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The estimation of pack-ice motion in digital satellite imagery by matched filtering Collins, Michael John 1987

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T H E E S T I M A T I O N O F P A C K - I C E M O T I O N IN D I G I T A L S A T E L L I T E I M A G E R Y B Y M A T C H E D F I L T E R I N G by M I C H A E L J O H N C O L L I N S B.Sc.Eng. (Survey) , The Univers i ty of New Brunswick , 1981 A T H E S I S S U B M I T T E D IN P A R T I A L F U L F I L 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 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 S T U D I E S D E P A R T M E N T O F O C E A N O G R A P H Y We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A M a r c h 1987 © Michae l John Col l ins , 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I 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 Oceanography The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date April 10 1987 DE-6(3/81) Abstract This thesis addresses the problem of computationally estimating the motion of pack ice in sequential digital satellite images. The problem is posed in terms of linear filter theory and is solved by minimizing the error variance. The intuitive use of cross correlation and edge detection are shown to flow naturally from this approach. The theoretical framework also allows a geometric intuition into the action of the filter which is not possible through ad hoc methods. The noise corrupting the filtering process is investigated and the filter is implemented through both a first order method common to image processing, and a more sophisticated second order approach from computational vision. The class of imagery for which the filtering system is appropriate is discussed and the images chosen for the experiments are shown to be representative of this class. The experimental results reveal the power of the system in estimating ice motion, and some analysis of the derived motion is performed by comparison to a simple theory of wind-driven ice motion. The failings of the system are discussed and improvements are suggested. ii Contents Abstract ii List of Tables iv List of Figures v Acknowledgement vi 1 Introduction 1 1.1 Arc t i c Sea Ice 1 1.1.1 Phys ica l Extent 1 1.1.2 Dr i f t Character ist ics 3 1.1.3 Genera l Character ist ics of Ice Dynamics 6 1.1.4 T h e Role of Remote Sensing 7 1.2 Out l ine of The Research 9 1.2.1 Prev ious Work 9 1.2.2 Organizat ion of the Thesis 10 2 Theory 12 2.1 Statement of the Prob lem 12 2.2 Der ivat ion of the O p t i m u m F i l te r 13 2.3 Subopt ima l Match ing 20 2.4 Summary 22 3 Implementation 23 3.1 Questions of Noise 23 3.1.1 A Noise Mode l 24 3.2 The Whi ten ing F i l te r 27 3.2.1 The F i rs t Order F i l te r 31 3.2.2 The Second Order F i l te r 32 3.3 The Ma tch ing F i l te r 35 3.3.1 O p t i m u m Window Size 37 3.3.2 E r ro r Correct ion 38 i i i 4 Experiments 41 4.1 A V H R R data 41 4.1.1 Results 45 4.2 S A R data 54 4.2.1 Results 56 5 Discussion 59 5.1 E s t imat ion versus Detection 59 5.2 Confidence Estimates 60 5.3 Computat iona l Efficiency 61 5.4 System Evaluat ion and Future Research 62 B ibliography 65 A Review of Two Important Noise Procosse-s 80 A . l Gaussian Stationary White Noise 80 A.2 Gauss ian Stationary Markov Noise 84 B A V H R R Graphic Results 86 C S A R Graphic Results 129 iv List of Tables 4.1 The channel band widths of the A VH on the NO A A satellites used in this study 42 4.2 A V H R R images used in this study 47 B . l comparison between the u a and u m measurements for image pair 1 (June 05 - 06 1984), where e is the magnitude of the error vector and a is its angle 99 B.2 comparison between the u „ and u m measurements for image pair 1 (June 06 - 07 1984), where e is the magnitude of the error vector and a is its angle 104 B.3 comparison between the u a and u.,n measurements for image pair 3 (June 11 - 12 1984) 109 B.4 comparison between the ua and u.,n. measurements for image pair 4 (June 12 - 13 1984) 114 B.5 comparison between the u n and um measurements for image pair 5 (July 30 - 31 1985) 119 B.6 comparison between the u a and u m measurements for image pair 6 ( A u -gust 06 - 07 1985) 124 v List of Figures 1.1 Arctic sea ice extent in February (top) and September (bottom) based on 75% concentration limits from Hibler (1980) 3 1.2 300-month time series of the depature from the monthly means of the area covered by arctic sea ice, showing unsmoothed values (a) and 24-month running means (b). from Walsh and Johnson (1971) 4 1.3 Ice drift in the Arctic ocean. The arrows indicate the average direction of ice movement in each region. Several locations are marked by letters: A-The New Siberian Islands; B-the Bering Strait; C-Franz Joseph Land; D-Spitzbergen and P-North Pole, from Pounder (1965) 5 3.1 Autocorrelation functions Rn for AVHRR images (top) and SAR images (bottom) 26 3.2 A small portion of rev. 1439 showing some significant image features (top). Profiles showing brightness gradient from image and operator responses to the image features (bottom) 30 3.3 Mask used for a generic 3 x 3 operator 32 3.4 One dimensional slice through the (a) impulse response and (b) fre-quency response of the V2g filter 35 4.1 Spectral albedos over snow and melt ponds: (a) dry snow (b)wet new snow, (c) melting old snow, (d) partially refrozen melt pond, (e) early season melt pond (10 cm deep), (f) mature melt pond (10 cm deep) on MY ice, (g) melt pond on FY ice (5 cm deep), (h) old melt pond (30 cm deep) on MY ice. Grenfell and Maykut (1977) 43 4.2 Spectral albedos over bare sea ice: (a) frozen MY white ice, (b)melting MY white ice, (c) melting F Y white ice, (d) melting F Y blue ice. Grenfell and Maykut (1977) 44 4.3 If the wind stress r 0 balances TW + C, then a larger stress r'a turned to the right of ra is required to balance Tw + C + V-o\ Thorndike and Colony (1982) 49 B.l AVHRR band 1 - June 05 1984 87 B.2 AVHRR band 1 - June 06 1984 88 vi B.3 A V H R R band 1 - June 07 1984 89 B.4 A V H R R band 1 - June 11 1984 90 B.5 A V H R R band 1 - June 12 1984 91 B.6 A V H R R band 1 - June 13 1984 92 B.7 A V H R R band 1 - July 30 1985 93 B.8 A V H R R band 1 - July 31 1985 94 B.9 A V H R R band 1 - August 06 .1.985 95 B.10 A V H R R band 1 - August 07 1.985 96 B . l l Automatically measured displacements from. June 05-06 1984 97 B.12 Manually measured displacements from June 05-06 1984 98 B.13 Displacements predicted from theory for June 05-06 1984 along with the 24 hour averaged isobars for same period. . 100 B.14 Residual displacements (see text) for June 05-06 1984 along with the 24 hour averaged isobars for same period 101 B.15 Automatically measured displacements from June 06-07 1984 102 B.16 Manually measured displacements from June 06-07 1984 103 B.17 Displacements predicted from theory for June 06-07 1984 along with the 24 hour averaged isobars for same period 105 B.18 Residual displacements (see text) for June 06-07 1984 along with the 24 hour averaged isobars for same period 106 B.19 Automatically measured displacements from June 11-12 1984 107 B.20 Manually measured displacements from June 11-12 1984 108 B.21 24 hour averaged isobars for June. J 1-12 1984 I l l B.22 Automatically measured displacements from June 12-13 1984 112 B.23 Manually measured displacements from June 12-13 1984 113 B.24 Displacements predicted from theory for June 12-13 1984 along with the 24 hour averaged isobars for the same period 115 B.25 Residual displacements (see text) for June 12-13 1984 along with the 24 hour averaged isobars for the same period 116 B.26 Automatically measured displacements from July 30-31 1985 117 B.27 Manually measured displacements from July 30-31 1985 118 B.28 Displacements predicted from theory for July 30-31 1985 along with the 24 hour averaged isobars for the same period 120 B.29 Residual displacements (see text) for July 30-31 1985 along with the 24 hour averaged isobars for the same period 121 B.30 Automatically measured displacements from August 06-07 1985 122 B.31 Manually measured displacements from August 06-07 1985 123 B.32 Displacements predicted from theory for August 06-07 1985 along with the 24 hour averaged isobars for the same period 125 B.33 Residual displacements (so;e text) for July 30-31 1985 along with the 24 hour averaged isobars for the same period 126 vii B.34 Sample cross-correlation ['unction for typical window for the June 11-12 1984 127 B . 35 Sample cross-correlation function for typical window for the June 11-12 1984 128 C l S E A S A T S A R - revolut ion 1439 - October 5 1978 130 C . 2 S E A S A T S A R - revolut ion .1.482 - October 8 1978 131 C.3 Locat ion of S E A S A T S A R image pair used in this study 132 C.4 Manua l l y measured ice displacements computed f rom S A R pair (cour-tesy of D .A . Rothrock, Universi ty of Washington) 132 C.5 Manua l measurements of ice displacement, (a) shows displacements as vectors, (b) shows displacements as a deforming grid (see text) , square denotes outl ine of undeformed grid 133 C.6 Convolut ion of Sobel operator with rev. .1.439 134 C.7 Convolu t ion of Sobel operator with rev. 1482 135 C.8 Convolut ion of V 2 g operator wi th rev. 1439 136 C.9 Zero crossings from rev. 1439 137 C I O Convolu t ion of V 2g operator wi th rev. 1482 138 C . l l Zero crossings f rom rev. 1482 139 C.12 Automat ica l l y measured displacements from first order operator exper-iment 140 C.13 Automat ica l l y measured displacements from second order operator ex-periment 141 C.14 Errors in the first order experiment 142 C.15 Errors in the second order experiment 143 C.16 Detai l 1 from the lower left corner ol rev. 1439 (top) and rev. 1482 (bottom) 144 C.17 Deta i l 2 f rom the upper right corner of rev. 1439 (a) and rev. 1482 (b). 145 vi 11 Acknowledgements I would l ike to acknowledge the support of my superviser, B i l l Emery, whose un-flagging enthusiasm for this project was essential to its f rui t ion. R o n Ninn is gave me the idea to pursue this topic and also provided me wi th a great deal of support dur ing my first year of study. Peter Lawrence and J i m Riemer of the Dept . of Electr ical Engineer ing and Par is Vachon of Oceanography were always ready to discuss the nu-ances of digi ta l filtering. B o b Woodham introduced me to computat ional v is ion and stressed the importance of the theoretical approach which has, in no smal l way, in -spired my work. Drew Rothrock of the Univers i ty of Washington introduced me to the communi ty of ice mot ion researchers and our discussions were invaluable. The research described in this thesis was performed at the Laboratory for Computa t iona l V is ion in the Dept . of Computer Science and the Satell i te Oceanography Laboratory in the Dept . of Oceanography and I thank D a n Razzel l of the former and Denis Laplante and P a u l Nowlan of the latter for their assistance w i th many technical details. D o n Hodgins and D o n Dunbar of Seaconsult L t d . helped to obtain and process the data necessary for the product ion of the geostrophic wind field. I must also acknowledge the tireless efforts of M a r k M a j k a wi thout whom this thesis would st i l l be a series of TfjXerror messages. M y parents were always support ive of my return to school. F ina l ly , wi thout the love and fr iendship of Ramona I would be adrift. ix Chapter 1 Introduction 1.1 Arctic Sea Ice A remote frozen blanket of sea ice covers approximately 10 % of the northern hemi-sphere and 13 % of the southern hemisphere and modifies their fundamental character accordingly. It displays dramatic spatial and temporal variability and yet it is only in the last twenty years that a concerted effort by oceanographers, glaciologists and material scientists has lead to a quantum leap forward in our understanding of the behavior and properties of this inscrutable material. The harsh environment of the polar and sub-polar regions severely restrict the spatial and temporal coverage which may be obtained through surface-based sea-ice experimental programmes. Yet it remains a scientific priority to monitor these regions due to their profound effect on global weather systems, on oceans and on climate. The advent of satellite-borne remote sensing has provided a solution to this problem and a time-series of the ice data may be built up with a variety of sensors. 1.1.1 Physical Extent The maximum amount of ice covering the Arctic has been estimated to be 15 x 106 km 2 (Walsh and Johnson, 1979; Nazarov, 1963) with a seasonal variability of 20-25 % (maximum in February, minimum in August) as shown in figure 1.1. An interesting feature of this figure is the presence of ice year-round off the northeastern coast of 1 CHAPTER 1. INTRODUCTION 2 Greenland. Th is is due to its being the main channel of ice flow out of the A rc t i c basin. T h e extent of the ice may vary substantial ly f rom year to year as can be seen in figure 1.2. The general trend is that the ice extent has increased over the last twenty years which appears to agree w i th surface air temperatures ( Walsh, 1970). The extent of ice on the nor th slope of A laska was investigated by Barnett (1979), who produced a time-series of ice severity, a stat ist ical ly derived quantity. Wh i le he attempted no explanat ion of the var iabi l i ty of this index, he noted a strong correlat ion between the severity and the atmospheric pressure changes over the durat ion of his study. Rogers (1978) studied the interannual var iabi l i ty of ice in the Beaufort Sea and found that the air temperature, in the form of thawing degree days, was the parameter wh ich correlated most highly w i th the summert ime ice margin. CHAPTER 1. INTRODUCTION 3 F igure 1.1: A r c t i c sea ice extent in February (top) and September (bottom) based on 75% concentrat ion l imits f rom Hibler (1980). 1.1.2 Drift Characteristics The general drift of ice in the Arct ic basin is well known and the circulat ion consists of a central gyre and a transpolar drift stream, as shown in figure 1.3. The recirculating nature of the Beaufort gyre makes it possible for ice to survive the summer melt and stay in the basin for many years which explains why some of the thickest ice is found in this region. Dr i f t rates in the Arc t ic can vary over days, months and years, but a representative figure could be taken from, for instance, Dunbar and Wittman (1962) - about 6 km/day . Th is figure falls between extreme values of 20 k m in one day and many weeks of no CHAPTER 1. INTRODUCTION 4 Figure 1.2: 300-month time series of the depature from the monthly means of the area covered by arctic sea ice, showing unsmoothed values (a) and 24-month running means (b). from Walsh and Johnson (1971). CHAPTER 1. INTRODUCTION 5 Figure 1.3: Ice drift in the Arc t ic ocean. The arrows indicate the average direction of ice movement in each region. Several locations are marked by letters: A - T h e New Siber ian Islands; B-the Ber ing Strai t ; C-Franz Joseph Land ; D-Spi tzbergen and P-Nor th Pole, f rom Pounder (1965). CHAPTER 1. INTRODUCTION 6 mot ion (Thorndike and Colony, 1979). Dunbar and W i t t m a n have observed that it took ten years for an ice is land to make a complete circuit of the Beaufort gyre. Whi le the drift of ice stations gives us general drift characteristices, more complete information on ice drift has been gained f rom the U.S. Navy navigat ion satellites (Thorndike and Colony, 1979), and laser surveying equipment (Hibler, 1980). These methods show that velocity variat ions are greatest at low frequencies (0.2 d a y - 1 ) , but are also stat ist ical ly significant at 2.0 d a y - 1 . Th is high frequency var iabi l i ty is more pronounced in the summer and has been explained by McPhee (1978) as inert ial oscil lations result ing f rom a balance between the inert ia of the ice and the Cor io l is force. These 12 hour oscil lations are prominent in a typical seasonal velocity spectrum f rom the summer (Thorndike and Colony, 1979). 1.1.3 General Characteristics of Ice Dynamics Recent research i n sea ice dynamics has substant ial ly improved our understanding of the nature of ice drift . Sea ice, that is, ice of sea or igin (pack ice) rather than land origin (icebergs and ice islands), forms a floating three-dimensional cont inuum that moves in response to stresses exerted on it by the wind f rom above and the sea f rom below. Th is cont inuum is also able to sustain substant ial internal stresses which result f rom external forcing. It is clear that to understand the physics of ice drift and deformation we must first understand the way ice interacts w i th its environment. The study of ice interact ion in the central ice pack was the pr imary objective of the Arc t i c Ice Dynamics Jo in t Exper iment ( A I D J E X ) (Maykut, Thorndike and Untersteiner, 1972; Untersteiner, 1978) whi le the Marg ina l Ice Zone Exper iment ( M I Z E X ) is focused on the complex dynamics at the various ice edges or marginal ice zones. Sea ice dynamics deals w i th the way momentum is transferred through the sea-ice-atmosphere system. It is generally agreed that the large scale dynamics of this complex system can be characterized by the fol lowing elements (Hibler, 1984, 1980): 1. A momentum balance describing the ice drift , inc luding: air and ocean stresses; CHAPTER 1. INTRODUCTION 7 Corio l is force; internal ice stress; inert ial forces and ocean current effects. 2. A n ice rheology wh ich relates the stress experienced by the ice to the deformation and strength of the ice. 3. A n ice strength which can be found in studies of the mechanical properties of the ice (Mellor, 1988; Michel, 1978), but for model l ing purposes is often a funct ion of the so-called ice thickness distr ibut ion (Hibler, 1979). 