radar JPL Jet Propulsion Laboratory L Q Linear Quantization M C M Maximize Coherence Magnitude M F V Maximize Fringe Visibility M R C Minimizing Residue Count M S E or mse mean square error R M S or rms root mean square SAR Synthetic Aperture Radar SIR-C Shuttle Imaging Radar - Mission C SLC Single Look Complex SNR Signal to Noise Ratio V M P Verification Mode Processor xvi Acknowledgment First I would like to thank Dawn for putting up with me during this project, especially during periods of stress — when frustration, ridiculously long hours, and violent mood swings were all too common. Secondly, I would like to thank my supervisor, Ian Cumming, for his encouragement and advice throughout this work, and frequent efforts "beyond the call of duty" on my behalf. I would also like to express my appreciation to the members of the U B C Radar Remote Sensing group, for providing a group atmosphere that was a pleasure to work in. Of this group, special thanks go out to Mike Seymour, for his hours of patient instruction in the finer points of SAR Interferometry, and the use of his interferometry software. Finally, I am also grateful for the support and technical advice received from Melanie Dutkiewicz and Ian Burke of MacDonald Dettwiler and Associates. For financial support, I am indebted M D A and to the National Science and Engineering Research Council of Canada for their support through an NSERC Graduate Scholarship. xvii .... but life is short and information endless. Abbreviation is a necessary evil, and the abbreviator's business is to make the best of a job which, although intrinsically bad, is still better than nothing. — Aldous Huxley xviii 1 INTRODUCTION The following document is a thesis report on research carried out towards a Master of Applied Science degree at the University of British Columbia Department of Electrical Engineering. The work was carried out under the MacDonald Dettwiler and Associates (MDA) / National Sciences and Engineering Research Council (NSERC) Industrial Research Chair, in co-operation with M D A . The work was part of the European Space Research and Technology Center (ESTEC) contract 10737/94/NL/JG, WO 03, for which Saab Ericsson Space (SES) was the prime contractor, and MacDonald Dettwiler the subcontractor. The results of the study were presented successfully at ESTEC on July 7,1995 [ 1,2,3], at the International Geoscience and Remote Sensing Symposium (IGARSS) on July 14,1995 [4], and at the Progress in Electromagnetics Research Symposium (PIERS) on July 24,1995 [5]. 1.1 O b j e c t i v e s o f t h e R e s e a r c h The objective of this study was to quantitatively determine the impact of the Flexible Block Adaptive Quantization (FBAQ) SAR data reduction algorithm on SAR Interferometry and SAR Wave Mode, two demanding applications of SAR images. The F B A Q algorithm is a lossy data 1 Chapter 1: Introduction reduction algorithm developed by M D A for use with the Advanced Synthetic Aperture Radar (ASAR) to be launched aboard the European Space Agency (ESA) satellite ENVISAT in 1998. Previous work by M D A had established F B A Q as a viable SAR signal data reduction algorithm primarily through SAR signal data and image analysis. However, in many cases it is difficult to predict the impact of encoding on applications of the SAR images simply from image quality measures. Thus, this study sought to extend the work of the previous studies to determine the impact of F B A Q on two important applications of SAR images: SAR Interferometry, a process whereby precision Digital Elevation Models (DEMs) can be generated from S A R images, and Wave Mode processing, whereby ocean wave qualities such as wavelength, direction and relative strength are derived from the spectra of a SAR ocean image. The goals of the work were as follows: 1. Determine the feasibility of performing interferometric SAR (InSAR) processing using data which had been encoded using F B A Q . This included a quantitative error analysis of the major steps in the InSAR processing (registration, calibration, and phase unwrapping), and measurement of terrain height estimation error due to F B A Q encoding. 2. Based on results from 1, determine the level of data reduction acceptable for the generation of precision DEMs using SAR Interferometry. 3. Quantitatively determine the impact of F B A Q encoding on ocean wave parameters generated using the ESA Wave Mode processing algorithm. 2 Chapter 1: Introduction 4. Based on the results of 3, determine if quality of Wave Mode results is acceptable compared to the current standard of Wave Mode data. To meet the above goals, actual satellite SAR data was encoded using various reduction ratios, and then used to generate Interferometric and Wave Mode results. The quality these results, compared to those generated using data with no encoding, were used to determine the impact of the F B A Q algorithm. 1.2 Outline of Report This thesis report is composed of the following: • Chapter 2: Explains the basic principles of SAR image formation pertinent to the data encoding work. • Chapter 3 explains the theory of F B A Q data encoding, and the operational F B A Q system. • Section I (Chapters 4 to 10) detail the theory, methodology, experimental results and conclusions of the SAR Interferometry study. • Section II (Chapters 11 to 14) detail the theory, methodology, experimental results and conclusions of the Wave Mode study. • Chapter 15 summarizes the conclusions of the study. • Appendix A examines the method of data preconditioning used in the study. • Appendix B describes the impact of ships in a SAR Wave Mode scene. 3 Chapter 1: Introduction • Appendix C examines the quality of Wave Mode data at lower data reduction levels. • Appendix D investigates through simulation the impact of saturated data on F B A Q data encoding quality and quality measurement. 4 2 PRINCIPLES OF SYNTHETIC APERTURE RADAR In order to understand some of the techniques and terminology used later in the report, a brief introduction to synthetic aperture radar (SAR) is required. As a complete treatment of SAR would fill several volumes, the introduction will be limited to those concepts needed to understand the ensuing data encoding work. 2.1 B a s i c s o f S y n t h e t i c A p e r t u r e R a d a r Synthetic aperture radar is a coherent microwave imaging system capable of producing high resolution images of terrain and target objects [7]. A n active sensor, SAR provides day/night, all-weather sensing capabilities that make it particularly useful for monitoring areas of persistent cloud cover (e.g. rainforests), determining wave conditions under storm systems, making mission critical observations (e.g. disaster monitoring), and monitoring temporal change (i.e. same region viewed several days in succession). In addition, SAR data provides information not contained in optical satellite images, which may be exploited for certain applications. A good example of this is the process of SAR interferometry, in which the phase of the received microwave signals of two SAR images can be used to determine terrain elevation. 5 Chapter 2: Principles of Synthetic Aperture Radar Figure 2-1 shows a simplified imaging geometry for a SAR sensor. As shown in the diagram, S AR is a side-looking sensor. The sensor moves in a direction called the azimuth direction, while transmitting pulses of microwave energy in the slant range direction. For non-squinted SAR, the slant range direction is perpendicular to the direction of motion. Figure 2-1 Simplified SAR imaging geometry For spaceborne SAR systems, image formation is a two step process, which is generally divided between the on-board and ground station systems. The first step is the acquisition of raw signal data. Signal data is acquired through the transmission of microwave pulses to the Earth, and the coherent reception of the energy reflected back from the ground. The transmitted energy is in the form of a wide bandwidth linear frequency modulated pulse called a chirp. As the satellite moves along its trajectory, chirp returns from the ground are stored for each position along the azimuth track. Thus, the SAR signal data can be viewed as a two-dimensional array of microwave 6 Chapter 2: Principles of Synthetic Aperture Radar energy returns, indexed by slant range and azimuth time. Note that due to the spreading of return echoes and movement of the sensor, the SAR signal data is not an "image" in the visual sense. In fact, the signal data is difficult to distinguish from random Gaussian noise. The second step in image generation is to convert the SAR signal data into a visual image. For spaceborne systems, this step generally occurs on the ground, after the signal data has been downlinked to Earth1. To create the visual image (or SAR image data), the signal data is input into a SAR processor. Several different techniques have been developed to correlate the SAR signal data into an image, the most common of which is the Range/Doppler algorithm. In all techniques, however, matched filtering operations are applied in both the slant range and azimuth directions to "compress" the received pulses to high resolution. Pulse compression in the range direction is accomplished through the matched filtering of the phase of the wide bandwidth chirp signals, the same concept used in traditional radar systems. Pulse compression in the azimuth direction is achieved through matched filtering the azimuth phase returns of the signal, using knowledge of the phase history a target on the ground would experience as it is swept by the radar beam. The output of the SAR processor is a two dimensional visual image of the region the sensor passed over. Bright areas in the magnitude of the image correspond to regions of strong radar reflectivity on the ground, and dark areas correspond to regions of low reflectivity (note that because the SAR is a coherent sensor, the processed SAR image data has both a magnitude and phase component). A n important characteristic of SAR images is that they are prone to a multiplicative noise called speckle. Speckle occurs when the phases of reflected energy add constructively or destructively, creating a "salt and pepper" granularity on the image. Speckle is often reduced 1. ERS-1 and ERS-2 support an option which allows partial image formation on-board the satellite. This op-tion is rarely invoked, however, primarily due to quality concerns. 7 Chapter 2: Principles of Synthetic Aperture Radar through averaging techniques before the images are utilized in remote sensing applications. Images which have been averaged to reduce speckle are termed multi-look images, while those which have not are called single-look images. Finally, it is worth emphasizing that due to the processing performed during image formation, SAR image data and SAR signal data have very different statistical properties, particularly in dynamic range, sample to sample correlation, and overall redundancy [6]. Thus, data reduction of SAR image data is quite different from data reduction of SAR signal data, and different techniques may be optimal for the different data types. Since this study is concerned with on-board data reduction for spaceborne applications, we will limit our focus to the reduction of SAR signal data. 2.2 S A R S i g n a l D a t a R e d u c t i o n SAR signal data is quite volumous. A single scene of approximately 100 km in range by 100 km azimuth requires several hundred megabytes when quantized to 8-bits/sample. Thus, downlink bandwidth and on-board storage capacity are limiting factors in the design of operational spaceborne SAR systems. In many systems the swath width is either data rate limited, or the dynamic range of the system is sacrificed by reducing the number of bits/sample [7]. SAR signal data reduction is thus important to lessen the severity of such restrictions. 8 Chapter 2: Principles of Synthetic Aperture Radar 2.2.1 Characteristics of SAR Signal Data In order to choose an appropriate signal data reduction technique, the statistics of the signal data itself must be considered. ENVIS A T SAR signal data has the following statistical properties [6]: 1. Complex data, sampled at baseband 2. Zero-mean circular Gaussian distribution 3. Slowly changing variance in both slant range and azimuth 4. Zero I, Q, sample correlation 5. Low inter-sample correlation 6. Digitized to 8-bits/sample (81, 8Q) 7. Some degree of saturation of the distribution due to hard hmiting of the analog to digital converter 8. Data spectrum is relatively constant over both range and azimuth Due to the low sample to sample correlation, lack of systematic patterns in the data, and relatively large bandwidth with respect to the sampling rate, many traditional spatial data reduction algorithms such as predictive coding or transform coding are not optimal for SAR signal data. Thus, to date, most work in on-board data reduction has focused on various forms of Vector Quantization (VQ), Block Adaptive Quantization (BAQ), or some combination of the two [6 -12]. While both techniques have been shown to deliver comparable quality, only versions of B A Q have ever been implemented on a spaceborne system. This is primarily due to the importance of 9 Chapter 2: Principles of Synthetic Aperture Radar simplicity for space qualified systems. The B A Q encoder is far simpler and faster than the V Q encoder, meaning less use of satellite resources while in orbit, and less chance of failure since operational constraints such as clock speed can be relaxed. In the case of ENVISAT, both V Q and B A Q were re-examined by M D A during the initial algorithm selection process [13]. However, due to the complexity of V Q in the face of little performance gain, the technique ultimately developed, Flexible Block Adaptive Quantization (FBAQ), is a variant of B A Q rather than V Q [13,14]. 10 3 DATA ENCODING THEORETICAL DEVELOPMENT As previously mentioned, F B A Q is a variant of Block Adaptive quantization (BAQ) first used by N A S A for the SAR sensor aboard the Magellan mission to Venus [8]. In turn, the B A Q algorithm is based on the concept of the minimum mean square error quantizer, first put forth by Max in 1960 [15]. In the following sub-sections, the background theory and implementation details of the M D A F B A Q algorithm are presented. 3.1 Minimum Mean Square Error Quantization In its simplest sense, the theory of minimum mean square error quantization recognizes the fact that the thresholds and reconstruction levels of a quantizer should be set based upon the probability density function of the incoming data. Quite simply, threshold and reconstruction level density should be greatest near the "center of mass" of the probability distribution, i.e. where the majority of the data is expected to lie. Thus, the traditional Uniform quantizer, with linearly spaced threshold and reconstruction levels, is only optimal if the incoming data has a Uniform distribution. Data which is not Uniformly distributed (such as SAR signal data) can thus be represented more efficiently (in the mean squared error sense) if optimal threshold and reconstruction levels are used. 11 Chapter 3: Data Encoding Theoretical Development To determine the optimum thresholds, fy, and optimum reconstruction levels, for a given probability density function, p(u), we must minimize the function e = E[(u-u')2] = J (u-u')2p(u)du , (3-1) where e is the mean squared error, defined as the expected value of the square of the difference between the scalar random variable w, and the reconstruction of this signal u'.As shown, this can also be expressed using the integral form of expected value, for an L-level quantizer [15]. If equation (3-1) is re-written as L e " '<« -r , ) 2 | > («)«/«, (3-2) i - l the equations needed to solve for the threshold and reconstruction levels can be obtained by differentiating it with respect to and r^ and setting the results equal to zero [15]. Doing so yields tk = * 2 * 1 , (3-3) and mK+ 1 J up(u)du J p(u)du 12 Chapter 3: Data Encoding Theoretical Development which must be solved simultaneously given the boundary values of tj, tL+1, and the probability density function p(u). In practice this is accomplished numerically using iterative methods. 3.1.1 Properties of the Minimum Mean Square Error Quantizer There are three properties of the minimum mean square error quantizer which are important when considering it as a SAR data reduction method [15]: 1. The quantizer output is an unbiased estimate of the input. Thus: E[u] = E[u']. 2. The quantization error is orthogonal to the quantizer output. Thus the quantizer noise T| is uncorrected with the quantizer output, which means we can express the following: u = H ' + T | , (3-5) and E[(u-u')2] = E[u2]-E[(u')2] . (3-6) Since aJ" > 0, equation (3-6) implies that the quantizer output power is reduced by the amount of the quantization noise power. 3. The threshold and reconstruction levels are scalable. Thus, it is only necessary to design minimum mean square quantizers for zero mean, unity variance 13 Chapter 3: Data Encoding Theoretical Development distributions. To illustrate, if tk and rk are the threshold and reconstruction levels for zero mean, unity variance distribution, tk = u+ofy and rk=[i+ork are the threshold and reconstruction levels of the same distribution but with a mean of u. and variance of a 2 . 3.2 M i n i m u m M e a n S q u a r e d E r r o r Q u a n t i z a t i o n f o r S A R s i g n a l D a t a : B l o c k A d a p t i v e Q u a n t i z a t i o n Having developed the theory behind minimum mean square error quantization, the next step is to apply the theory to SAR data. As previously discussed, SAR data can be characterized by a zero mean, circular Gaussian distribution with a slowly changing variance in both the slant range and azimuth directions. Due to the Gaussian nature of the data, it is thus reasonable to assume that it can be more efficiently represented (i.e. in less bits/sample) if quantized using optimal Gaussian thresholds and reconstruction levels rather than Uniform quantization. If we assume the data to be zero mean, then the only variable that needs to be determined to calculate all threshold and reconstruction levels is the variance of the data. However, as mentioned, this value is non-stationary within the data set. To overcome this problem, the data is first quantized to 8 bits/sample using a standard Uniform quantizer. B A Q then divides the 8-bit SAR signal data up into blocks, under the assumption that the statistics within each block are stationary. Choosing the size of the block to use is a trade-off, as the block must be small enough that the variance of the SAR data is essentially constant within the block, but large enough that an adequate estimate of the variance can be made. Thus, for each block of data, the following steps occur during B A Q encoding/decoding: 14 Chapter 3: Data Encoding Theoretical Development 1. Estimate the variance of the data, 2. Calculate the optimum thresholds, 3. Quantize the data using the optimum thresholds to less than 8 bits/sample, 4. Transfer the variance estimate and the quantized data to ground, 5. On the ground, use the variance estimate to calculate the optimum reconstruction levels, and reconstruct the quantized data. The amount of variability within the SAR data depends on the system parameters of the SAR system, and thus the choice of block size is sensor dependent. For practical implementation, a block usually consists of a number of samples taken from a single range line, since using more than one range line necessitates storing several full lines of SAR data in memory before B A Q processing can commence. With 5616 8-bit complex samples for each E N VIS A T range line, storing more than one range line quickly becomes computationally prohibitive, requiring long buffers and significant time delays in processing. Previous studies at M D A determined that the algorithm was very insensitive to block sizes between 64 and 128 samples [14]. The second major difficulty in adapting the theory of minimum mean square to SAR signal data is the problem of actually calculating the standard deviation of the data (hereafter referred to as the rms value of the data) on-board the satellite. For a zero mean signal, the rms value o, can be calculated as: (3-7) 15 Chapter 3: Data Encoding Theoretical Development where N is the number of samples in the block, and and Qt are the real and imaginary components of the i'th sample. However, this calculation requires the use of square and square root operators, which are difficult to implement in hardware, and involve the use of approximations. To overcome this problem, the statistic used to determine the rms value is average magnitude, which for an 8-bit saturated Gaussian distribution can be related to rms value through the equation [14]: 126 mean\l\ = mean\Q\ = 127.5- (3-8) i-0 Using the estimate of (3-8), it is thus possible to map the mean magnitude estimate of I and Q to the estimated threshold value using a look up table. 3.3 F B A Q During the original development of the B A Q algorithm, only one data reduction ratio was available, that from 8-bits/sample to 2 bits/sample. The F B A Q algorithm use all the methodology of the B A Q algorithm described above, but takes it one step further by allowing a flexible choice of reduction ratios. Four ratios are supported: 8-bit to 2-bit, 8-bit to 3-bit, 8-bit to 4-bit, and a pass-through mode (no encoding). The data reduction levels are selectable by mission control during satellite operation, and are essentially implemented by adding more look up tables to the B A Q algorithm to support the different reduction ratios. Results of the previous M D A studies concluded that different reduction ratios were appropriate for SAR data intended for different uses. These results can be summarized as follows: 16 Chapter 3: Data Encoding Theoretical Development • 2 bits/sample: good visual quality, all image features well reproduced, no mis-registration. Recommended for applications dependent on visual quality and particularly wide swath and quick-look applications • 3 bits/sample: excellent image quality, radiometry and point target responses well preserved, no mis-registration effects. Recommended for most users. • 4 bits/sample: best image quality, radiometry and image statistics very close to 8-bit data levels, low levels of phase error (weighted phase error approximately 4 degrees). Recommended for precision uses such as interferometry. Ln addition to the flexibility in reduction ratio, a final enhancement was added to F B A Q not found in the traditional B A Q . This enhancement relates to property number 2 of the minimum mean square quantizer which states that a minimum mean square quantizer by nature is not power conserving, but loses power between input and output. In some applications of SAR data (such as Wave Mode spectral analysis) losing power due to F B A Q may be undesirable. Thus, F B A Q supports a decoder option which scales each block of decoded data by a scalar factor to equalize the input and output rms levels. This equalization, of course, comes at a small expense in mean square error performance, and is only recommended for certain applications where power conservation is desirable. 17 Section I: SAR Interferometry Study 4 REPEAT-PASS SAR INTERFEROMETRY Interferometric SAR (InSAR) is the process whereby the phase differences of two overlapping SAR images of a common scene are used to obtain new information about the terrain features within the scene. This study focuses on using satellite InSAR in the across-track geometry, a technique through which the generation of large scale Digital Elevation Models (DEMs) from space is possible. To estimate terrain elevation using interferometry, the SAR images of a scene must be acquired from slightly different across-track locations, separated by a distance called the baseline. While airborne InSAR systems utilize a fixed baseline and can obtain two simultaneous SAR images by employing two sensors aboard a single aircraft, this is not true of current SAR satellites. For spaceborne InSAR, a process called repeat-pass interferometry is employed, in which two images of a scene are acquired by the same satellite flying in a parallel repeating orbit. Hence, images are acquired several days apart, rather than simultaneously, and the baseline length is variable depending on the orbit parameters. For ERS-1, the closest repeat is approximately three days, and baselines are typically in the order of tens to hundreds of meters. A baseline of 175 to 19 Chapter 4: Repeat-Pass SAR Interferometry 200 meters is considered optimal for ERS-1 [18,29], but no technique is currently available to ensure such a baseline occurs. To produce DEMs, InSAR utilizes the fact that a phase change between the two images for a given pixel will be proportional to a elevation change in the terrain. The phase differences between the two images is measured by registering the images to a sub-pixel level, then multiplying one image by the complex conjugate of the other to form an interferogram. The interferogram phase is thus a map of phase difference at each pixel location in the registered images. Terrain height estimates can then be calculated from the interferogram phase if the geometry of the InSAR system is known accurately. In practice, for satellite InSAR, several known elevation points on the ground (called tie points) are needed in order to calibrate the system to the degree of accuracy required for useful elevation estimates. 