GEOPHYSICAL ASPECTS OF SOIL MAPPING USING AIRBORNE GAMMA-RAY SPECTROMETRY by Stephen David Billings fulfsiultibmhnmee insttit se Ad retqhueti hrfeeo mre onft s DegreoDefo ctooPrfh ilosophy eD ScieCrop of partment n ces FacultoAyfg riculture ThUen iversitoSyfy dney ,WSN 2 006,A ustralia Octobe1r9 98 i Statement This thesis is an account of my own research undertaken during the period March 1995 to October 1998, while I was a full-time student in the Department of Crop Sciences at the University of Sydney. Except as otherwise indicated in the acknowledgements and in the text, the work described is my .nwo This thesis has never been submitted to another university or similar institution. Stephen David Billings Canberra October 1998 ii TCARTSBA enrobriA yar-ammag yrtemortceps si a etomer gnisnes euqinhcet taht si ylgnisaercni gnieb desuot tnemelppus ro ecalper gnitsixe seuqinhcet rof lios .gnippam syar-ammaG morf evitcaoidar yacedfo potassium, uranium and thorium in the soil are counted with a large crystal detector mounted in a low-flying aircraft. However, even with recent developments in multichannel processing techniques, there are difficulties in creating accurate images from these noisy, blurred and unevenly sampled rev O.atad eh tgnimoc se difficulties is the scope of this thesis. ,tsriF a citegrene-onom ledom fo yar-ammag yrtemortceps si depoleved taht stnuocca rofrotceted shape and aircraft movement during data collection. The detection probability depends on gamma-ray energy, and the area and thickness that the detector projects in the direction of a source. The resulting geometrical model is approximately radially symmetric and predicts a reduced measurement footprint compared to a model that ignores detector shape. The aircraft movement introduces asymmetry, which can be significant when there is a low rati o of aircraft height to distance travel ltnemerusae mgniru dde . Second, linear error propagation is used to trace the initial Poisson counting error through the various processing operations applied to a gamma-ray spectrum. The method returns est mi ates of relative error for surveys flown at diff e r ent heights and with different isotope concentrations and background contamination. The optimal Wiener filter for deconvolution of radiometric data is derived and used to estimate the spatial resolution of radiometric surveys. The analysis reveals that radiometric surveys are usually adequately sampled along-lines but under - .meh tssorc adelpmas ,drihT I poleved a lareneg ygolodohtemw hich I call the Arbitrary Basis Function FBA( ) ,krowemarf that unifies many differ e nt types of interpolation method ( radial basis functions , splines ,nigirk gdna sinc functions). An FBA consists of an optional polynomial and a sum of translations of a fixed basis function multiplied by a set of weights. The FBA emarf krow nac etadommocca yportosina dnanac be used for exact or smooth interpolation. Further, it c au se s the interpolated surfaces to inherit certain desirable properties from the basis function . The princ ipal impedi ment for practical application of the method is the computer memory and time required to solve the resulting large and de nse matrix equations. A three-step process for reducing these requirements is proposed, and the first step (an iterative matrix solver) is impl e nem ted on a small radiometric survey. iii Next , I evired eht Fourier transform of an FBA surface. Trad it ionally, the Fourier transformation required to deconvolve radiometric data would be calculated by a Fast Fourier Transform of an imaged version of the data. This ra ises several important considerations (edge mismatch, survey spag , quadrature app roximation, etc .) regarding the relationship between the continuous transformation of theory and the discrete version of practice. The FBA approach, by contrast, can be used to overcome many of these problems, as it defines an interpolant that extends over all space. Further, the Fourier transform of an FBA surface can be calculated exactly on an arbitrary grid, and has a particularly simple form that facili tates computation and subsequent analysis. Lastly, I show that convolution or deconvolution of an FBA surface gives another FBA surface with the same set of weights but a different basis function. This means that the deconvolution of radiometric data can be entirely encapsulated by the new ba sis fu nction, thus obviating the need to calculate the Fourier transform of the data. The major difficulty with the method is calculating the new basis function, which can be complex for singular bases, such as splines. Spline deconvolution htiw a yllaidarmys metric Wiener filter is suc c yllufsse detnemelpmi dna na etamixorppadohtem developed for the non-radial case. FBA can be used for interpolation, Fourier transformation, and convolution/deconvolution foyna type of data regardless of the sampling configuration (regular or irregular, sparse or dense, etc.). This thesis demonstrates the potential of the FBA ygolodohtem no a llams dataset, in this case a subsection of the Je gnolam radiometric survey. Application of the FBA krowemarf ot regralsyevrus