1.1.4 The Role of Remote Sensing Satell i te borne remote sensing has made major contr ibut ions to our knowledge of the sea ice cover of the polar oceans and is now a standard operat ional tool in both polar regions. The sensors current ly in use al l make unique contr ibut ions to our knowledge of the ice and conversely they al l have drawbacks which l imit their usefulness. A n y future operat ional system should make use of some combinat ion of two or more satell ite data. The T I R O S - N series of operat ional weather satellites operated by the Nat ional Oceanic and Atmospher ic Admin is t ra t ion ( N O A A ) use an Advanced Very H igh Res-olut ion Radiometer ( A V H R R ) which operates in the visible and Infrared (IR) bands. Th is sensor has proven to be extemely useful for studying the ice over scales of hundreds of ki lometers. A t present there are two such satellites in orbit and images of the arctic may be collected several times per day. Wh i le the pixel resolution of approximately 1.5 k m excludes studies of smaller scale dynamics, the imagery affords a synoptic picture of a large region. Th is imagery combined w i th some geophysical forcing data provides an excellent research tool . The major drawback of these sensors is that both the visible and IR bands w i l l not see through clouds and the former is also restricted to daylight use. These restrict ions are severe especially in the arctic which is often cloudy and has l i t t le or no l ight f rom November to M a r c h . In spite of these restr ict ions, however, the Satel l i te Oceanography Laboratory at U . B . C . has a total of 104 separate image pairs that it has collected since it began receiving arctic data in late 1983. One sequence CHAPTER 1. INTRODUCTION 8 of images August 1985 covers 10 consecutive days in 10 images, and has been used to study the year ly ice-arching phenomenon in Amundsen Gu l f (Collins, 1986). Th is strongly suggests that this type of sensor is essential to arct ic research. Microwave frequencies may also be sensed passively and the passive microwave sensors are enjoying increasing populari ty. The Electr ica l ly Scanning Microwave R a -diometer ( E S M R ) on board N A S A ' s Nimbus-5 satell ite and the Scanning Mul t i f re-quency Microwave Radiometer ( S M M R ) on board the Nimbus-7 have extremely large coverage and are not affected by clouds or darkness. They have proven very useful for mapping ice concentrations in a single scene and variations in the extent of sea ice over both polar regions over several scenes. The very large footprint, or ground resolution of the sensor (~ 25km) has confined its usage to global studies. The sensor wh ich has been touted by many as having the greatest potent ial for sea ice research is imaging radar, especially Synthet ic Aper ture Radar ( S A R ) . Features w i th in a S A R image appear bright to the extent that they are strong scatterers of the pulses of microwave energy emit ted by the S A R system. Sea ice features are characterized by a wide range of scattering coefficients so that there is usual ly good contrast between mult iyear ice (ice which has survived at least one summer) , first year ice (which has not yet been through a summer), new ice (ice wh ich is less than 10 c m th ick) , ridges and rubble fields (piles of ice blocks and fragments result ing f rom convergence or shear between ice floes) and open water. The sensit ivity of S A R to variat ions in the ice/water surface scattering combined w i th it 's fine spatial resolution makes it very useful for ice type characterization as well as for studies of ice kinematics and dynamics. The S A R has the added advantage of being an all-weather all-season system. A part icular disadvantage of the system is that it has proven useful only in smal l regions since it lacks the useful synoptic scale of the N O A A satellites. The fine spat ial resolut ion scales of the S A R have rarely been matched by observations of forcing phenomena, mak ing it of l imi ted use in smal l scale ice dynamics studies. The resolution of the S A R has, however, made it the most useful data set for t racking ice mot ion in CHAPTER 1. INTRODUCTION 9 succesive images. 1.2 Outline of The Research The study of sea ice dynamics has been greatly enhanced by remote sensing. Ea r l y sea ice studies involv ing remote sensing used photographic images only (Nye and Thomas, 1974; Nye 1975a, 1975b), which involved extemely t ime-consuming photogrammetr ic techniques in order to y ie ld quant i tat ive results. Recent studies have al l used digital imagery wh ich lends itself very well to purely computat ional analysis. Th is thesis seeks to construct a theoretical framework w i th in wh ich a purely computat ional pack-ice mot ion est imat ion system may be developed. Th is framework is the well-developed theory of matched filtering. The mot ion est imation problem is first posed as a filtering problem which is then solved by opt imiz ing a part icular filter performance characterist ic common in est imat ion problems. The derived filters are equivalent to intui t ive and thus far, ad hoc, techniques. The filtering system is tested on several A V H R R image pairs and one S A R image pair. 1.2.1 Previous Work The problem of est imat ing sea ice mot ion f rom remotely sensed imagery has t radi t ion-al ly been solved v ia manua l , hence very t ime-consuming, procedures. 1 Th is manual method was first appl ied to S A R imagery by Hall and Rothrock (1981). The approach taken was to match recognizable features in the ice field which was imaged f rom two successive orbits. The S A R data used was opt ical ly processed and hence was subject to resolut ion degradation and geometric distort ions incurred dur ing the processing. The displacement errors result ing f rom the use of this imagery were reduced by Leberl, Raggam, Elachi and Campbell (1983) by using radargrammetr ic methods to improve posit ional accuracy. Th i s latter study, however, st i l l relied on the manual t racking of 1Hibler, Tucker and Weeks, 1975; Crowder, McKim, Ackley, Hibler and Anderson, 1974; Shapiro and Burns, 1975 CHAPTER 1. INTRODUCTION 10 ice features. In fact the authors went so far as to say that as more accurate data became available, it was the operators ability to recognize ice features that would be the lim-iting factor on measurement accuracy. This assertion has been ignored by subsequent attempts to automate the procedure yet as we shall see is very true. A subsequent paper by Curlander, Holt and Hussey (1985) used digitally corre-lated SAR data which substantially reduced radiometric and geometric errors. The measurement of ice displacements was facilitated by the use of digital data; however, the system still relied completely on the visual identification of ice features by an operator, and despite the claim of limited operator intervention, the displacements be-tween the images used in the present study required ten hours of operator time. Fily and Rothrock (1985,1986) completely automated the measurement procedure by using cross correlation which they borrowed from the cloud tracking work of Leese and his co-workers (Leese and Epstein, 1963; Leese, Novack and Clark, 1971, 1972; Phillips and Smith, 1972). However, while Leese et al computed their cross correlations using fast Fourier transforms (FFT), Fily and Rothrock computed them directly. They used a coarse-to-fine strategy to successively increase the accuracy of the displacement mea-surements while drastically reducing the computations. This approach was also adopted by Vesecky, Samadami, Daida, Smith and Bracewell, 1986a, 1986b, 1987. They have also investigated the effects of rotation and shear on the area correlation method and have successfuly applied feature matching techniques to SEASAT SAR images. Some of the results from this thesis have already been published in the first and, to date, the only published ice motion study using A V H R R imagery (Ninnis, Emery and Collins, 1986). 1.2.2 Organization of the Thesis Chapter 2 poses the motion estimation problem as a filtering problem and derives the optimal filter by solving an appropriate optimization problem. This chapter also gives a brief review of some other sub-optimal matching schemes and uses the theory CHAPTER 1. INTRODUCTION 11 of abstract funct ion spaces to provide some geometric in tu i t ion into the act ion of the opt imal and sub-opt imal filters and defines a simple relat ionship between them. Chapter 3 deals w i th the implementat ion of the derived filter. It refers to a disussion of the relevant features of white vs. Markov noise included in appendix A and proposes a noise model to be used in the study. Based on the previous noise discussion the noise model is shown to agree w i th the data. The noise model used is implemented as a two-stage filter for the S A R data where the first stage is an edge enhancing prefilter. A n analyt ic form for this prefilter is derived and two implementat ions of it are discussed in 3.2. The chapter closes w i th a discussion of the implementat ion of the matching filter. Chapter 4 reports the results of the experiments and is div ided into a discussion of the A V H R R results and of the S A R results. The S A R results are compared to a manual ly measured gr id of displacements and the results are analysed. The A V H R R results are compared to a some scattered manual ly measured displacements and the ice mot ion derived f rom the geostrophic w ind. These results are analysed to reveal some important properties of the mot ion of the ice. Chapter 5 addresses some issues raised by the research, reviews the general results and suggests directions for future work. Chapter 2 Theory 2.1 Statement of the Problem The prob lem wh ich we wish to solve is the automated measurement of the field of mot ion of A rc t i c pack ice in sequential satellite images. We demand that the automated system which is to be implemented is an opt imal solut ion to an appropriate est imat ion problem. We begin w i th two discrete image functions m'(x, y), imaged at t ime to, and f(x,y), imaged at t ime ti, of finite domain, whose range is [0,255]. The images are seperated in t ime by an amount, At = t i — t0, that wi l l allow al l significant changes to be modeled as translat ions. Th is is a constraint both on the t ime interval between images and on the image pai r themselves since many image pairs wi l l be unsuitable for this technique due to smal l scale rotations or a large amount of deformation. F i rs t we focus on a part icular area in the ice pack by operat ing on m'(x, y) w i th a boxcar funct ion wh ich is zero outside some region and uni ty inside. Th is operat ion yields a subimage, m(x,y), which is reduced in spat ial extent. We also assume that, while somewhere w i th in the funct ion f'(x,y) there is an instance of m(x,y), the former funct ion is corrupted w i th addi t ive stat ionary Gauss ian noise, n(x,y). We may narrow the region of / ' in which we are interested by operat ing on this funct ion w i th a second window funct ion. Th is subimage is denoted / . Hence we have the fol lowing two signals of interest: • subimage 1 m(x,y) imaged at t ime to 12 CHAPTER 2. THEORY 13 • subimage 2 f(x,y) = m(x + x0,y + yo) + n(x,y) imaged at t ime ti where (xo,yo) is the displacement of the ice w i th in / ' over the t ime interval, At. A n important assumption which we use is that the mot ion of the ice w i th in / ' is homoge-nous, ie. that the ice is moving at the same speed and direct ion. A s shal l presently be seen this is equivalent to imposing a large signal-to-noise rat io ( S N R ) . The prob lem to be solved is: find the filter funct ion, h(x,y), such that when it is convolved w i th f(x,y) the max imum value of the output wi l l y ie ld the best estimate of the displacement of m(x,y). Fur ther , we wish the estimate to be optimal, that is we wish to min imize the error, e, between the true displacement, (as, y), and the estimated displacement, (x, y). The most commonly used performance cr i ter ia i n est imation problems is the mean-squared error, E(e2). It is assumed that the error, which is a random var iable, has zero mean, E(e) = 0, hence the mean square error is the variance of e. T h e matched fi lter is a general result and no restictions are placed on the colour, or frequency d is t r ibut ion, of the noise. In the case of white noise the filter is simply a correlator. Due to the l ineari ty of the filter the coloured noise result may be broken into two filters: the first whitens the noise and the second is a correlator. T h e matched filter is certainly not the only solut ion used in detecting image signals of known form. Another body of subopt imal matching techniques, each based on a part icular measure of match or mismatch, are briefly reviewed. T h e section closes w i th a brief summary on the key features of the theory to be carr ied forward into the implementat ion. 2.2 Derivation of the Optimum Filter A t the filter input there may appear a combinat ion of the signal of known form m(x, y) and the noise n(x, y) wh ich is a stat ionary random process. Hence at the filter input CHAPTER 2. THEORY 14 we have: f(x, y) = m(x, y) + n(x, y) (2.1) This signal is passed through a linear filter U with transfer function H(u, v), where u, v are spatial frequencies or wavenumbers. At the output of H we have: <p(x, y) = //(x, y) + u(x, y) (2.2) where /x(x, y) is the result of passing the useful signal m(x,y) through )i, and u(x, y) is the result of passing the noise n(x,y) through M. We shall now derive the optimal signal parameter estimation filter which to a large extent follows McGillem and Svedlow (1977). The derivation begins by expanding fi(x, y) in a second-order Taylor series about the true lag position (x, y) H{x,y) ~ /x(x,y) +px{x,y)[x - x] + (ty{x,y)[y - y] (2.3) +fixv{x,y)[x - x][y -y} + \fxxx{x,y)[x - x]2 where 5/z(x,y) w>y) = —Q-.— (2.4) x-i,y-y Here we have assumed that (x — x) and (y — y) are small enough so that all higher order terms may be neglected. The necessary condition for a maximum at (x, y) is d»{x,y) = dp{x, y) = Q dx dy substituting (2.3) and (2.5) into (2.2) we obtain <p(x,y) = fi{x,y) + fix{x,y)[x - x] + fiy(x,y)[y -y} + vxy{x,y)\x - x][y - y) (2.6) +^xx{x,y)[x - x]2 + ^ t f y (z ,y)[y - y]2 + v{x,y) The filter that we seek will generate a maximum at the estimated lag position (x, y), hence we shall introduce the necessary condition for a maximum at this point d<p(* ,y^ = fixy{x,y)[y - y] + fJ.xx{x,y)[x - x] + i/x(x,y) = 0 d (x )  (2'7) ^ a ' = »xV{x,y)[x - x] + fiyy{x,y)[y - y] + vy{x,y) = 0 ay CHAPTER 2. THEORY 15 if we arrange these equations in terms of (x — x) and (y — y), we can write the error in the lag as follows (x — x) = ^"xyUv ^vyUx (y-y) = * x v U x ^ y l^xxl^yy fJ-xy where the arguments (x, y) and (x, y) have been left out for compactness of notation. The variance of the error may be found by taking the mathematical expectation of (x — x)2 and (y — y)2 where it is assumed that E[x — x] = E[y — y] = 0 /g. _ ~p t4ytf - IVxyrlyvWy [HxxVyy ~ »ly\2 VlyVl - 1ilxy»xxP~Vy~ + vLtf (2.9) [Vxxrlyy ~ fily? These relationships are greatly simplified if we assume fj,xy = 0. In this case the variance expressions become 1/2 Pxx (2.10) (&-y) 2 = - f McGillem and Svedlow (1977) have suggested that a reversible linear transformation may be applied to the (x,y) coordinate system so that in the new coordinate system ((xi,yi),say), the term nxy{xi,yi) = 0 this new coordinate system and the reverse transformation can be applied to return to the original space. Hence there is no loss of generality in assuming fj,xy = 0. We may rewrite (2.10) in their equivalent integral forms (x — x)5 //fJh(a^)H x^y)Rxx{(x - x,/? - y) dad/3 dxdy \jjh(a,j3)mxx(x - a,y - 0) dad/3 / / / / h(a,(3)h(x,y)Ryy(a - x,(3-y) dad/3 dxdy (y - y)2 = J J J J r , , T-J (2.11) jyJh(a,f3)mVy(x - a,y - /3) dad/3 CHAPTER 2. THEORY 16 where R(x, y) is the autocorrelat ion funct ion of the input noise n(x, y), the subscripts denote par t ia l differentiation w i th respect to the corresponding variables. G iven these expressions for the variance one would l ike to find a filter, h(x, y), that minimizes (x — x)2 and (y — y ) 2 . The derivat ion may be stated in the fol lowing equivalent form (Luenberger, 1969). minimize : 1(h) = I h(a,(3)h(x,y)Rxx(a — x, j3 — y) dad/3dxdy rf a a ] 2 (2-12) subject to : J(h) = \ I h(a, j3)mxx(x — a,y — /?) dadf3 — K\ where K\ is a constant and 1(h) and J(h) are functionals of the filter h(x,y). Th is is a constrained min imizat ion problem and is refered to as the isoperimetric problem in the calculus of variat ions (Weinstock, 1974). T h e method of solut ion is to form an augmented funct ional , which is the sum of 1(h) and a parametr ic constant, A, times J(h) and then minimize the augmented funct ional. T h e appropriate funct ional in our case is L(h) = 1(h) - XiJ(h) (2.13) = IJjJ[h(a,l3)h(x,y)Rxx(a-x,l3-y) - A i / i ( a , P)h(x,y)mxx(x - a,y - (3)mxx(x - x,y - y)\dad/3 dxdy to simpl i fy the notat ion let us make the fol lowing subst i tut ion Q(x,y,a,(3) = Rxx(a - x,/3 -y) - \xmxx(x - a,y - (3)mxx(x - x,y - y) (2.14) then we may rewrite the augmented funct ional as L(h) = HIjh(a,/3)h(x,y)Q(x,y,a,/3) dad(3dxdy (2.15) The necessary condi t ion for L(h) to be a m in imum is that the variation of the aug-mented funct ional be zero (ie. that L is stat ionary). Hence 6L(h) = HI![6h(a, {3)h(x, y) + h(a, (3)Sh(x, y)]Q(x, y, a, /3) da d(3 dx dy (2.16) = 2//{/ Jh(a>P)Q-(a>P>x>y) dadp}6h(x,y) dxdy = 0 CHAPTER 2. THEORY 17 the latter result is due to the symmetry of Q w i th respect to a , /? and x,y. If (2.16) is equal to zero then J Jh{a,(3)Q{ct,0,x,y)dad(3 = 0 (2.17) by expanding Q in (2.17) we have j[Rxx(x - a,y - /3)h{a,/3) dad0 = X^m^x -x,y-y) (2.18) Th is can be solved by taking the Four ier Transform of bo th sides: fjf