4.1 I n S A R E q u a t i o n s A simplified across-track InSAR configuration is shown in Figure 4-1. In this figure, two SAR system antennas are portrayed separated by a baseline distance denoted by B. The baseline is tilted at an angle a with respect to the horizontal, where the horizontal is defined as a tangent plane to the Earth geoid at the point of interest. S AR1, at height h above the reference plane is arbitrarily chosen as the geometric reference for the interferometric process. The point of interest on the surface is point P, which is at an elevation of z from the surface. Knowing the range (r), we see that in order to determine the elevation of P one must determine the angle 0, called the interferometric 20 Chapter 4: Repeat-Pass SAR Interferometry nadir angle for pixel P. However, note that if only SARI was considered, point P could be interpreted at being at any point along the arc of radius r, such as point A . Thus, a single sensor SAR cannot determine the height of point P since it is impossible to distinguish the interferometric nadir angle of P from that of any other point which may be at distance r from the radar. of radius r+8 geoid tangent Figure 4-1 Satellite across-track InSAR geometry However, when a second sensor acquires the same scene from a slightly different viewing angle, it is possible to isolate P, and hence determine the elevation at P. To accomplish this we must determine the extra slant range distance of the second sensor (8 ). In SAR interferometry, this is accomplished through measurement of the phase difference between the two images. To understand this, consider the phase of SARI (§{) which can be expressed as: 21 Chapter 4: Repeat-Pass SAR Interferometry <t>, = arg{S1} = 2 r ( y ) + < | W , 1 (4-1) where X is the wavelength of the SAR system, and §noisei accounts for noise sources of the signal, including system noise, and (if employed) lossy data reduction noise. Similarly, the phase ofSAR2(<t>2) is: • a = arg {S2} = 2 (r + 8) ( y ) + <t>„0I,e2 (4-2) By subtracting equation (4-1) from (4-2) above, it can be seen that the phase difference between the images (<P) is directly proportional to the extra distance in slant range (8) travelled by the signal from SARI plus additional noise terms: * - 4>2 " f l = (x) 8 + Koise + temporal + ^baseline C 4 " 3 ) The tynoiseterm m (^-3) is the combined effect of the noise terms from (4-1) and (4-2). The ^temporalterm m m ^ s equation accounts for noise added to the phase difference due to changes in the ground reflectors between image acquisition times. For ERS-1 the time between acquisition is a minimum of three days, and significant scene changes may occur during this period (movement of reflectors, ground freezing, precipitation, etc.) which will cause temporal decorrelation. The ^baseline term accounts for phase noise due to the differing viewing angle of the scene. Changing 22 Chapter 4: Repeat-Pass SAR Interferometry the viewing positions of the satellites causes phase noise because the interaction of ground scatters with the SAR signal will be slightly different when the microwave signal comes from a different angle. The difference in phase interaction due to these terms not exactly cancelling out is termed baseline decorr elation, since it grows with increasing baseline. When the baseline becomes greater than the critical baseline (about 1.1 km for ERS-1 and flat topography), the difference in viewing angle of the two sensors becomes too extreme and baseline decorrelation dominates the InSAR processing, making height estimation impossible.1 However, assuming a reasonable baseline length such that §baseune is small and also assuming relatively small contributions from other noise sources, we can see from equation (4-3) that the phase shift for pixel P is proportional to the distance 8. Thus we can determine 8 from the phase difference measurement. In interferometry, this phase difference is obtained by multiplying one image by the complex conjugate of the other pixel by pixel, to form the interferogram. Once 4> has been measured and 8 calculated, 0 can be determined using the geometry of Figure 4-1. Applying the law of the cosines to the triangle formed by S A R I , SAR2, and P results in: sin ( a - 6 ) -  (r*S)l~;-Bl (4-4) Now that 0 is known, determining the height of pixel P is a simple matter of trigonometry which yields the result: 1. Recent work by Gatelli et al. [22] has shown that for baselines less than the critical baseline it is possible to reduce the effects of baseline decorrelation by spectral filtering during range compression of the raw data. This is a topic of current research and was not implemented in this study. 23 Chapter 4: Repeat-Pass SAR Interferometry z = h-rcosQ (4-5) Thus, using InSAR, every pixel in an image is fixed in three dimensions: range, azimuth and elevation, instead of only two dimensions as found in conventional SAR. 4.2 Interferometric Processing Steps In order to implement the InSAR theory, satellite InSAR processing can be divided into several basic steps. These steps are explained in the following sub-sections: 4.2.1 Obtain Two Single Look Complex Images of a Scene This is accomplished in spaceborne interferometry during two repeat orbits of the satellite. The images must be processed carefully using a phase preserving SAR processor. 4.2.2 Register the Images For interferometry it is desirable to register the images within a fraction of a pixel. Several different techniques have been proposed and used successfully to calculate the shifts in slant range and azimuth needed to register SAR images accurately. These methods include: • Maximizing Correlation of Image Magnitude: Subsets of the images (image chips) are extracted at regular intervals across the scene. The chips are shifted minutely (fractions of a pixel) with respect to each other and a correlation function of image magnitude computed for each small shift. The local maximums are used as the actual shift values. 24 Chapter 4: Repeat-Pass SAR Interferometry • Maximizing Coherence Magnitude: Similar to above, except a different correlation function is employed which takes into account interferometric phase correlation, rather than just the image magnitude. Coherence magnitude will be discussed in detail in following sections. • Maximizing Fringe Visibility: Also similar to above except small interferograms are formed instead of calculating a correlation value. Fringes refer to the periodic 2 n wrapped phase bands which are characteristic of the phase of raw interferograms. These fringes should be most visible (i.e. least noise) when the images are exactly registered. Thus, a series of small shifts are applied and the peak of the Fourier transform of the phase fringes used as the quality measure. • Minimizing Residue Counts: Again, images chips are shifted with respect to each other. Residues are phase discontinuities which should be fewest when the images are exactly registered. Phase residues are counted for each small shift and the local minimums taken as the shift estimates. Once the required registration shifts have been estimated, they are implemented using quadratic interpolation (to determine the shift needed at each pixel) and a bank of FLR shift filters. 4.2.3 U p s a m p l e t h e I m a g e s Complex multiplication in the signal domain is equivalent to convolution in the frequency domain, which has the effect of doubling the interferogram bandwidth from that of the images. 25 Chapter 4: Repeat-Pass SAR Interferometry Thus, images should be upsampled by a factor of two in each dimension before complex conjugate multiplication is performed, in order to avoid aliasing. 4.2.4 F o r m a n I n t e r f e r o g r a m The interferogram is formed by the complex conjugate multiplication of the two registered images. The phase of each pixel of the interferogram is the phase difference between corresponding pixels in the registered images. The raw interferogram is "wrapped" meaning there is a 2TC ambiguity in the phase. This gives rise to phase fringes in the interferogram. The properties of interferograms are discussed in more detail in the following section. 4.2.5 F l a t E a r t h C o m p e n s a t i o n Even if the ground is perfectly flat, the raw interferogram phase will contain fringes in the range direction due to the change in slant range difference between corresponding image pixels for increasing range. This is a systematic effect which can be calculated from the InSAR geometry and subtracted from the interferogram. This is often called removal of the "range phase ramp" or "flat Earth fringes". Once the range phase ramp has been removed, the fringe pattern of the interferogram should correspond very closely to the pattern of contour lines found on a conventional topographic maps. i.e. quick rises in elevation are characterized by close, narrow fringes, and large flat areas will be characterized by wide constant phase regions. In some scenes, an azimuth phase ramp may also exist due to changes in the parallel component of baseline across the azimuth extent of the image [28]. This can be removed in a similar manner to the range phase ramp. 26 Chapter 4: Repeat-Pass SAR Interferometry 4.2.6 S m o o t h t h e I n t e r f e r o g r a m The raw interferogram is smoothed by averaging and down-sampling small blocks of the complex pixels. This is done to reduce statistical variations and noise found on the raw interferogram in anticipation of phase unwrapping and the generation of height estimates. The amount of averaging performed is a trade-off between accuracy and resolution of the height estimates. 4.2.7 U n w r a p t h e I n t e r f e r o g r a m Unwrapping the phase of an interferogram is the process of removing the 2n ambiguity from the phase fringes. This step is necessary if absolute height estimates are to be made from the interferogram in order to generate a Digital Elevation Model (DEM). This is often a difficult step in the InSAR processing due to phase discontinuities (called residues) caused by noise and terrain effects. Several techniques for dealing with residues have been proposed in the literature [30]. For spaceborne InSAR, relative low SNR levels mean that some level of manual intervention is often necessary during the phase unwrapping process, especially in difficult terrain such as mountains where fringe density is very high. In addition, some areas may be impossible to unwrap due to a high concentration of residues. Such areas must be removed from the phase unwrapping process and later estimated by interpolation or using existing topographical data from other sources. 4.2.8 P e r f o r m H e i g h t C a l c u l a t i o n s ( C a l i b r a t i o n ) Height estimates (or InSAR calibration) are made using equations (4-3) to (4-5). In practice, due to imprecise knowledge of the satellite orbit, points of known elevation on the ground (called 27 Chapter 4: Repeat-Pass SAR Interferometry tie points) are required to refine the estimation of parameters such as baseline length and phase offset in order to obtain accurate terrain elevation estimates. A minimum of three tie points is generally required, though using more increases the calibration accuracy [28]. 4.2.9 R e - s a m p l e t o G r o u n d R a n g e C o - o r d i n a t e s . As with regular SAR images, the interferogram must be re-sampled from slant range to ground range co-ordinates for mapping applications. 28 5 THEORETICAL DEVELOPMENT OF INSAR ANALYSIS In this section we develop the theory behind the interferogram quality measures used in the study in order to understand the impact F B A Q will have on height estimation uncertainty. The interferogram phase is a map of phase difference (modulo 2K) VS . slant range and azimuth distance. As mentioned, it is derived by multiplying one complex SAR image by the complex conjugate of another image of a common scene that has been carefully registered to the first. Thus, the pixel values, F(iJ) of an interferogram can be mathematically expressed as the multiplication of each pixel in one image, Sj(ij), by the complex conjugate of the corresponding pixel in the other image S2(i,j). While the phase of the interferogram (<P) is proportional to difference in slant range (8), there are also other factors which affect it. It can be shown [18] that variations in the interferogram phase (AO) can be linearly approximated as 5.1 Interferogram Quality + n2n + 3>g (5-1) 29 Chapter 5: Theoretical Development of InSAR Analysis where Ar is the change in slant range, Az is the change in elevation, and Arj corresponds to a coherent movement of all scatterers within the resolution cell between image acquisitions. Bn is the baseline distance normal to the slant range direction. The first term in equation (5-1) is the systematic "range phase ramp" which is eliminated by the flat Earth correction outlined in Section 4.2.5. The second term is the most important for topographical mapping since it relates elevation change to interferogram phase change. The third term accounts for coherent temporal changes in the scene occurring between the two image acquisitions. Such changes could occur due to coherent movement of the scatterers or changes in their electromagnetic properties. The fourth term accounts for random phase noise. Sources may be from non-coherent temporal change, speckle, atmospheric effects, and thermal noise. The fifth term in the equation accounts for the 2n ambiguity of the interferogram phase. It is this term which is removed through phase unwrapping. Finally, the last term, ® q , is phase quantization noise contributed by the F B A Q algorithm. It is this term which we wish to isolate for this study. 5.1.1 S o u r c e s o f P h a s e N o i s e In the above analysis of interferogram phase, all factors are systematic (i.e. correctable) except for the phase noise and quantization noise terms. Since interferogram phase values are directly proportional to terrain elevation, a primary concern in interferometry is the amount of this uncorrectable phase noise. High levels of phase noise in the interferogram will result in inaccurate height estimates, and may make procedures such as phase unwrapping difficult, i f not impossible. 30 Chapter 5: Theoretical Development of InSAR Analysis Phase noise can arise from several different sources during InSAR processing. The primary sources (in no particular order) are: a) system noise (radar receiver noise and speckle) b) quantization noise (from 8 bit quantization — generally kept below the system noise level) c) Phase noise from data reduction d) temporal distortion (changes in scene characteristics between acquisitions) e) baseline decorrelation (changes in phase scattering center due to change in viewing angle from one acquisition to the next) f) SAR processing errors g) mis-registration of images In this study, data reduction phase noise was isolated by subtracting interferograms formed by images encoded to 4-, 3-, and 2-bits/sample from a reference interferogram (hereafter called the original interferogram) which was formed using un-encoded images. Special care was taken to ensure that the processing of the images (SAR and InSAR) was identical for all encoding levels, thus ensuring accurate comparison. The exact experimental methodology is outlined in Chapter 6. The two primary quality measures used in the study are phase noise and coherence magnitude. 5.2 P h a s e N o i s e M e a s u r e m e n t The relation between phase noise and height estimation uncertainty can be illustrated by differentiating equations (4-3) to (4-5) with respect to O. Doing so yields: Chapter 5: Theoretical Development of InSAR Analysis dz . JdQ) rsin9 ( X X2® } „ 5* - r w n f l \ 3 * J - cos(a-9)l4Ss + r B ^ J ( 5" 2) The inclusion of r (on the order of 850 km) in the denominator of the second term within the brackets of this equation makes it negligible compared to the first term, and the expression can be simplified as: dzmjL( s i n 9 ) ( 5 . 3 ) d<b 4 7 t B U o s ( a - e ) J ^ From the InSAR geometry, the baseline component normal to the center look direction (0C) can be termed Bn, and the component parallel to the center look direction termed Bp. From this definition it can be shown [28] that Bn - Bcos (a-0 c) (5-4) Assuming a small range of look angles, as is the case for scenes processed in this study, equation (5-3) can now be rewritten as: /TA.sin0c " n Thus, if the level of phase error added to an interferogram due to data reduction techniques can be measured (c/O), the resulting error in height estimates (dz) can be determined using (5-5). It can be seen that for a given ERS-1 InSAR pair with phase error dQ>, the dominate factor in determining the amount of height error is the normal baseline component. A larger baseline will 32 Chapter 5: Theoretical Development of InSAR Analysis mean less height estimation error for a given phase error. However, with a larger baseline the level of phase error on the interferogram will increase due to baseline decorrelation, and larger baselines are also prone to aliasing error in mountainous terrain. Based on these effects, an optimum ERS-1 baseline of 125 to 175 meters was proposed by Hagberg et al. [18], and a 200 m baseline was deemed optimal by Zebker et al. [29]. Equation (5-5) also confirms the intuitive assumption that higher levels of data compression, which cause higher levels of phase error, will result in higher levels of height estimation error in the final InSAR product. The relation between phase error, baseline length, and height estimation error is shown in Figure 5-1, where typical ERS-1 parameters of r = 850 km, X = 0.0566 m, and 9 C = 23 degrees have been assumed. Note that while the phase error dependence is linear, the baseline dependence is not, meaning dramatic increases in the height estimation error per phase error for shorter baselines. Height estimation error dependence on baseline length and phase error rms phase error (degrees) 0 0 normal baseline length (m) Figure 5-1 Height error as a function of baseline length and interferogram phase error 33 Chapter 5: Theoretical Development of InSAR Analysis 5.2.1 E r r o r D i s t r i b u t i o n The previous F B A Q studies performed by M D A [13,14] measured levels of phase error added to several images due to F B A Q encoding. As a rough analysis of expected interferogram phase error, one can consider the average phase error measurements from these studies to infer the average phase error which will be associated with an interferogram produced by them. Assuming phase encoding error to be zero-mean, Gaussian, and independent between the two images (also shown in [14]), the variance of the phase error on the raw interferogram due to encoding should be the sum of the variances of the phase error on the images used to produce it. That is: °tgram = </<V + °?2 • O 6 ) Where oigram is the standard deviation of F B A Q phase noise for the interferogram and oqj and oq2 are the standard deviations of phase noise for the images used to form it. 1 Thus, assuming both images used to form the interferogram have the same phase error rms value (or close to the same) the expected average phase errors of the raw interferograms can be 1. Note that for a zero mean distribution and sufficient number of samples standard deviation equals RMS. 34 Chapter 5: Theoretical Development of InSAR Analysis calculated from the average phase error measurements of the previous study. The results of such calculations are displayed in Table 5-1. Table 5-1 Average Phase Error Estimates for Images and Raw Interferograms (before phase smoothing) 4-bit encoding 3-bit encoding 2-bit encoding Average RMS phase error for images from previous study (deg.) -12 -20 -33 Average expected RMS phase error for interferograms (deg.) -17 -28 -47 It is important to remember, however, that the above phase errors are for the raw interferogram. In practice the interferogram is smoothed in order to reduce phase noise. The word average has been stressed throughout this analysis. This is because the phase error added to the images due to F B A Q encoding is not uniformly distributed across the image. Specifically, areas of the image with high signal to noise ratios (SNR), characterized by bright areas in the detected images, are less affected by phase encoding noise than those areas with low SNR. Thus, for a given SAR image, phase errors due to F B A Q encoding will be most heavily concentrated in the sections of the image corresponding to dark areas in the magnitude image. The reason for this effect is illustrated in Figure 5.2, which depicts two arbitrary image pixels in the complex plane. Pixel 1 has a magnitude vector (Ml ) which is quite small. This would correspond to a dark area of the SAR scene (low SNR). Pixel 2 is from a bright (high SNR) portion of the scene, and has a magnitude vector of M2. For ease of illustration both pixels lie on the real 35 Chapter 5: Theoretical Development of InSAR Analysis axis. The analysis can be extended to vectors with imaginary components without loss of generality. The dotted circle represents the region the vectors M l and M2 may occupy after F B A Q encoding error is added to each pixel in the Cartesian domain (real/imaginary). Note that the error added to M2 is greater than that added to M l . This is because F B A Q quantizes larger samples more coarsely than smaller ones (the graph is not to scale and the level of error has been exaggerated for purposes of illustration). M l ' and M 2 ' are the new magnitude vectors computed after F B A Q encoding/decoding. The phase error due to F B A Q for pixel 1 and pixel 2 is given by 01 and 02 respectively. Note that even with the much greater error added to M2, the phase error at M l , 01, is still substantially larger than 02. Thus it can be seen that for a given level of encoding error added in the Cartesian plane, the phase error in the polar plane is larger for dark (low SNR) pixels than bright (high SNR) pixels. Figure 5-2 Relation of F B A Q encoding phase error to magnitude. Note 01 > 02. Of course, it may theoretically be possible to choose threshold and reconstruction levels that produce a uniform level of phase error for all magnitude sizes, but the current version of F B A Q is not designed in this manner. The idea would be complicated by the fact, however, that equalizing the phase error in the raw data domain would not necessarily guarantee equalization of the phase 36 Chapter 5: Theoretical Development of InSAR Analysis error in the image domain since the dynamic range of the magnitude of the data is greatly expanded in the image domain (while phase error must remain between -180 and 180 degrees). The effect illustrated in Figure 5-2 is very important to SAR interferometry, since areas of images with low SNR already contain high levels of phase noise which can make them undesirable for InSAR processing. In practice, such areas are usually removed from the phase unwrapping process, then estimated using interpolation. Thus, it is expected that the phase noise added to the images (and hence the interferogram) due to the F B A Q algorithm will be distributed in such a way that most phase errors wil l be concentrated in those areas which already contain high levels of phase error, and the areas appropriate for InSAR processing (those with relatively little phase noise initially) wil l be less effected by the encoding process. Thus, the actual impact of FBAQ encoding on parts of the interferogram appropriate for DEM generation should be much smaller than the average error predictions would indicate. The measure of coherence magnitude provides another method of assessing the noise content, and hence quality, of interferograms. The complex coherence (y) between two images S1 and S2 used to create the interferogram is defined as: 5.3 Coherence Magnitude 7 = (5-7) where E { # } represents the expected value and * represents complex conjugation. 37 Chapter 5: Theoretical Development of InSAR Analysis The coherence magnitude ry| varies between 0 (no correlation) and 1 (complete correlation), and the coherence phase (arg^yS^* )) is simply the phase of the interferogram. The coherence magnitude will be 1 if arg{SjS2* } is the same for all pixels within the averaging region. The addition of phase noise in the region has the effect of increasing the standard deviation of arg{5;52*}, which in turn lowers the value of coherence magnitude. Thus, coherence magnitude provides a means of assessing the amount of phase error associated with an interferogram (i.e. pixels of the image with low levels of phase noise will have high coherence magnitude values, those with high phase error will have low coherence magnitude). For InSAR, this implies that areas with high coherence will be those most appropriate for processing as they have less phase noise and will thus yield better height estimates. Areas with low coherence magnitude wil l yield poor height estimates, and will have low fringe visibility, meaning phase unwrapping may be impossible. The relationship between coherence magnitude and standard deviation of phase error is derived in the following section. 5.3.1 M a x i m u m L i k e l i h o o d C o h e r e n c e In order to reduce statistical noise, a common technique is to average the interferogram pixels. In InSAR processing this is called smoothing the interferogram and is an important step before phase unwrapping or height estimation. Smoothing reduces the level of phase noise associated with a given level of coherence magnitude. For interferograms, it was shown by Rodriguez [19], that the optimal estimator in the maximum likelihood sense is to spatially average the complex signal values, called "taking looks". Here, the "number of looks" is equal to the 38 Chapter 5: Theoretical Development of InSAR Analysis number of complex interferogram pixels averaged. For the case of n looks, the expression for complex coherence becomes: 5 > i ( 0 V ( 0 (5-8) 7 = 1 Elv>|2Xlv>p i -1 i - i Using equation (5-8), Lee et al. [20] have shown that the probability density function for a smoothed interferogram can be expressed as a function of the size of the averaging region (n), and the coherence magnitude as: r ( n + ^ ) ( i - l 7 l V p n - W 2 ) « />(*) 7T7T72+ 2K (^(">1:1/2;P2))> -n«Z<n . (5-9) 27ir («) ( i -3 2 ) where 3 = |yicos(<D-9). (5-10) and F(n,l;l/2;3 2 ) is the Gauss hypergeometric function. Note the pdf of (5-9) depends only on the number of looks and the coherence magnitude of the data. The peak of the distribution will be located at 0=8, which is the mean phase for the averaging region. As 9 varies, the center point of the distribution will shift, but the shape of the pdf will remain the same. Recall that a single fringe on the interferogram is defined by 8 going from -7t to n over some spatial distance. 39 Chapter 5: Theoretical Development of InSAR Analysis Figure 5-3 show the theoretical pdf of interferogram phase for 10 look averaging and coherence magnitude varying from 0.2 to 0.8, assuming 8=0. The distribution of points about the mean is due to phase uncertainty of the estimator, and thus assuming a mean phase uncertainty of zero, the curves of Figure 5-3 can be viewed as phase error distributions. At a coherence magnitude of zero the pdf is a completely uncorrelated uniform distribution. As the coherence increases, however, the pdf becomes narrower and Gaussian-like in shape. Theoretically at a coherence magnitude of 1 the pdf would be a delta function (i.e. no phase uncertainty). By multiplying equation (5-9) by <P2 and numerically integrating with respect to <P, we can determine the variance of the pdf, which is a measure of the phase uncertainty of the estimated mean phase value. Figure 5-4 shows the theoretical plots of coherence magnitude vs. the standard deviation of phase error after smoothing for single look, 10-look and 20-look calculations (no averaging, 10 pixels averaged and 20 pixels averaged respectively). These plots provide a way to utilize coherence magnitude to measure phase error. It can be seen that two factors govern the level of phase uncertainty: coherence magnitude level, and number of looks. The standard deviation of the phase estimate is lower for higher coherence magnitude, and for a given level of coherence magnitude, the phase error decreases with increasing number of looks. 