would require assembling several fast algorithms that currently exist in isolation. vi STNEMEGDELWONKCA The greatest source of encouragement, advice and inspiration I received during this thes is was f ro m aG yrr asweN m of DSTO. He showed a keen interest in a PhD project for which he rec ei devon official recognition and that was peripheral to his major field of study. All the material from half aw yht rough Chapter 4 to the end of the thesis has its roots in ideas I discussed with Garry . knahT oy u very much for all your assistance. gniruD eht esruoc fo ym DhP I saw yrev etanutrof ot eb detaicossa htiw ORISC dnaL dna ,retaWohw dedivorp ym htiwc omputer support and an office at Black Mountain in Canberra. Alan Marks in particular was always ready with a word of encouragement and I benefited tremendously from his endless source of contacts in the geophysics field. Alan always told me that once I got a PhD I’d evah na noinipo no gnihtyreve dna on .sdneirf m’I gnitiaw ot ees tahwsneppah . elihW ta ORISCI worked under the supervision of Brian Tunstall. He was always prepared to spend time discussing all sorts of different issues and I learnt a lot from our many lengthy conversat .snoi I saw yrros otees nairB evael ORISC erofeb I dehsinif eht .siseht knahToy u ot enoyreve esle ta ORISCgnidulcni Andrew Bell , Chris Barnes nry Byu Gdn adle Hxel A,pukci Pffoe G,hcti Frete P, e . knahT uoy ot ym lapicnirpus pervisor , Brent Jacobs, who took over the helm after my initial supervisor, Craig Pearson , left the department. Brent was always ready to give advice and provided emos yrev evitcurtsnoc sweiver fo lareves fo ym .sretpahc evetS streboR emaceb ymetaicossa rosivrepus rof eht tsal raey fo ym ,DhP dna I detifeneb morf ruo ynam snoisses htiw rekram nepdna white board. Steve also reviewed a lot of the mathematical material in the thes .si knahT uoy otym t ow honours supervisors, M mlocladirbmaS ge and Brian Kennett who kindly revi dewe emos foeht chapters in the thesis and were always ready to provide advice. The Australian Geological Survey Organisation flew the gnolameJ radiometric survey specifically for my PhD research. Thank you to all in the airborne mapping group and Brian Minty in particular. Brian was always prepare d to assist with advice on every aspect of gam am y.ayrr-temortceps serugiF 1, 2 and 3 in Chapter 2 were reproduced with permission from an electronic version of Brian’s thesis. He contributed the spectral components for calculating live-time (Appendix C), calibration data (Appendix D) and a Monte Carlo code for assessing radon error (Appendix E). The ins piration for Chapter 5 was provided by kciR nostaeB ohw I desrevnoc htiwlevisnetxe yyb E-mail and who financed a one month trip to visit him in Christch .hcru I osla evah kciR otnaht krof v introducing me to Matlab, which is a fantastic environment for testing new ideas and methods. Discussions over E-mail with Jens Hovgaard resulted in the geometrical detector model derived in Chapter 3. He also supplied the Monte Carlo modelling results that were used to validate the model. nI rebmeceD 6991 I tem nhoJ ,renruT ohw saw tuoba ot ecnemmoc sih sruonoH raey ta eht University of Canberra. We decided to merge our projects together, with John concentrating on the soil mapping and myself on the geophysics. Some very fruitful sessions of fieldwork followed during which we had an excellent time . Thanks for everything John, not the least for those 500 odd soil horizons you analysed back in the laboratory ( sorry I didn’t use them in the thesis!). naD ,ydomraC Grant Koc h and Step hen Griffin also assisted with the fieldwork and their snoitubirtnoc era yllufetarg .degdelwonkca naD saw syawla ydaer dna gnilliw ot edivorp emoshcum needed moral support during the PhD. No offence Dan, but I think you have the largest rep ertoire of dab sekoj fo enoyna I wonk dna( nalA skraM !)seerga I tnrael a tol morf ynam yhtgnelenohp conversations with Stephen Griffin, including the practicalities of geophysical surveys, the utility of Fourier .sreyal pnoin Uybgu Rnailartsu Asuoma f,yltnatropm itso mspahre pdn a,smrofsnar t nO a lanosrep ,level I dluow ekil ot knaht ym owt doog ,sdneirf miJ palnuD dna oderflA .ohcamaC I enjoyed the morning sessions at the A UN miJ htiw .myG oderflA dedivorp elpma noitaripsni rofem to finish, as I bet him that I would be first to hand in the thesis. Unfortun ,yleta I tsol yb tuoba araey at which point the main incentive to finish was to avoid the painful “Have you handed in yet”? I would like to extend my appreciation to the Land and Water Research and Development Corporation for a PhD scholarship and also their support and encouragement. knahT uoy ot eht sedlyW-gnolameJ snialP gnireetS ,eettimmoC dna ieR remeuB dna naI htimSni particular, for funding the soil sampling used in Supplement 1. During the course of our fieldw kroni eht sedlyW-gnolameJsnialP Bob and Jan McPhillam y yrev yldnik dewolla su ot yats ni a egattocno rieht .ytreporp knahT uoy ot ,meht yasdniL llaB dna divaD ylligneP for letting us survey their zi bru on oskcoddap a te sekib-dau qerr - up. ,yllaniF tub ta eht pot fo ym ,tsil I dluow ekil