JRxx(x - a , y - /3)h[a,/3)e-2^ux+v^ dxdy = Jjmxx(x-x,y~ y^-K^+w) dxdy the frequency domain equivalent of this is - 4n2u2Sn(u,v)H{u,v) = -4Tr2u2\1K1M*{u,v)e- j2^u'x+v'^ (2.20) solving this for the filter transfer funct ion H(u,v) we obtain H{u, v) = XlK1e- i2'^+^ ™^U,V) (2.21) Sn(u,v) which is the defini t ion of the matched filter derived earlier mut ip l ied by an arbi t rary constant factor X\Ki. Hence the filter which minimizes the error variance along the (2.19) x direct ion is the matched filter. The min imizat ion of (t/ — y ) 2 follows the same path where we are min imiz ing (from 2.11) lA = J!JJh{a, (3)h{x,y)Ryv{a - x, p - y) dad/3 dxdy (2.22) subject to t4v = [ / / H a , /3)myy(~x - a,~y - (3) da d/3] = K2 (2.23) The min imizat ion problem yields the fol lowing solut ion for H(u,v) H(u,v) = X2K2  M ^ : J . e (2.24) Sn[u,v) CHAPTER 2. THEORY 18 This is the transfer function of the Matched Filter which simultaneously minimizes the error variance in both the x and y directions. The physical meaning of (2.24) is quite simple, the larger the amplitude spectrum of the useful signal and the smaller the power spectrum of the noise in the frequency interval (u, v) —* (u + du, v + dv) the more the optimum filter H passes the frequencies in this interval. If the noise is white, that is its power spectral density is constant Sn{u,v) — So = constant This means the noise spectrum is evenly distributed over all frequencies. The optimum filter corresponding to the white noise case called the Matched Filter, (although this name is often applied to the coloured noise case as well). The transfer function is given by H (u, v) = e- j(UXo+VVo)M*{u, v) (2.25) hence H(u, v) is completely determined by the form of the signal m(x,y) to which it is matched. Equation (2.25) is obtained by making the substitution c = So in equation (2.24). The impulse response of the matched filter defined by (2.25) is h{x,y) = 1-j Je j{ux'vy)H{u, v) du dv (2.26) = _Lyy gi[«(*-*o),»(v-yo)iM(_Uj _ „ ) d u d v or h(x,y) = m(x - x0,y - yQ) (2.27) hence the action of the matched filter is described by the following /i(x, y) = j j f{x', y')m{x' -x + x0,y'-y + y0) dx' dy' (2.28) If we consider (2.1) the matched filter forms the cross-correlation between the useful signal m(x, y) and the input function f(x, y). It happens that the signals we are dealing CHAPTER 2. THEORY 19 with have finite domain, thus m is zero outside some bounded region A. We also assume for the moment that m is small compared to / , although it will become clear later that this is a necessary condition for the correct calculation of the correlation. We thus compute (in abbreviated notation) / / mf for all spatial lags of m with respect to / . By the Schwartz inequality we have m{x,y)f(x + a,y + 0) da d/3 . . , A , , < 1 (2.29) VJ J A m 2 ( x , y )  d x d y J J Af2(x>y) dxdy This is the normalized cross correlation which is the form of the matched filter we shall use. It should be noted that while / / m2 is constant, / / f2 changes with each lag position and must be recomputed. A further normalization is accomplished by removing the mean value from each image signal before (2.29) is computed. This makes the (2.29) a normalized cross covariance function which in statistical parlance is the Pearson product moment correlation(Brownlee, 1960). The normalization of the correlation function makes the matching measure more robust against bright spots in / . These bright spots heavily influence the correlation by dramatically increasing the sum of products in a small area. However, while the normalization can reduce these effects it cannot remove them completely. A better understanding of the mismatch error variance can be obtained by substi-tuting the filter transfer function into our original expressions for the error variance (2.11) and transforming to the frequency domain. [x — x) = {y - yf 6v2p + 6v*p 6v2p (2.30) CHAPTER 2. THEORY 20 where 6u 11 S„(u,v) II \M(u,v)\2 du dv -1/2 Sn(u,v) — effective signal bandwidth of the input signal i n the x direct ion Sv — 47T \M(u,v)\'' Sn{u,v) ,-1/2 (2.31) du dv II |Af(u,t,)|- du dv Sn(u,v) = effective signal bandwidth of the input signal in the y direct ion and | M ( u , V ) | 5 du dv (2.32) Sn(u,v) = output signal-to-noise rat io for correlated noise Here the variance is expressed in terms of the S N R and a quant i ty known as the effective or mean squared bandwidth. We may simpl i fy (2.30) i f we assume that fixy is zero. Th i s assumption would apply if the spectral shape of the noise and the useful signal was s imi lar and could be modeled as differing by only a constant (McGillem and Svedlow, 1976). The variance expressions may then be wr i t ten: (x — x ) 2 " and (y - y) = (2.33) 8u2p 6v2p Hence for a given S N R there is an inverse relationship between bandwidth and variance. Th is is a reaff irmation of the classic tradeoff between the spatial extent and bandwidth of a signal. 2.3 Suboptimal Matching Whi le the matched filter is the opt imal matching system it is not wi thout i t 's com-peti tors. In fact there are many ways of measuring the degree of match or mismatch between two image functions m and / over some region A. The most popular measures (Rosenfeld and Kak, 1982; Ballard and Brown, 1982) are as follows: CHAPTER 2. THEORY 21 1. M a x i m u m of the Absolute Differences ( M A D ) ma,xA \m — f\ 2. S u m of the Absolute Differences (SAD) I!A\m-f\ 3. M e a n Square Difference ( M S D ) These expressions are al l distance measures or metrics. The measure is computed for each for each lag posi t ion and the m in imum is chosen as the correct match loca-t ion. The i r use is embraced by a field known generally as Template Matching and the pr inc ipal mot ivat ion for using them is to reduce the computat ional load impl ied by the matched filter. If the image functions are viewed as points in an inf initely dimensioned function space, then the distance between two points in any part icular space are measured wi th the metr ic under ly ing that space. The metrics underly ing the three distance measures given above as well as the matched filter are members of the same class of metrics, the Lp Norms. Dvornichenko, (1981) made a study of the effects of changing the metric under ly ing cross correlat ion. He found that for analyt ic functions representit ive of image signals bo th the M A D and S A D are unsuitable measures. T h e M A D is used very infrequently in practice. The S A D is much more popular and is the cornerstone of the Sequential Similarity Detection Algorithm (SSDA) (Barnea and Silverman, 1972). Th i s a lgor i thm reduces the number of h igh accuracy calculations by randomly selecting the areas of the search image to be compared w i th the reference image. In the S S D A if the accumulated absolute differences exceeds some threshold before al l calculat ions w i th in it have been completed , processing for that part icular lag posit ion ceases and the number of pixels examined is recorded as a rat ing for that lag posi t ion. Af ter a l l possible positions have been examined, the posit ion w i th the highest rat ing is declared the best match. B o t h a constant and a monotonical ly decreasing(from the centre of A) threshold were used. Use of the S S D A resulted in significant savings in computat ion t ime for the registration of images f rom the N O A A meteorological satellite I T O S - 3 (Barnea and Silverman, 1972). However the match measure defined the S A D CHAPTER 2. THEORY 22 is unsuitable for most imagery (Dvornychenko, 1981). The M S D match may be rewrit ten as: / / ( m - / ) ! = / / ^ + /// !-2//m/ (2.34) the first te rm represents the reference image energy which is a funct ion of the part icular reference, not of lag posit ion and thus remains constant unt i l a new reference is chosen. The second te rm is the search image energy and is constant for each lag posi t ion. Hence max imiz ing / f(m — f)2 is equivalent to min imiz ing / / mf for a given / / m2 and / / f2. Thus we see that the M S D measure is equivalent to the matched filter for the white noise case. In this chapter we pose the problem we wish to solve as a est imation problem. The solut ion to this problem is an opt imal filter which is obtained by min imiz ing the error variance. The transfer funct ion of this opt imal filter is shown to be: For the case of whi te noise the filter is a correlator. For the case of coloured noise the filter, due to i t 's l inearity, may be broken into two filters. The first is an inverse filter that shapes the signal spectrum according to the power spectrum of the noise. Th is step yields a white noise signal and is, hence, called a whitening filter, Hw(u,v), and the second step is a correlator, Hm(u,v), hence, 2.4 Summary H{u,v) = X2K2e- j2<u'x+vyy> M*{u,v) Sn(u,v) (2.35) H{u,v) = Hw(u,v)Hm(u,v) Chapter 3 Implementation 3.1 Questions of Noise Noise is a somewhat ambiguous term when standing on its own. It may be conveniently defined as some process, either random or structured, wh ich is corrupt ing or masking a part icular process or funct ion of interest. For instance the snow wh ich appears in television pictures is representative of a whole class of noise and its removal was one of the first appl icat ions of digi tal signal processing (Graham, 1962). The noise belonging to this class is comprised by random uncor rec ted addit ive errors whose formal character izat ion is Gaussian stationary white noise. In this study the noise wh ich is of interest, and that which is modeled is the noise wh ich corrupts the systems abi l i ty to detect the mot ion of the ice. Th i s is fundamental ly different f rom the noise which corrupts the imaging process. In the case of A V H R R this latter type of noise corrupts the radiometers abi l i ty to sense the ice and is well modeled as addi t ive and white. In the case of S A R this noise is due to the coherent nature of the processing and is referred to as speckle noise. Th is noise being a funct ion of range is mul t ip l icat ive (the brighter the region the noisier it is) and is potent ial ly very dangerous to the detection system. However the digi ta l processor developed at the Jet Propu ls ion Laboratory includes a means for signif icantly improv ing this noise s i tuat ion (Curlander et al, 1982; Porcello et al, 1976) and the low pass fi l tering used in this study further improves the noise (Lee, 1981). Hence at the 200m resolution 23 CHAPTER 3. IMPLEMENTATION 24 used the effects of the speckle are effectively nonexistant and the image noise may be modeled as in the A V H R R . T w o important theoretical models of noise are briefly reviewed in Append ix 1. B o t h models allow a complete descript ion of the noise w i th second order statistics and both y ie ld analyt ic expressions for the autocorrelat ion funct ion wh ich is subsequently compared to that f rom the images themselves. 3.1.1 A Noise Model The noise model used in this study is simi lar to that used in a series of papers f rom the Laboratory of Appl icat ions in Remote Sensing ( L A R S ) at Purdue Univers i ty (McGillem and Svedlow, 1976,1977; Anuta and Davillou, 1982) on the registrat ion of L A N D S A T agr icul tural scenes. In these studies the subimage at t ime ti is modeled as signal, while the subimage at t ime £2 is modeled as signal plus addit ive noise, where the temporal changes between t\ and £2 are assumed to be contained in the noise. Th i s means that samples of the noise may be obtained by comput ing / - m a t the correct match locat ion between them. F igure 3.1 show the autocorrelat ion functions, Rn, of f — m, where / — m is the average of many samples of / — m between the two images. It is clear that Rn f rom the two A V H R R image pairs approximates a delta funct ion which suggests a white noise model is appropriate for this imagery. F igure 3.1 (bottom) shows Rn f rom the S A R image pair as well as the funct ion specified by the Markov model derived in Append ix 1.1. R{Tx,Ty) = ^ e x p l - a ^ l ^ l + ay\Tv\} (3.1) The correspondence between the two curves strongly suggests that the Markov model is appropriate for the S A R imagery and hence the image signal must be whitened w i th the noise spectral density. It should be noted that in comput ing the averaged difference image, f — rn, the ergodic assumption was made. Th is assumption equates spatial and ensemble averages CHAPTER 3. IMPLEMENTATION 25 and allows for the computat ion of certain stat ist ical averages such as the mean and autocorrelat ion funct ion f rom a single finite segment of data. CHAPTER 3. IMPLEMENTATION 26 x profile 01 magnitud y profile J -31 31 June 11-12 LEGEND June 12-13 X profile y 1 \ o •0 3 y profile . *j ~B I', u A § /;' / / \ \ \ \ -31 3 31 LEGEND markov model image diffs Figure 3.1: Autocorre la t ion functions Rn for A V H R R images (top) and S A R images (bottom). CHAPTER 3. IMPLEMENTATION 27 3.2 The Whitening Filter The transfer function of the optimal estimation filter was found in chapter 2, and was given by: H{u, v) = ce->2*(uxo+vv°) ^T / " ' ^ . (3.2) If Sn(u,v) satisfies e < Sn < M for some e > 0 and M < co and the noise auto-correlation function satisfies \Rn\ < co, then Sn is given by the following (Woods, 1984): = ciid^oF (3-3) In fact by modeling the noise as a Gaussian stationary Markov process we avoid all the difficulties of inverse filtering that are implied by (3.3) since the inverse of its spectral density is well defined. If we assume that the Markov process is seperable then the spectral density of the noise is given by the Fourier transform of (3.1): - 4apK substitution of this expression into (3.2) followed by factorization into complex conju-gates yields the whitening filter: Hw(u, v) = 1/—5(a/3 + Pj27ru + ctj2-KV - 4ir2uv) (3.5) IKy Otp the spatial domain representation of this is: d i _ d a a/M(z,y) + /?£ + a& 2 It is clear that this filter acts as a differential operator and in what follows I shall focus on this basic function rather than on attempting to discretize the function. Differential operators have been standard tools in image processing for locating edges or bright-ness changes in digital imagery. SAR imagery is, in general, replete with edges that correspond to significant features in the ice pack, such as: 1. The boundary between open water or very thin ice (nilas) and the actual ice floes CHAPTER 3. IMPLEMENTATION 28 2. boundaries between mult iyear ice and younger ice 3. locations of pressure ridges and rubble fields w i th in the ice floes. T h e S A R imagery used in this study is f rom an L-band S A R whose features generally tend to be of type 1 or 3. The discr iminat ion between thick first year and older ice has only been successful w i th higher frequency S A R such as X and C-band or a combinat ion of X / C and L-band S A R (Lyden et al, 1984; Burns et al, 1984; Holmes et al, 1984). A n example of the S A R signatures of types 1 and 3 features may be seen in figure 3.2 Hence by operat ing on the imagery w i th a differential operator, the second stage of the matched filter operates on the significant image features rather than on the raw brightness values. Differential operators may be broken roughly into two classes (Horn, 1986): 1. F i rs t order operators (a) brightness gradient 2. Second order operators (a) squared gradient 9(x, y) -(b) Lap lac ian (c) quadrat ic var iat ion -(fi)(SMS)' ™ T h e brightness gradient and the Lap lac ian are both l inear while the other two are bo th non-l inear. The brightness gradient is the only direct ional operator which CHAPTER 3. IMPLEMENTATION 29 produces results which depend on the orientation of the edge, implying a need for the application of a series of orientation-specific operators at any given point. The second order operators are all rotationally symmetric, treating edges at all angles equally, however of the three only the Laplacian retains the sign of the brightness difference across the edge meaning that it alone would allow the reconstruction of the original image from the edge image. The brightness gradient and the Laplacian have become the most popular differential operators for very different reasons. CHAPTER 3. IMPLEMENTATION 30 Figure 3.2: A small portion of rev.1439 showing some significant image features (top). Profiles showing brightness gradient from image and operator responses to the image features (bottom). CHAPTER 3. IMPLEMENTATION 31 A n important characterist ic of these two operators is their frequency response or transfer funct ion given, in the x direct ion, by: brightness gradient C?i(u,v) = -j2nuF(u,v) (3.11) Lap lac ian G2(u, v) = - 4 T T 2 ( U 2 + v2)F(u, v) (3.12) It is evident that the the transfer funct ion of G\ increases l inearly whi le that of G2 increases exponential ly. Th i s has an important impact on the discrete implementat ion of the operators. It should be remembered that the noise model discussed in §3.1.1 is assumed to represent the temporal changes between the images; in fact, each image is also corrupted w i th white noise. Wh i le both G\ and G2 enhance the high frequency components of the image noise, G2 has a tendency to get lost in the noise especially when its impulse response has smal l support (spatial extent). One of the earliest publ ished edge detector was a 2 x 2 implementat ion of G i by (Roberts (1965), this operator along w i th several smal l support ( 5 x 5 and less) implementat ions of Gi (Prewitt, 1970; Abdou and Pratt, 1979; Shaw, 1979) have shown that it is robust against noise in many appl icat ions. 3.2.1 The First Order Filter The 3 x 3 Sobel edge detector given a reasonable S N R has been shown to be an accurate measure of edge presence (Davis, 1975; Abdou and Pratt, 1979; Kittler, 1988; Iannino and Shapiro, 1979). The operator is defined in terms of two 3 x 3 masks wh ich measure the edge signal in both the x and y directions. If the 3 x 3 mask is centered on the pixel defined in figure 3.3 as e, then the edge signals in the x and y directions at the pixel e are defined by: ex = [c + Kf + i] - [a + Kd + g] , , ey = [a + Kb + c}-[g + Kh + i]  { } CHAPTER 3. IMPLEMENTATION 32 a b c d e f g h i F igure 3.3: Mask used for a generic 3 x 3 operator, the magnitude of the edge signal is: E = y/'l +  el (3-14) while the edge angle is given by: 7 = arctan — (3.15) ex where K has been shown to be a funct ion of the angle (Kittler, 1988) and varies between 2.0 and 3.0. In this study we use edge magnitude only and use the standard value of K = 2.0 (Ballard and Brown, 1982). The discrete implementat ion of the whitening filter based on the Sobel operator involves the computat ion of E at each pixel in the images defined in chapter 2 as, / and m. The response of the Sobel operator to a typical one dimensional image slice is shown in figure 3.2. 