40 Chapter 5: Theoretical Development of InSAR Analysis Probability Density Function for Interferogram Phase, 10 look smoothing phase error (rad) Figure 5-3 Probability density functions of 10 look interferogram phase for coherence magnitudes 0.2,0.4,0.6, and 0.8. Multi-look Phase Standard Deviation vs. Coherence Magnitude 1201 1 1 1 1 1 1 1 1 coherence magnitude Figure 5-4 Interferogram phase standard deviation vs. coherence magnitude for 1,10, and 20 looks 41 Chapter 5: Theoretical Development of InSAR Analysis For example, a data set with a coherence magnitude of 0.8 will have a phase uncertainty of about 52 degrees for the single look image, but only about 10 degrees in the 10 look case. It is worth noting, that equal movement between points on the single look curve (to which the phase noise in interferometry is initially applied since interferograms are formed with single look images before smoothing is performed), such as between 0.8 and 0.7 and between 0.3 and 0.2 coherence magnitude levels, does not imply equal levels of phase error being added to the interferogram. Rather, the change in coherence magnitude due to the addition of a given amount of phase noise will depend on the initial level of coherence magnitude. A small amount of phase noise added to an area of high coherence (such as 0.8) will cause a relatively large drop in coherence magnitude, while the same amount of phase noise added to an area of low coherence wil l not cause an appreciable change in the coherence magnitude level. Thus, much more phase noise would have to be added to cause the coherence magnitude to move from 0.3 to 0.2 than from 0.8 to 0.7. Of course, F B A Q phase noise is added to the images used to form the interferogram, not to the interferogram itself. However, a strong relationship exists between the SNR level of the images (which we have previously shown to be an important factor in the level of F B A Q phase noise added to the image) and the coherence magnitude of the interferogram. Assuming that temporal and baseline decorrelation are minimal for a scene, the primary source of decorrelation in the interferogram wil l be receiver noise and speckle, which are directly related to the SNR of the original images used to form the interferogram. Thus, areas of low SNR in the images wil l produce areas of low SNR within the interferogram, which in turn will have a low value of coherence magnitude. The relationship between SNR and coherence magnitude can be expressed as [23] Chapter 5: Theoretical Development of InSAR Analysis i T l - i / (5-H) i](l + SNR1 ) (l+SNR2 ) where SNRj and SNR2 are the SNR values for the images used to form the interferogram. Of final note in this discussion of coherence magnitude, is the fact that the theoretical formulae for the interferogram phase pdf are derived assuming perfect interferogram pixel independence. This is not entirely true for real interferograms. In addition, the curves are derived assuming a perfectly flat topography. In practice, when coherence magnitude levels are calculated it is extremely difficult to avoid including phase variations due to the terrain of the scene within the smoothing block. These effects will cause the smoothing effectiveness to deviate from theoretical levels. In practice it has been empirically observed that for real interferograms the level of phase variance will be higher than that predicted by theory [21,27,28]. However, it has been shown experimentally that the phase can still be modelled using the theoretical curves, but using an effective number of looks that is less than the number of pixels used in the averaging region. For example, for a 20 pixel averaging region, experiments have shown the effective number of looks for real ERS-1 data to fall between 8 and 12 looks [27,28]. 5.3.2 O t h e r U s e s o f C o h e r e n c e M a g n i t u d e Assessing the degree of change in coherence magnitude due to F B A Q encoding is important for reasons other than those directly associated with interferometry. In addition to providing an estimate of interferogram quality, other applications of coherence magnitude have been found related to feature classification and are a subject of current research. These include: Chapter 5: Theoretical Development of InSAR Analysis • Terrain characteristics: Various terrain features will exhibit varying degrees of coherence. For example, forested areas have been shown to exhibit lower coherence magnitude on average than grasslands primarily due to temporal decorrelation and volume scattering effects found in forests [18]. • Ground Temperature: Hagberg et al. [ 18] and Zebker et al. [21 ], have observed that ground temperature changes between image acquisition can have a dramatic effect on coherence magnitude. This is believed due to changes in electromagnetic properties of the ground scatterers. • Ice Movement: Coherence magnitude measurements can be used to gain insight on the movement of ice sheets in the arctic. High coherence areas tend to correspond to sections of the ice pack where little or no movement has occurred between acquisitions, while low coherence can indicate ice motion. Differential InSAR techniques (2 or more interferograms compared) or along track InSAR measurement (zero baseline, temporal separation only) can be used to estimate the actual ice displacement. 5.4 Summary In this section we have presented the theoretical framework of the study. We have shown that F B A Q encoding of SAR data is expected to cause an increase in phase noise on the interferogram due to the fact that it causes an increased phase noise on the images used to form the interferogram. Higher data reduction ratios will give rise to higher phase noise. The amount of phase noise incurred on a region of the interferogram will depend on the local SNR of the images used to 44 Chapter 5: Theoretical Development of InSAR Analysis produce it. In turn, this phase noise will result in height estimation errors in the digital elevation model produced from the interferogram, the magnitude of which will be strongly influenced by the InSAR baseline. Spatially, the errors are not uniformly distributed across the images, but are greater in areas of low SNR. Since areas of low SNR in the images will correspond to areas of low coherence magnitude in the interferograms, it is expected that the majority of large phase errors on the interferogram wil l be confined to these low coherence regions. The change in coherence magnitude caused by this effect will depend on the magnitude of the F B A Q phase noise and the initial value of coherence magnitude of the region in question. Since the areas of high coherence are the most important for interferometric applications, it is expected that the impact of F B A Q encoding on D E M generation wil l be smaller than average phase error measurements would indicate, since lower levels of phase noise will be located in high coherence regions. Thus, it is important to consider the impact of the encoding noise due to F B A Q within the context of the initial level of phase noise found in the region of interest. 45 6 INSAR EXPERIMENTAL METHODOLOGY The preceding chapters of this report outline the general processing steps which are involved in satellite interferometry. Interferometric processing is a dynamic research topic and new processing techniques are constantly being developed to improve the quality of interferograms. In this study, a conscious effort was made to keep the interferometric processing as straight-forward as possible. This was done for several reasons. Firstly, it was imperative that the processing be performed in such a way that it could be verified as consistent for every image, regardless of encoding level. Thus, all processing errors (such as mis-registration error and SAR processing errors) would be the same in each image and the encoding error could be isolated by comparison to the original (un-encoded) interferogram. Secondly, it was desirable that the processing reflect the basic techniques of interferometry and thus the study would be of use to the majority of interferometric researchers, not just those using specialized techniques. Finally, the large number of images to be processed necessitated an efficient processing scheme. This section examines in detail the actual methodology used in this study to create interferograms using ERS-1 data, and to assess the effects of the F B A Q algorithm on interferogram quality, and D E M height estimation. The methodology can be divided into several distinct steps which are explained in the following sub-sections, and is summarized by Figure 6-1. 46 Chapter 6: InSAR Experimental Methodology SAR Signal Data A 1. C SAR Signal Data B 1 Data Pre-conditioning 1 Encode/Decode Signal Domain Analysis SAR Process SAR Process SAR Process SAR Process c Image Domain Analysis 3 Register to A Register to A Form Original Interferogram Form Encoded/Decoded Interferogram c Raw Interferogram Analysis 1 Flat Earth Correction, Coherence and Smoothing Flat Earth Correction, Coherence and Smoothing c A Smoothed Interferogram and Coherence Magnitude Analysis 1 Relative height error estimate Relative height error estimate c Height Estimate Error Analysis Figure 6-1 Row Chart of Experimental Methodology 47 Chapter 6: InSAR Experimental Methodology 6.1 D a t a S e l e c t i o n Lti order to model the effect of data encoding on ENVIS A T , it was necessary to use data with properties similar to those of the future system. These properties are dependent on such things as imaging geometry, transmitter power, wavelength, polarization, and level of quantization. Data with properties very close to those expected for ENVIS A T can be simulated from data from the current ESA SAR satellite ERS-1 [16]. Thus, ERS-1 data was selected for use in this study. For satellite interferometry, images must be obtained by repeat orbit over a given scene. Therefore, an appropriate InSAR data set is one in which the two images have been acquired in relatively quick succession (3 or 6 days apart). Considerations for choosing the particular scene for study within a data set included the range of SNR of the image, and fringe visibility and coherence magnitude levels. To verify the theory outlined in the previous chapter, it was desirable to study scenes which included both areas of high and low coherence, generated from images with varying levels of SNR. 6.2 D a t a P r e - c o n d i t i o n i n g In order to model 8-bit, zero mean ENVISAT data, the 5-bit, 15.5 mean ERS-1 data must be pre-conditioned to 8-bit zero-mean format. The algorithm to accomplish this was developed in the previous data encoding studies performed by M D A , and is outlined in detail in document [16]. It essentially involves subtracting the 5-bit mean, adding a small amount of noise to the data set, reducing the saturation levels of the histogram, and expanding the data to 8-bit format. The algorithm was validated in [16] where it was shown to provide data which adequately modeled the expected ENVISAT data. 48 Chapter 6: InSAR Experimental Methodology However, since interferograms are formed by taking the phase difference of two images, the technique is especially sensitive to any type of data distortion, and there was concern that interferometry would be impossible after preconditioning. For this reason, the methodology of pre-conditioning the data was re-examined in this study. However the results of the pre-conditioning study indicated that interferometry was still possible. The analysis is presented in Appendix A . After pre-conditioning the data to 8 bit, zero mean format, the data was encoded/decoded using the F B A Q algorithm. Encoding was performed using the software environment developed for ESTEC by M D A during the Data Encoding Techniques Study (DETS) [13,14]. For the interferometry study, the block size was set at 1 azimuth by 126 range samples. This size was previously determined to be the optimal size for SAR images based on the requirements of the satellite On-Board Data Reduction system [14]. Using this block size, the data for both images was encoded/decoded to 4,3, and 2, bits/sample.1 After encoding/decoding the signal data, the reconstructed data was compared to the original data to statistically evaluate the effects of algorithm. Following encoding/decoding, the SAR data was processed into single look complex images. The images were formed using the DETS SAR processor, a fast convolution processor contained 1. Although other block sizes of 1x63 samples and 1x84 samples were used in the previous study [14] for 4-bit and 3-bit encoding respectively, these values were due to downlink restrictions which have since been re-moved. The block size of 1x126 was shown to be superior due to its higher compression ratio and better es-timation of local data variance. 6.3 Data Encoding/Decoding 6.4 SAR Processing 49 Chapter 6: InSAR Experimental Methodology within the DETS environment and used during previous data encoding projects. Image Registration Once the single look complex images were formed, they were registered using specialized InSAR software not incorporated into the DETS environment. In order to isolate the effects of encoding from mis-registration error, the shifts in azimuth and range needed to register the images were calculated using the original (un-encoded) images only. Then, the same calculated shifts were applied to both the original images and the 4,3, and 2 bit encoded/decoded images. This procedure makes the assumption of no spatial shift of images due to data encoding, a fact which was demonstrated in the previous studies [13,14]. However, it is certainly true that the noise added due to data encoding may affect algorithms used to estimate the registration shifts needed for SAR interferometry. To investigate this possibility a separate experiment was performed, the results of which are presented in Chapter 8. The method used to register the original images was a four step process. Throughout the process, one image was chosen as the base image and remained unchanged. A l l resampling and shifting was done on the second image. The four steps in the image registration were: 1. Visual Bulk Shift Estimation: Bulk shifts were estimated visually by taking pixel measurements of distinctive features and by visual inspection of fringe quality for various integer shift sizes. The second image was then shifted to roughly correspond to the base image. This method obtained an accuracy within 2 or 3 pixels in the range and azimuth directions using only integer shifts. 50 Chapter 6: InSAR Experimental Methodology 2. Refinement of Bulk Shifts: A program was run which extracted small chips from the image, oversampled them by a factor of 2 and calculated required shifts based on maximizing the correlation function of the image magnitude between image chips. This step yielded shift levels to an accuracy of half a pixel. These shift estimates were then used as input to the final fine registration program as an initial estimate of shift value. 3. Fine Shift Estimation: Fine shift estimation involved implementing the registration criteria outlined in sub-section 4.2.2. Fine shifts were estimated separately using three different criteria: Maximizing Fringe Visibility (MFV), Maximizing Coherence Magnitude (MCM), and Minimizing Residue Counts (MRC). The method which yielded the best interferogram was chosen for registration. A l l programs followed the same basic algorithm: a) First, image chips of size 64 by 64 complex samples were extracted at regular intervals throughout the image. The extracted chips were then oversampled via a 2-D FFT, with care taken to ensure the center frequency of the data was taken into account. b) Then, the chip from the "re-sampled" image was shifted relative to the reference chip and an interferogram formed for each shift. For each interferogram formed at each shift, the shift criteria was applied. These were a coherence magnitude calculation, a peak FFT magnitude 51 Chapter 6: InSAR Experimental Methodology measurement, and a residue count for M C M , M F V and M R C algorithms respectively. c) For each algorithm, after the calculations for all relative shifts had been performed, the final shift between the two image chips was determined by quadratic interpolation of the shift criteria calculations. This shift (both in range and azimuth) along with the center position of the chips were written to a file for use in the resampling software. 4. Resampling the Image: The final step in registering the images was to apply the shift estimates to the "re-sampled" image in order to register it to the reference image. This was accomplished by using a least mean square algorithm along with the calculated shifts file to determine shift and stretch parameters. After the parameters have been determined, the input image file is resampled to l/50th of a pixel using a set of 15 point FIR filters. The quality of the interferograms produced using the 3 criteria were assessed by computing the average coherence magnitude of the scene. The estimates which produced the highest overall level of coherence magnitude were adopted for use in the remainder of the study. In practice, due to the small size of the scenes used in this study and the fact that the shifts were found to be very consistent across the images, a linear shift implemented in the frequency domain delivered the same level of coherence as non-linear shifting. Since linear shifting resulted in a considerable saving in processing time it was adopted for the study. 52 Chapter 6: InSAR Experimental Methodology 6.5 Upsampling and Interferogram Formation The next step in the process was to upsample the registered images by a factor of 2 in both range and azimuth then multiply one image by the complex conjugate of the other. The phase of the product forms the raw interferogram. This was done for the images from each encoding levels to yield interferograms for the original (un-encoded), 4-bit, 3-bit, and 2-bit encoding levels. 6.6 Raw Interferogram Analysis The raw interferogram was analyzed using the standard DETS evaluation software. Of particular interest was the phase error due to encoding for the 4-bit, 3-bit and 2-bit raw interferograms. Histograms and statistics of the global raw interferogram phase error (phase error over the entire raw interferogram) were generated. Rat Earth fringes were removed by generating the expected flat Earth fringes based on the geometry of the InSAR system and subtracting that phase from the interferogram. If an azimuth ramp was detected it was also removed. The raw interferogram was smoothed using Maximum Likelihood averaging. This smoothed interferogram was then analyzed using the DETS evaluation software. Statistics were calculated for the global phase error, and histograms were produced. 6.7 Flat Earth Compensation 6.8 Smoothed Interferogram Analysis 53 Chapter 6: InSAR Experimental Methodology 6.9 Coherence Analysis A complex coherence calculation was performed on the raw interferogram using the Maximum Likelihood estimation criteria (equation (5-8)) . The magnitude of this image is thus the coherence magnitude of the interferogram, and the phase is the interferogram smoothed phase. The complex coherence image formed was then evaluated in several different manners: 1. Average Value: The average coherence magnitude of the entire image for each encoding level was calculated and compared to the baseline. 2. Error Images: a) Coherence Magnitude: A coherence magnitude error image was formed by subtracting the original coherence magnitude image from the encoded/ decoded coherence magnitude image. This showed which areas of the image were experiencing the greatest change in coherence magnitude. Mean and rms change in coherence magnitude were calculated and histograms produced to try to understand the nature of the change in coherence magnitude due to encoding. b) Phase: A phase error image was formed by subtracting the base-line interferogram phase from the encoded/decoded interferogram phase. This showed which areas of the interferogram were experiencing the largest amount of phase change due to encoding. Mean, absolute mean and rms values of phase error were calculated and histograms produced. 54 Chapter 6: InSAR Experimental Methodology 3. Change relative to initial coherence magnitude: To test the hypothesis that areas of high coherence would be less affected by encoding error than those of low coherence, each pixel in the image was monitored for 3 values: it's initial (un-encoded) coherence value (measured in the base-line interferogram), the change in coherence due to encoding (measured from the coherence magnitude error image), and phase error (measured from the phase difference image). Plotting the error values vs. the un-encoded coherence magnitude illustrated how different levels of coherence are effected by the F B A Q encoding. Of particular interest were those areas with coherence magnitude of 0.3 and greater since they are most appropriate for D E M generation. 4. Bin Analysis: Finally, the pixel changes obtained above were placed into bins based on their initial coherence magnitude values. Ten bins were formed over the 0 to 1.0 range of coherence magnitude (each bin 0.1 wide). For each bin, rms and mean changes in coherence magnitude and phase were calculated and histograms obtained to better illustrate the effects of F B A Q encoding on different coherence levels. 6.10 D E M E r r o r A n a l y s i s Using equation (5-5), height error estimates were made for the smoothed phase interferograms. To estimate the impact of F B A Q phase noise on D E M height estimates, two methods were employed. In the first, the variance of the height error contributed by F B A Q for calculated levels of original (un-encoded) coherence magnitude was added to the theoretical 55 Chapter 6: InSAR Experimental Methodology variance of height estimation uncertainty calculated using the curves of Figure 5-4. In the second, coherence magnitude changes were used to estimate the change in D E M accuracy due the F B A Q encoding. Both methods have advantages and disadvantages. The one which yielded the higher (i.e. worse) error figure was adopted as the "upper bound" on F B A Q impact on D E M accuracy. This analysis will give the "worst case" impact of F B A Q on D E M accuracy, since the original height error calculated using the curves of Figure 5-4 do not take into account errors introduced from phase unwrapping and calibration (i.e. this study assumes zero uncertainty in estimation of InSAR parameters such as baseline length, orbit height, etc.). 6.11 P h a s e U n w r a p p i n g E r r o r Through the derivatives of the InSAR equations, it was possible to determine the impact of F B A Q phase error on D E M height estimation without unwrapping the phase of the interferogram. In addition, the complexity and degree of manual intervention required make phase unwrapping a difficult source of error to keep consistent from encoding level to encoding level. Thus, phase unwrapping was not attempted in this study. However, in order to investigate the impact of F B A Q encoding on phase unwrapping, the number of phase residues in both the original and encoded interferograms were monitored. Phase residues are the limiting factor in the effectiveness of conventional phase unwrapping algorithms, and a significant change in the number of residues will affect the unwrapping process. The results of this portion of the study are given in Chapter 9. 56 7 INSAR EXPERIMENTAL RESULTS This section presents the main results of the study obtained using ERS-1 SAR data. Two InSAR pairs were selected for the study. The first is the Sardinia, Italy InSAR pair which consists of two ERS-1 data acquisitions taken six days apart, during orbits 241 and 327 which occurred on August 2,1991 and August 8,1991 respectively. The second is the Toolik Lake, Alaska InSAR pair which consists of 2 scenes obtained six days apart during orbits 943 and 1029 which occurred September 20,1991 and September 26,1991 respectively. As explained in the previous sections, in order to evaluate the impact of F B A Q encoding on interferometry, the data was processed first without any encoding. The images and interferograms produced using this data will be termed the original, or un-encoded data throughout the remainder of this report. The data to which F B A Q encoding was applied will be termed 4-bit, 3-bit and 2-bit encoded data. 7.1 Data Selection The choice of the Sardinia data set for the study of F B A Q encoding errors for interferometry was motivated by several reasons. Firstly, the Sardinia data is the test set for the FRINGE 57 Chapter 7: InSAR Experimental Results interferometry study group, and is thus relatively well documented, and has been proven suitable for interferometric applications. Secondly, the data exhibits a high degree of SNR variability which was deemed important to assess the dependence of F B A Q encoding phase error on SNR. Thirdly the scene is well known to be a very challenging InSAR scene to process due to the mountainous terrain, thus use of this scene will illustrate the effect of F B A Q encoding on scenes where the InSAR processing is difficult. Finally, data set was readily available at the outset of the study and could thus be processed in a timely fashion. The Toolik Lake scene was selected because, in contrast to the Sardinia data set, it exhibits relatively uniform radiometry and a consistently high degree of coherence magnitude. It is also a well documented scene as it was the subject of intensive processing by the Jet Propulsion Laboratory [21]. Where Sardinia represents a very challenging InSAR pair due to lower coherence levels and mountainous terrain, Toolik represents a far more "ideal" InSAR scene, with relatively flat terrain and high coherence magnitude levels. In order to choose a specific 2048 by 2048 sample sub-scene from the data sets for the study, several candidates were roughly processed into interferograms and evaluated for range of coherence values and fringe visibility. The Sardinia sub-scene eventually used in this study was chosen due to the wide distribution of SNR within the single-look complex (SLC) images, and coherence magnitude in the interferogram. The Toolik Lake sub-scene used was chosen due to its high level of coherence magnitude and relatively flat terrain. The relevant processing parameters for the ERS-1 Sardinia and Toolik Lake scenes are listed in Table 7-1. 