ot knaht ym ,efiw Janee for all her support and encouragement, especially during the last few months while I was writing up. The writing of a thesis can be a difficult time and her understanding during this time helped me enormously. iv ELBAT SFTONETNOC NOITCUDORTN I 1RETPAHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1 Quantifying the influence of survey geometry on spatial resolution and assay uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 ,noitalopretn iro fsdohte mdevorpm if onoitatnemelpm I2.2 Fourier transformation and deconvolution of radiometric data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Thesis layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 GNIPPA MLIO SN ISNOITAREDISNO CLACISYHPOE G 2RETPAHC YRTEMORTCEP SYAR-AMMA GENROBRI AGNISU . . . . . . . . . . . . . . . . . . . . . . . . 8 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 yrtemortcep syar-amma gf oslatnemadnu F2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 sya ramma gf osecruo slaruta N1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 muiroh tdn amuinar u,muissato pf oyrtsimehcoida R1.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . 10 muiroh tdn amuinar u,muissato pf oyrtsimehcoe G2.1.2 . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.3 Disequilibrium in uranium and thorium decay series . . . . . . . . . . . . . . . . . . . . . 14 2.2 Sources of background radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Interactions with matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4 Interference with the signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 sya ramma gf onoitcete D5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 gniyevru syar-amma genrobri A3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.1 The effect of aircraft height on the measured count rates . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 Calibration of radiometric surveys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 syevru syar-amma gf ognissecor P3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 yrtemortcep syar-amma genrobri agnis ugnippa mlio S4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.1 Existing techniques for soil mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2 Soil information required for land evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.3 Parametric yevru slio s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 yrtemortcep syar-amma ggnis ugnippa mlio ssuoiver P4.4 . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5 Problems with current mapping techniques: A case study . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.1 Soil sampling stratified by airborne radiometrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 stluse rf oyrammu S2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 iiv 6 Geophysical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 oita resion-ot-langi swo leh tgnivorpm I1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6.2 Assay uncertainty and spatial resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 6.3 Interpolation of data to a regular grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.4 Compensation for the smoothing effect of height . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 YRTEMORTCEP SYAR-AMMA GENROBRI AF OLEDO M A 3RETPAHC . . . . . . . . . 46 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 langi sya ramma geh tgnilledo M2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 larene g A1.2 citegrene-onom ledo m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ledo meh tgniyfilpmi S2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.3 Derivation of existing results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3 A geometrical detector model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.1 The solid angle of a rectangular detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.2 Incorporating variation in the efficiency of a rectangular detector . . . . . . . . . . . . . . . . 57 ledo mf onosirapmo C3.3 l ed results to experimental data . . . . . . . . . . . . . . . . . . . . . . . . . 62 4 The point spread function in airborne gamma ray spectrometry . . . . . . . . . . . . . . . . . . . . . 63 4.1 The point spread function of a 16.8 litre airborne detector . . . . . . . . . . . . . . . . . . . . . . 64 5 Comparison to Monte Carlo simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6 Contributing area calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 7 Incorporation of the platform velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 ENROBRI AN IYTNIATRECN UDN ANOITULOSE R 4RETPAHC SYEVRU SYAR-AMMAG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 2 Propagation of errors during processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.1 Background to error propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.2 Tracing the error introduced by each processing step . . . . . . . . . . . . . . . . . . . . . . . . . 