3.2.2 The Second Order Filter The abi l i ty of the first order filter to wi thstand lower S N R ' s is somewhat gratuitous since numerical differentiation is not, in general, robust against noise. In fact numerical differentiation is known to be an ill-posed problem (Torre and Poggio, 1984; Poggio and Torre, 1984). A n important cr i ter ion for defining well-posed problems is that the solut ion depends continuously on the ini t ia l data (this is equivalent to saying that the solut ion is robust against noise). Th is cr i ter ion is violated in numerical differentiation CHAPTER 3. IMPLEMENTATION 33 wh ich can be disrupted by smal l amounts of noise. The method for transforming an il l-posed problem into one which is well-posed is known as regularization (Tikhonov and Arsenin, 1977). T h e theory of regularizat ion has been appl ied to edge detection by Torre and Poggio (1984) who found that the appropriate regularizing operator was a low-pass filter. Th i s is an intui t ive result since f rom looking at (3.11) it would appear that a low-pass filter would be necessary to stabil ize the frequency response of the second order differential operator. Th is is also a very general result and the specific shape of the filter is constrained by two issues: 1. intensity changes in the imagery are generally local ized in the spatial domain. 2. a single operator cannot detect intensity changes at different spat ia l scales and it is therefore desirable to restrict the scale at wh ich intensity changes take place in the output of the filter. These two issues are confl ict ing but may be opt imal ly satisfied by the Gaussian smoothing funct ion (or roll-off low-pass filter) whose impulse response, g, and transfer funct ion, G are given (in one dimension) by: g{x) = K 1 e x p { - x 2 / 2 a 2 } = K2exp{-Ks2o2uj2} where K\, K2 and K3 are scaling constants. W h e n this filter is combined w i th the Lap lac ian funct ion we have the so-called V2g operator given, in two dimensions, by, V2g(x, y) = K[2 - (r 2 /<r 2) ]exp{-r 2 /2<7 2} (3.17) where r 2 = x2 + y2 and i f is a scaling constant. Th i s filter is an approximately bandpass filter which detects edges of a part icular resolution. The one dimensional profile of this filter is shown in figure 3.4. T h e band of frequencies passed by the filter may be selected by changing the value of cr2. B y using part icular values for this parameter a number of wavenumber chan-nels may be produced. Each channel corresponds to a part icular filter, and hence CHAPTER 3. IMPLEMENTATION 34 edge, resolution. Th i s approach has been the core of the theory of computat ional v is ion spearheaded by the late Dav id M a r r of M. I .T. (Marr, 1976,1977,1982; Marr and Poggio, 1977,1978; Marr and Hildreth, 1979; Poggio, 1985). A quant i ty which is used as a measure of zero-crossing strength is the slope (Grim-son and Hildreth, 1985). It is defined to be the magnitude of the gradient of the bandpassed images at the locat ion of the zero-crossing. Thus slope = | V ( V 2 £ ® 7)| (3.18) where I denotes the image. The zero crossings were located using an efficient predicate-based a lgor i thm described in Huertas and Medioni (1986). Th i s a lgor i thm inspects each 3 x 3 neighborhood in the convolved image and matches it to one of eleven allowable zero crossing predicates which define a total of 24 edges positions T h e contrast and w id th of the intensity change may be computed expl ic i t ly using the slopes of the zero-crossings f rom operators w i th two different resolutions. In this study a single resolution was used whose frequency response matched the spatial resolution of the intensity changes of interest. The value of a used in the study was \ / 2 which is a tradeoff between high bandwid th (lower number of filter coefficients) and low al iasing error (due to the sampl ing of a continuous function) (Clark and Lawrence, 1984)- The edge magnitudes for the S A R images are shown in Append ix C where they have been scaled between 0 and 255 and shown as grey scale images. T w o recent papers have highl ighted an important feature of the zero crossings of the V 2 0 filter (Berzins, 1984; Clark, 1986). It has been shown that some of the edges asserted by the filter do not correspond to any significant changes in image intensity. These edges have been referred to as phantom edges and C la rk provides an excellent analysis of these features. He shows that if the image funct ion is modeled as a white normal ly d istr ibuted funct ion, the phantom edges are fewer by a factor of three and a half and weaker by a factor of eight than the authentic edges. The model used to represent the image in order to derive these quanti tat ive results is not part icular ly well-suited to most images. However it is evident that the strength of the authentic CHAPTER 3. IMPLEMENTATION 35 J Figure 3.4: One dimensional slice through the (a) impulse response and (b) frequency response of the V 2g filter. edges is much greater than that of the phantom edges and the classification function proposed by Clark was not implemented in this study. 3.3 The Matching Filter As mentioned in chapter 2 the matching system systematically divides the images into smaller subimages where it is assumed that the motion of the ice within each subimage is homogeneous. This methodology has previously been applied to the detection of cloud motion (Leese and Novack, 1971) and updating automatic guidance systems (Lo and Gerson, 1979; Mostafavi, 1979; Steding and Smith, 1979a). While the approach may seem somewhat brute-force, this systematic division is what makes the system fully automatic. It removes the necessity for a decision mechanism as to which piece of ice or ice feature to track and which to ignore. These kinds of decisions are either made by the operator (Curlander and Holt, 1985), in which case the system is no longer automatic, or they are computational. The process of making decisions computationally involves compiling a list of rules and as regards ice motion this would require a better knowledge of radar and radiometer ice signatures and of the geographic variability of CHAPTER 3. IMPLEMENTATION 36 ice dynamics than is presented available. A general discussion of these issues is included in chapter 5. The cross correlat ion is made computat ional ly more efficient by using the convolut ion theorum which equates convolut ion in the spatial domain to mul t ip l icat ion in the wavenumber domain (Cooley et al, 1967), m(x,y) ® f{x,y) = M*(u,v)F(u,v) Th is is a standard result which has been harnessed many times in correlat ion- based image registrat ion (Anuta, 1970) and cloud-motion t racking (Leese and Novack, 1971). However care must be exercized in using this theorem since the Four ier transform assumes the signal is periodic and convolutions implemented in this way are subject to wrap-around error i n wh ich the correlat ion coefficients at each spat ial lag wi l l be biased w i th the exception of zero lag (Blackman and Tukey, 1958). Th is problem is easily rectif ied, however, by choosing signal segments of different lengths. The matching system processes the imagery according to the fol lowing sequence of events: 1: The image m(x , y) is subdiv ided into Nf x Nf pixel filter windows, mw(x, y), and the succeeding image f{x,y) is subdiv ided into corresponding Ns x Ns p ixel search windows, fw{w,y). 2: The window means were subtracted and the variances were computed. In the case of fw{x,y) we have an array of variances corresponding to each locat ion of mw wi th in / „ , . 3: The filter and search windows are padded w i th zeros to the next highest power of 2 and the F F T ' s of mw and / „ , (denoted Mw and Fw resp.) are computed. 4: The product , Rmf(u,v) — M*w{u,v)Fw{u,v) is formed and the inverse F F T , rmf, is computed. 5: The unbiased por t ion of rmj (correlations which correspond to posit ions where mw lies entirely w i th in /„ ,) is extracted and the max imum value w i th in this unbiased array is located. CHAPTER 3. IMPLEMENTATION 37 3.3.1 Optimum Window Size T h e choice of the filter window size has an important effect on the performance of the system. The size must balance two opposing cr i ter ia: it must be small enough to be increase the probabi l i ty that the mot ion wi th in the window is pure t ranslat ion; and it must be large enough to ensure a good dist inct ion between a false match and a correct match. T h e window sizes used were based on a smal l amount of experimentat ion w i th different values and is therefore rather subjective. However, since the ice mot ion is potent ia l ly discontinuous and the scales of rotat ion and deformation are unknown a more analyt ic approach such as found in a group of papers f rom Systems Cont ro l Inc 1 is not possible. Th is latter research was devoted to automat ical ly updat ing unmanned airborne vehicle guidance systems and derived opt imal filters and windows for registrat ion based on a single rotat ion angle appl ied to the whole scene, where this rotat ion was predictable. T h e A V H R R filter window used agrees w i th some error analysis done by Fily and Rothrock (1986b) whi le the S A R filter window agrees w i th some exper imentat ion wi th window sizes done by Vesecky et al, (1986b). The filter window sizes used are as follows: • A V H R R Nf — 22 x 22 pixels • S A R Nf = 30 x 30 pixels The choice of the search window size is based on the max imum amount of ice mot ion expected in the region. Th is is a difficult issue to resolve since the magnitude and var iabi l i ty of ice mot ion varies according to season and geographic locat ion. However the fact that this method approximates al l mot ion as translations combined w i th the assumption that mot ion wi th in the filter windows is homogeneous restricts the use of this system, as presently implemented, to winter scenes of the central ice pack. It is well known that the two major forces in this regime are the wind and water stress 1 Mostafavi and Smith, 1978a,b; Smith and Mostafavi, 1979; Mostafavi, 1979,1981; Steding and Smith, 1979a,b CHAPTER 3. IMPLEMENTATION 38 (Rothrock, 1979; Hibler, 1984). Of these, the former especially causes the the ice to react quickly to its influence. Hence a knowledge of the prevai l ing current system and surface pressure field would considerably improve the systems abi l i ty to ini t iate itself (Holt, 1986). In the absence of this information a max imum amount of ice mot ion was input. Th i s value was chosen to be reasonable both in terms of the amount of computat ion it impl ied and the number of different image pairs to which it could be appl ied. A m a x i m u m magnitude of 4 k m per day was used which translates into the fol lowing search window sizes: • A V H R R Ns = 32 x 32 pixels • S A R Ns = 128 x 128 pixels It should be noted at this point that another force which plays a major role in the dynamics of the ice cover is the complex and poor ly understood interact ion of the ice w i th itself. Th is force depends not only on the mater ial characterist ics of the ice, wh ich vary considerably according to the age of the ice, but also on the presence of an immovable object such as a coastl ine. The increased influence of this force over the dynamics of the ice implies a much larger potent ial for discontinuous mot ion which in tu rn w i l l elude the detection system. 3.3.2 Error Correction A s has been discussed earlier, the matched filter can often give erroneous results due to its dependence on the energy of the image signals. The prewhitening of the S A R images wh ich produces an edge image means that, in this case, the filter is dependent on edge energy rather than on image energy. The false matches stem f rom the fact that the matched filter correlates areas only. These false matches may be detected in a post-processing step which examines the fol lowing evidence: 1. global statistics of the correlation coefficients; CHAPTER 3. IMPLEMENTATION 39 2. consistency in the local field of mot ion. W i t h regard to the first, the mean and standard deviat ion of the correlat ion coefficients are computed and a vector is considered in error if it is more than one standard deviat ion f rom the mean. Consistency in the local displacement field is also interpreted in terms of stat ist ical deviat ion. F i rs t we compute the mean and standard deviat ion of the x and y displacements in the 5 x 5 neighborhood around the vector being processed. A displacement is considered inconsistent if i t deviates by more than one standard deviat ion f rom the mean. The displacement tagged as incorrect by either of the above decisions is recom-puted by using the displacements of its eight nearest neighbors as queues in searching the or iginal unbiased correlat ion array. The max imum correlat ion coefficent is cho-sen f rom the 5 x 5 neighborhood surrounding each search queue. Th is process yields eight coefficients f rom which the largest is chosen as the replacement for the erroneous displacement. It was found that the recomputat ion process was most effective when implemented in two passes, the first pass used both decisions whi le the second pass used only the mot ion consistency decision. Th is two pass process al lowed the system to recover f rom certain noisy neighborhoods which would normal ly have escaped f rom the first pass. Hence the matching system consists of three steps, the matching step and two recomputat ion steps. In some cases the deformation of the ice cover wi l l be too great and the system's estimate of the ice mot ion wi l l be poor. In this study the qual i ty of the correlation coefficient was estimated statist ically. In order to estimate the stat ist ical significance of the correlat ion coefficients some simple analysis, s imi lar to Emery et al (1986), was performed. T h e spat ial decorrelation scales of the imagery were first derived by aver-aging the autocorrelat ion functions f rom all of the Nf x Nf windows in the imagery and finding the distance f rom the central peak to the nearest zero-crossing. The total number of independent points or, effective degrees of freedom, N*, of the correlation computat ions, was obtained by d iv id ing the total number of observations by the decor-CHAPTER 3. IMPLEMENTATION 40 relation scale. Finally N* is used to compute the correlation coefficients which are significant at the 95% significance level. The null hypothesis, that the population cor-relation coefficient is zero, is tested by computing the following statistic (Brownlee, 1960): !