58 Chapter 7: InSAR Experimental Results Table 7-1 ERS-1 Parameters for scenes used in the study Parameter (symbol) Sardinia Toolik Lake wavelength (X) 0.0566 m 0.0566 m look angle (0C) at center swath 22.4 degrees 24.0 degrees slant range sampling interval (Ar) 7.905 m 7.905 m baseline normal to slant range (B n) 126 m 40.4 m baseline parallel to slant range (B p) 65 m 98.2 m slant range distance to center range pixel (rc) 844 km 858 km 7.2 D a t a P r e - c o n d i t i o n i n g The 5-bit ERS-1 data was pre-conditioned to reflect the expected properties of 8-bit ENVIS A T data. Signal pre-conditioning was re-evaluated for this study due to concerns that the method of pre-conditioning performed in the earlier studies [13,14] would introduce too much phase error to make interferometric processing possible. This hypothesis was discovered to be false as average coherence magnitude levels changed by only 0.02. Results of the investigation are presented in Appendix A . Once the data has been preprocessed into 8-bit format, it is considered the "true" original ENVISAT data set, and all error measures are taken with respect to it. 7.3 D a t a E n c o d i n g / D e c o d i n g Using the DETS environment, the Sardinia raw data was encoded/decoded using the F B A Q algorithm with a block size of 126 range samples by 1 azimuth sample. Encoding was done to 4-59 Chapter 7: InSAR Experimental Results bit, 3-bit and 2-bit levels. After encoding/decoding, the signal data was analyzed using various statistical measures of data quality found in the DETS software. Some of these statistical measures are found in Table 7-2 and 7-3 including the average signal to quantization noise ratio (ASQNR), and average phase error. Table 7-2 Signal Data Statistics for Sardinia orbit 327 and Toolik Lake orbit 943 Sardinia Toolik Lake Encoding Level 4-bit 3-bit 2-bit 4-bit 3-bit 2-bit ASQNR, real (dB) 20.2 14.4 9.2 19.5 14.2 9.1 RMS phase error (deg) 6.8 11.5 18.1 6.8 11.5 18.1 mean abs. phase error (deg.) 4.4 8.1 14.1 4.4 8.1 14.0 Table 7-3 Signal Data Statistics for Sardinia orbit 241 and Toolik Lake orbit 1029 Sardinia Toolik Lake Encoding Level 4-bit 3-bit 2-bit 4-bit 3-bit 2-bit ASQNR, real (dB) 20.0 14.3 9.1 19.6 14.2 9.1 RMS phase error (deg) 6.8 11.5 18.1 6.8 11.5 18.1 mean abs. phase error (deg.) 4.4 8.1 14.1 4.4 8.1 14.1 A l l of the above measurements in Tables 7-2 and 7-3 fall within the range expected from results of the previous F B A Q study [14], indicating proper functioning of the F B A Q algorithm. Chapter 7: InSAR Experimental Results 7.4 S A R P r o c e s s i n g Following encoding/decoding, the data was processed into single look complex images using the DETS SAR processor. Special care was taken during processing that all encoding levels were processed using the same focusing parameters. The resulting detected image for Sardinia orbit 241 is shown in Figure 7-1. Figure 7-1 also shows the associated absolute phase error for the 3-bit encoding level. The phase errors have been magnified by a factor of 10 in order to emphasize texture, meaning all errors greater than 25.5 degrees appear as a bright white pixel after conversion to an 8-bit display. Note, that as hypothesized, the majority of phase errors appear in areas corresponding to the dark (low SNR) sections of the detected image. This is further illustrated in Figure 7-2 which shows a scatter plot of phase error due to three bit encoding vs. magnitude of the original (un-encoded) image and error in the real part of the image vs original coherence magnitude. Note also that as explained in Chapter 5, the error in the real part of the image increases with increasing magnitude and the phase error decreases (the same trend occurs for the imaginary component of the image). The trend is also observed in the Toolik Lake scene shown in Figure 7-3, though it is far more difficult to observe visually in the images. This is because the Toolik Lake scene lacks the extreme SNR contrasts of the Sardinia scene. However, if one plots the real error and phase error vs. original magnitude as in Figure 7-4, the general trend is clearly the same as that of Sardinia. 61 Chapter 7: InSAR Experimental Results detected image 3-bit encoding absolute phase error (xlO) Figure 7-1 Single look complex image of Sardinia and corresponding phase error due to 3-bit encoding scaled to show detail (orbit 241, slant range vertical, azimuth horizontal) Phase error due to 3 bit FBAQ vs. Original Magnitude (orbit 241) Ui - ° ° I ' • ' • ' • • • • i h ! original (un-encoded) magnitude original (un-encoded) magnitude Figure 7-2 Sardinia: Phase error and real error due to 3 bit F B A Q encoding vs. magnitude. 62 Chapter 7: InSAR Experimental Results detected image 3-bit encoding absolute phase error (xlO) F i g u r e 7-3 Single look complex image of Toolik and corresponding phase error due to 3-bit encoding scaled to show detail (orbit 943, slant range vertical, azimuth horizontal) Phase error due to 3 bit FBAQ vs. Original Magnitude (orbit 943) o<... original (un-encoded) magnitude Real error due to 3 bit FBAQ vs. Original Magnitude (orbit 943) original (un-encoded) magnitude Figure 7-4 Toolik Lake: Phase error and real error due to 3 bit F B A Q encoding vs. magnitude. Chapter 7: InSAR Experimental Results The SLC images were also analyzed using the same criteria as that for the signal data. The results of analysis for both steps were compared to results obtained in the previous M D A study. The results for both scenes (orbits 241 and 327) are presented in Tables 7-4 and 7-5. The F B A Q algorithm performed on a comparable level to the results obtained in the previous study. Note that a higher level of phase error is noticeable in the Sardinia processed images. This is to be expected since Sardinia contains large areas of low SNR, which we have shown contain higher levels of phase error, while Toolik Lake does not. Table 7-4 SLC Image Statistics of Sardinia orbit 327 and Toolik orbit 943 Sardinia Toolik Lake Encoding Level 4-bit 3-bit 2-bit 4-bit 3-bit 2-bit ASQNR, real (dB) 21.5 15.6 10.3 20.5 15.1 10.0 RMS phase error (deg.) 13.2 23.3 37.2 10.9 18.7 30.8 mean abs. phase error (deg.) 6.5 12.8 23.1 5.6 10.4 18.9 Table 7-5 SLC Image Statistics of Sardinia orbit 241 and Toolik orbit 1029 Sardinia Toolik Lake Encoding Level 4-bit 3-bit 2-bit 4-bit 3-bit 2-bit ASQNR, real (dB) 21.2 15.4 10.1 20.5 15.1 10.0 R M S phase error (deg) 14.0 24.2 38.3 10.9 18.7 30.8 mean abs. phase error (deg.) 6.9 13.4 23.9 5.6 10.4 18.8 64 Chapter 7: InSAR Experimental Results 7.5 I m a g e R e g i s t r a t i o n Image registration was performed using the techniques outlined in Chapter 6. For Sardinia, the image acquired during orbit 327 was arbitrarily chosen as the base image, and image 241 chosen as the image to be re-sampled. For Toolik Lake, image 943 was selected as the base image, and image 1029 was registered to it. It was found that best results (highest average coherence) were attained when the shifts were estimated using the Minimizing Residue Criteria for Sardinia, and with the Maximizing Fringe Visibility Criteria for Toolik Lake. For the main portion of the study, it was required that all processing be performed using exactly the same parameters in order to ensure a fair comparison. For this reason, all subsequent images for 4-bit, 3-bit and 2-bit encoding were registered using the exact same shifts as those for the original (un-encoded) scene. The issue of how the shift estimation procedure is affected by F B A Q is addressed in Chapter 8 of this report. Images were first registered using the non-linear re-sampling criteria outlined in Chapter 5. However, in practice it was found that the same level of coherence could be obtained for these particular InSAR scenes by performing precise linear shifts. This is probably due to the relatively small size of the S L C image and the level of consistency of the shift values across the image. As well, linear shifts performed using a phase shift in the frequency domain can be made more accurately than those performed with an interpolator since the interpolators wil l be unable to obtain a perfect "sine function" response [28]. This method was thus adopted due to its considerable computational time savings. 7.6 I n t e r f e r o g r a m F o r m a t i o n a n d A n a l y s i s After registration, the images were oversampled and the interferogram was formed. The raw (no smoothing) interferograms were analyzed using the analysis software found in the DETS 65 Chapter 7: InSAR Experimental Results package. The RMS, mean and mean absolute phase errors for the entire raw interferogram are summarized in Table 7-6. The expected value of the average RMS phase error (found by adding the variances of the phase error on each SLC image as discussed in Chapter 5) is shown in brackets. The measured phase errors on the raw interferograms corresponded well to the expected results, verifying that the F B A Q phase noise is well modelled as an independent Gaussian distribution. Histograms of interferometric phase error for the raw interferograms are shown in Figure 7-5 and 7-6 for the 4-bits, 3-bits, and 2-bits/sample encoding levels of Sardinia and Toolik Lake. Table 7-6 Average Phase Error Analysis for Raw Interferogram Sardinia {expected value} Toolik Lake {expected value} Encoding Level 4-bit 3-bit 2-bit 4-bit 3-bit 2-bit R M S phase error (deg.) 18.8 {19.2} 32.6 {33.6} 50.7 {53.4} 15.3 {15.4} 26.1 {26.1} 42.2 {43.6} mean absolute phase err. (deg.) 10.2 19.8 34.6 8.45 15.7 28.0 mean phase err. (deg.) -0.05 -0.06 -0.03 -0.01 0.03 0.02 Raw Interferogram phase error to Sardinia scone from 4-Wt FBAQ Raw Interferogram phase error tor Sardinia scene from 3-bH FBAQ Raw Werfcrogram phase error tor Sardinia scene from 2-blt FBAQ phase error (degrees) phase error (degrees) phase error (degrees) Figure 7-5 Sardinia: Histograms of raw interferometric phase error for 4-bit, 3-bit, and 2-bit. 66 Chapter 7: InSAR Experimental Results Raw Interferogram phase error tor Toolik scene from 4-bil FBAQ Raw Interferogram phase error for Toolik scene from 3-bflFBAQ Raw Interferogram phase error for Too* scene from 2-ttt FBAQ -100 0 100 phase error (degrees) -100 0 100 phase error (degrees) Figure 7-6 Histograms of raw interferometric phase error for 4-bit, 3-bit, and 2-bit encoding for the Toolik Lake scene. 7.7 F l a t E a r t h C o m p e n s a t i o n Flat Earth fringes were removed by subtracting this component from the images. The flat Earth fringes were calculated using the first term in equation (7-1): 4nBn In addition, an azimuth ramp was detected in the Toolik Lake scene and removed. 7.8 S m o o t h e d I n t e r f e r o g r a m A n a l y s i s The raw interferograms were smoothed using maximum likelihood averaging over a 2 range by 10 azimuth block size. The phase error was analyzed using the standard DETS analysis software. Figure 7-7 shows the magnitude and phase of the smoothed interferograms. A thorough analysis of the smoothed interferogram phase follows in the "Coherence Analysis" sub-section. 67 Chapter 7: InSAR Experimental Results Magnitude Phase Sardinia Magnitude Toolik Lake Figure 7-7 Smoothed original interferograms: magnitude and phase. Note how phase follows terrain contours. 68 Chapter 7: InSAR Experimental Results 7.9 C o h e r e n c e M a g n i t u d e A n a l y s i s The complex coherence of the raw interferogram was calculated. As mentioned previously, the phase of the complex coherence is the smoothed interferogram phase, while the magnitude is called coherence magnitude. Due to the severe change in elevation found in the Sardinia scene, the coherence magnitude calculation can be affected by the phase variation caused by terrain slope. The effect is particularly important for Sardinia in the range direction. To limit this effect, a block size of 2 range samples by 10 azimuth samples was used to determine each complex coherence pixel, which yields approximately square pixels in the smoothed interferogram with respect to ground range.1 Terrain effects in the relatively flat Toolik Lake scene were considered small, but a 2 by 10 block size was still used for the coherence magnitude calculation in order to facilitate useful comparison between the two data sets. As mentioned, since the interferogram pixels are not entirely uncorrelated, the 2 by 10 sample averaging results in an effective smoothing of about 10 looks [21,27,28]. 7.9.1 A v e r a g e C o h e r e n c e M a g n i t u d e A n a l y s i s The coherence magnitude images for the original and encoded/decoded images are shown in Figure 7-8 and 7-9 for Sardinia and Toolik Lake, and the average results for the scenes summarized 1. Note that the disadvantage of using a smaller block size for coherence magnitude is an increase in bias error and stand-ard deviation of the estimator. These effects, however, are significant mainly at low coherence levels. Since this study was primarily concerned with the higher coherence levels appropriate for DEM generation, the reduction in terrain effects through use of the 2 by 10 block size was considered of greater importance. Chapter 7: InSAR Experimental Results in Table 7-7. It can be seen that, as expected, the coherence magnitude was found to change to a greater extent with greater data reduction. Table 7-7 Average coherence magnitude results Measurement Sardinia Toolik Lake Original 4-bit 3-bit 2-bit Original 4-bit 3-bit 2-bit average C M a 0.480 0.478 0.471 0.446 0.761 0.757 0.744 0.698 a. CM = Coherence Magnitude 70 Chapter 7: InSAR Experimental Results Original 4 bits/sample 3 bits/sample 2 bits/sample Figure 7-8 Sardinia coherence magnitude images for the original (un-encoded), 4-bit, 3-bit and 2-bit encoding levels. 71 Chapter 7: InSAR Experimental Results Original 4 bits/sample 3 bits/sample 2 bits/sample Figure 7-9 Toolik Lake coherence magnitude images for the original (un-encoded), 4-bit, 3-bit and 2-bit encoding levels. 72 Chapter 7: InSAR Experimental Results It is difficult to assess the degradation in coherence magnitude directly from the from the images in Figures 7-8 and 7-9. Figures 7-12 and 7-13 present the absolute difference coherence magnitude images which provide a better indication of the effect of the F B A Q encoding. Notice that the highest RMS change takes place in the areas corresponding to low coherence of the original interferogram. The histograms of the original and error coherence magnitude images are included in Figures 7-10 and 7-11. As can be seen in the histograms, the mean shift in coherence magnitude is negative, with a greater negative shift for higher data reduction, and an increasing standard deviation for higher data reduction. H i s t o g r a m o f C o h e r e n c e M a g n i t u d e f o r 8 a r d l n l a , m e a n « 0 . 4 8 0 1 H j a ^ o j d r a m o f C o h e r e n o e M a g n i t u d e C h a n g e f o r S a r d i n i a , 4 - b i t F B A Q , m e a n - - 0 . 0 0 2 4 2 r 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 c o h e r e n c e m a g n i t u d e 0 . 8 0 . 9 1 H i s t o g r a m o f C o h e r e n c e M a g n i t u d e C h a n g e f o r S a r d i n i a , 3 - b f t F B A Q , m e a n - - 0 . 0 0 9 2 1 4 0 0 0 r H i s t o g r a m o f C o h e r e n c e M a g n i t u d e C h a n g e f o r S a r d i n i a , 2 - b l t F B A Q , m e a n •> - 0 . 0 3 4 0 9 0 0 0 r - 0 . 2 0 0 . 2 0 . 4 c h a n g e In c o h e r e n c e m a g n i t u d e 0 . 6 0 . 8 - 0 . 4 - 0 . 2 O 0 . 2 0 . 4 c h a n g e In c o h e r e n c e m a g n i t u d e 0 . 6 0 . 8 Figure 7-10 Histograms of original coherence magnitude and 4-bit, 3-bit, and 2 bit change in coherence magnitude due to F B A Q for Sardinia. 73 Chapter 7: InSAR Experimental Results H i s t o g r a m o f C o h e r e n c e M a g n i t u d e f o r T o o l i k L a k e , m e a n • 0 . 7 6 1 1 H i s t o g r a m o f C o h e r e n c e M a g n i t u d e C h a n g e f o r T o o l i k L a k e , 4 - b i t F B A Q , m e a n • 1 6 0 0 0 r - 0 . 2 - 0 . 1 0 0 .1 0 . 2 c h a n g e In c o h e r e n c e m a g n i t u d e H i s t o g r a m o f C o h e r e n o e M a g n i t u d e C h a n g e for T o o i l k L a k e , 3 - b f t F B A Q , m e a n - - 0 . 0 1 7 6 1 2 0 0 0 r H l s t o g r a m o f C o h e r e n o e M a g n i t u d e C h a n g e f o r T o o i l k L a k e , 2 - b i t F B A Q , m e a n • l O O O O i -- 0 . 2 - 0 . 1 0 0 .1 02 c h a n g e In o o h e r e n c e m a g n i t u d e 0 . 3 0 . 4 - 0 . 4 - 0 . 2 0 0 . 2 c h a n g e In o o h e r e n c e m a g n i t u d e 0 . 4 0 .6 Figure 7-11 Histograms of original coherence magnitude and 4-bit, 3-bit, and 2 bit change in coherence magnitude due to F B A Q for Toolik Lake. 74 Figure 7-12 Absolute coherence magnitude difference images (scaled by a factor of 1000) for Sardinia. Chapter 7: InSAR Experimental Results 4 bits/sample 3 bits/sample 2 bits/sample Figure 7-13 Absolute coherence magnitude difference images (scaled by a factor of 1000) for Toolik Lake. Chapter 7: InSAR Experimental Results 7.9.2 A v e r a g e P h a s e A n a l y s i s Figure 7-14 displays the smoothed interferogram phase of the Sardinia scene. Note, as expected, the best fringe visibility corresponds to those regions with high coherence magnitude values. As with the coherence magnitude plots, it is difficult to see the effects of encoding until the error images in Figure 7-15 are examined. Figure 7-16 displays the absolute phase error of the interferograms (multiplied by a factor of 10 for display purposes). Referring back to Figure 7-8, it can be seen once again that the majority of the phase error is concentrated in the regions of low coherence magnitude in the original interferogram. Since height error is linearly related to phase error through equation (5-5), these areas will have more height error added to them. Figures 7-16 and 7-17 display the smoothed interferogram phase and phase error for the Toolik Lake scene. The average phase error measurements for over the entire interferograms are summarized in Table 7-8. Table 7-8 Average phase error statistics for smoothed interferogram (2x10 smoothing) Sardinia Toolik Lake 4-bit 3-bit 2-bit 4-bit 3-bit 2-bit RMS error (deg) 13.0 22.8 36.9 3.6 6.8 13.0 mean absolute error (deg) 6.2 12.2 22.6 2.5 4.7 9.0 std. deviation of error (deg) 13.0 22.8 36.9 3.6 6.8 13.0 mean error (deg) 0.04 0.05 0.16 0.008 -0.003 0.004 As can be seen, the interferogram phase error due to encoding has been reduced through the smoothing operation from the levels found in the raw interferogram. Note that the smoothing 77 Chapter 7: InSAR Experimental Results operation was far more effective for the Toolik Lake scene than for Sardinia. This is to be expected, since the smoothing operation is most effective in the high coherence regions and Toolik Lake has much more data in these regions than Sardinia (as evident by the histograms in Figures 7-10 and 7-11. Histograms of phase errors for the smoothed interferograms are shown in Figure 7-18 and 7-19. Like the phase error associated with the images, and the raw interferograms, the phase error of the smoothed interferogram due to encoding is zero mean, Gaussian shaped, with a standard deviation dependent upon the level of data reduction. 78 Chapter 7: InSAR Experimental Results rag Original 4 bits/sample 3 bits/sample 2 bits/sampl Figure 7-14 Sardinia interferogram phase images for original, 4-bit, 3-bit, and 2 bit F B A Q Chapter 7: InSAR Experimental Results 4 bits/sample 3 bits/sample 2 bits/sample Figure 7-15 Sardinia phase error due to F B A Q encoding (multiplied by a factor of 10). 80 Chapter 7: InSAR Experimental Results Figure 7-16 Toolik Lake interferogram phase images for original, 4-bit, 3-bit, and 2 bit F B A Q Chapter 7: InSAR Experimental Results 4 bits/sample 3 bits/sample 2 bits/sample Figure 7-17 Toolik Lake phase error due to F B A Q encoding (multiplied by a factor of 10). 82 Chapter 7: InSAR Experimental Results Histogram of Smoothed Intsrferogram Phase Error due to 4-btl FBAQ tor Sardinia Hlstorjram of SmooMlrterteror/am Phase Era Histogram of Srrartrri Interferogram Phase Error rjue to 2-rjft FBAQ IwSarrJnla 2.5r -100 0 100 phase error (degress) -100 0 100 phase error (degress) 100 0 100 200 phase error (degrees) Figure 7-18 Histograms of smoothed interferogram phase error due to F B A Q for 4-bit, 3-bit and 2-bit encoding levels for Sardinia. Histogram of Smoothed Interferogram P t ^ Error due to 4-bti FBAQ tor Tool* Lake Histogram of Smoothed Interferogram Phase Error due to 3-blt FBAQ for Tooilk Lake Histogram of Smoothed Interferogram Phase Error due to 2-bit FBAQ for Toolk Lake 4.5r -100 0 100 phase error (degrees) -200 -100 0 100 200 300 phase error (degrees) Figure 7-19 Histograms of smoothed interferogram phase error due to F B A Q for 4-bit, 3-bit and 2-bit encoding levels for Toolik Lake 7.9.3 P h a s e N o i s e a n d C o h e r e n c e C h a n g e v s . O r i g i n a l C o h e r e n c e The previous images show that the level of degradation due to F B A Q depends on the original level of coherence of the interferogram. To further investigate this, individual pixels were 83 Chapter 7: InSAR Experimental Results monitored for their original coherence magnitude, phase error due to encoding, and change in coherence magnitude due to encoding. It was found, as expected, that areas which originally showed high coherence had less RMS phase error added to them, and experienced a smaller RMS change in coherence magnitude than those with low coherence values before encoding/decoding. The plots of phase error vs. original coherence magnitude are shown in Figures 7-20 and 7-21. From these, it is clear that low coherence regions are more adversely affected by phase errors due to F B A Q encoding than high coherence regions, and that the amount of phase error added depends on the data reduction level. Phase error vs. original coherence mag. for 4 bit FBAQ e e.i o.i o.a 0.4 OA at o.7 o.a o.g i original (un-encoded) coherence magnitude Phase error vs. original coherence mag. for 3 bit FBAQ o o.i o.i o.a 0.4 o.s o.a o.7 o.a at 1 original (un-encoded) coherence magnitude Phase error vs. original coherence mag. for 2 bit FBAQ o o.i e.1 o.a a4 as a* 0.7 o.a 0.0 1 original (un-encoded) coherence magnitude Figure 7-20 Scatter plots of interferogram phase error due to F B A Q encoding for Sardinia. 84 Chapter 7: InSAR Experimental Results 200 Phase error vs. original coherence mag. for 4 bit FBAQ <L> T 3 g 4 3 --200 T r i i 0 0.1 Oi OJ 0.4 O.S 0.6 0.7 0.8 0.9 1 original (un-encoded) coherence magnitude Phase error vs. original coherence mag. for 3 bit FBAQ 1 200 0 0.1 0.2 0.3 0.4 O.S 08 0.7 O.S 0.8 1 original (un-encoded) coherence magnitude Phase error vs. original coherence mag. for 2 bit FBAQ cS -a -200 • I , 1 •» .', ,v.'." 0 0.1 0.2 0.3 0.4 0.S 0.8 0.7 O.S 0.S 1 original (un-encoded) coherence magnitude Figure 7-21 Scatter plots of interferogram phase error due to F B A Q encoding for Toolik Lake. As can be seen in Figure 7-21 the trend in the data is more difficult to detect for the Toolik Lake scene because it has very few pixels in the low coherence regions (below about 0.3). The trend is still present, however, as will be demonstrated further in the following sections. Figures 7-22 and 7-23 shows the change in coherence magnitude due to encoding vs. original coherence magnitude. Note the lower bound of slope -1 clearly visible in the low coherence region. 85 Chapter 7: InSAR Experimental Results This bound is due to the fact that coherence magnitude is a positive value which cannot fall below zero (i.e. if originally coherence magnitude is 0.2, the largest possible drop is -0.2). This bound tends to skew the mean change in coherence at low coherence magnitude in the positive direction. A less visible upper bound also exists. Change in coherence mag. for 4 bit FBAQ vs. original coherence mag. original (un-encoded) coherence mag. Change in coherence mag, for 3 bit FBAQ vs. original coherence mag. 0 0.1 0.2 0.3 0.4 0.E 0.4 0.7 0.8 0.0 1 original (un-encoded) coherence mag. Change in coherence mag. for 2 bit FBAQ vs. original coherence mag. •^ c 1 1 i 1 1 I I original (un-encoded) coherence mag. Figure 7-22 Scatter plot of change in coherence magnitude due to F B A Q encoding vs. original coherence magnitude for Sardinia. 86 Chapter 7: InSAR Experimental Results 8P « Change in coherence mag. for 4 bit FBAQ vs. original coherence mag. o o .S O f 00 6 •0.5 T 1 upper bound Q Q.I 0.2 0.9 0.4 O.S 0.9 0.7 original (un-encoded) coherence mag. 0.8 0.9 1 ei) 0,5 E •8 6 o -0.5 Change in coherence mag, for 3 bit FBAQ vs. original coherence mag. 1 -*v£A lower bound I 8P a •8 u .9 u •1 0.1 0.2 0.3 0.4 O.S O i 0.7 0.6 0.9 1 original (un-encoded) coherence mag. Change in coherence mag, for 2 bit FBAQ vs. original coherence mag. — -L T upper bound 0 0.1 1 0.2 0.3 0.4 O.S 0.6 0.7 0.9 0.9 original (un-encoded) coherence mag. Figure 7-23 Scatter plot of change in coherence magnitude due to F B A Q encoding vs. original coherence magnitude for Toolik Lake. 7.9.4 B i n A n a l y s i s The final step in the interferogram analysis was to place the phase noise and change in coherence magnitude values gathered above into bins based on their original coherence magnitude values. The coherence magnitude range of 0 to 1.0 was divided into ten 0.1 width bins. Each bin 87 Chapter 7: InSAR Experimental Results was analyzed for RMS variation, mean, mean absolute, and standard deviation, and histograms were produced for each bin. Representative histograms of phase error for Sardinia with 3 bit encoding for bins 3 and 8 (original coherence of [0.2 -> 0.3), and (0.7->0.8) respectively) are shown in Figure 7-24. The trend shown in Figure 7-24 confirms earlier observations that the standard deviation of the phase error and coherence magnitude change increase with decreasing original values of coherence magnitude. This trend was observed for both scenes at all encoding levels, and is summarized in Tables 7-9 and 7-10, and the plots of Figure 7-25. Histogram of phase error due to 3 bit FBAQ for bin 3 1500 -i 1 1 i 1 1 r -200 -150 -100 -50 0 50 100 150 phase error (degrees) Histogram of change in coherence magnitude due to 3 bit FBAQ for bin 3 . j i i_ -50 0 50 100 Histogram o l c h a n g e in c o h e r e n c e magnitude due to 3-bit FBAQ for bin 8 1500r -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 change in coh. mag. -0.2 -0.1 ),1 02 0.3 0.4 0.5 change in coh. mag. Figure 7-24 Histograms of phase error and coherence magnitude change due to encoding for 3 bit F B A Q . Bin 3 is original coherence magnitude values 0.2 to 0.3, bin 8 is original coherence magnitude values 0.7 to 0.8. 88 Chapter 7: InSAR Experimental Results Table 7-9 Summary of bin analysis for Sardinia b i n l bin 2 bin 3 bin 4 bin 5 bin 6 bin 7 bin 8 bin 9 bin 10 Original avg. CM a 0,0665 0.1553 0.2531 0.3521 0.4509 0.5497 0.6480 0.7438 0.8351 0.9153 # elements 3381 10148 16694 22440 27090 28571 24702 15900 4872 202 4 bits RMS phs. err. 55.23 26.68 15.33 9.51 6.69 5.12 3.96 3.05 2.37 1.64 mean phs en- -1.28 0.27 0.26 0.038 0.029 0.020 0.052 -0.036 0.029 0.055 RMS ACM 0.061 0.052 0.050 0.047 0.043 0.037 0.032 0.025 0.017 0.011 mean ACM 0.027 0.0089 0.0014 -.0023 -.0043 -.0052 -.0059 -.0057 -.0050 -.0032 3 bits RMS phs. err. 72.57 48.49 30.00 21.16 14.85 11.06 8.45 6.22 4.76 3.54 mean phs err 1.03 0.75 -0.21 -0.083 -0.081 0.048 0.11 0.049 -0.020 0.29 RMS ACM 0.113 0.091 0.085 0.083 0.080 0.074 0.067 0.055 0.042 0.029 mean ACM 0.070 0.030 0.0070 -.0059 -0.015 -0.019 -0.023 -0.022 -0.020 -0.015 2 bits RMS phs. err. 86.85 69.30 53.51 40.74 30.29 23.06 17.38 12.76 9.65 7.64 mean phs en- 3.34 1.1 -0.65 0.28 -0.10 0.11 0.18 0.11 0.13 0.11 RMS ACM 0.179 0.136 0.120 0.122 0.129 0.136 0.135 0.123 0.105 0.076 mean ACM 0.132 0.069 0.021 -0.018 -0.043 -0.064 -0.074 -0.074 -0.068 -0.054 a. CM is coherence magnitude 89 Chapter 7: InSAR Experimental Results Table 7-10 Summary of bin analysis for Toolik Lake b i n l bin 2 bin 3 bin 4 bin 5 bin 6 bin 7 bin 8 bin 9 bin 10 Original avg. CM a 0.0655 0.1600 0.2578 0.3548 0.4577 0.5573 0.6566 0.7447 0.8483 0.9214 # elements 68 254 658 1717 4023 9859 22413 44772 56905 13331 4 bits RMS phs. err. 54.01 23.45 11.77 8.40 5.91 4.76 3.80 3.03 2.41 1.91 mean phs err -5.62 0.167 -0.116 0.121 0.056 0.040 -0.021 -0.012 0.016 -0.001 RMS ACM 0.059 0.052 0.051 0.048 0.043 0.038 0.033 0.026 0.018 0.011 mean ACM 0.032 0.013 0.0085 0.0020 -.0003 -.0017 -.0032 -.0044 -.0045 -.0040 3 bits RMS phs. err. 77.44 46.24 23.20 21.20 16.44 11.62 8.98 7.14 5.72 3.57 mean phs err -9.55 4.84 -1.36 0.277 0.046 -0.187 0.037 -0.02 0.023 -0.019 RMS ACM 0.123 0.087 0.088 -.0838 0.0757 0.071 0.063 0.052 0.038 0.026 mean ACM 0.089 0.031 0.015 -.0039 -0.010 -0.015 -0.018 -0.020 -0.018 -0.016 2 bits RMS phs. err. 87.15 68.32 49.67 36.33 24.34 18.61 14.01 10.95 8.66 6.83 mean phs err 8.71 4.32 -0.066 0.25 -0.22 -0.14 -0.06 0.02 0.02 0.02 RMS ACM 0.182 0.145 0.125 0.128 0.131 0.132 0.125 0.113 0.0943 0.071 mean ACM 0.146 0.083 0.027 -0.009 -0.037 -0.058 -0.068 -0.071 -0.066 -0.054 a. CM is coherence magnitude 90 Chapter 7: InSAR Experimental Results (a) (b) (c) s o eo ro I eo s o t 4 0 1 3 0 so 10 o R M S phase noise added due to F B A Q as a function of original coherence magnitude .1 i 1 1 1 1 1 1 1 o X £ o 1 ardlnla "oollk La c e i . . >0!<. . . . L / | s > * s » < . . V . 3i-blt L . . C T > ^ . . J 4 -b l t 3 8 i i i i • i 0.1 0 . 2 0 . 3 O.A O.S o.e 0 . 7 original coherence magnitude o.s o .s Mean change In coherence magnitude due to F B A Q 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 O . B original coherence magnitude R M S change In coherence magnitude due to F B A Q 0 . 3 0 . 4 O . S o.e 0 . 7 original coherence magnitude Figure 7-25 (a) RMS encoding phase error, (b) mean coherence magnitude change and (c) RMS coherence magnitude change vs. original coherence magnitude. 91 Chapter 7: InSAR Experimental Results The plots of Figure 7-25 summarize the results of the bin analysis which are as follows: • RMS phase error added to the interferogram due to F B A Q decreases with increasing original coherence magnitude level. • The RMS change in coherence magnitude decreases with increasing original coherence magnitude. This is due in part to the fact that more phase error is added in these regions and to the fact that the standard deviation of the coherence magnitude estimator increases with decreasing coherence magnitude [18]. The erratic nature of the plots at low coherence levels is due to the effect of the lower bound observed in the scatter plots of Figures 7-22 and 7-23. As expected, it is most noticeable in the 2-bit case which has wider distribution and thus hits the bound earliest. • The trend in the mean change in coherence magnitude curve shows a positive bias at mid to low range coherence levels. This is in part due to the lower bound of Figure 7-22 and 7-23, but is also due to to coherence magnitude estimator bias, the effect of which will be discussed in detail in the following sub-sections. From the plots of Figure 7-25 it is evident that the curves for both scenes show good agreement, with the Toolik Lake scene generally exhibiting slightly lower levels of phase error. This is probably due to a greater level of terrain effects entering the smoothing block region for the Sardinia scene. Overall however the agreement is exceptional considering that these are two very different scenes in terms of terrain and coherence level. The agreement is particularly good for 4-92 Chapter 7: InSAR Experimental Results bit encoding with coherence magnitude greater than about 0.35 which is important because we will be focussing on that region in the following sub-section. 7.10 I m p a c t o n D E M G e n e r a t i o n In order to assess the impact of F B A Q encoding on D E M generation, it is necessary to compare the height error contributed by F B A Q encoding to the original height estimation uncertainty inherent in interferometric techniques due to random phase noise other than that contributed by F B A Q . The original phase estimate uncertainty can be related to coherence magnitude through the graph of Figure 5-4, which shows the standard deviation of phase for a given coherence magnitude level and number of looks. The phase uncertainty can then be converted to height uncertainty through equation (5-5), the derivative of the InSAR equations with respect to interferogram phase. When accounting for the additional noise due to F B A Q , two approaches are possible which are outlined in the next two sub-sections. 7.10.1 A d d i t i o n o f V a r i a n c e s If we assume the phase noise due to system noise and decorrelation effects is independent of the phase noise due to encoding, and we assume that both are Gaussian distributions with zero mean, the sum of the variances of the phase noises can be added in to yield a total phase uncertainty statistic. While this is the approach often taken by researchers when measuring the effects of additive phase noise, it does have limitations. The most obvious is the fact that a true Gaussian distribution extends to ± °°, while the phase noise distributions are truncated at ± n. Note also, that while the assumption of zero mean, Gaussian phase noise is a good approximation of the F B A Q 93 Chapter 7: InSAR Experimental Results phase noise, it is not entirely valid for all levels of coherence magnitude for the original phase noise. As the plot in Figure 5-3 illustrates, as coherence magnitude decreases, the original noise distribution is trending towards a uniform distribution rather than Gaussian (it reaches a uniform distribution at a coherence magnitude of zero). As coherence magnitude levels approach one, the distribution tends towards a delta function. However, for values of coherence magnitude we are interested in for practical D E M generation (between about 0.35 and 0.95) the probability density function of the original phase error will take the Gaussian shape shown in Figure 5-3, and the assumption of a Gaussian distribution should be a reasonable approximation. Once the total phase uncertainty is calculated, it can then be converted into a total height uncertainty using equation (5-5). Comparison of this number to that of the theoretical original RMS height uncertainty calculated using the curves of Figure 5-4 and equation (5-5) provides an estimate of the overall impact of F B A Q encoding on D E M generation. This analysis was applied to both scenes, and the results presented in Table 7-12 and 7-13 and summarized with Figures 7-26 and 7-27. When examining the results, it is important to consider the effect of the baseline on the calculation. The Toolik lake scene has a much shorter baseline than the Sardinia scene, and as such all incremental height errors will be larger for the Toolik Lake scene than Sardinia. However, because of the shorter baseline length, the RMS height uncertainty in the original interferogram will also be larger for the Toolik Lake scene. Thus, it is more appropriate to consider the percentage error measurements (included in parentheses in the table), as they are not affected by the baseline. 94 Chapter 7: InSAR Experimental Results Table 7-11 Estimates of total RMS height uncertainty for Sardinia (normal baseline = 126 m) bin 4 (0.3-0.4) bin5 (0.4-0.5) bin 6 (0.5-0.6) bin 7 (0.6-0.7) bin 8 (0.7-0.8) bin 9 (0.8-0.9) bin 10 (0.9-1.0) Original RMS height uncertainty (m) 8.83 6.40 4.63 3.39 2.52 1.81 1.19 4-bit FBAQ RMS height uncertainty (m) {absolute increase (m)} (percent increase) 9.03 {0.20} (2.3%) 6.54 {0.14} (2.2%) 4.74 {0.11} (2.4%) 3.48 {0.09} (2.6%) 2.59 {0.07} (2.8%) 1.87 {0.06} (3.3%) 1.24 {0.05} (4.2%) 3-bit FBAQ RMS height uncertainty (m) {absolute increase (m)} (percent increase) 9.79 {0.96} (10.8%) 7.06 {0.66} (10.3%) 5.13 {0.50} (10.8%) 3.79 {0.40} (11.8%) 2.81 {0.29} (11.5%) 2.05 {0.24} (13.2%) 1.39 {0.20} (16.8%) 2-bit FBAQ RMS height uncertainty (m) {absolute increase (m)} (percent increase) 12.03 {3.2} (36.2%) 8.83 {2.43} (38.0%) 6.54 {2.51} (41.2%) 4.87 {1.48} (43.6%) 3.59 {1.07} (42.5%) 2.65 {0.84} (46.4%) 1.94 {0.75} (63%) Table 7-12 Estimates of total RMS height uncertainty for Toolik Lake (normal baseline = 40.4 m) bin 4 bin 5 bin 6 bin 7 bin 8 bin 9 bin 10 (0.3-0.4) (0.4-0.5) (0.5-0.6) (0.6-0.7) (0.7-0.8) (0.8-0.9) (0.9-1.0) Original RMS height uncer- 30.30 21.72 15.65 11.44 8.40 5.94 3.96 tainty (m) 4-bit FBAQ RMS height 30.85 22.11 15.99 11.74 8.66 6.17 4.16 uncertainty (m) {absolute increase (m)} {0.55} {0.39} {0.34} {0.30} {0.26} {0.23} {0.20} (percent increase) (1.8%) (1.8%) (2.2%) (2.6%) (3.0%) (3.8%) (5.0%) 3-bit FBAQ RMS height 32.38 23.17 16.85 12.47 9.29 6.73 4.66 uncertainty (m) {absolute increase (m)} {2.08} {1.45} {1.20} {1.03} {0.89} {0.79} {0.71} (percent increase) (6.9%) (6.7%) (7.7%) (9.0%) (10.6%) (13.3%) (17.9%) 2-bit FBAQ RMS height 39.43 27.53 20.30 15.02 11.33 8.45 6.17 uncertainty (m) {absolute increase (m)} {9.13} {5.81} {4.65} {3.58} {2.93} {2.51} {2.22} (percent increase) (30.1%) (21.1%) (29.7%) (31.3%) (34.9%) (42.2%) (56%) 95 fer 7: InSAR Experimental Results DEM height estimation uncertainty for Sardinia 14, 1 , , , , n l 1 1 I i I I I 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 original coherence magnitude Figure 7-26 Total RMS height uncertainty vs. original coherence magnitude for Sardinia. DEM height estimation uncertainty for Toolik Lake original coherence magnitude Figure 7-27 Total RMS height uncertainty vs. original coherence magnitude for Toolik Lake Chapter 7: InSAR Experimental Results From the results of the analysis, it is evident that 3-bit and 2-bit encoding add a non-negligible error level to the D E M estimates, and should not be recommended for high accuracy D E M generation. The 4-bit F B A Q performs well however, with an RMS error increase of less than 5 percent for all levels of coherence for both scenes. 7.10.2 C h a n g e i n C o h e r e n c e M a g n i t u d e M e a s u r e m e n t The second approach to estimating the impact of F B A Q phase noise on D E M accuracy is to measure the change in coherence magnitude at each level of original coherence. This change can then be subtracted from the original value to yield the new coherence magnitude level. The height error associated with this new coherence level can then be determined through the curves of Figure 5-4 and equation (5-5) as before. The difference in height uncertainties will be the height uncertainty added due to F B A Q . This approach has the advantage of not requiring the assumption of Gaussian distributions, but unfortunately has practical difficulties due to the very small changes in coherence magnitude which we are attempting to measure, and the bias error associated with coherence magnitude estimation, which is explained in the following sub-section. 7.10.2.1 C o h e r e n c e M a g n i t u d e B i a s E r r o r The bias error of a coherence magnitude estimator depends on the block size (number of looks) used to estimate the coherence level, and the actual level of the coherence magnitude being estimated. For example, bias errors are lower for the 4 by 20 look estimator than a 2 by 10 estimator, but in both cases the level of bias is very non-linear with coherence magnitude level, and its effects only become significant at low levels of coherence magnitude. Since our study focuses 97 Chapter 7: InSAR Experimental Results on coherence levels of about 0.35 and greater we are not strongly affected by the bias in measuring absolute measures of coherence magnitude and using the 2 by 10 block size is a good trade-off because it allows us to minimize terrain induced error. However, when measuring the mean change in coherence magnitude, the bias can become a problem. This is because the change in coherence magnitude is a distribution rather than an absolute value, as illustrated by the histograms of Figure 7-24. Thus, while the mean change in coherence magnitude may be relatively small, some pixels may change over an extent great enough that a significant bias term (compared to the mean change being measured) will be included in the difference measurement. To illustrate with an extreme example, consider a pixel which has an initial estimated coherence magnitude of 0.8. The bias at 0.8 is approximately 0.0021. Assume now, that due to encoding noise, the coherence magnitude of the pixel drops to 0.2. The bias level at 0.2 coherence magnitude is approximately 0.07 (much greater than at 0.8). This means that our measured change in coherence magnitude of -0.6 is really a difference of -0.668. Thus our difference measure has been biased positively due to the estimator bias by 0.068. Such a bias level is important when attempting to measure small changes in coherence magnitude. Note that this effect will be more noticeable as original coherence magnitude decreases, and when the distribution of change in coherence magnitude is wider (as for 2-bit encoding). This trend is apparent when we consider the mean change in coherence magnitude curves for Sardinia and Toolik Lake in Figure 7-25(b). The positive trend in the curves in the mid range coherence levels is being caused by the combined effect of the inclusion of estimator bias in the difference measurement, and by the lower bound of the plots of Figure 7-22 and 7-23. 1. Bias levels were determined through simulation, and closely match those determined in [18] 98 Chapter 7: InSAR Experimental Results We can not remove the effect of the lower bound, but we can reduce the impact of estimator bias by using a larger block size for our estimator. However, the effect of extreme terrain precludes the use of a larger block size for the Sardinia scene. The Toolik Lake scene is relatively constant phase, however, and we were thus able to obtain more accurate coherence magnitude change measurements for this scene using a larger block size. The measurements for bins 4 to 10 are presented in Table 7-13. Table 7-13 Change in coherence magnitude measurements for Toolik Lake measured using 4 by 20 coherence magnitude estimator bin 4 (0.3-0.4) bin 5 (0.4-0.5) bin 6 (0.5-0.6) bin 7 (0.6-0.7) bin 8 (0.7-0.8) bin 9 (0.8-0.9) bin 10 (0.9-1.0) Mean change in C M for 4-bit -0.0041 -0.0040 -0.0036 -0.0041 -0.0041 -0.0041 -0.0034 Mean Change in C M for 3-bit -0.017 -0.019 -0.019 -0.020 -0.020 -0.018 -0.013 Mean Change in C M for 2-bit -0.039 -0.056 -0.068 -0.073 -0.072 -0.063 -0.048 Note the agreement with the 2 by 10 estimator is good in the upper bins. This makes sense as the upper bins should be less affected by bias error entering the difference measurement. In the middle bins, we begin to see the 4 by 20 estimator deliver change measures more negative than the 2 by 10 estimator, indicating a decreased influence of estimator bias. It is important to note, however, that estimator bias cannot be completely eliminated in practical InSAR measurements, so a small amount will still exist even for the 4 by 20 estimator. This is most apparent in the 2-bit case which begins to shows trend towards positive values again in bins 6 to 4. Of course, the 2-bit 99 Chapter 7: InSAR Experimental Results case wil l also begin encountering the first effects of the lower bound in that region of coherence as well, due to its wider distribution of coherence magnitude change as was observed in Figure 7-25(c) and 7-23. The measurements of coherence magnitude change using the 4 by 20 estimator indicate a relatively constant level of change for all initial coherence levels. This is consistent with our earlier results showing that higher levels of phase noise are added to lower coherence regions, and the observation that more phase error is required at low coherence levels to create the equivalent change in coherence magnitude. Using the measured change in coherence, the Toolik Lake scene D E M height estimation errors were re-evaluated. The results are presented in Table 7-14 and Figure 7-28. Table 7-14 Estimates of total RMS height uncertainty for Toolik Lake using change in coherence magnitude measurements (normal baseline = 40.4m) bin 4 (0.3-0.4) bin 5 (0.4-0.5) bin 6 (0.5-0.6) bin 7 (0.6-0.7) bin 8 (0.7-0.8) bin 9 (0.8-0.9) bin 10 (0.9-1.0) Original R M S height uncertainty (m) 30.30 21.72 15.65 11.44 8.40 5.94 3.96 4-bit F B A Q R M S height uncertainty(m) {absolute increase (m)} (percent increase) 30.68 {0.38} (1.2%) 21.98 {0.26} (1.2%) 15.83 {0.18} (1.1%) 11.58 {0.14} (1.3%) 8.50 {0.10} (1.2%) 6.04 {0.10} (1.7%) 4.05 {0.09} (2.3%) 3 bit F B A Q R M S height uncertainty (m) {absolute increase (m)} (percent increase) 31.96 {1.66} (5.5%) 23.09 {1.37} (6.3%) 16.64 {0.99} (6.3%) 12.20 {0.76} (6.6%) 8.95 {0.55} (6.6%) 6.39 {0.45} (7.6%) 4.33 {0.37} (9.4%) 2 bit F B A Q R M S height uncertainty (m) {absolute increase (m)} (percent increase) 34.19 {3.89} (13%) 26.03 {4.31} (19.8%) 19.58 {3.93} (25%) 14.42 {2.98} (26%) 10.56 {2.16} (26%) 7.58 {1.64} (28%) 5.29 {1.33} (34%) 100 Chapter 7: InSAR Experimental Results DEM height estimation uncertainty for Toolik Lake original coherence magnitude Figure 7-28 Total RMS height uncertainty vs. original coherence magnitude for Toolik Lake derived using measured change in coherence magnitude. The results of this analysis show the same trends as the previous addition of variances analysis though with lower levels of error. This is encouraging as it suggests that F B A Q adds even lower levels of error to D E M than would be predicted by adding the variances of the noise sources. The conclusions of this analysis are the same however, with 4-bit F B A Q recommended for high accuracy D E M applications as opposed to 3-bit or 2-bit. Although the above analysis suggests that the impact of F B A Q encoding wil l be lower than estimates obtained by adding the variances on the phase noise, we will adopt the higher estimates of the adding of variances for the remainder of this study. In this way, we will treat the results of the adding of variances as an "upper bound" on the error caused by F B A Q encoding. 101 Chapter 7: InSAR Experimental Results 7.10.3 O v e r a l l I m p a c t o n D E M H e i g h t U n c e r t a i n t y The final step in the analysis of the impact of F B A Q encoding on D E M accuracy is to calculate the average increase in D E M height estimation uncertainty caused by F B A Q encoding. This is the number generally quoted when discussing D E M accuracy. To estimate the average uncertainty in the original data, we must consider the distribution of the coherence magnitude values of the data. Pixels with low coherence magnitude will contribute more error to the calculation than pixels with high coherence magnitude. Again, we will confine our analysis to pixels with a coherence magnitude of 0.35 and greater, and we will use the "upper bound" error estimates obtained from the adding of variances analysis. Three calculations are presented in Table 7-15 below. The first is for the Sardinia scene, the second is for the Toolik Lake scene, and the third is for a hypothetical scene with the coherence magnitude distribution of Toolik Lake but with a normal baseline length of 200 m rather than 40.4. The purpose of the third calculation is to illustrate the effect of F B A Q encoding on the D E M of an "ideal" ENVISAT interferogram. Here we have defined "ideal" as being a scene with a coherence magnitude distribution of Toolik Lake (very high), but with a baseline length in the range more appropriate for D E M applications. For brevity we will only consider the 4-bit F B A Q case since we have dismissed the 3-bit and 2-bit encoding as inappropriate for high precision D E M generation. 102 Chapter 7: InSAR Experimental Results The average height accuracy before and after 4-bit F B A Q , assuming nominal 10 look smoothing, are shown in Table 7-15: Table 7-15 Average D E M RMS height estimation uncertainty Sardinia Toolik Lake "ideal" E N V I S A T Original R M S height uncertainty (m) 5.14 8.61 1.74 RMS height uncer-tainty after 4-bit F B A Q (m) 5.26 8.85 1.79 Absolute increase (m) 0.12 0.24 0.05 Percentage increase 2.3% 2.9% 2.9% Thus we observe from Table 7-15 that the average increase in D E M height estimation uncertainty is less than 3% for both the Toolik Lake and Sardinia scenes, and the "ideal" scene (with coherence levels of Toolik Lake but with a longer baseline) would suffer an increase in height estimation uncertainty of only about 5 centimeters against an initial uncertainty of about 2 meters. It is worth noting that in the above analysis we have only considered pixels with an average coherence magnitude of greater than 0.35. This constitutes 99.4 % of the pixels of the Toolik Lake scene, but only 80.3 % of the Sardinia scene. This implies that more areas of the Sardinia scene would be unsuitable for D E M generation than for the Toolik Lake scene. 7.10.4 T h e I m p a c t o f C a l i b r a t i o n E r r o r To place the results derived above into proper perspective, one must consider the effect of inaccurate calibration (estimation of baseline geometry, altitude of satellite, etc.) on D E M accuracy. The dominate source of error in calibration is due to inaccurate estimation of the baseline 103 Chapter 7: InSAR Experimental Results tilt angle (a). By differentiating the InSAR equations with respect to a we can express the increase in height estimation error due to tilt estimation error as [29]: 8z = rsineSa (7-26) Note that r is on the order of 850,000 m for ERS-1 and ENVISAT orbit altitude. Thus, assuming an average value of 0 of 23 degrees, we see that in order to obtain an R M S height estimation error of less than 5 m, the baseline tilt angle must be known to within 1.5xl0"5 radians R M S . This level of accuracy is not possible using conventional orbit estimations, however it may eventually be possible using differential Global Positioning Satellite (GPS) techniques. According to Zebker et al. [29], GPS techniques promise to deliver the relative satellite positions to an accuracy of about 3 mm. For a 200 m baseline, this translates into a baseline tilt estimation accuracy of 1.5xl0"5 radians. Shorter baselines would suffer greater error. Longer baselines would suffer less error, however, as we have discussed previously, baselines longer than 200 m are not optimal for ERS-1 and ENVISAT, particularly for scenes with hilly terrain. Currently, in order to obtain accurate results using repeat pass satellite InSAR, an array of points on the ground of known elevation (called tie points) must be used to refine the InSAR calibration. Iterative techniques are then used to estimate the InSAR geometry from the known elevation points. The accuracy of the resulting estimation will be dependent on the accuracy of the tie points, the distribution of the tie points within the image, and the number of tie points employed in the calculation, and wil l thus be very much scene dependent. However, to give an indication of the level of error encountered, we cite simulation work performed by Ian Joughin of the University of Washington [28]. Using ERS-1 SAR data with laser altimeter derived tie points, and a normal baseline of 200 m, the number of tie points used was varied between 20 and 122. Joughin found 104 Chapter 7: InSAR Experimental Results the standard deviation of the resulting height estimation errors to vary between about 6 m and 2 m respectively. From these results, if we assume a "typical" level of height estimation error due to calibration uncertainty to then be approximately 3 m rms (corresponding to about 75 laser altimeter tie points in the Joughin experiment), the overall impact of F B A Q on the "ideal" ENVISAT case can be calculated as in Table 7-16,'by adding the variances of the error sources. Table 7-16 Example of the Impact of Calibration Error (for the "Ideal" ENVISAT case) Original 4-bit F B A Q Initial RMS height uncertainty due to phase noise alone (m) 1.74 1.79 Nominal RMS height uncertainty due to calibration error (m) 3.0 3.0 Total R M S height uncertainty (m) 3.47 3.49 Absolute increase in RMS height uncertainty due to F B A Q (m) 0.02 Percent increase in RMS error due to F B A Q 0.58% From Table 7-16 we see that the impact of F B A Q is still further reduced when calibration error is considered, to about 0.6 %. The example was given to provide a rough idea as to how calibration error will impact the overall results, and it should be recognized that the level of calibration error may be either smaller or greater for individual scenes. Note that this analysis is still missing a final source of error, that due to phase unwrapping error. The impact of F B A Q on phase unwrapping will be investigated in Chapter 9. 