^E2 (3.19) This value is referred to a £ table with (n — 2) degrees of freedom. Chapter 4 Experiments 4.1 A V H R R data The A V H R R images used in this study were received and processed at the Satell i te Oceanography Laboratory in the Department of Oceanography at the Universi ty of B r i t i sh Co lumb ia . The raw satellite data was earth-located and geometrically rectified using h igh qual i ty ephemeris data suppl ied by the Uni ted States Navy who indepen-dently track the N O A A satellites. The satellite orbit is modelled as a ellipse. The radius of the earth is calculated for the centre of the region of interest which is assumed to be local ly spherical. The spherical pixel coordinates are then mapped to a plane using the Lamber t conformal conic ( L C C ) map projection w i th one standard paral lel . The final correct ion step is a translat ion of the image based on the locat ion of one ground control point ( G C P ) . Th is navigat ion procedure was described by Emery and Ikeda (1984) where a one pixel accuracy was reported. The resolution of the navigated pixels is 1.1 k m at nadir and changes slowly away f rom this point. There is some spatial d istort ion at great distances f rom the nadir, however these situations were avoided in this study. T h e navigated digital images cover 870 x 870 k m (512 x 512 pixels) and are centered at 72°N, 127°W. The frequency bands sensed by the A V H R R instrument are shown in table 4.1. The image sequences used in this study are shown in table 4.2. Dur ing the winter months (October - May ) the smal l amount of dayl ight necessitates the use of the thermal 41 CHAPTER 4. EXPERIMENTS 42 channel wavelength (jura) comment 1 0.58 - 0.68 visible - red 2 0.73 - 1.10 near-infrared 3 3.55 - 3.99 extremely noisy - not used 4 10.3 - 11.3 thermal- infrared 5 11.5 - 12.5 thermal- infrared Table 4.1: The channel bandwidths of the A V H R R on the N O A A satellites used in this study. infrared (IR) channels to view the ice cover. Images f rom this per iod often exhibit uni form pixel intensities over the ice pack reflecting the near un i form temperatures of the blanket of snow. The presence of leads are the only clearly dist inguishable feature in winter scenes due to the great difference in the thermal signatures of ice/snow and water. Because the computat ional method developed in this study is essentially an area correlat ion, it is not useful for studying most winter scenes of the ice. T h e onset of the summer melt season is characterized by a decaying snow cover, the exposure of bare ice and the format ion of melt ponds over much of the ice surface. The albedo of the exposed sea ice, which relates direct ly to the amount of solar radiat ion backscatter, depends most strongly on the internal structure of the near-surface layers. There are substant ial differences in the overall internal structure between mult i-year ( M Y ) ice and first-year ( F Y ) , which is ice that has not undergone a summer melt season. In contrast to the relatively high albedo of the bare ice, melt ponds are low albedo areas where a large amount of solar radiat ion is absorbed. Me l t ponds reach a max imum extent short ly after the disappearance of the snow cover, when they may cover up to 50% of the ice. Fol lowing this max imum, pond coverage on the mult iyear ice decreases as some ponds dra in and others deepen. Grenfell and Maykut (1977) have broken bare first year ice into two categories: i) melt ing white ice (ice whose surface lies above the local water table) and ii) blue ice (ice saturated w i th , but not covered by, melt CHAPTER 4. EXPERIMENTS 43 oi 1 1 1 1 ' 1 4 0 0 6 0 0 8 0 0 1 0 0 0 W A V E L E N G T H (nm) Figure 4.1: Spectra l albedos over snow and melt ponds: (a) dry snow (b)wet new snow, (c) mel t ing old snow, (d) part ial ly refrozen melt pond, (e) early season melt pond (10 c m deep), (f) mature melt pond (10 cm deep) on M Y ice, (g) melt pond on F Y ice (5 c m deep), (h) old melt pond (30 cm deep) on M Y ice. Grenfell and Maykut (1977). CHAPTER 4. EXPERIMENTS 44 4 0 0 6 0 0 8 0 0 WAVELENGTH (nm) Figure 4.2: Spectral albedos over bare sea ice: (a) frozen MY white ice, (b)melting MY white ice, (c) melting FY white ice, (d) melting FY blue ice. Grenfell and Maykut (1977). water). Surface conditions on MY ice differs from F Y ice in two respects: i) melts ponds cover a smaller area but are substantially deeper and ii) white ice, rather than blue ice, covers the unponded areas due to greater surface relief and better drainage (Grenfell, 1979). Grenfell and Maykut made extensive high resolution measurements of the albedo over FY ice, MY ice, melt ponds and snow. These measurements are summarized in figures 4.1 and 4.2. It is evident from these curves that the magnitude and shape of the spectral albedo function depends strongly on the amount of water in the near-surface layer of the ice. Two features which impact on this study are the significant overlap in albedo between: i) unfrozen melt ponds and blue ice and ii) frozen melt ponds and FY white ice. The image intensities in band 1 are proportional to the albedo of the underlying surface. Their absolute values are not significant because the signal has been attenuated by the atmosphere in an unknown way. Grenfell and Maykut suggest that in the absence CHAPTER 4. EXPERIMENTS 45 of leads the regional albedo may be approximated by a simple combinat ion of melt pond /b lue ice and white ice albedos. The albedo of leads is much lower than either of these two categories and they can saturate a pixel even their size is smaller than the resolut ion of the instrument. The band 1 images used in this study show great variat ions in pixel intensity over the ice pack. These variat ions reflect the differing amounts of melt water at the surface of the ice and is robust over the t ime periods used. 4.1.1 Results Veri f icat ion of the automat ical ly measured displacements (u 0 ) was achieved through comparison to: (a) manual ly measured displacements ( u m ) and (b) displacements predicted by a theory of wind-dr iven ice mot ion (u p ). The graphic results of this section are a l l contained in Append ix B . The first data set was gathered by si t t ing at an image display system and measuring the displacement of invariant features (IFs) in the ice. T h e IFs are sharp intensity changes (edges) which are perceived by the operator as being significant features in the ice pack. A t the scale of the imagery these ice pack features are most often large leads, since ridge lines and rubble fields are never 1 k m wide. B y quickly and repeatedly v iewing each image in a pair we can easily perceive the optical flow of the scene. Th is is a qual i tat ive perception of the field of mot ion which the human v isual system can perform under remarkable condit ions. In order to measure the motion field necessary for scientific and operational purposes, however, we must choose one pixel which appears to be an IF and find it in the next image. The accuracy of such a measurement is very difficult to estimate. The operator obtains an excellent sense of the orientat ion of the displacement f rom the opt ical flow. A conservative estimate of the error in the manual measurements in this study is 0.5 pixels, half that of the navigat ion. In many cases, especially when the IF is robust, the measurement is essentially errorless. In cases when the IF was not as dist inct, the displacement was measured a number of t imes and the mean of these measurements was recorded. In CHAPTER 4. EXPERIMENTS 46 these cases the standard error was very smal l (less than 0.1). Th i s apparently small measurement uncertainty, however, does not correctly reflect the ambigui ty which often exists in mak ing measurements in the central pack. The images of the summer ice pack contains a great deal of spatial var iabl i ty due to the aforementioned variabi l i ty of surface condit ions. There are often very large areas of the pack where there are no leads and no other physical features which produce the intensity discontinuit ies that a human operator needs to make accurate manual measurements. The fact that an operator has such diff iculty measuring mot ion in these situations underl ines the ut i l i ty of the automated system. The density of the manual measurements is very much a funct ion of how many significant edges could be measured w i th at least the same accuracy as the navigat ion. Th is led to a sparse dist ibut ion of um . It was felt that due to this pauci ty of data the interpolat ion of the irregularly distr ibuted data onto a regular gr id would introduce too much error into u m . The comparison between u 0 and u m is therefore carr ied out in a series of tables which are shown in Append ix B . F r o m these tables we may cul l the fol lowing representative figures: • \um\ — \ua\ = 0 pixels in 48% of the measurements • |«m| — \ua\ < 1 pixels in 94% of the measurements It is evident f rom these figures that the automatic measurements represent the manual measurements to w i th in the uncertainty imposed by the navigat ion and the manual measurement. It should be emphasized that u a and u m are two very different mea-surements. We have no way of manual ly measuring spatial correlat ion. A s wi l l be investigated more closely in the next section, small amounts of mot ion shear could yield a discrepancy between u a and u m . If the pixel chosen as the IF is on one side of a shear and the cross-correlation f rom the automatic scheme contains the shear, then u„ and um w i l l differ. Because of the resolution of the imagery this difference be smal l . A s mentioned earlier the area correlation is only appropriate in the central ice pack CHAPTER 4. EXPERIMENTS 47 Date T ime ( G M T ) Satell i te Pass # June 05 1984 21:50 N O A A 7 15227 June 06 1984 23:17 N O A A 7 15242 June 07 1984 23:05 N O A A 7 15256 June 11 1984 22:16 N O A A 7 15312 June 12 1984 22:04 N O A A 7 15326 June 13 1984 22:04 N O A A 7 15340 Ju l y 30 1985 21:32 N O A A 9 3251 Ju l y 31 1985 02:11 N O A A 8 12161 August 06 1985 21:58 N O A A 9 3350 August 07 1985 21:47 N O A A 9 3364 Table 4.2: A V H R R images used in this study. where the mot ion of the filter windows may be approximated as translat ions. For this reason the values of u a are not shown at the edge of the ice pack which is a M I Z nor are they shown for Amundsen Gu l f (between Banks Island and Cape Par ry ) . The results of the method are also not shown in the top left hand corner of the images since the qual i ty of the images in this area is seriously degraded by dropouts (scan lines dur ing wh ich there were reception problems). The image pairs analysed were: 1: June 05-06 1984 2: June 06-07 1984 3: June 11-12 1984 4: June 12-13 1984 5: Ju l y 30-31 1985 6: August 06-07 1985 and are shown in Append ix B . It should be noted that the images shown in the figures CHAPTER 4. EXPERIMENTS 48 have been enhanced, by a contrast stretch (Gonzalez and Wintz, 1977) to br ing out the ice features. T h e fields of u 0 and u m for each pair are shown in Append ix B . A r e a correlat ion methods have often been accused of y ie lding a broad correlation funct ion, mak ing the selection of a single max ima difficult. F igure B.34 and B.35 shows two typ ica l examples of the correlation funct ion f rom the th i rd pair. Wh i l e there is a certain amount of spat ial stucture to these functions, the peaks clearly rise above the background level result ing in well-defined max ima. A s discussed in Chapter 3, a simple test was performed to determine the stat ist i -cal significance of the cross-correlation coefficients. The effective degrees of freedom is computed using the decorrelation length scales derived f rom the averaged autocor-relat ion functions of the imagery. These functions were very nearly isotropic w i th a decorrelat ion length of 10 pixels (about 15 km) . In the 22 x 22 window used, there were 484 data points, hence there were 48 effective degrees of freedom. F r o m a t table we find that coefficients greater than 0.24 are stat ist ical ly significant at the 95% level of confidence. In the plots of u 0 shown in Append ix B , vectors that are below the significance level are plotted in gray, while those above are in black. One of the pr imary uses of the synoptic field of ice displacements derived from remotely sensed imagery is in verifying the results of dynamic models of ice mot ion (Coon, 1980). The A I D J E X project, which investigated the mot ion and deformation of the ice pack in the Beaufort Sea, used pr imar i ly L A N D S A T imagery, suppl imented by some N O A A and aircraft data. The model l ing group wi th in A I D J E X used the 80 m resolut ion L A N D S A T imagery to generate an ice thickness dist r ibut ion for the model area and also to test the momentum balance and const i tut ive relations used in the model . Unfor tunately the spatial and spectral resolution of the N O A A A V H R R imagery does not allow us to produce the accurate thickness d is t r ibut ion required by the A I D J E X model and its close relative, the I N T E R R A ice model (Lea vitt, Sykes and Wong, 1981; Hall, 1980). The force whose funct ional form is known the least and whose spat ial and temporal CHAPTER 4. EXPERIMENTS 49 Figure 4.3: If the wind stress r 0 balances TW + C, then a larger stress r'a turned to the right of ra is required to balance TW + C + V-o*. Thorndike and Colony (1982). variability is the highest arises due to the spatial variations in the state of stress o in the ice cover. Th is force is given by V-cr and, although it cannot be measured directly, it can be estimated indirectly as a residual if the other terms in the force balance are known (Rothrock, Colony and Thordike, 1980). Thorndike and Colony (1982) investigated the following linear relationship between the ice velocity u, the geostrophic wind G and the mean ocean current c u = AG + c + e (4.1) where A is a complex constant and the vectors u, G,c and e are represented as complex numbers. In this equation, e represents that part of the ice motion which is neither constant nor a linear function of the geostrophic wind. It includes the effects of mea-surement error, internal ice forces, time dependent currents, etc. T h e complex constant A consists of a scaling factor and a turning angle, A = \A\e-" T h e validity of the relationship was tested against observations of the Arct ic data buoy array deployed by the Polar Science Centre of the University of Washington. Using two CHAPTER 4. EXPERIMENTS 50 ful l years of ice mot ion and geostrophic wind data, Thorndike and Colony (1982) found that after removing the mean current, the geostrophic w ind accounted for 70% of the variance of the ice velocity. Us ing shorter t ime series they investigated the seasonal and geographic differences of \A\ and 0. They found a marked seasonal difference as follows, 0.0077 e~*5° winter, spring 0.0105 e~m° summer 0.0080 e~ i6° fall A = Th is seasonal difference is most l ikely a combinat ion of two effects: • the relat ionship between the surface wind and the geostrophic w ind changes in the summer w i th the surface stress on the ice being 10-20% larger and 6° further to the right given the same geostrophic w ind (Albright, 1980), • internal ice stress play a larger role in the dynamics of the ice cover in the winter. T h e net effect of these stresses is to dissipate energy and oppose ice velocity as shown in figure 4.3. W i t h this added effect, a larger w ind stress and a smaller turn ing angle is required to balance the forces (Thorndike and Colony, 1982). There were no geographic variat ions observed except near coastlines where, again, two effects could exist. • One of the data buoys drif ted near the Canad ian Archipelago where the correla-t ion between the variance of the ice velocity and G dropped to less than 0.5. It was postulated that ice stress gradients play a large role near coastlines and that (4.1) may only be applied in regions 400 k m f rom the coast. • Several of the buoys drif ted near the east coast of Greenland where the ice-wind correlat ion again became smal l . The cause was believed to be the large mean and time-dependent currents which exist in the area. T h e ice velocities predicted by Thornd ike and Colony were subtracted f rom the automat ical ly measured ice displacements in order to compare the residuals w i th the CHAPTER 4. EXPERIMENTS 51 known mean current in the Beaufort Sea and the ant ic ipated effect of the internal ice stresses based on an inspect ion of the imagery. The surface pressure charts were obtained f rom the Arc t i c weather office in E d m o n -ton. These charts contained 24 hour averaged isobars which are drawn on the plots of the predicted ice mot ion and the residuals shown in Append ix B . A regular grid of surface pressures and geostrophic winds were computed at Seaconsult L t d . using standard techniques. The error in the geostrophic w ind estimates are approximately 2-4 m /s (Thorndike and Colony, 1982). In his study of geostrophic winds for A I D J E X , Albright (1980) used only those winds in excess of 5 m/s since below this level the uncertainty approaches 100%. Another problem w i th winds less than 5 m/s is that the assumptions that led to (4.1) break down and the relationship becomes non-l inear. The last consideration is the effect of orographic steering of the winds. Th is is an extremely impor tant effect for in the western Beaufort Sea due the presence of the Brooks Range of mountains wh ich affects the surface w ind wi th in a region of at least 50 k m f rom the coast (Dickey, 1961; Kozo and Robe, 1986). Our study area is far removed f rom this range of mountains which ends abrupt ly at the M c K e n z i e R iver . The topography of the southern main land is quite flat and wi l l have no effect on the w ind . The southern t ip of Banks Island (Nelson Head) contains one peak whose height is at the lower l imit of topographic influence and steering may be a factor in this area. It is highly unlikely, however, that any steering which may be present wi l l affect the winds over the pack and orographic effects are therefore ignored. In his extensive study of the circulat ion of the Canada Bas in , Newton (1973) con-f i rmed, through his computat ion of the dynamic height, that the long-term mean flow is the ant icyclonic Beaufort gyre. He also computed a simple steady, frictionless vor-t ic i ty balance wh ich he compared to the dynamical ly more complex numerical model of Gait (1973) to conclude that the wind stress cur l was the important dr iv ing mech-anism. The mean atmospheric c irculat ion over the Canada Bas in is dominated by a high centered at approximately 78°N, 140°W. Newton computed the magnitude of the CHAPTER 4. EXPERIMENTS 52 wind stress cur l and found that the predominant feature was a region of large nega-t ive cur l west of the Canad ian Archipelago. The transfer of the wind's momentum to the ocean is heavi ly damped by the ice cover and the magnitudes of the flow varies f rom 1 c m / s near the centre of the gyre to 3-5 c m / s near Banks Island. The centre of the gyre according to Newton's dynamic height is approximately 76°N, 145°W which is about 150 k m northwest of the images shown at the begining of Append ix B . The current at any point w i l l , of course, contain a component of the stat ionary mean flow and a component f rom each of a variety of t ime dependent flows. The latter class of flows are dominated by highly energetic mesoscale eddies. These eddies are strongly barocl inic w i th a subsurface velocity max imum in the steep density gradient between 30 and 200 m (Manley, 1981). It has been shown that the kinet ic energy of the central A rc t i c ocean resides almost entirely in these eddies w i th the mean flow contr ibut ing less than 1%. Wh i l e the eddies are usually under-sampled, their diameter appears to be 10 k m , and it has been estimated that they cover 10-20% of the western Arc t i c ocean (Hunkins, 1981). Wh i le the eddies of the At lan t ic have their velocity max ima at the surface, the Arc t i c eddies have their max ima at a much deeper level (about 120 m) due to the presence of the ice cover. Whi le these eddies lose some of their momentum and vort ic i ty to the ice cover, the time scale of this transfer is much longer than that represented by the image sequences used in this study. Hence any ocean circulat ion component of the residual field wi l l represent the mean flow. The ant ic ipated flow is that of the lower right corner of the gyre, that is south past Banks Island, turning clockwise and then westerly. T h e first image sequence (June 5 /6 /7 1984) is characterized by large pressure gradi-ents giv ing rise to a predicted mot ion field towards the northwest. The ice has obviously responded to this per iod of strong w ind forcing as is evidenced by the fields of u 0 . The residual field shows a pat tern which resembles the mean ocean c i rculat ion w i th mag-nitudes away f rom the coast that correspond to those observed (3-5 cm/s) (McPhee, 1980). Since the area immediately west of Banks Island is not a region of mean flow CHAPTER 4. EXPERIMENTS 53 intensif ication (Newton, 1973; Semtner, 1986), the large residual magnitudes in this region are most l ikely a manifestat ion of the internal ice stress. T h e observed ice mot ion may be resolved into a component paral lel to the coast (northward) and a component perpendicular to the coast (westward). It is well known that ice cannot support tensile stress (Mellor, 1973), hence the perpendicular component is unopposed. The parallel component gives rise to a shear stress which is opposed by V-a. A n inspect ion of the images shows that the northwest corner of Banks Island is ju t t ing out into the ice field and hinder ing the northward component of ice mot ion. Th is region is also characterized by the largest residual magnitudes. There is a large band of open water separating the pack f rom the southern coastl ine and the residuals in this area are l ikely due to the mean surface circulat ion of the ocean. T h e second sequence of images (June 11/12/13 1984) is characterized by very weak pressure gradients. The pressure field is so weak in the June 11/12 pair that it con-founded the interpolat ion software and the predicted ice motions are not given. The very smal l geostrophic w ind field is below the threshold necessary for (4.1) to be useful. The predicted mot ion is shown for June 12/13, but, again, the magnitudes of the uncer-tainties have risen to the level of the computed winds, as is evidenced by the spurious vectors. T h e measured mot ion is in the form of two gyres which are almost closed. The pressure system shows some north-eastward mot ion as the gradients strengthen in this direct ion. T h e ice gyre centre (taken to be the region of no motion) also shows signs of moving in a north-eastward direct ion. It would appear that , in spite of the light geostrophic winds, the ice is strongly coupled to the atmospheric pressure systems on these short t ime scales. The residual field is rather difficult to interpret since, given the arguments used in the previous paragraph, the direct ion appears to be opposite f rom what is expected. Since the computed geostrophic winds are of the same order as their uncertainty, the computat ion of the residuals may not be a useful exercise. T h e next pair of images (July 30/31 1985) is separated by only 4.5 hours, and contains the k ind of weather system that strikes fear into the heart of anyone involved CHAPTER 4. EXPERIMENTS 54 in summert ime offshore Arc t i c act iv i ty ( ie.dr i l l ing rigs). The pressure gradients and the geostophic winds are very large and the latter flows south-easterly, pushing the entire pack onto the main land. In the short t ime interval between images the ice moved 1 k m towards the east. The ice in the v ic in i ty of Amundsen Gul f , which is more unconsol idated than the pack, appears to respond more direct ly to the wind forcing, and the smal l residuals in this area suggest that the wind is dominat ing the force balance. T h e residuals over the pack are quite uni form and show no hint of the mean ocean c i rculat ion. The perpendicular (southward) component of the wind results in a compressive stress i n the ice cover while the parallel (eastward) component gives rise to a shear stress. T h e residuals show that V-cr is act ing in opposi t ion to the former, wh ich is a reasonable result since it would be the larger of the two. A longer interval may have shown a west-ward component as wel l . The last sequence (August 6 /7 1985) contains a region of fair ly h igh pressure gradi-ents beside one which is almost homogeneous. The predicted mot ion in this latter area is, as expected, useless. The pattern of the residuals in the central pack again reflects the mean c i rculat ion w i th magnitudes that are reasonable except along the western-most isobar and at the southern extremity of the measured field. The mot ion of the ice may again be resolved into a parallel (shear stress producing) and a perpendicular (tensile stress producing) component. Wh i le the northward residuals do not agree wi th what is expected f rom the 24 hour averaged forcing, it may be a delayed or continuing response to a previous atmospheric forcing event. 4.2 S A R data The S A R imagery used in this study was imaged by S E A S A T , a satell ite dedicated to observing the global oceans w i th a number of microwave sensors (Jordan, 1980). S E A S A T , in spite of a very short l ifespan (June 28 - October 10, 1978), generated a data set so large that it is st i l l being evaluated. The S A R was a 1.27 G H z (L-band) radar, wh ich, while not being an opt imal frequency for sea ice type discr iminat ion, CHAPTER 4. EXPERIMENTS 55 proved quite useful. The images used in this study are f rom revolutions 1439 (October 5) and 1482 (October 8) and are centered at about 74.5°N, 174°W (figure C.3) . The images cover ~ 100 x 100 k m containing about 4000 x 4000 pixels at a resolution of 25 m per p ixe l . The images were provided by the Jet Propuls ion Laboratory where they were earth located and geometrically and radiometr ical ly rectified ( Curlander, 1982; Curlander et al, 1985). It was felt that the resolution of 25m was unnecessarily high and the images were filtered w i th a hal f-band low-pass filter (Reimer, 1986; Clark and Lawrence, 1985) and sub-sampled unt i l they were in a 512 x 512 pixel format. These images have a 200 m resolut ion and improved noise statistics (Lee, 1981; Porcello, Massey, Innes and Marks, 1976). T h e images were then filtered wi th the first and second order differential operators described in chapter 3 (figures C.6-C.11) and submit ted to the mot ion detection system. For a winter scene of sea ice, the radar scattering coefficient which describes the backscattered radar signal is known to be control led by comparable contr ibut ions of surface roughness, volume refractive-index inhomogeneities and surface morphology (Carsey, 1985). Prev ious studies using various frequency bands dur ing winter and spr ing, have shown that frequncies above 9 G H z (X - and K-band) have the abi l i ty to discr iminate between between different ice types, while frequencies below 1.5 G H z (L-band) do not (Onstott and Gogineni, 1985). Returns in the latter band, however, easily dist inguished between flat areas of the pack and more prominent features such as pressure ridges and rubble fields (Onstott, Moore and Weeks, 1979). The water surface in the leads has a very weak return and is easily dist inguishable in the two images used in this study. In some cases, however, w ind roughened water wi l l have a very high return, and the signature of large leads, wi l l change dramat ical ly, due to a change in the local w ind field. The image pair used did not fal l into this category and the image intensities wh ich reflect the return strength are thought to be invariant over the 3 day interval between images. CHAPTER 4. EXPERIMENTS 56 T h e manual ly computed displacements, u m (shown in figure C.4) were computed by Univers i ty of Washington scientists using software described in Curlander et al (1985). The density of these measurements (779 in all) allowed for a relat ively accurate interpolat ion to a regular grid (figure C.5) which was performed by Fily and Rothrock (1985, 1986) who estimated the error in u m to be ± 3 pixels (0.075 km) . 4.2.1 Results The automat ical ly measured displacement fields f rom the first order and second order operators are shown in figures C.12 and C.13. The error in these fields is found by s imply comput ing e = u m - u a at each gr id point. The field of errors is shown in figure C.14 (first order) and figure C.15 (second order). It is evident that the second order filter improves upon the results of the first order filter, if only sl ightly. A further improvement may wel l be gained by using authentic zero crossings only as discussed in chapter 3. Another possibi l i ty which warrants further investigation is using both the magnitude and direct ion of the zero-crossings. F igure C.5 represents the ice mot ion as a deforming two-dimensional gr id where the displacements are appl ied to each vertex in the gr id. Th is plot gives a different, and in many ways a more relavent sense of the ice mot ion, since it describes the scene as a deforming sol id rather than disconnected displacements. One characterist ic of the the mot ion wh ich is immediately evident f rom looking at figure C.5 is that the entire scene has rotated by about 4°. The lack of any systematic element in the fields of e demonstrates the robustness of the matching system against this amount of rotat ion. Wh i l e this amount of rotat ion would be smal l for the rapidly deforming M I Z it is a very reasonable value for the perennial ice cover (Colony and Thorndike, 1981, 1984), where this method may be appl ied. The second interesting feature of the field of mot ion in C.5 is that there are several regions which rotate as solid bodies as evidenced by the lack of gr id deformation w i th in them. These regions are separated by narrow corridors of deformation or mot ion shear. A n inspection of the raw images wi l l show that the CHAPTER 4. EXPERIMENTS 57 deformation corresponds to areas where leads are either opening or closing in a highly discontinuous fashion. F igure C.15 shows that the errors in the matching system are l imi ted to these regions of deformation. Th is observation was also made by Fily and Rothrock, (1986) in their study of the same S A R image pair. It is appropriate to examine two of these areas more closely to determine the nature of these deformations and why area correlat ion is not always an effective technique. The first area, denoted detail 1 (figure C.16), is f rom the lower left of the images and gives rise to the line of error vectors which the matched filter d id not improve. Though it is certainly not obvious f rom looking at the images, detai l 1 contains two ice fields moving as sol id bodies. The 25 k m . long floe marked A in figure l i b and lead E are w i th in the northern field while lead C is in the southern field. The different motions of these two fields has jo ined a number of disconnected leads into the long lead denoted D in l i b . Th i s long lead terminates in lead B wh ich appears to remain undeformed. Th is is a classic case of a discontinuous ice mot ion field which is so common in A rc t i c pack ice. It is clear that both matched filtering and cross correlat ion are able to estimate the solid body mot ion on either side of D. It is , however, inevitable that the windows of the area correlat ion w i l l straddle the discontinuity and produce erroneous estimates. In the manual measurement, features along the edges of the two mot ion fields were chosen and the discontinuity was resolved. Since D was formed f rom a number of leads in C.16a, the dig i ta l edges delineating the ice-water boundaries of these leads remain in C.16b. It is these digi ta l edges which must be detected, extracted and matched if an automated system is to successfully resolve the mot ion. Detail 2, shown in figure C.17, is f rom the upper right corner of the images and exhibits a very different k ind of deformation field. Rather than a simple discontinuity ,this detai l shows three fields moving in different directions. These directions are shown roughly in C.17b. The smal l scale rotat ion of the ice in the upper left of the detai l opened up lead J , whi le the mot ion of the field to the immediate right has greatly reduced the size of lead I and closed leads K and L. The discontinuity between the CHAPTER 4. EXPERIMENTS 58 two r ight hand fields is easily seen by tracking the feature marked G wh ich is wi th in the middle field, and floe F wh ich belongs to the right hand field. The main result of this discontinuous mot ion is the enlargement of lead H. T h e edge detection step was able to improve the performance of the area correlat ion by enhancing features in the ice pack wh ich remained intact. These features include: (a) the many ridge lines running through the ice which remain in spite of the leads opening between them and (b) the many smal l leads whose shape remains unaltered. In every region of the mot ion discont inuity in the imagery the matched filter was able to enhance significant features in the ice, either ridge lines or lead boundaries, and dramat ica l ly reduce the errors induced by match ing the raw image intensities. However to ful ly resolve discontinuous mot ion the matched filter system must be suppl imented by a system that can extract and match features. The stat ist ical significance of the correlation coefficients was also computed for the S A R vectors. The decorrelation length scale of 3 pixels (about 600 m) was observed. The total number of data points was 30 x 30 = 900, hence the effective degrees of freedom is approximately 265. F r o m this we find that al l coefficients greater than 0.13 are significant at the 95% level. Th i s test is, however, incapable of tagging erroneous vectors. Th is is i l lustrated in figures C.14 and C.15 where vectors whose coefficients are below the significance level are plotted in gray. Chapter 5 Discussion 5.1 Estimation versus Detection In this study the problem of comput ing ice mot ion was posed as a signal est imation problem and the matched filter was found to be an opt imal solut ion. However, the matched filter is more often found in connection w i th signal detection problems (Scwartz and Shaw, 1975, Stark and Tuteur, 1979), and it is therefore necessary to compare the two prob lem types and just i fy our choice. Signal detection is characterized by the common problem of detecting the presence or absence of a part icular type or class of signals in the presence of noise. The theory of detect ion concerns itself w i th either maximiz ing the probabi l i ty of detecting a signal when it appears or min imiz ing the probabi l i ty of mistaking noise for signal when it is absent. One or other of these cr i ter ia must be chosen since both cannot be opt imized simultaneously. The opt imizat ion leads to a likelihood ratio which in turn leads to a decision threshold wh ich is used to determine whether or not the signal is present in the input data. Signal est imat ion involves est imating the values of a signal f rom a given data sample. The signal , in this case, is known to be present but due to noise and a variety of other obscuring phenomena the data samples scatter about the actual signal values. A very simple example of est imat ion is l ine-fi t t ing. In this case the line is the signal whose slope and intercept are sought such that it best fits the data. Es t imat ion theory requires 59 CHAPTER 5. DISCUSSION 60 the defini t ion of a quanti tat ive measure of goodness-of-fit. The most commonly used measure, and that which is just i f iable on stat ist ical grounds, is the mean-square error. The problem could have been posed as a detection problem in which case the opt imal system would have been derived by maximiz ing the probabi l i ty of detecting a signal when a signal is present while keeping the false-alarm probabi l i ty at a tolerable level. Th is is simi lar to a radar-type detection problem where the false-alarm probabi l i ty can be specified based on the physics of the problem. The specif ication of such a parameter in the present problem has no physical meaning and it was decided to avoid this s i tuat ion. 5.2 Confidence Estimates In the present study the confidence of the mot ion estimates was evaluated in two different ways. In the A V H R R imagery the vector fields were smooth and no post-processing was necessary. The output of the system, namely the cross-correlation coefficients, represent the est imation error. In both the S A R and A V H R R the 95% confidence threshold was computed for the coefficients and vectors wh ich fell below this level were flagged. Th is process was, however, incapable of finding vectors which were obviously wrong. In the S A R case the first mot ion estimates were quite noisy and a post-processing step was inst i tuted to smooth the field. A s was stated two cr i ter ia were used to find bad vectors; deviat ion f rom the global statist ics of the coefficients and deviat ion from the local statist ics of the mot ion. No interpolat ion was done in this step, the original correlat ion funct ion was searched and the replacement vector was selected f rom it. Another possible means for evaluating quanti tat ive confidence in the vectors is the shape of the correlat ion funct ion around the peak. Wh i le several people have shown plots of this funct ion for i l lustrat ive purposes (Smith and Phillips, 1972; Vesecky et al, 1986a, b; Emery et al, 1986), none have attempted to analyse them. Th is is not to say that it is not on people's minds (Burns, Curlander, Vesecky - personal