105 8 REGISTRATION ACCURACY In the main experimental work of the study described in Chapter 7, all scenes were registered using the shift parameters estimated from the Original (un-encoded) data. This was necessary in order to ensure that F B A Q encoding noise was isolated from the Original for each encoding level. However, it is a valid concern that the encoding noise contributed to the SAR images by F B A Q may make the estimated shift parameters themselves less accurate when the estimation procedure is performed on encoded data. To investigate this, the registration shifts were re-estimated for each pair of encoded images. This was accomplished by running the estimation program used on the original data once again, but this time on the encoded data (using exactly the same input parameters as was used for the original data). As explained in Chapter 7, the Minimizing Residue Count criteria was apphed to the Sardinia data, and the Maximizing Fringe Visibility criteria was applied to the Toolik Lake scene. New interferograms were formed using the new shift estimates, and the average coherence magnitude levels of the interferogram were then calculated to determine if any degradation in 106 Chapter 8: Registration Accuracy interferogram quality had occurred due to use of the new shift parameters. The results are presented in Table 8-1. Table 8-1 Change in coherence due to use of shift parameters obtained from encoded data. Sardinia Toolik Lake Encoding Level 4-bit 3-bit 2-bit 4-bit 3-bit 2-bit Coherence magnitude from old shifts 0.4777 0.4709 0.4461 0.7574 0.7435 0.6974 Coherence magnitude from new shifts 0.4802 0.4722 0.4462 0.7574 0.7435 0.6974 mean difference +0.0025 +0.0013 +0.0001 0 0 0 From Table 7-1 it is evident that no change at all occurred for the Toolik Lake scene, indicating that for a high coherence scene like Toolik Lake, the impact of the F B A Q encoding noise on the registration algorithm was negligible and did not affect the accuracy of the shift estimates. The change in the average shift estimates for 4-bit encoding was 0.00012 in the range direction and 0.00006 in the azimuth direction. For 2-bit encoding the average change in shift estimates from those estimated for the original data was 0.00069 in the range direction and 0.00029 in the azimuth direction. The Sardinia scene, however, presented different results. Surprisingly, a net increase in coherence magnitude was observed for all encoding levels. These results indicate one of two possibilities. Either a small shift was being introduced to the images due to F B A Q encoding, or the shift estimates determined using the encoded data were actually more accurate than the shift estimates determined using the original images. ' 107 Chapter 8: Registration Accuracy To test the above possibilities, interferograms were formed from the original images, but using the shifts derived from the 4-bit, 3-bit, and 2-bit encoded images. The result was an increase in coherence magnitude for the original interferogram over previous levels. This indicates that there was not a shift introduced by the F B A Q (which confirms results from previous studies [14]) but in fact the shift estimates determined using the encoded data provided better registration than those of the original images. The results are displayed in Table 8-2. Table 8-2 Coherence Magnitude of Sardinia Original (un-encoded) data Coherence Magnitude Using shifts from Original images 0.4801 Using shifts from 4-bit images 0.4827 Using shifts from 3-bit images 0.4815 Using shifts from 2-bit images 0.4803 To understand the above results one must appreciate the registration difficulties that a relatively low coherence scene like Sardinia presents. For a high coherence scene like Toolik Lake, the registration functions used to determine the shift estimates have very strong, well defined maximums or minimums, meaning very accurate shift estimation is possible. Sardinia, however, is a scene which is difficult to register, as the high density of fringes, high variation of fringe direction, and low coherence make the maximums or minimums far more difficult to locate. A possible explanation for the observed results may lie in the registration method used for Sardinia, which was minimizing the number of phase residues. Residues counting yields an integer 108 Chapter 8: Registration Accuracy value, which means in areas of low residue counts it may be difficult to accurately determine the minimum, and hence the required shift. If more residues are present, however, the minimum location may become better defined and hence yield a more accurate shift estimation in that area. This idea is illustrated in Figure 8-1. Thus, it is conceivable that residue counting wil l yield better shift estimates in noisy data. Certainly, in this study, residue counting yielded the best shift estimates out of the methods tried for the relatively noisy Sardinia scene, while it did not for the relatively high coherence Toolik Lake scene. A more thorough study of this effect is beyond the scope of this study. Regardless, through the above experiment it was shown that it is possible to register the F B A Q encoded data accurately, even for difficult scenes like Sardinia. shift locations Initial Estimate with few residues. shift locations Estimate with more residues Figure 8-1 Hypothetical effect of adding more residues to an area of low residue concentration. 109 PHASE UNWRAPPING Phase unwrapping was not attempted in this study. The reasons for this are as follows: • The purpose of phase unwrapping is to provide absolute phase (and hence height) measurements. However, absolute height measurements were not required in this study. It was possible to obtain useful measurements of height error due to F B A Q without the phase unwrapping step. • Phase unwrapping is a difficult, time consuming operation, which often requires manual intervention in order to obtain acceptable results. This is particularly true for a mountainous scene with relatively low coherence levels like Sardinia. The manual intervention will be different depending on the phase residue densities and their locations within the interferogram, thus it is very difficult to devise a phase unwrapping method which would be consistent across all encoding levels. This consistency is important if meaningful results are to be derived by comparison to the original data. 110 Chapter 9: Phase Unwrapping However, the importance of assessing the effect of F B A Q on phase unwrapping was recognized, and this section of the report details experimental results designed to assess that effect. 9.1 P h a s e R e s i d u e s Many different algorithms have been proposed to accomplish phase unwrapping [30]. However, for all algorithms we are aware of, a fundamental difficulty is dealing with phase discontinuities within the interferogram called phase residues. Phase residues are detected when the circular integration about a region of the interferogram is not equal to zero. Phase residues are created by 2 primary sources: 1. Phase noise on the interferogram 2. Rapid changes in topography that exceed the maximum detectable change of the InSAR system (as defined by system parameters). It has been shown in this study that F B A Q encoding is a source of phase noise. Therefore, in order to assess whether phase unwrapping will be more difficult due to F B A Q , the number of residues on the interferogram before and after encoding were monitored. The results are presented in Table 9-1, where the number of residues have been expressed as a percentage of the total number of pixels in the interferogram, and thus represent a measure of "residue concentration". Ill Chapter 9: Phase Unwrapping Table 9-1 Total number of residues in the smoothed interferogram Total number of phase residues for Sardinia Total number of phase residues for Toolik Lake Original 7.8% 0.04% 4-bits/sample 8.1% 0.04% 3-bits/sample 8.5% 0.07% 2-bits/sample 10.4% 0.13% Table 9-1 illustrates the differences between the Toolik Lake and the Sardinia scene more dramatically than any previous figure or table. Where the Sardinia scene contains an almost eight percent phase residue concentration in the original data, the Toolik Lake scene contains 0.04 %. Based on these numbers, we would expect phase unwrapping in the Toolik Lake scene to be very straight-forward, whereas phase unwrapping in the Sardinia scene would be difficult, and probably confined to only certain areas of the scene. No increase in the number of residues was observed for the Toolik Lake scene due to 4-bit F B A Q encoding, although the change in the noise pattern on the interferogram did cause the locations of phase residues to change. However, based on these results it is unlikely that phase unwrapping would be made any more difficult due to 4-bit F B A Q encoding. The Sardinia scene experienced a 3% increase in phase residues due to 4-bit encoding, however, it was noted that the majority of these were added inside the low coherence regions of the interferogram which already contained numbers of residues prohibitive to phase unwrapping. This makes sense in light of the previous section which showed that more phase noise is added to 112 Chapter 9: Phase Unwrapping the low coherence sections than to others. Thus, the relatively small increase in residues would not be expected to increase phase unwrapping difficulty in those areas where it was possible in the original scene. Such areas wil l be the sections of the residue maps where the residue density is low (see Figure 9-1). While 4-bit F B A Q caused minimal change in residue numbers, the increases in residues for the 3-bit and 2-bit encoding levels were more substantial for both the Toolik Lake and Sardinia scenes, and may serve to increase the difficulty of phase unwrapping. Figures 9-1 and 9-2 show the residue maps for Sardinia and Toolik Lake respectively. Based on the above results it is the conclusion of this experiment that phase unwrapping should not become appreciably more difficult due to 4-bit F B A Q encoding. In fact, in high coherence regions (as good or better than Toolik Lake) the effect of encoding of phase unwrapping should be negligible. 113 Chapter 9: Phase Unwrapping Original 4-bits/sample 3-bits/sample 2-bits/sample Figure 9-1 Phase residue maps for Sardinia. Each dark point is a phase residue. Chapter 9: Phase Unwrapping Original 4-bits/sample 3-bits/sample 2-bits/sample Figure 9-2 Phase residue maps for Toolik Lake. Each dark point is a phase residue 115 10 INSAR CONCLUSION AND RECOMMENDATIONS In this study, the effect of F B A Q encoding on the practical application of satellite repeat pass interferometry has been investigated. The results of the study can be summarized as follows: • The impact of F B A Q on interferometric accuracy is through the addition of phase noise to the images used to form the interferogram. F B A Q was shown to add phase error to the encoded images, the magnitude of which was dependent on the data reduction level chosen (more error for greater data reduction), and was inversely proportional to the local SNR of the image pixels. • The variance of the F B A Q phase error on the interferograms produced using the F B A Q encoded images was found to be the sum of the variances of the F B A Q phase error on the images themselves. • Assuming a minimal amount of temporal and baseline decorrelation, the pixel brightness (SNR) of the images can be related to the coherence magnitude of the interferogram. Under this assumption, F B A Q phase error was found to be inversely related to interferogram coherence magnitude level. 116 Chapter 10: InSAR Conclusions and Recommendations • InSAR processing was possible for all F B A Q encoding levels including 2-bits/ sample. Registration programs were not hampered by the F B A Q encoding noise, even in the difficult to register Sardinia scene. • The decrease in D E M RMS height accuracy due to 4-bit F B A Q encoding was less than 3 percent of the initial uncertainty of the Original data. This was shown to represent an increase of about 5 centimeters in RMS height uncertainty for a D E M which initially had 1.74 meter RMS height uncertainty. This statistic is presented as an "upper bound" on the error, as it was derived by adding the variances of the F B A Q and Original data error measurements. The second analysis, monitoring the change in coherence magnitude, yielded 4-bit F B A Q encoding error in the 1.5% range. Both analysis methods were shown to have advantages and disadvantages. • The percentage error is further reduced (in some cases to a fraction of a percent) when RMS height error due to calibration error is included in the Original height uncertainty estimate. Thus the figure of 3% represents a worst case error. An example performed to illustrate the effect of calibration error showed the F B A Q percent error dropped to 0.6 % when calibration error was considered, corresponding to an average increase of 2 cm RMS for 3.47 meter RMS initial accuracy. • The "upper bound" average decrease in accuracy for the 3-bit and 2-bit F B A Q is on the order of 7-12 % and 30-45 % respectively, again not including calibration 117 Chapter 10: InSAR Conclusions and Recommendations error. This level of error is not recommended for precision InSAR mapping, though may be useful for large scale applications where precision is not a priority. • The mean change in coherence magnitude due to encoding was found to be relatively constant across the interferograms. This is consistent with the observation that higher coherence areas receive less phase errors than low coherence areas. The mean change in coherence for 4-bit, 3-bit and 2-bit encoding levels were approximately -0.0040, -0.020 and -0.065 respectively. The RMS change in coherence magnitude was shown to increase with decreasing coherence level. RMS change in coherence was higher for higher data reduction levels. • Phase unwrapping should not be made more difficult due to 4-bit F B A Q encoding, particularly for a scene with relative high levels of coherence. The increase in the number of phase residues was minor even for the low coherence Sardinia scene (approx. 3 % increase) and there was no increase in phase residues for the high coherence Toolik Lake scene due to 4-bit F B A Q . Phase unwrapping may be more difficult in the 3-bit and 2-bit cases, as higher residue counts were observed for these levels. It was noted, however, that in all cases the majority of phase residues tend to be added to low coherence regions of the interferogram, which in many cases are already inappropriate for phase unwrapping 118 Chapter 10: InSAR Conclusions and Recommendations Thus, it is the recommendation of this study that 8-bit to 4-bit F B A Q be employed as the maximum data reduction level for data targeted for precision InSAR mapping applications. The 3-bit and 2-bit applications may find use in lower precision InSAR mapping, or for data not intended for interferometric processing. 119 Section II: SAR Wave Mode Study 11 WAVE MODE SYNTHETIC APERTURE RADAR The term "Wave Mode" refers to the operating mode of the A S A R sensor in which small images (on the order of 10 km by 6 km) called "vignettes" are acquired over oceanic regions of the Earth at intervals of 100 to 200 km along track. The occurrence of ocean waves within the footprint of the SAR serve to modulate the radar reflectivity measured by the SAR sensor, causing the wave system to be visible within the detected vignette. By taking the power spectrum of the vignette, information about the wave system direction and wavelength is revealed. This data can be delivered to users within 24 hours to assist in decision making for marine shipping, offshore exploration, and weather forecasting. Oceanographers and meteorologists may also track the evolution of swell wave systems through examination of a series of power spectra. The ability of S AR to penetrate rain and cloud cover makes it particularly well suited for determining ocean conditions under adverse weather. Since Wave Mode data is often acquired far from the nearest satellite ground-station, the raw data must be stored on-board the satellite until downloading is possible. Thus, data reduction techniques are important in order to reduce the volume of raw data which must be stored, and to minimize the down-link period. Currently, one of two possible methods is employed to accomplish 121 Chapter 11: Wave Mode Synthetic Aperture Radar data reduction on ERS-1 and ERS-2. The first is to simply truncate the 5-bits/sample S A R data to 2 bits/sample. The second method is to perform range compression (the first step in the Range/ Doppler image formation algorithm) on-board the satellite. Performing range compression results in a "throw-away" region which can be eliminated from the raw data set, thus reducing the data volume. The range compressed data is then quantized to 4 bits/sample for storage. Both methods of data reduction incur a certain amount of degradation. In the first method, quantization of the data to 2 bits/sample obviously results in a reduction in precision from the 5-bit case. For the case of on-board range compression (OBRC), the process of range compression can result in a dramatic increase in the dynamic range of the data (>20 times) in the range direction, which is difficult to quantize accurately using only 4 bits/sample. For ENVISAT, it has been proposed that F B A Q encoding be employed for Wave Mode data reduction. While 4-bit F B A Q has been proposed as the default data reduction level for Image Mode data, 2-bit F B A Q has been proposed as the default for Wave Mode data, primarily due to on-board storage limitations. Because of technical problems, on-board range compression wil l not be included as an option on ENVISAT. 11.1 ESA Wave Mode Algorithm The processing of SAR vignettes into power spectrum form which best reflect ocean wave properties is a subject of current research. In order to maximize the utility of this study to ESA and current users of ESA Wave Mode products, this study will use the processing steps currently employed by ESA to create SAR Wave Mode products. The processing steps, as outlined in ESA document ER-TN-ESA-GS-0341 [31] are as follows: 122 Chapter 11: Wave Mode Synthetic Aperture Radar 1. Form an intensity image by squaring the pixel values of the detected, multi-looked vignette. 2. Compute the mean of the vignette and subtract it from all pixels. 3. Weight the data with a Hamming window. 4. Re-compute the image mean and subtract it from all pixels. 5. Zero pad the data to 512 by 512 pixels. 6. Perform a512by512 two-dimensional Fast Fourier Transform (FFT) and square law detect the spectral values. 7. Estimate the ocean clutter by computing the mean of a 25 by 25 pixel region below 40 m wavelength. 8. Transform the spectrum from Cartesian to polar co-ordinates. 9. Divide the wavelengths in the regions from 90 m to 1100 m into 12 logarithmically spaced bins (nominal wavelength 100 m to 1000 m) and the directions of 0 to 180 degrees with respect to the satellite track into 12 linearly spaced bins as shown in Figure 11-1. 10. Average the pixels within each bin, normalize by the maximum spectra bin, multiply by 255. 11. Display in 8-bit colour on polar grid. Chapter 11: Wave Mode Synthetic Aperture Radar The final output is a colour coded polar map of 288 bins representing the smoothed wave spectrum. The symmetrical nature of the spectrum means that only 144 of the bins are unique. The output format of the ESA product is shown in Figure 11-1. Figure 11-1 ESA Wave Mode Product Layout 124 Chapter 11: Wave Mode Synthetic Aperture Radar Note that the ESA product is but one method among many employed by oceanographers to process the vignette into a power spectrum product. Other methods include removing a 2-D low frequency trend from the data as opposed to just the DC value, using many small chips from a SAR image and then averaging their power spectrums together (either coherently or non-coherently, and dividing the squared vignette pixels by their mean value to place a greater focus on relative modulation levels within the image [33 to 38]. Also, for the E S A product, no attempt is made to transform the SAR spectrum into a "true" ocean spectrum via non-linear methods such as that proposed by Hasselmann [32]. 125 1 2 WAVE MODE EXPERIMENTAL METHODOLOGY The objective of the experiment was to quantify the effects of 2-bit F B A Q on the wave spectrum of SAR vignettes produced using ESA methods. This was accomplished using the Data Encoding Techniques Study (DETS) software previously developed for for data encoding studies at M D A [13], which was expanded during this study to include Wave Mode processing and analysis software. The reference for the study was ERS-1 data, pre-conditioned to simulate the expected 8-bit, zero mean format of ENVISAT data. The processing steps in the study are summarized in Figure 12-1 and were as follows: • Select 5-bit ERS-1 Image Mode scenes (no data reduction performed on-board the satellite) of ocean regions which exhibit visible wave patterns. • Pre-condition the data to 8-bit, zero mean format using methods from the previous F B A Q studies. These methods are described in detail in [16]. • Encode/decode the raw data using the 2-bit F B A Q algorithm. • Perform a statistical analysis of the reconstructed raw data by comparing it to the original 8-bit data. Chapter 12: Wave Mode Experimental Methodology • Process both original and reconstructed data into multi-look, detected SAR images using the DETS SAR processor. • Extract a chip of data from identical regions of the original and reconstructed images. • Obtain the 512 by 512 pixel power spectrum of the image chips using ESA Wave Mode processing methodology steps 1 to 7 outlined in Chapter 11. At this point the power spectrum is in Cartesian co-ordinates and has not been subjected to any averaging. The Detailed Analysis of Wave Spectra was applied at this point as described in the next sub-section. • Complete the formation of the ESA Wave Product using steps 8 to 10 outlined in Section 3. The ESA Wave Product Analysis was applied at this point to access the changes in the final ESA Wave Product due to F B A Q encoding. 12.1 Evaluation Criteria The signal domain raw data and image domain data were evaluated using the standard DETS analysis software found in the DETS environment. This includes statistical measures of data quality such as Average Signal to Quantization Noise Ratio (ASQNR), rms error, and mean absolute error, along with histogram generation. This analysis was useful as an intermediate check of F B A Q algorithm performance. The Wave Mode power spectrum evaluation consisted of 2 steps: 1. Detailed Analysis of Wave Spectra 2. ESA Wave Product Analysis 127 Chapter 12: Wave Mode Experimental Methodology SAR Process Wave Mode Process (steps 1 to 7) Wave Mode Process (steps 8 to 10) SAR Signal Data (of ocean scene) Wave Mode Process (steps 1 to 7) Signal Domain Evaluation Image Domain Evaluation Detailed Analysis of Wave Spectra Wave Mode Process (steps 8 to 10) ESA Product Analysis Figure 12-1 Flow chart of experimental methodology 128 Chapter 12: Wave Mode Experimental Methodology 12.1.1 Detailed Analysis of Wave Spectra The first analysis of the wave spectrum took place in the Cartesian domain without any averaging. Thus, this is an analysis of the "raw" spectrum. Before analysis, each spectrum (both Decoded and Original) was normalized by its mean value in order to focus on relative modulation levels within the spectrum. Note that for the Final ESA Product the spectrum is normalized by the highest spectrum peak, which achieves much the same effect as normalizing by the mean. Once the spectrum had been normalized by the mean, the following analysis steps were performed: • Error Spectrum Generation: An error spectrum was generated by subtracting the spectrum of the reconstructed data from that of the original. Viewing the error spectrum gave insight into the spatial distribution of errors caused by encoding. • Statistical Analysis: Standard statistical measures (rms error, mean absolute error, mean error etc.) were applied to the spectrums of the original and reconstructed images and the error spectrum. • Spectral Peak Location and Magnitude: Of primary importance in the ESA Wave Mode product is the location and magnitude of spectrum peaks. The original and reconstructed Wave Mode spectrums were interpolated in order to accurately determine peak size and location. Only peaks between 90 and 1100 meters were considered since this is the region used in the E S A Product. • Clutter analysis: In the context of Wave Mode, clutter is a measure of high frequency energy found in the Wave Mode spectrum. This is measured by Chapter 12: Wave Mode Experimental Methodology calculating the mean value of a 25 by 25 bin region of the spectrum below 40 m wavelength. Clutter levels in the Original and Decoded spectra were measured and compared. • Integrated Wave Energy Analysis: Another useful measure in wave analysis is the total energy of a given wave system, found by integrating the volume under the spectral peaks corresponding to the wave system. To perform this measure, a section of the spectrum — corresponding to where the majority of the peaks for a given wave system were located — was integrated. Original and Decoded measures were compared. 12.1.2 E S A W a v e P r o d u c t A n a l y s i s In order to accurately gauge the effect of F B A Q encoding on the final E S A product as a whole, the remaining steps in the ESA methodology were performed and the resulting E S A Wave Product analyzed. The following steps were applied: • Statistical Analysis: Statistical error measures such as RMS error and mean absolute error were used to asses the effect of encoding on the ESA product. • Bin Error: The percent error of each of the 144 bins in the final product was calculated. The average percent error (with respect to 8-bit data) was used to assess the quality of the spectral product. • Spectral Peak Location Error: Since the ESA product scales data by the peak bin value, a magnitude analysis of the spectral peak would be meaningless. 130 Chapter 12: Wave Mode Experimental Methodology However the location of the peak value was monitored to assess whether F B A Q caused any shift compared to the 8 bit data. 12.2 L i n e a r Q u a n t i z a t i o n C o m p a r i s o n In the above analysis, all error measures have been taken with respect to the original 8-bit data. However, using 8-bit data is not an option in Wave Mode due to storage limitations on-board the satellite. In order to provide a more meaningful comparison, the scenes used in this study were also reduced to 2 bits/sample using a linear quantizer (thresholds spaced evenly on the range -127 to 127). This was done in order to simulate the quality of Wave Mode data currently available using ERS-1 and ERS-2, which is reduced to 2 bits/sample by simply truncating the Image Mode data to 2-bits/sample, effectively the same as linear quantization. A l l of the same analysis procedures were applied to the linearly quantized data (hereafter referred to as LQ data) as were applied to the F B A Q data. 131 13 WAVE MODE EXPERIMENTAL RESULTS The following section details the experimental results obtained using data from ERS-1. Three ERS-1 SAR ocean scenes which showed pronounced wave systems were chosen for evaluation, and all scenes were encoded/decoded to 2-bits using F B A Q encoding. The decoder was set for rms equalization (as explained in Chapter 3 of this report). The F B A Q block size used was 126 range samples by 1 azimuth sample, the recommended block size for F B A Q [14]. Linear Quantization (LQ) to 2-bits/sample was performed outside the DETS environment for comparison purposes. After decoding, the scenes were processed into multi-look, detected images using the DETS SAR processor. The pixel spacing on the final images was approximately 25 by 25 meters. From these images, a 370 by 370 pixel chip was extracted around a region of interest, which was processed using ESA wave mode techniques outlined in the previous sections. The wave spectrum images produced by these methods were then analyzed using criteria also described in the previous section. The results of the comparison are presented in the following subsections. For purposes of discussion the un-encoded 8-bit data will be referred to as the Original data, and the 2-bit F B A Q encoded/decoded data will be referred to as Decoded data. 132 Chapter 13: Wave Mode Experimental Results 13.1 P r o c e s s e d I m a g e s a n d W a v e S p e c t r a The following figures show the processed SAR images used in the study and their corresponding wave spectra. A white square is used to mark the approximate location from which the image chip was extracted for wave mode processing. For comparison purposes, the F B A Q decoded scene is shown as well. Following the images, the 512 by 512 point (no smoothing) wave spectra for both Original and Decoded vignettes are displayed along with the Difference Spectrum (Original Spectrum - Decoded Spectrum). The histogram of the SAR signal data used to generate the Original is also shown, as it will be important in the Discussion of Results section of the report. The first ERS-1 scene was acquired on June 15,1993 (orbit 10023, frame 783), off the coast of Portugal. The raw data set contains a relatively high level of saturation. The particular subset of this data isolated for study contains strong wave patterns in two directions, as well as several ships and ship wakes. The second ERS-1 scene was also acquired on June 15,1993 during orbit 10023, but from adjacent frame 765 (approximately 100 km to the south). This scene shows a very bright ship, and strong wave pattern in one direction. Raw data saturation levels are relatively low. The final ERS-1 scene used in the study was acquired on December 4,1994 off the coast of Newfoundland, Canada, (orbit 17708, frame 909). It features relatively high frequency wave action in one direction. The raw data for this scene is highly saturated. 