communication). CHAPTER 5. DISCUSSION 61 Vesecky et al have shown the effect of window size on the shape of the funct ion, where, as might be expected, the larger the window the smoother and broader the funct ion. The computat ion of certain representative shape characterist ics wi l l certainly provide more informat ion w i th which we can evaluate the goodness of the vector estimate. It may prove to be more useful than simply evalut ing the peak itself, however, it wi l l certainly be at the expense of computer t ime. 5.3 Computational Efficiency The computat ions by the system described in this study are performed at one resolution only. The scheme is made more efficient by using the F F T ; however, the system is extremely computat ion intensive. The system wi l l provide substant ial savings in t ime for A V H R R appl icat ions but is not yet pract ical for S A R images due to the necessity of very large search windows. T h e hierachical system used by Fily and Rothrock (1985, 1986) and Vesecky et al (1986a, 1987) does provide a more efficient means of comput ing cross-correlations for S A R imagery. In this system the image is low-pass filtered and sub-sampled a number of t imes thus producing an image pyramid. The mot ion est imation is first performed on the coarsest resolution images and the vectors are used to guide the est imation at the next highest level. Th is passing of informat ion between levels means that the sizes of the filter and search windows may be greatly reduced in size leading to much less computat ions and large savings in t ime. The cr i t ical stage is the first level since errors at this level can propogate to the next level and, due to the much smaller search space, produce incorrect estimates. Another important consideration wh ich has not been direct ly addressed in the aforementioned studies is al iasing due to incorrect low-pass filtering and sub-sampling. The low-pass filter must be a half-band filter in order that the result ing image may be sub-sampled correctly. A part icular ly efficient implementat ion of this is given by Clark and Lawrence (1984). CHAPTER 5. DISCUSSION 62 5.4 System Evaluation and Future Research The major assumptions bui l t into the system are that the mot ion w i th in each filter window may be approximated by pure translations (ie no discontinuities) and that the rotat ion of each window is smal l . Whi le these assumptions are restrict ive, they are obeyed in the major i ty of scenes of the central ice pack. The smal l number of A V H R R images used are felt to be approximately representative of typical pack ice condit ions and further testing would not disprove the success of the filter in detecting the major i ty of the ice mot ion for this appl icat ion. The S A R image pair is quite well-behaved, however, and it is not known, due to the paucity of data, whether it is t ru ly representative of central pack-ice condit ions. The results w i l l presently be examined to determine methods of improvement, but it is evident that the matched filter should form the core of any larger integrated system. Th is conclusion has recently been confirmed by N A S A ' s ice motion algorithm working group (Holt, 1986). Another impor tant considerat ion is that the system not only automates what a human operator can do but provides measurements of ice mot ion in areas of the image where it is extremely diff icult to obtain manual measurements. Improvements to the system may be suggested by looking at where the two main assumptions break down. The pure translat ion assumption w i l l fai l (i) at the ice edge, (ii) at locations of mot ion shear and (iii) where the ice mot ion exhibits divergence or convergence. Since leads manifest themselves as edges in the digi tal imagery a possible method for resolving the disparate mot ion of these features is to use feature-matching instead of area-matching in these areas. A recent thesis by Szczechowski (1986), was successful at detecting significant edges in A V H R R imagery using relaxat ion techniques, wh ich at least demonstrates that these features can be extracted. There have been several studies involv ing edge extract ion and matching in remotely sensed imagery (Little, 1980; Medioni and Nevatia, 1984; Majka, 1982), but the features extracted have a l l had a strongly l inear nature, in contrast to the highly deformed edges in the CHAPTER 5. DISCUSSION 63 S A R imagery. The A V H R R edges are generally more linear and these methods may be appl ied more direct ly to this imagery. Vesecky et al (1986a) have been quite successful at extract ing and matching edges in the S A R image pair used in this study. However the derived field duplicates the area correlat ion and does not provide any more information regarding the exist ing mot ion discontinuit ies. The edges wh ich exist between the rigid plates are highly deformed and it does not appear hopeful that any edge detection a lgor i thm could extract any edge segments in this area that are long enough to be useful for match ing purposes. A possible approach may be some sort of local texture analysis to y ie ld , for instance, the spatial d is t r ibut ion of ice versus open water (Fily and Rothrock, 1985) since ice in highly deforming areas wi l l be heavily interspersed w i th open water. A cue that significant deformation is tak ing place would be large changes is the rat io of ice to open water. The assumption of smal l rotations wi l l fai l at or near the M I Z which has been iden-tified as the most variable and dynamic region of the entire ice pack. B u t rotations which wi l l cause an area-correlation to fai l may also occur dur ing storms exhibi t ing large w ind stress cur l . Wh i le cross-correlation is str ict ly va l id only for pure transla-t ion, smal l rotat ions can st i l l be handled. Vesecky et al (1986b) have experimented wi th the effect of rotat ion on the correlation funct ion and found that for a 32 x 32 window, the autocorrelat ion peak falls f rom 1.0 at 0 degrees rotat ion to about 0.6 at 5 degrees and 0.2 at 10 degrees. One way to make the area correlat ion robust against large rota-tions is to compute and match rotat ion- and scale-invariant stat ist ical moments w i th in the filter and search windows (Hu, 1962; Sadjadi and Hall, 1978). The presence of rotat ion generally signals the presence of deformation which preempts the systematic decomposit ion of the image into windows. A more appropriate approach may be to segment the image based on some texture measure and compute the aforementionaed statist ics for these homogeneous regions. Vesecky et al (1986b) manual ly extracted ice floes f rom other S E A S A T S A R imagery and computed the series of moments advocated by Wong and Hall (1978). These moments, however, were very ambiguous measure-CHAPTER 5. DISCUSSION 64 merits of the under ly ing scene and were unable to discr iminate between the correct ice floe and other incorrect scenes. The image processing system at the Jet Propu ls ion Laboratory presently uses a human operator to identify polygons wi th in the scene which move as r igid pieces. The next step is to perform this step computat ional ly. A useful way to cue the whole system to potent ial ly dangerous rotations is to extract the stongest edges in the scene and match these. If these features rotate signif icantly then there should be some sort of decision mechanism to send the processing in a direct ion wh ich is more robust against rotations and deformations. A stand-alone system wi l l have a decision network as its spine. It w i l l have a knowledge base containing various types of informat ion: ice type signatures, last detected posit ion of the ice edge, etc and a pallette of matching methods f rom which to choose. The core of the system wi l l be the matched filter and those scenes wh ich are relatively undeformed wi l l y ie ld the densest gr id of observations. Bibliography A B D O U I. E . and W . K . P R A T T (1979) Quant i tat ive design and evaluat ion of en-hancement/ thresholding edge detectors. Proceedings of the IEEE, 67, 753-763 A B R A M A T I C J . F . (1983) Two-dimensional digi tal signal processing. In: Funda-mentals of Computer Vision, J . F . Abramat i c , editor, Oxford Univ ivers i ty Press, 27-56 A H L N A S K . and G . W E N D L E R (1979) Sea ice observations by satell i te in the Ber ing, Chukch i and Beaufort Sea. Sixth International conference on Port and Oceans under Arctic Conditions, 313-320 A L B R I G H T M . (1980) Geostrophic wind calculations for A I D J E X . 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C O L O N Y (1979) Large scale ice motions in the Beaufort Sea dur ing A I D J E X A p r i l 1975 - A p r i l 1976. In: Sea Ice Processes and Models, R. S. P r i t cha rd , editor, Univers i ty of Washington Press T H O R N D I K E A . S. and R. C O L O N Y (1982) Sea ice mot ion in response to geostrophic winds. Journal of Geophysical Research, 87, 5845-5852 T O M V . T . , G . K . W A L L A C E and G . J . W O L F E (1983) Image registration by a stat ist ical method. Proceedings of the SPIE, 332, 240-203 T O R R E V . and T . P O G G I O (1984) On Edge Detection Massachusetts Institute of Technology, Artificial Intelligence Laboratory, A I M e m o N o 768, (unpublished document) T U R I N G . L. (1960) A n introduct ion to matched filters. IRE Transactions, IT-6, 311-320 77 U N T E R S T E I N E R N . (1978) The Arc t i c ice dynamics jo int experiment. Glaciological data: Arctic sea ice Part I, Wor ld D a t a Centre A for Glaciology, Colorado, 25-32 V E S E K E Y J . , R. 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(1977) The incorporat ion of ice stat ion data into a study of recent A rc t i c temerature f luctuations. Monthly Weather Review, 105 W A L S H J . E . and C . M . J O H N S O N (1977) A n analysis of Arc t i c sea ice flucuations 1953-1977. Journal of Physical Oceanography, 9 W E E K S W . F . (1981) Sea Ice: the potential of remote sensing. Oceanus, 24 , 39-48 W E E K S W . F . and S. F . A C K L E Y (1982) The growth structure and properties of sea ice. C R R E L T R 82-1, U .S . Co ld Regions Research and Engineer ing Laboratory, (unpubl ished document) W E I N S T O C K R. (1974) Calculus of Variations, Dover Publ icat ions, 326 pp W O N G R. Y . (1978) Sequential scene matching using edge features. IEEE Transac-tions, A E S - 1 4 W O N G R. Y . (1979) Rada r to optical scene matching. Proceedings of the SPIE, 186 , 108-114 W O N G R. Y . and E . L. H A L L (1978) Sequential hierarchical scene matching. IEEE Transactions. C - 2 7 , 359-366 W O N G R. Y . and E . L. H A L L (1978) Scene matching by invariant moments. Com-puter Graphics and Image Processing, 8, 16-24 78 W O O D S J . W . (1984) Image detection and est imation. In: Digital Image Processing Techniques, M . P . E k s t r o m , editor, Academic Press Y A G L O M A . M . (1962) Stationary Random Functions, Prent ice Ha l l 79 Appendix A Review of Two Important Noise Processes A . l Gaussian Stationary White Noise The most important characteristic of both types of noise reviewed in this appendix is that they are random processes. It is well known that random processes are, formally speaking, an indexed family of random variables ~x.(i,j) (Papolous, 1984). This family of random variables is completely characterized by its n t h order probability distribution functions, which, in general, may be a function of the indicies i and j, which in this case are associated with spatial position. At each point in the image the noise is described by a particular probability distribution function P(x;i,j) = probability^',./) < x] (A.l) where x(i,j) denotes the random process and for a specific x(i,j) is a random variable. The density function is given, as usual, by: . dP(x;i,j) P{x;t,j) = — ^ — The function P(x; i,j) is called a first order distribution function of the process x(z, j). Given two spatial positions and (12,32)1 * n e interdependence between the two random variables x(t'1,j'i) and x(z ' 2 , y 2 ) is given by their joint probability distribution 80 funct ion: P{x1,x2;i1,j1,i2,J2) = p r o b a b i l i t y ^ ' ! , j i ) < x i 5 x ( i 2 , y 2 ) < x2,] (A.2) Th is is called a second order d istr ibut ion of the process x ( t ,y ) . The corresponding density funct ion is given by: / . . . . . d2P(xux2;i1,j1,i2,j2) p[xi,x2;ii,j1,i2,j2) = dxxdx2  Simi lar ly the n t h d is t r ibut ion funct ion is given by P ( x i , . . . , x „ ; . . .,in,jn) = p r o b a b i l i t y ^ ' ! , .71) < xu ...,x.(in,jn) < xn]. (A.3) for any n and «i , j i , . . . , in, jn. In what follows I shal l make the assumption of continuous random variables i n giving stat ist ical expressions, not ing that the discrete extension is straightforward. Wh i l e a complete descript ion of a random process requires a knowledge of its n t h order probabi l i ty functions this is not usually possible. In this case we may char-acterize the random process by stat ist ical averages of the random variables. Such averages are called ensemble averages and consist of the moments of the process where the m t h moment of the first order probabi l i ty d ist r ibut ion of x is defined by (Davenport and Root, 1958): poo E[xm(i,j)\= xmp{x;i,j)dx (A.4) J oo where E denotes mathemat ical expectat ion. C lear ly the moment of interest is the first moment wh ich defines the mean of the process: E[x{iJ)]=mx{iJ) (A.5) In general the mean of the product of two random variables is not equal to the product of the mean (ie. E[xijXki] ^ E[xij\E[xki)). If this is the case, however, the random variables are said to be linearly independent or uncorrelated. 81 Another useful statistical measure is the first order mth central moment of the process defined by: = E[{x{t,j) -mx{i,j))m] (A.6) = f£i x-mx{i,j))mp{x;i,j) dx The central moment of interest is given by m = 2 which defines the variance (denoted axij) °f ^he process. While the first order moments (ie. mean and variance) are im-portant characterizations of a random process they provide only a small amount of information about the process. A more precise description of a random process is pro-vided by its second order moments (Yaglom, 1962). The second order first moment is also known as the autocorrelation function and is given by: R{iiJi,i2,J2) = £[x(»i,.7i),x(»i,.7i) < xi,] (A-7) = /~  xix2P(xi,x2;ii,ji,i2,j2) dxx dx2 Similarly the second order second central moment is known as the autocovariance function and is defined by: C{h,ji,i2,j2) = E[(x(ii,ji) - m r (» i , ji))Gx(*2,.fe) - mx(i2,j2))} g. = -R(» l»J l , » 2 i ^ ) - " » * ( * i . i i ) ) » w * ( * 2 » ^ ) ) The autocorrelation function measures the dependence between different values of a random process at different points in space. Hence it describes the spatial variation of a random signal. As has been pointed out, the statistical properties generally vary in space. In a stationary process, however, the statistical properties are invariant to a shift in spatial origin. Stationarity of order one means the first order density function is independent of spatial position that is p(x;i,j) = p(x), ie E\x.(i,j)] - mx = constant E[(x.(i,j) - mx)2] = cr2 (A.9) If we denote spatial difference by {a,0), stationarity of order two must be such that: p{x1,x2]i1,jui2,J2) = p{xi, x2; a,0) (A.10) 82 Hence the autocorrelat ion and autocovariance depend only on spatial difference: R{a,P) = E[x{i + a,j + P)x{iJ)] = R{-a,-B) C{a, P) = E[{x{i + a , j + 0)- mx)(x(.\ j) - m x ) ] { A ' L l ) A random process is called stat ionary in the strict sense if i t is stat ionary of order n, that is its n t h order probabi l i ty density remains the same if the indicies i,j are shifted along the spatial axes. Th is implies that al l first order d is t r ibut ion functions are ident ical (ie. do not depend on i and j) and al l second order d is t r ibut ion functions depend only on the difference i — a and j — ft. For most pract ical situations the random processes that are encountered are not stat ionary in the strict sense, yet their first order moments are constant and the auto correlat ion functions satisfy 3.11. In this case the processes are said to be stat ionary in the wide sense. Hence wide sense stat ionary processes are staionary of order two. It should be noted that the aforementioned moments do not specify the random process completely unless the probabi l i ty d is t r ibut ion functions describing the process are al l Gauss ian. For independent random variables which may not have Gaussian distr ibut ions we may appeal to the central limit theorum to impose Gauss ian statistics on our problem (Yaglom, 1962). F ina l l y i f a l l the random variables in a random process are uncorrelated then we have (Oppenheim and Schafer, 1975) m*  = ° (A 12) R{k,l) = 6{k,l)  [ A - l ) where 6(k, I) is the Dirac delta funct ion defined as usual. Th is leads to a flat power spectrum (2.15) and the process is referred to as white noise. A s noted by Yaglom (1962), (2.15) corresponds to an unbounded power spectrum and hence infinite average noise power, thus white noise is useful only as a mathematical ideal izat ion. The ut i l i ty of this ideal izat ion stems f rom the fact that the frequency response of any physical ly realizable system is always zero outside some finite interval. Hence when a random process n(x,y), whose spectral density Sn{u,v) is constant over some finite region A, 83 appears at the input of such a system there is no need to describe in detai l how Sn(u, v) falls off at high frequencies and we may treat it as being constant. A.2 Gaussian Stationary Markov Noise W h e n a stochastic process is Markov many problems are considerably simpli f ied. Recal l that the character izat ion of random processes normal ly requires the knowledge of the n t h order probabi l i ty d istr ibut ion functions for each random variable. However the assumption that the variables are jo int ly Gaussian reduces the descript ion to simply a knowledge of the mean and covariance. The Markov property makes a s imi lar reduction in effort possible. In what follows the notat ion is simplif ied by using one dimensional processes to develop the ideas, where the extension to two dimensions is a straight forward extension. A lso several assertions regarding the Markov process are given wi thout proof, these proofs may be found in most standard texts on stochastic processes (Papolous, 1984; Melsa and Sage, 1973). A stochastic process is called Markov for al l i,j, where i and j are subscripts denoting spat ia l posi t ion, and i,j = l,...,n if, P{xn|Xn-l, Xn-2, • • • , Xi) = p(xn\xn-i) (A.13) ie. the condi t ional d ist r ibut ion of x n given al l past states (x„_!,..., xx) equals the condit ional d is t r ibut ion assuming only the most recent state(x„_x). A central characterist ic of a Markov process is that jo int probabi l i ty densities may be expressed as products of so-called transition probability densities, ie. p(xi, x 2 , . . . , xn) - p ( x n | x„_i ) p ( x„_i | x n _ 2 ) . . . p ( x n|xi ) p(xi) (A.14) that is the jo int d is t r ibut ion of Xi,...,x n is determined in terms of p(xi) and the transi t ional densities p ( x n | x n _ i ) . In a Markov process the past is independent of the future under condit ion of the present. Th i s means that if n > r > s then, assuming xr, the random variables xn and 84 xs are independent of one another ie. p(xn,xa\xr) = p(xn\xr)p(xr\xa) (A.15) A Markov process is called homogeneous if the condit ional density, p ( x n | z n _ i ) , is independent of n. A Markov process is called stat ionary if it is homogeneous and al l the random variables, x „ , have the same density functions. If a process x ( £ ) is Gaussian Markov w i th zero mean then its autocorrelat ion satisfies •R(*s.