133 Chapter 13: Wave Mode Experimental Results Scene 1 (Coast of Portugal) Figure 13-1 Scene 1 off coast of Portugal: A . Original 8-bits/sample. B . Decoded 2-bits/sample 134 Chapter 13: Wave Mode Experimental Results V (a) t azimuth t azimuth t azimuth (b) Scene 1: Raw Data Histogram -50 0 50 quantized vaue (C) (d) Figure 13-2 Scene 1 wave spectra: (a) Wave spectrum of 8-bit original, (b) Decoded 2-bit F B A Q wave spectrum, (c) Absolute Difference Spectrum, (d) Histogram of pre-conditioned (Original) data. 135 Chapter 13: Wave Mode Experimental Results Scene 2 (Coast of Portugal) Figure 13-3 Scene 2 off coast of Portugal: A . Original 8-bits/sample. B . Decoded 2-bits/sample FBAQ 136 Chapter 13: Wave Mode Experimental Results 136 Chapter 13: Wave Mode Experimental Results Chapter 13: Wave Mode Experimental Results Scene 3 (Coast of Newfoundland) Figure 13-5 Scene 3 off coast of Newfld.: A . Original 8-bits/sample. B . Decoded 2-bits/sample F B A Q 138 Chapter 13: Wave Mode Experimental Results Chapter 13: Wave Mode Experimental Results 13.2 R a w D a t a C o m p a r i s o n Before beginning the comparison of wave spectra results, it is useful to first consider the error measures associated with the raw data directly after encoding to see how they compare to the wave spectrum errors which will be presented later on, and to examine the differing characteristics of the scenes used in the study. Measures of Original mean square, mean squared error, and Average Signal to Quantization Noise Ratio (ASQNR) 1 for the real portion of the data are presented in Table 13-1. The measurements for the imaginary portion were very close to the same. Essentially, the mean square of the original data gives a measure of signal energy in the original data, the mean squared error due to encoding is the value which F B A Q is designed to minimize, and the ASQNR is the value of their ratio expressed in decibels. From Table 13-1, a number of important trends are visible. Firstly, the three scenes differ substantially in the level of energy they contain, with Scene 3 being the largest. Secondly, the mean squared error and ASQNR of the F B A Q encoded data is always higher than that of the L Q data, particularly in the case of Scene 2. These results will be analyzed further in the Discussion of Results section of the report, following the presentation of the wave spectra analysis in the following sections. Table 13-1 Raw Data Statistics Scene 1 Scene 2 Scene 3 Encoding Method F B A Q L Q F B A Q L Q F B A Q L Q Original Mean Square 2876 1914 3180 Mean Squared Error 298.1 343.1 223.6 342.5 316.8 342.8 A S Q N R (dB) 9.84 9.23 9.32 7.47 10.0 9.67 1. All ASQNR measurements are taken with respect to the 8-bit data 140 Chapter 13: Wave Mode Experimental Results 13.3 D e t a i l e d A n a l y s i s o f W a v e S p e c t r a In order to closely assess the effect of F B A Q encoding on the wave spectra, the ESA Wave Mode processing chain was broken before the spectrum was averaged into the 144 bins of the ESA Wave Product. This "raw" (no averaging) spectrum was then compared to the spectrum obtained from the original 8-bit data. It is this "raw" spectrum which is shown in Figures 13-2,13-4 and 13-6. Breaking the processing chain meant that the spectrum normalization step (the last step of the ESA Wave Mode processing algorithm) had not yet been performed. To compensate for this, and to thus focus on relative modulation levels of the wave systems, the spectra were normalized by their respective mean values before analysis. Following the normalization, several analysis measures were applied to the "raw" 512 by 512 spectrum. The results are summarized in Table 13-2 and in the following sub-sections. 13.3.1 N o r m a l i z e d S p e c t r u m S t a t i s t i c s Measures of RMS error, mean absolute error, and mean error were taken. The F B A Q encoding was found to deliver lower levels of both RMS and mean absolute error measures, particularly in the case of Scene 2. The mean error was found to be zero for all cases. 13.3.2 I n t e r p o l a t e d S p e c t r a l P e a k A n a l y s i s For ocean studies, of primary importance is the location and relative magnitude of the wave spectrum peaks. The former can be used to determine the wave propagation direction, while the latter can be inverted to provide information on relative wave energy and height. To investigate the 141 Table 13-2 Detailed Analysis ofWave Spectra Results Scene 1 Scene 2 Scene 3 F B A Q L Q F B A Q L Q F B A Q L Q Normalized Spectrum Statistics RMS error 0.56 0.59 0.59 0.80 0.43 0.44 Mean absolute error 0.28 0.29 0.32 0.39 0.27 0.27 Mean error 0.00 0.00 0.00 0.00 0.00 0.00 Interpolated Spectral Peak Analysis Average absolute percent error in spectral peaks (top 20 peaks) 5.7% 6.1 % 7.2% 11.5% 4.9% 5.2% Average movement of spectral peaks (bins) 0.11 0.11 0.09 0.035 0.042 0.05 Average error in estimated wave-length of spectral peaks (m) 2.1 2.9 0.25 0.10 0.07 0.05 Average error in angular location of spectral peaks (degrees) 0.16 0.22 0.066 0.03 0.014 0.02 Integrated Wave Energy Analysis Percent error (1st wave train) -7.4 % -9.4 % -8.8 % -11.6% -4.2 % -4.5 % Percent error (2cd wave train) -5.2 % -5.9 % - - - -Clutter Analysis Percent change in mean clutter level +5.4 % +2.2% -1.1% +10.3% -3.5% +1.7% 142 Chapter 13: Wave Mode Experimental Results effect of encoding on spectrum peak location and magnitude, the normalized wave spectrum (512 by 512 points) interpolated by a factor of 8 to locate peaks to within 0.125 of an FFT bin. The highest 20 peaks of the decoded and original spectra between 90 and 1100 meters were compared to establish the accuracy of the encoding algorithm in preserving the spectrum shape. The results of the analysis for both 2-bit F B A Q and 2-bit L Q are tabulated in Table 13-1 as the interpolated spectral peak analysis. The size of the 20 largest peaks were monitored and the absolute percent error calculated for each. Table 13-2 displays the average of the 20 values in the row entitled Average absolute percent error in spectral peaks. Examining these error levels we see that the average absolute error in the relative magnitudes of the 20 largest peaks with respect to the 8-bit data is between 4.9 and 7.2 percent for F B A Q while between 5.2 and 11.5 percent for LQ. F B A Q delivered lower error levels in all cases, and the greatest improvement occurred for Scene 2. The locations of the peaks were monitored in several different manners. First, the change in spectral location was measured in the Cartesian co-ordinate system. The average shift observed for the 20 highest peaks measured in terms of FFT bins is tabulated under Average movement of spectral peaks in Table 13-2. Both F B A Q and LQ performed well in maintaining peak location compared to the 8-bit data, the highest average movement being observed in Scene 1 at a value of 0.11 FFT bins. The second measure of location movement was performed after converting the "raw" Cartesian spectrum into polar co-ordinates of angular displacement and wavelength. Change in location of the 20 highest spectral peaks were again monitored, this time measured in terms of angular error and wavelength error. The average results of the top 20 peaks between 90 meters and 143 Chapter 13: Wave Mode Experimental Results 1100 meters are presented in Table 13-1 under Average error in estimated wavelength of spectral peaks, and Average error in angular location of spectral peaks. As in the previous location analysis, both F B A Q and L Q were found to deliver a high level of spectral location accuracy. Changes in angular location were in the range of 0.2 degrees at maximum, and average wavelength estimation was between .07 m and 2.1 meters. Note that the relatively higher level of wavelength and angular estimation error in Scene 1 is a direct result of the longer wavelength system within the scene, as spectral peaks located in the longer wavelength regions of the spectrum (near the center) incur a higher wavelength and angular estimation error for the same shift in FFT bins. To summarize, the interpolated spectral peak results revealed that error in the relative magnitude of spectral peaks for 2-bit F B A Q were consistently on the order of 6 %, which is an improvement over L Q techniques which had relative magnitude errors as high as 11.5 %. Spectral peak location errors were extremely small, and should be negligibly small when some level of spectral smoothing is applied 13.3.3 I n t e g r a t e d W a v e E n e r g y A n a l y s i s A variation of the above analysis is to measure the integrated wave energy. For each of the Decoded spectra the portion of the spectrum containing energy associated with a wave train was integrated and compared to the same area integrated in the Original. The percent error of each integration was calculated and is tabulated in Table 13-2 under Integrated Wave Energy Analysis. Note that Scene 1 contained two distinguishable wave trains, while Scene 2 and Scene 3 each contained only one. 144 Chapter 13: Wave Mode Experimental Results The integrated wave energy analysis showed good agreement with the error measures of the integrated peak analysis. F B A Q error was on the order of 4.2 to 8.8 percent error for F B A Q and L Q error ranged between 4.5 to 11.2 percent error. F B A Q delivered lower percent error levels for all scenes and all wave systems, with the greatest degree of improvement (about 3%) realized for Scene 2. 13.3.4 C l u t t e r A n a l y s i s Finally, a clutter analysis was performed by computing the mean of a 25 by 25 sample chip in the highest frequency (shortest wavelength) section of the spectrum. The percentage change in mean clutter level is shown in the last row of Table 13-2. No clear pattern is observable for the three scenes used in this study, though it should be noted that the mean clutter level was well below the peak levels of the spectrum by about 30 dB on average for all of the scenes used. 13.4 E S A W a v e P r o d u c t A n a l y s i s The major goal of this project was to determine the effect of encoding on the E S A Wave Mode Product, which is a smoothed wave spectrum display used by meteorologists and oceanographers in near real-time decision making. The format of the ESA Wave Mode Product was discussed in Section 3. The final step in the spectrum analysis done during this study was to calculate the bin values which appear on the ESA Product for both the original (8-bit) and 2-bit F B A Q spectrums. These values were then compared to determine what level of degradation was caused to the spectrum display due to the encoding process. Percentage error, rms error, and peak 145 Chapter 13: Wave Mode Experimental Results location error were the primary evaluation criteria. The results of this analysis are displayed in Table 13-3. As can be seen from Table 13.3 the results of the ESA Wave Product Analysis are consistent with those of the detailed spectrum analysis. No shift in peak location occurs in the ESA Wave Product due to encoding for either L Q or F B A Q . The percentage error of the product bins in relatively consistent for FBAQ, varying between about 10.7 to 13.4 percent. For L Q , however, the percentage error in bin magnitudes was found to be more scene dependent, varying between 11.6 and 20.4 percent. This is due to the fact that L Q does not adapt to the shape of the signal data histogram or the changes in radiometry of the signal data, a fact that will be investigated further in the following sub-section. Note that the percentage error measure in Table 13-3 is the average absolute percentage error for the whole ESA Wave Product (all 144 bins) — it is not the same as the interpolated peak analysis of Table 13-2. In the ESA Wave Product, many of the peaks measured in the interpolated peak analysis have been averaged together to create a single peak in one of the E S A Wave Product bins. Since the ESA Wave Product is normalized by this peak (for the Original and Decoded respectively) the percentage error at the peak is 0 by definition. Table 13-3 E S A Wave Product Analysis Results (bin magnitude values range from 0 - 255) Scene 1 Scene 2 Scene 3 F B A Q L Q F B A Q L Q F B A Q L Q R M S error (magnitude) 2.69 2.63 6.24 7.42 8.57 9.32 Mean absolute error (magnitude) 1.36 1.43 4.31 5.89 5.63 5.95 Average percent error 11.3% 13.3 % 13.4 % 20.4 % 10.7 % 11.6% 146 Chapter 13: Wave Mode Experimental Results 13.5 D i s c u s s i o n o f R e s u l t s In the preceding sections it was demonstrated that 2-bit F B A Q encoding delivers lower relative magnitude errors in the Wave Mode spectrum than 2-bit LQ, and negligible peak location error. Most importantly, the F B A Q data exhibits a high degree of consistency in the level of error incurred due to data reduction across all three scenes studied, while the L Q encoding was found to be more variable, with much higher error levels occurring in Scene 2 than in the other two scenes. The reasons for the consistency of the F B A Q with respect to that of the L Q is due to the adaptive nature of F B A Q that allows it to adjust to the characteristics of the SAR signal data. The most important of these characteristics which affects F B A Q performance was found to be the Saturation Level of the raw data, the results of which are explained in the following sub-sections. 13.5.1 S a t u r a t i o n L e v e l From Tables 13-1 and 13-2 it can be seen that the greatest improvement in F B A Q performance with respect to L Q occurs in Scene 2, while in Scene 3 the differences are much smaller. Upon examination of the histograms of the signal data (see Figures 13-2(d), 13-4(d), 13-6(d)) we see that this trend roughly corresponds to the degree of saturation (i.e. the size of the "spikes" at -127 and 127) for each of the three scenes. From an intuitive standpoint this would seem to make sense since we know that F B A Q has been optimized for a Gaussian distribution, while L Q would be optimal for a Uniform distribution. As the saturation level increases, the data distribution becomes broader with higher "spikes" at -127 and 127, and the distribution begins to look more Uniform in shape rather that Gaussian, leading to decreased improvement due to F B A Q . 147 Chapter 13: Wave Mode Experimental Results To verify this, Scene 1 was modified to further reduce the saturation levels of the histogram. This involved removing the spikes in the histogram, adding a small amount of exponential noise to the ends of the histogram then re-truncating the data (detailed explanations of the saturation reduction procedure can be found in [16]). The new histogram for the low saturation Scene 1 is shown in Figure 13-7. The scene was then re-processed using the same steps as before for both 2-bit F B A Q and 2-bit L Q . The new ASQNR levels and ESA Wave Product error levels are shown in Table 13-4. As can be seen, the reduction in the saturation level had a dramatic effect on the linear quantization, driving its ASQNR performance down to levels similar to that of L Q for Scene 2, and raising the percentage error in the ESA Wave Product to 16.4 %. The ASQNR performance of the F B A Q encoding also dropped slightly. The reason for this is explained by examining the original mean square and mean squared error statistics also listed in Table 13-4. Note that, as expected, the original mean square of the saturation reduced data is lower than for the saturated data. Correspondingly, the mean squared error for the F B A Q encoding is also lower in the low saturation case and higher in the saturated case. This is also as expected. Now, if the data statistics were perfectly Gaussian (ie. no saturation "spikes") the ratio of mean squared error to original mean square should stay constant for F B A Q — meaning constant ASQNR [15]. However, when the histogram saturates at -127 and 127 we artificially limit the mean squared error relative to total power of the signal (ie. the total power of the signal is less than what it would be without saturation, but the mean squared error is less than what it would be without saturation by 148 Chapter 13: Wave Mode Experimental Results a larger amount). This has the effect of raising the ASQNR as measured against the 8-bit data. The effect is explained in detail by simulation results included in Appendix D. Scene 1: Raw Data Histogram Saturation Reduced quantized value Figure 13-7 Histogram of Scene 1 original raw data with reduced saturation levels Table 13-4 Performance of F B A Q vs. L Q for Scene 1 with reduced saturation Scene 1 (high saturation) Scene 1 (reduced saturation) F B A Q L Q F B A Q L Q Raw Data Statistics Original mean square 2876 1853 Mean squared error 298.1 343.1 209.8 341.5 ASQNR (db) 9.84 9.23 9.46 7.34 E S A Wave Product Error Average percent error 11.3 % 13.3 % 11.4% 16.4% It is worth noting that the lower levels of saturation exhibited by the above scene and Scene 2 of the study probably better reflect the type of histogram expected for ENVISAT data than the highly saturated Scene 1 and Scene 3. With the appropriate gain setting, the 8-bit A D C of ENVISAT should saturate less often than the 5-bit A D C of ERS-1. Thus, while distributions such 149 Chapter 13: Wave Mode Experimental Results as those shown for Scene 3 will still be possible, narrower distributions such as that of Scene 2 should be more common, meaning F B A Q should outperform L Q by a large margin in most cases. This is illustrated by the graphs presented in Figure 13-8 and Figure 13-9 which summarize results obtained with simulated SAR data. In the graph of Figure 13-8, the A S Q N R performance of 8-bit L Q , 2-bit F B A Q , and 2-bit L Q are shown for Gaussian input of varying standard deviation. A S Q N R performance here is measured against the "true" continuous Gaussian input. In Figure 13-9, the same graph is presented, though this time the 2-bit L Q and 2-bit F B A Q performance is measured with respect to the 8-bit data (input to the 8-bit quantizer), as the A S Q N R figures presented in this study are. The 8-bit quantizer performance of Figure 13-9 is still measured against the continuous distribution and is included for reference purposes only. From Figure 13-8 we see that the 8-bit quantizer achieves best performance between a standard deviation of about 15 to 45. The 2-bit LQ, however, achieves best performance in the range of 50 to 80. Obviously these values are incompatible. The advantage of F B A Q is clearly evident as it extends the maximum of the 2-bit L Q curve into the full operating region of the 8-bit quantizer. Thus, with F B A Q both the 8-bit quantizer and the 2-bit quantizer may be operated in their optimal regions, provided the data is scaled appropriately before entering the 8-bit quantizer. Figure 13-9 exhibits the same trend, though we see and increase in ASQNR performance at high saturation levels for both 2-bit LQ and 2-bit F B A Q . This is because the ASQNR measurements are taken with respect to the 8-bit data, which is now degraded by saturation effects. This effect is investigated in greater detail in Appendix D. In the region of optimal 8-bit performance however, we still see that 2-bit F B A Q would significantly outperform 2-bit LQ. 150 Chapter 13: Wave Mode Experimental Results Finally, the vertical dashed lines on the graph mark the locations of the average 8-bit standard deviation of the scenes used in this study. Note that the highly saturated scenes fall well outside the optimal range for the 8-bit quantizer, at a location were both the 2-bit F B A Q and 2-bit L Q perform at similar levels. This is an undesirable situation, as much information has been lost due to the 8-bit saturation. Scene 2 and the saturation reduced Scene 1 are located in a better region of the graph where the difference between 2-bit F B A Q and 2-bit L Q performance is greater. This corresponds to the results observed in the study, where the differences between 2-bit L Q and 2-bit F B A Q were seen to be greater for less saturated scenes. Ideally, however, the data would be located even further to the left on the graph, with a standard deviation of approximately 35. Thus, it is reasonably to hypothesize that properly scaled data in this region would experience a still greater benefit due to F B A Q than the data used in the study. ASQNR performance with respect to simulated continuous Gaussian input Scene 2 and Sat. Reduced Scene 1 \Scene 1 / ^ Scene 3 2-bit FBAQ | | | ' • • • / „ . . - - r - 1 1 — - 2-bit LQ " 0 10 20 30 40 50 60 70 80 90 100 standard deviation of continuous Gaussian input Figure 13-8 ASQNR performance measured with respect to the continuous input data for varying levels of histogram standard deviation. 45 40 35 30 ? 2 5 20 15 10 5 151 Chapter 13: Wave Mode Experimental Results 50 45 40 35 130 J25 J 20 15 10 5 ASQNR performance with respect to 8—bit Gaussian Input r 8-bl tLQ Scene 2 and Sat. Reduced Scene 1 \Scene 1 / Scene 3 2-bit FBAQ 2-bit LQ 20 30 40 50 60 70 standard deviation of 8-bit Gaussian Input Figure 13-9 ASQNR performance measured with respect to the 8-bit input data for varying levels of 8-bit histogram standard deviation (8-bit curve still measured with respect to continuous input). 13.5.2 U n d e r - E s t i m a t i o n o f A b s o l u t e S p e c t r u m M a g n i t u d e s Finally, an important aspect of F B A Q is the fact that due to its adaptive nature F B A Q always tends to under-estimate the absolute magnitudes1 of spectral peaks, while L Q may under-estimate or over-estimate depending on the saturation level of the data. For the three scenes studied in this project, the degree of under-estimation was greatest in Scene 2, followed by Scene 1 and then Scene 3 (i.e. greatest for the least saturated data). A pixel by pixel evaluation of each of the extracted image chips used for Wave Mode processing was performed to investigate this phenomena. Pixels greater that the median value were termed bright, and pixels less than the median value were termed dark. The analysis revealed that 75 % of bright 1. Absolute magnitude refers to the magnitude of spectral peaks before the normalization step where the spec-trum is divided by its mean value, or peak value in the case of the ESA Wave Product. 152 Chapter 13: Wave Mode Experimental Results pixels in the multi-looked, detected Scene 2 were mapped to lower levels due to F B A Q . In contrast, Scene 1 had 69 % mapped to lower levels, and Scene 3 had 64 % mapped to lower levels. Thus, the decrease in pixel brightness correlated well with the observed loss of energy. Extending the above analysis back to the signal domain, under-estimation of spectral peaks is believed due to the combined effect of the mapping of the highest value samples in the data to lower values while mapping the lowest values to higher values. The former occurs because the highest quantizer reconstruction level is always set lower than the highest data values. For example it is possible that a "bright" sample which originally had a value of 126 may be mapped to a new value of 96. A greater drop may occur for a less saturated histogram as thresholds and reconstruction levels will be set closer to zero. Conversely a "dark" sample will be shifted to a higher value because no reconstruction level is available right at zero for either F B A Q or L Q . Samples with original values near zero will thus be mapped to higher values due to F B A Q . This combined action has the effect of "squeezing" the energy of the scene into a more concentrated region, thus reducing the magnitude of spectrum peaks. Note that the overall energy of the scene may be the same for both Original and Decoded if the RMS equalization option of F B A Q has been invoked — but the energy has nonetheless been redistributed within the data set. Much the same effect occurs for linear quantization; however, since linear quantization does not adjust thresholds to match the data distribution, spectrum peaks may be both over-estimated or under-estimated. Under-estimation occurred in the highly saturated Scene 1 and Scene 3. Here, the fact that the histogram was already quite broad in shape meant that both L Q and F B A Q tended to reduce the wave energy in the data. However, in Scene 2, which has lower saturation and thus a much narrower histogram (ie. more data concentrated near 0) linear quantization served to expand 153 Chapter 13: Wave Mode Experimental Results the pixel values, and as a result an over estimation of spectral energy occurred with respect to the 8-bit data. Another example of this phenomena is that when the saturation levels of Scene 1 were reduced (see Section 13.5.1) F B A Q still under-estimated the absolute spectrum magnitude values, but L Q under-estimated the energy in the saturated case and over-estimated it when the saturation was reduced. The important point is that while both L Q and F B A Q may not reproduce the absolute spectral peak values precisely, F B A Q is at least consistent as it always under-estimates the peaks, and of course as shown earlier it more accurately preserves the normalized peaks. Though beyond the scope of this work, this fact would suggest that it may be possible to develop an algorithm which uses the statistics of the decoded data to compensate for the energy loss. For current uses, however, it is important to include a normalization step in the wave spectrum processing, either by the peak bin value (as for the ESA Wave Product) or preferably by the mean value (as done the interpolated peak analysis). 154 14 CONCLUSIONS AND RECOMMENDATIONS OF WAVE MODE STUDY In this study we have examined the effect of 2-bit F B A Q data reduction on the wave spectrum of Wave Mode SAR data, and specifically the ESA Wave Product. Comparisons were made between the results derived from the 2-bit F B A Q data, the original 8-bit data, and 2-bit linearly quantized data which simulates the current quality of 2-bit Wave Mode data. The results can be summarized as follows: • The error in spectral peak locations due to 2-bit F B A Q encoding is very minor. It is negligible when the spectrum is smoothed before analysis as is the ESA Wave Product. • The error in the magnitude of the normalized spectral peaks is not negligible compared to the 8-bit data, but is lower than spectra obtained from Wave Mode data employing 2-bit L Q . • It is important to normalize the wave spectra as F B A Q preserves the normalized magnitudes well but always tends to under-estimate the absolute magnitudes. L Q Chapter 14: Conclusions and Recommendations of Wave Mode Study may under-estimate or over-estimate depending on the shape of the SAR data histogram. Note that for the ESA Wave Product the spectrum is normalized by the peak bin as opposed to the mean. This is deemed somewhat less accurate than dividing by the mean value, but still has the desired effect of normalizing the spectrum energy. • F B A Q achieves the greatest gains over L Q when saturation levels of the data histogram are low. This is the case when the 8-bit quantizer is operating in its region of optimal performance. Overall, the use of 2-bit F B A Q encoding for Wave Mode data should provide E S A Wave Product spectrum estimates with greater accuracy in relative spectrum peak measurement than currently available from ERS-1 or ERS-2, and peak location accuracy comparable to that of 8-bit data. Improvements will be particularly noticeable for data with low saturation levels, which is expected for ENVISAT data opertating in the optimal region of the 8-bit quantizer. Note that a brief wave spectrum analysis using 4-bit F B A Q (the default for Image Mode) showed it to suffer only minor degradation (see Appendix C), thus very high accuracy wave analysis should still be possible using Image Mode data. 156 15 SUMMARY AND CONCLUSIONS In this study, the impact of the F B A Q SAR data reduction algorithm was quantitatively assessed for the SAR applications of Interferometry and Wave Mode. Real ERS-1 SAR data was used to experimentally determine F B A Q encoding error levels, by first modifying the data to ENVISAT format, then performing the necessary data processing to produce Interferometric and Wave Mode results. Simulations and theoretical explanations were used to support the conclusions of the experiments. 15.1 SAR Interferometry Study For InSAR, all 3 data reduction ratios of 8-bits/sample to 4 bits/sample, 8-bits/sample to 3 bits/sample, and 8-bits/sample to 2 bits/sample were evaluated. Using coherence magnitude and R M S phase error measurements as analysis tools, it was determined that, while it was possible to perform InSAR processing for all data reduction levels, the only data reduction level appropriate for precision InSAR mapping was the 8-bits/sample to 4-bits/sample. Such a level of compression caused an increase in D E M height uncertainty (over that of the 8-bit data) that was on the order of several centimeters, compared to inherent height uncertainties on the order of several meters. 157 Chapter 15: Summary and Conclusions Separate experiments verified that 4-bit F B A Q also caused no degradation in the accuracy of registration procedures during the InSAR processing, and also did not an appreciable increase in the number of phase residues, an important factor for the phase unwrapping process. 15.2 S A R W a v e M o d e S t u d y In the Wave Mode study, only the 8-bit to 2-bit F B A Q reduction was studied due to storage limitations on-board the satellite. To determine if the level of error introduced to the data due to F B A Q was acceptable, the F B A Q data was compared to data that had been linearly truncated to 2-bits/sample, one of the current methods used aboard ERS-1 and ERS-2 for Wave Mode data reduction. The study concluded that 2-bit F B A Q introduced negligible small shifts in the location of spectral peaks in the SAR Wave Mode Product, though did cause about a 10 % error in spectral bin magnitudes. This was found to be better than the current linear quantization method, which was found to have errors in spectral bin magnitudes up to as high as 20 %. The level of bin magnitude error for the linearly truncated data was shown to depend on the level of saturation of the 8-bit data, while the F B A Q encoder delivered more consistent error levels for all scenes tested due to the adaptive nature of the algorithm. 15.3 O v e r a l l C o n c l u s i o n s Based on the results of this study, and on previous work performed during the development of the algorithm, F B A Q SAR signal data reduction has been verified as a viable on-board data reduction system for the ENVISAT satellite. In particular, 4-bit F B A Q has been shown to be appropriate for precision InSAR processing, and 2-bit F B A Q appropriate for Wave Mode applications. 158 BIBLIOGRAPHY Data Reduction: [1] I. McLeod, I. Cumming, M . Dutkiewicz. F B A Q Extended Study Final Report (Volume 2 of 3): Wave Mode Study. M D A Document No. DC-TN-50-6905, June 19,1995. [2] I. McLeod, I. Cumming, M . Dutkiewicz. F B A Q Extended Study Final Report (Volume 3 of 3): SAR Interferometry Study. M D A Document No. DC-TN-50-6773, June 19,1995 [3] I. McLeod, I. Burke, I. Cumming, M . Dutkiewicz. F B A Q Extended Study Executive Summary. M D A Document No. DC-MA-50-6976, June 19, 1995. [4] I. McLeod, I. Cumming. On-Board Encoding of the ENVISAT Wave Mode Data. Proceedings oflGARSS '95, July 1995. pp. 1681-1683. [5] I. McLeod, I. Cumming, M . Seymour. Data Encoding Requirements of SAR Interferometry. Proceedings of PIERS '95, July 1995. pp. 154. [6] I. Cumming, M . Dutkiewicz, B . Akam. Study of synthetic Aperture Radar Data Encoding Techniques. ESTEC Image Processing Workshop, June 1991. [7] J. Curlander, R. McDonough. Synthetic Aperture Radar: Systems and Signal Processing. Wiley Series in Remote Sensing, Wiley & Sons, 1991. [8] R. Kwok, W. Johnson. Block Adaptive Quantization of Magellan SAR Data. IEEE Transactions on Geoscience and Remote Sensing, Vo l . 27, No. 4, July 1989. pp. 375 - 383. [9] K . Strodl, U . Benz, F. Blaser, T. Eiting, A . Moreira. A Comparison of Several Algorithms For On-Board SAR Raw Data Reduction. Proceedings oflGARSS '94, July 1994. pp. 2197-2199. 159 Bibliography [10] R. Bertoni, L . Di Paolo, R. Di Julio, F. Impagnatiello, F. Quaranta. Data Compression Device for SAR Application. Proceedings oflGARSS '94, July 1994. pp. 2194-2196. [11] M . Dutkiewicz, I. Cumming. Evaluation of the Effects of Encoding on SAR Data. Photogrammetric Engineering and Remote Sensing, Vol . 60, No. 7, July 1994. pp. 895-904. [12] G. Kuduvalli, M . Dutkiewicz, I. Cumming. Synthetic aperture Radar Signal Data Compression Using Block Adaptive Quantization. GSFC Space and Earth Sciences Data Compression Conference, 1994. [13] M . Dutkiewicz, I. Cumming. Data Encoding Techniques Study Final Report: SAR Study. M D A Document No. DC-RP-50-4040, January 17, 1992. [14] M . Dutkiewicz, G. Kuduvalli, I. Cumming. SAR Pre-Processing On-Board Study, Final Report, Vol . 1 of 2, Algorithm Definition, M D A Document No. DC-TN-50-5830, May 5,1994. [ 15] A . K . Jain. Fundamentals of Digital Image Processing. Prentice Hall Ltd., New Jersey, 1989. [16] M . Dutkiewicz, I. Cumming. SAR Pre-Processing On-Board: ERS-1 Data Pre-Conditioning, M D A Document No. DC-TN-50-5137, February 1, 1993. [17] G. Kuduvalli, I. Cumming. SAR Pre-Processing On-Board: Final Test Results, Algorithm Study. M D A Document No. DC-TN-50-5829, January 31,1994. Interferometry: [18] J. Hagberg. Repeat-Pass Satellite SAR Interferometry, Technical Report No. 170L, Department of Radio and Space Science, School of Electrical and Computer Engineering, Chalmers University of Technology, Goteborg, Sweden, February, 1994. [19] E. Rodriguez. Maximum Likelihood Estimation of the Interferometric Phase from Distributed Targets. IEEE Transactions Geosci. and Remote Sensing, 1991. [20] J.S. Lee, K . Hoppel, S. Mango, and A . Miller. Intensity and Phase Statistics of Multilook Polarimetric and Interferometric SAR Imagery. IEEE Trans, on Geosci. and Remote Sensing, Vo l 32, No. 5. Sept. 1994, pp. 1017-1027. [21] H . Zebker, C. Werner, P. Rosen, S. Hensley. Accuracy of Topographical Maps Derived from ERS-1 Interferometric Radar. IEEE Trans, on Geosci. and Remote Sensing, Vol 32, July 1994, pp. 823-836. 160 Bibliography [22] F. Gatelli, A . Guarnieri, F. Parizzi, P. Pasquali, C. Prati, and F. Rocca. The Wavenumber Shift in SAR Interferometry IEEE Trans, on Geosci. and Remote Sensing, vol. 32, no. 4, July 1994, pp. 855-865. [23] D. Just and R. Bamler. Phase statistics of interferograms with applications to synthetic aperture radar. Applied Optics, vol. 33, no. 20, July, 1994, pp. 4361-4368. [24] H . Zebker, J. Villasenor. Decorrelation in Interferometric Radar Echoes. IEEE Trans, on Geosci. and Remote Sensing, Vo l . 30, No. 5,1992, pp. 950-959. [25] F. L i , R. Goldstein. Studies of Multibaseline Spaceborne Interferometric Synthetic Aperture Radars. IEEE Trans, on Geosci. and Remote Sensing, Vol . 28, No. 1, Jan. 1990, pp. 88-97. [26] I. Joughin, D. Winebrenner, D. Percival. Probability Density Functions for Multilook Polarimetric Signatures. IEEE Trans, on Geosci. and Remote Sensing, Vol . 32, No. 3, May 1994, pp 562-574. [27] I. Joughin, D. Winebrenner. Effective Number of Looks for a Multilook Interferometric Phase Distribution. Proceedings oflGARSS 1994, pp. 2276-2278. [28] I. Joughin. Estimation of Ice Sheet Topography and Motion Using Interferometric Synthetic Aperture Radar. Ph.D. Thesis, University of Washington, March 1995. [29] H . Zebker, T. Fair, R. Salazar, T. Dixon. Mapping the World's Topography Using Radar Interferometry: The TOPSAT Mission. Proceedings of the IEEE, Vol 82, No. 12, December 1994, pp. 1774-1786. [30] J. Valero. A Survey of Phase Unwrapping Techniques, with Applications to InSAR. Technical Report (Draft Copy), University of British Columbia Radar Remote Sensing Group, May 27,1995. Wave Mode: [31] G. Brooker. U W A processing algorithm description, ESTEC/JW, Issue 1-1. ESTEC Document No. ER-TN-ESA-GS-0341, January 19,1994. [32] K . Hasselmann, S. Hasselmann. On the Nonlinear Mapping of an Ocean Wave Spectrum Into a Synthetic Aperture Radar Image Spectrum and Its Inversion. Journ. of Geophysical Research, Vo l . 96, No. C6, June 15, 1991, pp. 10,713-10,729. [33] Paris Vachon. Canada Center for Remote Sensing, Personal correspondence, Feb. 1995. 161 Bibliography [34] J.T. Macklin, R. Cordey. Ocean-Wave Imaging by Synthetic Aperture Radar: Results from the SIR-B Experiment in the N.E. Atlantic. IEEE Transactions on Geoscience and Remote Sensing, Vo l . 27, No. 1, Jan. 1989, pp. 28-35. [35] P. Vachon, H . Krogstad, J.S. Paterson. Airborne and Spaceborne Synthetic Aperture Radar Observations of Ocean Waves. Atmosphere-Ocean 32, 1994, pp. 83-112. [36] H . Johnsen. Multi-Look Versus Single-Look processing of Synthetic Aperture Radar Images with Respect to Ocean Wave Spectra Estimation. IGARSS '91 Proceedings, pp. 439-442. [37] G. Engen, H . Johnsen, H. Krogstad, S. Barstow. Directional Wave Spectra by Inversion of ERS-1 Synthetic Aperture Radar Ocean Imagery. IEEE Transactions on Geoscience and Remote Sensing, V o l . 32, No. 2, March 1994, pp. 340-352. [38] R. Beal, T. Gerling, D. Irvine, F . Monaldo, D. Tilley. Spatial Variations of Ocean Wave Directional Spectra From the Seasat Synthetic Aperture Radar. Journal of Geophysical Research, Vo l . 91, No. C2, Feb. 15,1986, pp. 2433-2449. [39] I. Jones. Real Time Processing Wave Spectra From Satellite SAR. Proceedings of First ERS-1 Symposium, Nov. 1992, pp. 735-737. 162 A A P P E N D I X A : D a t a P r e - c o n d i t i o n i n g The need to pre-condition ERS-1 data and the methodology for doing so is outlined in detail in document [16]. The following is a brief summary of the rationale and methodology outlined in that document, and the results of a test applied to the Sardinia InSAR data set. ERS-1 data is fundamentally different from future systems in one respect: It is quantized to 5-bits (on the range [0,31]) as opposed to 8-bits two's complement quantization. This creates several different statistical properties in the data which are summarized below: • the data mean is nominally 15.5 as opposed to zero • the data entropy of 5-bit data will be lower than that of 8-bit data • the frequency of saturation of 5-bit data is approximately 5% as opposed to 0.5% for 8-bit data • the quantization noise will be higher for 5-bit data, resulting in a slightly lower SNR than for 8-bit data A .1 R a t i o n a l e a n d M e t h o d o l o g y 163 Appendix A: Data Pre-conditioning The methodology for simulating 8-bit ENVISAT data from the ERS-1 data developed for the previous data encoding study is as follows: • Convert data to floating point representation and subtract the nominal mean of 15.5. • Reduce the data saturation level by adding exponential random noise to the clipped ERS-1 samples then re-clip the data at a higher value. • Increase the data entropy by adding uniform zero-mean random noise in the range [-0.5,0.5]. • Scale the data to 8-bits. Scaling factor is determined as (maximum value of 8-bit data) /(maximum value of saturation reduced "5-bit" data). • Convert the data from floating point to 8-bit zero mean format In document R-3, experimental results confirmed that the above methodology created a data set with properties similar to those expected for ENVISAT data. In particular it was noted that the average SNR of SAR signal data is of the order of 12 dB and thus the LSB of 5-bit data or the 4 LSBs of 8-bit data are entirely within the noise floor. Thus, adding the small amount of noise to the data in the procedure serves only to increase the entropy of the signal without significantly altering the noise distribution. Further, the standard deviation of the phase error between the original and pre-conditioned data was shown to be less than 5 degrees. However, since interferograms are formed by taking the phase difference of two images, the technique is especially sensitive to any type of data distortion, and there was concern that the level of distortion would make interferometric processing impossible. This concern proved to be false. Appendix A: Data Pre-conditioning A.2 Test Results Testing of the of pre-conditioning algorithm was performed exclusively on the Sardinia data set, as this work was done at the beginning of the study and the Toolik Lake scene had not yet been received. The tests consisted of processing three separate swaths of raw data, both in 5-bit and in 8-bit format (after pre-conditioning using ERSlpreproc, the image mode pre-conditioning program). The scenes were processed into interferograms and the coherence magnitudes of the 5-bit and 8-bit scenes compared. From the tests it was discovered that pre-conditioning the data from 5-bit, 15.5 mean to 8-bit, zero mean caused an average coherence drop of less than 0.02 in the resulting interferogram. This was considered an acceptable level of degradation considering that the study is comparative in nature and the advantages obtained by accurately modelling the expected format of the ENVISAT data. These advantages included: • The histogram of the data would have the proper characteristics of ENVISAT data, thus the algorithm would be performing on the type of data it was designed for. • The existing F B A Q simulation software could be utilized for the interferometry study. This avoided delays inherent to developing new software and ensured that interferometry results could be compared accurately to those obtained in the image mode study. • The previous DETS analysis software could be re-used, thus improving consistency between studies. 165 Appendix A: Data Pre-conditioning • The full compression ratio of 8-bits/sample to 4, 3, and 2-bits/sample could be examined, rather than only 5-bits/sample to 4, 3, and 2-bits/sample. Thus the data pre-conditioning used in the previous F B A Q study was again utilized for the interferometry study. 166 B APPENDIX B: The Impact of Ships on the ESA Wave Mode Algorithm A n extra area of investigation pursued in this study was to evaluate the effect of ships located within the vignette on the wave spectrum. This section of the study was first suggested due to concern that the range "streaks" caused by on-board range compression of data containing a ship would cause a degradation in F B A Q performance. Since the decision to remove the on-board range compression option from ENVISAT has been made, the effect of range "streaks" is no longer of concern. For non-range compressed data the point target response of a ship should be sufficiently diffuse in the SAR signal data that F B A Q performance will not be affected by it. However, in the course of evaluating the data several points concerning the importance of ships within the vignette were revealed, and are briefly presented here for the sake of interest. B.1 Theory The appearance of a ship within the boundaries of an vignette results in a small bright area of strong return. If we assume for the moment that the response caused by the presence of the ship is that of an ideal point target, the bright area can be characterized by a sine function in two 167 Appendix B: The Impact of Ships on the ESA Wave Mode Processing Algorithm dimensions. Using the E S A Wave Mode methodology, the detected vignette is first squared before the FFT is computed, and thus the input to the FFT operation will be a sine2. The Fourier spectrum of a sine function is a triangular function centered upon dc with a bandwidth inversely proportional to the width of the sine2 function. Thus, the effect of the ship wil l be to add a dc centered, wide bandwidth frequency response to the wave mode spectrum. By the law of superposition, the impact of this distortion on the usability of the wave mode spectrum is dependent upon the relative magnitudes of the ocean wave power to that of the ship. If the ship is very bright compared to the ocean background, and/or the power concentration of the wave energy small, the frequency response of the ship will far outweigh the frequency response of the ocean waves and wi l l obliterate any useful ocean information from the spectrum. If, however, the power spectrum of the ocean is very strong (i.e. from a very well defined regular wave train) and/or the ship is not strongly illuminated by the radar, the result will be an increase in the background noise of the spectrum, particularly at low frequency, but the ocean wave information will still be retrievable. It should be noted, that due to the use of the Hamming window in the ESA methodology, the position of the ship within the vignette is also crucial. If the ship appears at the edge of the vignette, its response will be muted by the tapering of the Hamming window. As the ship position moves closer to the center of the vignette however, it's impact on the wave spectrum becomes more pronounced. B.2 E x p e r i m e n t To investigate the effect of ships within an vignette, several ship scenes were evaluated from both Portugal data sets (the Newfoundland set contains no ships). Representative results taken from the second Portugal scene used in the main study are shown in Figure B - l . 168 Appendix B: The Impact of Ships on the ESA Wave Mode Processing Algorithm In the first vignette, the ship is clearly visible, but is located near the edge of the vignette. The tapering of the Hamming window thus modifies the impact of this ship on the wave spectrum, and a usable wave spectrum is produced. In the second example, the ship location has been shifted 120 pixels closer to the center of the vignette. The result is a dramatic distortion of the wave spectrum, in which the ocean wave information is completely hidden by the strong response of the ship. Figure B-2 displays the collapsed spectrums of the two vignettes. The collapsed spectrum in rows (or columns) is obtained by calculating the average value down each row (or column). Thus it serves as a way of examining the shape of the spectrum. As can be seen the frequency response of the ship when located near the center of the vignette in this particular scene is approximately two orders of magnitude stronger than the power concentration of the ocean wave spectrum. No useful wave information is distinguishable from the spectrum in this situation. The example portrayed in Figures B - l and B -2 was also observed to varying degrees in other vignettes containing ships. In all cases, the amount of distortion increased as the ship location was shifted towards the center of the vignette, though in some cases the wave spectrum was still visible despite the ship interference. The impact of this effect on ocean studies is of course dependent on the frequency in which ships appear in Wave Mode vignettes. A n in-depth investigation of the phenomena is beyond the scope of this study, but due to the potential of spectrum distortion, a ship detection scheme might be considered to warn oceanographers and meteorologists of which ocean spectrums are corrupted by ship noise. 169 Appendix B: The Impact of Ships on the ESA Wave Mode Processing Algorithm Figure B-l The Effect of Ship Location: (a) ship located near edge of vignette, (b) wave spectrum of (a); (c) ship located near center of vignette, (d) wave spectrum of (c). 170 Appendix B: The Impact of Ships on the ESA Wave Mode Processing Algorithm x 1 g'« Collapsed Columns of Imagette 1: ship near edge i 1 8 Collapsed Rows of Imagette 1 : ship near edge 100 200 300 400 fft bins 600 X <|Q" Collapsed Columns of Imagette 2 : ship near center x 1 g '9 Collapsed Rows of Imagette 2 : ship near center 300 fft bins 300 fft bins Figure B-2 Collapsed Views of the Spectra of Figure B - l . 171 C APPENDIX C: ESA Wave Product at Other FBAQ Reduction Levels As a matter of interest, and to verify trends related to encoding level, Scene 1 was briefly evaluated at 4-bit and 3-bit encoding levels. The results indicate that only very small wave spectrum degradation occurs at the 4-bit level (the default level for standard Image Mode) compared to the 8-bit data. The statistical analysis of the ESA Wave Product using 4-bit, 3-bit and 2-bit encoding is included in Table B - l . Table C-1 ESA Wave Product Bin Magnitude Error (bin magnitude values from 0-255) 4-bit 3-bit 2-bit RMS Error 0.74 1.59 2.69 Mean Absolute Error 0.34 0.74 1.36 Average Absolute Percent Error 3.1% 6.2% 11.3% 172 D APPENDIX D: ASQNR and Saturation A n interesting observation from the Wave Mode results is that ASQNR values are higher for scenes with higher levels of saturation (i.e. higher spikes at the ends of the histograms) than for unsaturated data. This was shown in Figure 13-9. Theoretically, ASQNR is a constant (ASQNR = 9.3003 for an ideal continuous Gaussian distribution and 2-bit quantization [15]) and therefore independent of the mean square value of the data. Thus, since ASQNR is the ratio of the mean square of the data divided by the mean squared error due to quantization, as the total energy of the Gaussian distribution increases the mean squared error (MSE) due to encoding must increase in proportion to it. However, in this study, ASQNR measurements are taken with respect to the 8-bit data, which may be saturated. When one places a limit on the maximum and minimum values of the data, creating a saturated distribution, the balance between the mean square of the original data and the M S E is disrupted. This effect is best understood by examining the results of a simple experiment illustrated in Figure D-1. First, a discrete integer Gaussian distribution was created with no saturation. This data was encoded/decoded using the 2-bit F B A Q algorithm. The statistics of the distribution and the mean 173 Appendix D: ASQNR and Saturation Simulated data — no saturation Simulated data — saturated at (-127,127) 600 data values Original Statistics for non-saturated data: mean square = 3018.3 RMS = 54.94 min value =-225 max value = 241 Error Statistics after Encoding/Decoding: ASQNR = 9.31 dB Total M S E = 353.88 max error =158.02 data values Original Statistics for saturated data: mean square = 2908.7 RMS = 53.93 min value = -127 max value =127 Error Statistics w.r.t. the 8-bit saturated data after Encoding/Decoding: ASQNR = 9.77 Total M S E = 306.64 max error = 45.5 -too -tn Error value vs. Original value before encoding - no saturation Total MSE = 353.88 ASQNR = 9.31 max errors 158.02. MSE=100.59 MSE=99.29 Original Value Error value vs. Original value - saturated at (-127,127) MSE=80.84 MSE = 76.23 / MSE=70.92 Total MSE = 306.64 ASQNR - 9.77 max error - 45.5 MSE=78.85 -tog o too Original Value Figure D-1 Summary of ASQNR Experiment demonstrating the effect of saturation on ASQNR measurements taken with respect to 8-bit data. 174 Appendix D: ASQNR and Saturation squared error and ASQNR due to encoding are listed Figure D-1. Then, the distribution was "made saturated" by mapping all values greater than 127 to 127 and all values less than -127 to -127. This data was also encoded using F B A Q , and again the statistics are presented in Figure D-1. Note the increase in ASQNR for the saturated data (9.77 vs. 9.31). It can be seen from the statistics that this is due to the fact that the M S E decreases substantially for the saturated data (from 353.88 to 306.64), while the mean square of the un-encoded data only decreases slightly from 3018.3 to 2908.7. To understand why the M S E has decreased for the saturated data, consider the plots of error distribution in Figure D-1. The error is distributed in 4 quadrants, created by the threshold values, and the mean squared error for each region is displayed on the plots. The most dramatic effect is shown in the outside regions where the extent of the error has been artificially limited by the saturation boundary. A decrease in M S E of approximately 20 % is evident in these areas. The central regions also have lower MSE values. This is because the RMS of the saturated distribution (which is the value used to set the threshold and reconstruction levels) is less than that of un-saturated data (53.93 vs. 54.94). Thus the error region between zero and the threshold value is smaller for the saturated distribution resulting in about a 5 % reduction in M S E in these areas. Thus, we observe that a lower MSE, and hence higher ASQNR measure, wil l result when taken with respect to saturated 8-bit data. This is not to imply that saturation of the data is a good thing — it is important to realize that the "true" ASQNR compared to the original continuous Gaussian distribution will actually be slightly lower in the presence of saturation. In the experiment of Figure D-1, for example, when ASQNR is measured with respect to the continuous input it was found to be 9.30 dB for the case of saturated input, slightly lower than the unsaturated case of 9.31 175 Appendix D: ASQNR and Saturation dB. This again reinforces the conclusion that special care should be taken in the scaling of ENVISAT data to ensure that the input power to the 8-bit quantizer falls within its optimum operating region. 176 

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