*a)-R ( * 2 . * i ) = J2(*s,« i ) f l(«2 ,ta) V * 3 > * 2 > * i - (A.16) If the process is stat ionary then, R{U - t2)R{t2 - h) = R{t3 - *i)J2(0) (A.17) or by subst i tut ing ts — t2 = A i and t2 — t\ = \ 2 we may wri te, R(\1 + A 2 ) = iE (A 1 ) J R(A 2 ) i 2 - 1 (0 ) . (A.18) The only continuous funct ion that satisfies (3.18) V A i , A 2 is an exponential of the form (Doob, 1953), R{r) = JE(0)e- e l TL (A.19) Hence the autocorrelat ion of a Gaussian stat ionary Markov process must be an expo-nent ial . Th i s is an important result which has been used successfuly in image processing where the two dimensional equivalent of (2.20) is used (Pratt, 1975) R(TX,TV) - K2 exp {-y/alr* + c$r*} (A.20) where K is an energy scaling constant and ccx, ay are scaling constants. In many cases the Markov process is assumed to be of seperable form in which case we may write R{TX,TV) = lf aexp{-a s|r,| + a „ | r w | } (A.21) 85 A p p e n d i x B A V H R R G r a p h i c R e s u l t s 86 Figure B.l: AVHRR band 1 - June 05 1984. APPENDIX B. AVHRR GRAPHIC RESULTS 88 Figure B.2: A V H R R band 1 - June 06 1984. APPENDIX B. AVHRR GRAPHIC RESULTS Figure B.3: A V H R R band 1 - June 07 1984. Figure B.l: AVHRR band 1 - June 11 1984. Figure B.2: A V H R R band 1 - June 12 1984. Figure B.3: A V H R R band 1 - June 13 1984. APPENDIX B. AVHRR GRAPHIC RESULTS 93 F igure B . l : A V H R R band 1 - Ju l y 30 1985. Figure B.2: A V H R R band 1 - Ju l y 31 1985. APPENDIX B. AVHRR GRAPHIC RESULTS Figure B.3 : A V H R R band 1 - August 06 1985. Figure B.l: AVHRR band 1 - August 07 1985. APPENDIX B. AVHRR GRAPHIC RESULTS 97 vector scale: 17.00 km t g i M* >> N H H H A \ \ \ N \ N N -,• H A ^^N. X» \ N N "i -i -i ^ N \ N. N > - i r_.^ N " \ N N "N -< \ \ \ \ \ N '/ . N N N N "-Y / \ A> N \ N \ / \ \ N \ N N S \ \ \ \ } J.. . >-t-— 1 V /.s sr / / W W .r-H %M l^t (\ ft!" <!t Figure B.2: Automat ica l l y measured displacements f rom June 05-06 1984. APPENDIX B. AVHRR GRAPHIC RESULTS 98 vector scale: 17.00 km !,ri« c / r . — ' l - - J L. J \=ZJ V? . J N N. \ \ \ N --a-, r i f r L t --"""TV r r& "• Figure B.3: Manually measured displacements from June 05-06 1984. APPENDIX B. AVHRR GRAPHIC RESULTS 99 measured displacement error e = u m -u a automatic manual X y X y X y a e -8 7 -6 6 -2 l 4 2.1 -3 0 -3 0 0 0 0 0.0 -6 4 -5 3 -1 l -3 1.4 -6 4 -6 3 0 I -7 0.5 -6 5 -6 3 0 0 0 0.0 -3 2 -3 2 0 0 0 0.0 -5 3 -4 3 -1 0 6 0.8 -5 6 -6 6 1 0 -5 -0.7 -7 7 -6 6 -1 1 0 1.4 -3 3 -4 3 1 0 -8 -0.8 -6 4 -6 5 0 -1 -6 -0.6 -7 6 -7 7 0 -1 -4 -0.7 -6 6 -6 7 0 -1 -5 -0.7 -4 4 -4 3 0 1 -8 0.7 -7 7 -6 5 -1 2 5 2.1 -7 7 -6 7 -1 0 5 0.7 -6 6 -6 6 0 0 0 0.0 -6 5 -6 6 0 -1 5 -0.7 -6 5 -7 6 1 -1 -1 -1.4 -8 6 -7 . 6 -1 0 4 0.8 mean errors X y a e -0.25 0.1 -0.4 0.25 Table B.l: comparison between the u a and um measurements for image pair 1 (June 05 - 06 1984), where e is the magnitude of the error vector and a is its angle. APPENDIX B. AVHRR GRAPHIC RESULTS 100 Figure B.4: Displacements predicted from theory for June 05-06 1984 along with the 24 hour averaged isobars for same period. APPENDIX B. AVHRR GRAPHIC RESULTS 101 measured displacement error e =u m -ua automatic manual X y X y X y a e -2 3 -2 l 0 2 30 1.4 -3 3 -3 2 0 1 11 0.6 -2 2 -2 2 0 0 0 0.0 -3 3 -3 2 0 1 11 0.6 -2 2 -2 1 0 1 18 0.6 -3 3 -4 3 1 0 8 -0.8 -2 2 -2 1 0 1 18 0.6 -4 4 -4 4 0 0 0 0.0 -4 3 -4 3 0 0 0 0.0 -4 3 -4 3 0 0 0 0.0 -4 4 -4 4 0 0 0 0.0 -2 0 -2 0 0 0 0 0.0 -4 4 -5 4 1 0 6 -0.7 -5 2 -5 3 0 -1 -10 -0.4 -5 5 -5 5 0 1 7 0.7 -5 3 -5 3 0 0 0 0.0 -4 0 -5 0 1 0 0 -1.0 -5 3 -6 3 1 0 4 -0.9 -5 3 -6 3 1 0 4 -0.9 mean errors X y a e 0.3 0.1 -15.6 0.0 Table B.l: comparison between the u a and u m measurements for image pair 1 (June 06 - 07 1984), where e is the magnitude of the error vector and a is its angle. APPENDIX B. AVHRR GRAPHIC RESULTS 102 vector, scale: 1—> 17.00 km \ \ \ \ TN. \ \ \. T \T 1 \ ^ \ T T X > > / T T J T\" H -.\ A x , \ r T j V v A H H -. ^ >V/-\ V -H\,H -1 A ^  \I ^ ^ \ v N N ' V V -V H -H ^ > X / ^ ^ J , , x > -4 H -. H -., X -c /" T V \ > \ ^ H H H \ T V r \ r r t,-\ \ -J ^  ^ \ \ "V>'' % // \ i I N VX^ , V V. v H \ X v -y y y s—..V y V V v S ^ x V V \ X V v IX , \ \ \ J / • V " " \ ^ 3 . . A l \ \ \ \ \ '•  •> ^^»—j/yJ*-^. 5 V "Jill? x X 6> \ \ \ \ \ \ \ \ \ \ \ Figure B.l: Residual displacements (see text) for June 05-06 1984 along with the hour averaged isobars for same period. 24 APPENDIX B. AVHRR GRAPHIC RESULTS 103 vector scale: 17.00 km N \ > N \ V V V N v \ V N \ V N \ V V V N N N V V H \ N N H " V —H •l E"- Til. ,.J - i-i. 1*L •W*'"-ri«.r r . , — r " • » J f !' H*' / L—"7" *,„->• i — tie w.-"' — \ jW i \ I I. I i { Figure B.2: Automat ica l ly measured displacements f rom June 06-07 1984. APPENDIX B. AVHRR GRAPHIC RESULTS 104 v e c t o r s c a l e : \ \ 17.00 km V + ft r~~ If 8' / A T ~ - i . .-L T"i p f H \ T'i, t' A. If 3 \ A / t1 \ V Figure B.3 : Manua l l y measured displacements f rom June 06-07 1984. APPENDIX B. AVHRR GRAPHIC RESULTS 105 Figure B.4: Displacements predicted from theory for June 06-07 1984 along with the 24 hour averaged isobars for same period. APPENDIX B. AVHRR GRAPHIC RESULTS 106 vector scale: 1— 17.00 km t I '—H.J I X* \ \ \ \ \ •\ \ \ \ \ ...... \ \ \ \ \ r \ "~xx \ \ \ T T KT T\I T T T \ T \ \ \ \ V / s \ > \ i i i i K i s v \N y / T T \ T T T f \ i T T 1 \ f XX \ I \ V T T T \ T T T T f, T T T T V V. / / \ " '  \ T T TV M n T M X \ " \ ^ N X..V * - T \ . t t f<r T ^ V.T \ X f h H H ^ T r r T T T T X T A A V / v >-v H H ' V T T T r j J T\T T T i V , - H H H H\.T- T -r -r V -r -r T-X-r ^ \ TTT X 7 T r \ r \ \ \ „ y X.^  ^ - >\ ^ ^ \ \ x../" X < ^ x . A V ^ , A / / X X \ \ X 'x \ t \ \ ..iii f t x . : . \ X. rt ft \ ^ , X \ X _ r J' •..!, 'i, \ jttTjiav X >^ ^"X^->-r/N X <iM X j ^ < \ r + r X X. X \ \ X ^---S ^ x, x, x \ \ \ v \ x x\. Figure B.5: Residual displacements (see text) for June 06-07 1984 along with the 24 hour averaged isobars for same period. APPENDIX B. AVHRR GRAPHIC RESULTS measured displacement error e = u m -u 0 automatic manual X y x y x y a e 1 - l 1 - l 0 0 0 0.0 2 -2 1 - l 1 -1 0 1.4 1 -1 1 - l 0 0 0 0.0 1 0 1 0 0 0 0 0.0 1 0 1 0 0 0 0 0.0 1 0 1 0 0 0 0 0.0 1 -1 1 -1 0 0 0 0.0 1 -2 1 -1 0 -1 18 0.8 -1 0 -1 0 0 0 0.0 1 -2 0 -1 0 1 0 1.0 1 0 1 0 0 0 0.0 1 -1 1 -1 0 0 0 0.0 0 3 0 0 1 0 1.0 0 2 0 1 0 1 0 1.0 0 -2 0 -1 0 -1 0 1.0 0 -1 0 -1 0 0 0 0.0 0 -1 0 -1 0 0 0 0.0 0 -1 0 -1 0 0 0 0.0 0 2 0 1 0 1 0 1.0 0 1 0 1 0 0 0 0.0 0 0 0 0 0 0 0.0 0 -1 0 -1 0 0 0 0.0 0 -1 0 -1 0 0 0 0.0 0 2 0 0 0 0 0.0 0 -1 0 -1 0 0 0 0.0 0 -1 0 -1 0 0 0 0.0 APPENDIX B. AVHRR GRAPHIC RESULTS 108 v e c t o r s c a l e : 17.00 km i- t- < <• h H h i- h J. h h K h X _t J. X X J. X X X X X X X X .1. V X X X ... i . >. X .1. Y V j. H Y V V j. H \ i . V V V H H V V V y V V \ N V V V \ V S- N- \ V V H \ V V V \ \ N V \ \ ~V -4 -^ -,(1=,. .—if "3 TP' r"'J" -1? y V .A-. A a Figure B . l : Automat ica l l y measured displacements f rom June 11-12 1984. APPENDIX B. AVHRR GRAPHIC RESULTS 109 vector scale: 1— 17.00 km I f sf a y / w 5 i 1. Jl C.Ji<^V*:':: v L, \ V C a a si:! e-i "h 17 HI i" *i j* Figure B.2: Manua l l y measured displacements f rom June 11-12 1984. APPENDIX B. AVHRR GRAPHIC RESULTS 110 0 -2 0 -1 0 1 0 1.0 0 2 0 1 0 1 0 1.0 0 2 0 2 0 0 0 0.0 0 2 0 2 0 0 0 0.0 -1 3 -1 2 0 1 8 1.0 0 1 0 1 0 0 0 0.0 0 -1 0 -1 0 0 0 0.0 0 2 -1 2 1 0 26 -0.2 -1 -1 0 -1 -1 0 45 -0.4 -1 2 -1 2 0 0 0 0.0 -1 1 -1 1 0 0 0 0.0 -2 2 -1 3 -1 -1 -27 -0.4 -1 0 -1 0 0 0 0 0.0 -2 2 -2 2 0 0 0 0.0 -2 1 -2 0 0 1 26 -0.2 mean errors X 0.0 y 0.1 a 2.3 |e| 0.1 Table B . l : comparison between the u a and u m measurements for image pair 3 (June 11 - 12 1984). APPENDIX B. AVHRR GRAPHIC RESULTS 111 \ \ TLJ-r r .Jiff rl — 7v \ ' U .-> f— . ' l ? ' [Ul . « ' ' f . / U X-Figure B.3: 24 hour averaged isobars for June 11-12 1984. APPENDIX B. AVHRR GRAPHIC RESULTS 112 vector scale: 17.00 km V V v v V V V V V V V V V V V V V v v \ \ \ \ \ \ \ \ V V Y Y Y V V \ x \ • A - " i v / f r i: / . . ' J ..... W "ft •";rv; \ I . r Figure B.4: Automatically measured displacements from June 12-13 1984. APPENDIX B. AVHRR GRAPHIC RESULTS 113 vector scale: 1— 17.00 km tr-^nj s ,— jr .t~"... i i. n j i ,BJ* T I t . f L v > > j r r 1" r •\ — * • { _ J Vs-. "i s . q^K? Figure B.5: Manually measured displacements from June 12-13 1984. APPENDIX B. AVHRR GRAPHIC RESULTS 114 measured displacement error e = u m -u a automat ic manual X y x y x y a |e 0 -1 0 - l 0 0 0 0.0 0 -2 0 -2 0 0 0 0.0 0 -2 -1 -2 l 0 26 -0.2 -1 -1 -1 -1 0 0 0 0.0 -2 0 -1 -1 -1 1 45 0.6 -2 -2 -1 -1 -1 -1 0 0.9 -1 1 -1 1 0 0 0 0.0 -1 2 -1 1 0 1 18 0.8 -2 0 -1 0 -1 0 0 1.0 -1 1 -1 2 0 -1 -18 -0.8 -1 0 -1 0 0 0 0 0.0 -1 0 -1 0 0 0 0 0.0 -2 1 -2 1 0 0 0 0.0 -2 3 -2 3 0 0 0 0.0 -2 1 -2 1 0 0 0 0.0 -1 2 -2 2 1 0 18 -0.1 -1 2 -2 2 1 0 18 -0.1 -3 3 -2 3 -1 0 -11 0.6 -2 2 -3 1 1 1 26 -0.7 -2 2 -2 2 0 0 0 0.0 mean errors X 0.0 y 0.0 a 6.1 |e| 0.1 Table B . l : comparison between the u a and u m measurements for image pair 4 (June 12 - 13 1984). APPENDIX B. AVHRR GRAPHIC RESULTS 115 vector scale: 17.00 km \ WW*? 4 i jJ$-J,1 N N ^O- N \ V v N ^ K W V N - ^ ^ N N Y Y N - , vv -K ^ - ^ N N v V x-^ \ n V ' v i / y \ _ V V ^x^ N X J X N X-V, \ V \ V N ^ V V V • N- N \ \ A,> N V ^ ' \ V A 1 < s X . ' V "x \ & \ I f ^ s ! "" *-r-gf I \ \ f fws-'"'.&f:' J \ 1 X . \ \ \ \ \ \ X Figure B.l: Displacements predicted from theory for June 12-13 1984 along with the 24 hour averaged isobars for the same period. APPENDIX B. AVHRR GRAPHIC RESULTS 116 v e c t o r s c a l e : 17.00 km r— *h <_ \ 1- h k k -r h - I - «^  AV^v. y * \- ^ r ^ S ^ A. -H ^ x t T V V «• I- I- A^  •• f c )- v V r I— I— i — i — " K I— \ A v-K "\ ^ ^ -\ ^ \ > "\ \ \ ^~ i V Y Ir h h I- ^ K" 1^  K- \ .:" .f / .r S f J I 1 / \ ' V v . y v v y -c • T Y* H T A T T ?* ' V h * T T I— h- Y" H <. T ^ T T I- Y~ A. A V v „ N. .1 X \ \ \ Figure B.2: Residual displacements (see text) for June 12-13 1984 along with the 24 hour averaged isobars for the same period. APPENDIX B. AVHRR GRAPHIC RESULTS 117 v e c t o r s c a l e : 17.00 km y . y TVS" A. A. * A. I— I— I— I— I— I— I— \ 1-V V V V h t-y y A A ;i -i- T T . " t- I- t- h h -i-^ ^ ^ h I- T T T y y y y y K y y y y > «. \ T -< -< -\ •\ <. T -< T y y ' y / \ i V V C 3 < y > L rJ '-3"' Figure B .3 : Automat ica l ly measured displacements f rom Ju l y 30-31 1985. APPENDIX B. AVHRR GRAPHIC RESULTS 118 vector scale: 1— 17.00 km SI " } ) / ? , — r - .,.r-,<iiJ-/ '-TV "XJ"'S^-A;l f!l. ^ & TO-**? s- ,-' "Pi) \ X. Mi, Figure B.4: Manually measured displacements from July 30-31 1985. APPENDIX B. AVHRR GRAPHIC RESULTS 119 measured displacement error e =u m -ua automatic manual X y X y X y a e 0 -1 1 - l -1 0 45 -0.4 1 0 1 0 0 0 0 0.0 2 -1 0 0 -1 26 0.2 2 0 0 0 0 0 0.0 1 0 1 0 0 0 0 0.0 1 0 1 . 0 0 0 0 0.0 -1 1 -1 l 0 0 0 0.0 1 0 1 0 0 0 0 0.0 1 0 1 0 0 0 0 0.0 1 0 1 0 0 0 0 0.0 1 -1 1 0 1 -1 45 0.4 -3 1 -1 -1 -2 45 1.6 1 0 1 0 0 0 0 0.0 0 1 -1 1 1 -45 -0.8 1 -1 1 0 0 -1 45 0.4 1 0 1 0 0 0 0 0.0 1 0 1 0 0 0 0 0.0 1 0 1 0 0 0 0 0.0 1 -1 1 -1 0 0 0 0.0 1 -3 1 -2 0 -1 8 1.0 -1 -2 0 1 0 1.0 1 0 1 0 0 0 0 0.0 1 0 0 -1 0 0 -1.0 1 -1 1 -1 0 0 0 0.0 1 0 1 0 0 0 0 0.0 mean errors X 0.0 y -0.2 a 7.5 |e| 0.1 Table B.2: comparison between the u a and u m measurements for image pair 5 (July 30 - 31 1985). APPENDIX B. AVHRR GRAPHIC RESULTS 120 Figure B.l: Displacements predicted from theory for July 30-31 1985 along with the 24 hour averaged isobars for the same period. APPENDIX B. AVHRR GRAPHIC RESULTS 121 vectpr scale:,1— J,7.p0 km \ S *\ / X J . v . x X X x \ \ \ \ \ \ \ X X. X v X x. ; X. • X . - - X * X ^ H * ^ X " x j " W X ^ X x. i. x . . X,S x X J X . V. -.J . X x ^ f V " x V+ » J. J- xV.J. J- J. X. r V X '"v. v. v-. 'V. V V X ,, 1 . . X . i X x X X x X x W ,/r X 'tis. ! !%fi* X X / \ X. Figure B.2: Residual displacements (see text) for July 30-31 1985 along with the 24 hour averaged isobars for the same period. APPENDIX B. AVHRR GRAPHIC RESULTS 122 vector scale: 1— 17.00 km tr\mj r i ^ . \ V V v \ v , , A 4. 4. * 4. A * * * * * H h * T T "\ T *r *r -r IS. i r t f ^ T rV / \ X x . Figure B.3 : Automat ica l ly measured displacements f rom August 06-07 1985. APPENDIX B. AVHRR GRAPHIC RESULTS 123 vector scale: 1 — 17.00 km L^AI..? / j ^ X / } i / . ixyx> v V v. \ f / / \ j ( Figure B.4: Manually measured displacements from August 06-07 1985. APPENDIX B. AVHRR GRAPHIC RESULTS 124 measured displacement error e = u m - u a automatic manual X y X y x y a e 1 -1 1 - l 0 0 0 0.0 1 -1 1 - l 0 0 0 0.0 1 -1 1 - l 0 0 0 0.0 0 -2 0 - l 0 -1 0 1.0 -5 -5 -4 -4 -1 1 0 1.4 -2 6 -2 6 0 0 0 0.0 -1 3 -1 3 0 0 0 0.0 0 -1 0 -1 0 0 0 0.0 0 1 0 1 0 0 0 0.0 1 0 1 0 0 0 0 0.0 1 0 1 0 0 0 0 0.0 1 0 1 0 0 0 0 0.0 1 -1 1 -1 0 0 0 0.0 1 0 1 0 0 0 0 0.0 0 1 0 1 0 0 0 0.0 0 2 0 2 0 0 0 0.0 -1 3 -1 2 0 1 9 1.0 -2 5 -2 5 0 0 0 0.0 -3 6 -3 6 0 0 0 0.0 -2 3 -1 4 -1 -1 -19 -0.5 -1 4 -1 4 0 0 0 0.0 -2 4 -2 5 0 -1 -5 -0.9 -3 7 -3 7 0 0 0 0.0 mean errors X y a e 0.0 0.0 -0.7 0.1 Table B.l: comparison between the u 0 and u m measurements for image pair 6 (August 06 - 07 1985). APPENDIX B. AVHRR GRAPHIC RESULTS 125 Figure B.5: Displacements predicted from theory for August 06-07 1985 along with the 24 hour averaged isobars for the same period. APPENDIX B. AVHRR GRAPHIC RESULTS 126 vector scale: 1 — 17.00 km u ^ . 1 ""~-Tw, J !K - ic* / r jf TV T " > H y. r T y <  A T T ^ T V r T A T A \ \ \ * A T \ \ \T A / ' \ V i '~;n I / 'Jit •*" -.1" .r" ji* K 1 \ f .-Si. V If \ \/> Figure B . l : Residual displacements (see text) for August 06-07 1985 along w i th the 24 hour averaged isobars for the same per iod. Figure B.2 : Sample cross-correlation function for typical window for the June 11-12 1984 Figure B.3 : Sample cross-correlation function for typ ical window for the June 11-12 1984 -Appendix C SAR Graphic Results 129 APPENDIX C. SAR GRAPHIC RESULTS 130 Figure C l : S E A S A T S A R - revolution 1439 - October 5 1978. APPENDIX C. SAR GRAPHIC RESULTS 131 Figure C.2 : S E A S A T S A R - revolution 1482 - October 8 1978. APPENDIX C. SAR GRAPHIC RESULTS 132 N O R T H L A T I T U D E W E S T L O N G I T U D E Figure G.3: Locat ion of S E A S A T S A R image pair used in this study. F igure C.4: Manua l l y measured ice displacements computed from S A R pair (courtesy of D . A . Ro th rock , Universi ty of Washington). APPENDIX C. SAR GRAPHIC RESULTS 133 Figure C.5: Manual measurements of ice displacement, (a) shows displacements as vectors, (b) shows displacements as a deforming grid (see text), square denotes outline of undeformed grid. APPENDIX C. SAR GRAPHIC RESULTS 134 Figure C l : Convolution of Sobel operator with rev. 1439. APPENDIX C. SAR GRAPHIC RESULTS 135 F igure C .2 : Convolut ion of Sobel operator w i th rev. 1482. APPENDIX C. SAR GRAPHIC RESULTS 1 3 6 APPENDIX C. SAR GRAPHIC RESULTS 137 F igure C l : Zero crossing magnitudes f rom rev. 1439. APPENDIX C. SAR GRAPHIC RESULTS 138 APPENDIX C. SAR GRAPHIC RESULTS 139 F igure C.3 : Zero crossing magnitudes f rom rev. 1482. APPENDIX C. SAR GRAPHIC RESULTS 140 Figure C l : Automatically measured displacements from first order operator experi-ment. APPENDIX C. SAR GRAPHIC RESULTS 141 vector scale: 1 — 4.00 km Figure C.2: Automatically measured displacements from second order operator exper-iment. APPENDIX C. SAR GRAPHIC RESULTS 142 vector scale: 1 — 4.00 km x i T J- y y y y A Y . T . . T \ V \ 1 v • • • •/ < 4 &A T r J. T A 1" " 7 * - -Figure C.3: Errors in the first order experiment. APPENDIX C. SAR GRAPHIC RESULTS 143 vector scale: . v N •N 4.00 km . s V V 1 , .,.: / T • X • j . ^ . 1 . • T T • • • • T V • ' ' T " " ' • / Y-T L . - J . , . . . J- * \ . • • • > \ T • v T X 1 • • * ^ r i n T ^ - A T . Figure C.4: Errors in the second order experiment. APPENDIX C. SAR GRAPHIC RESULTS 144 F igure C .5 : (bottom). Deta i l 1 f rom the lower left corner of rev. 1439 (top) and rev. 1482 APPENDIX C. SAR GRAPHIC RESULTS 145 Figure C.6: Detail 2 from the upper right corner of rev. 1439 (top) and rev. 1482 (bottom). 

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