81 2.3 Assay uncertainty under different conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3 Spatial resolution of radiometric surveys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.1 Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.2 The Optimal Wiener filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.3 Estimating the auto-correlation of the signal and noise . . . . . . . . . . . . . . . . . . . . . . . . 98 3.4 Spatial resolution of the gnolameJ yevru s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 I5.3 nfluence of assay uncertainty on spatial resolution . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.6 Influence of aircraft movement on spatial resolution . . . . . . . . . . . . . . . . . . . . . . . . . 107 viii 3.7 Effect of aircraft height on spatial resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 DN ANOITALOPRETN IRO FKROWEMAR FLARENE G A 5RETPAHC SMOOTH FITTING OF GEOPHYSICAL DATA . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 2 Exact interpolation using Arbitrary basis functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 3 Solution of the interpolation equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 3.1 Computational cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 3.2 Iterative methods for solving the matrix equations . . . . . . . . . . . . . . . . . . . . . . . . . . 119 3.2.1 Converting the interpolation equations to a positive-definite form by QR factorisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 3.2.2 sozcnaL iterative methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 3.3 Preconditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 3.4 Matrix-vector product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4 Generality of the Arbitrary Basis Function framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.1 Choice of basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.1.1 Radial and non-radial basis functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.1.2 Thin-plate splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.1.3 Tension splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.1.4 gnigirK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 4.2 Smooth fits to geophysical data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4.2.1 Regularising the solution with VCG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.2.2 Smoothing by using less nodes than data constraints . . . . . . . . . . . . . . . . . . . . 134 I5 nterpolation and smooth fitting of the gnolameJ yevru s . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.1 Exact interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.1.1 Exact thin-plate splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.1.2 Tension spline fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.2 Smooth fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 gnihtoom S1.2.5 spline fits to the thorium data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 5.2.2 Smooth fitting using the sinc function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.2.3 Spatial prediction by kriging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 skrame rgnidulcno C4.2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 ix EH T 6RETPAHC REIRUOF SISA BYRARTIBR AN AF OMROFSNAR T ECAFRU SNOITCNUF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 2 The Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 2.1 Definition of the Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 2.2 Discretisation and the 1-D Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 2.3 Fourier .ata dlacisyhpoe gD- 2f osmrofsnar t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 3 Fourier transforms and Arbitrary Basis Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 3.1 The Fourier transform of an Arbitrary Basis Function . . . . . . . . . . . . . . . . . . . . . . . . 160 3.2 The Fourier transform of a sinc interpolant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 3.3 The Fourier transform of a spline interpolant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 4 The Fourier transform of the gnolameJ radiometric data . . . . . . . . . . . . . . . . . . . . . . . . . . 167 5 Effect of basis function and solution method on the Fourier transform . . . . . . . . . . . . . . . . 172 6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 DN ANOITULOVNO C 7RETPAHC NOITULOVNOCED OF ARBITRARY SECAFRU SNOITCNU FSISAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 2 Basis transformation: An exact method for filtering FBA surfaces . . . . . . . . . . . . . . . . . . . 178 3 Weight transformation: An approximate method for filtering FBA surfaces . . . . . . . . . . . . . 181 4 Application of the FBA elpmax elanoisnemid-en oelpmi s ao tygolodohte m . . . . . . . . . . . . 184 4.1 Convolution with the sinc function as a basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 4.2 Convolution of curves defined by a cubic spline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 5 Extension to two-dimensional convolution and deconvolution . . . . . . . . . . . . . . . . . . . . . . 202 5.1 Thin-plate spline Basis transformation with radially symmetric filters . . . . . . . . . . . . 203 5.2 An approximate method for thin-plate splines and multiquadrics . . . . . . . . . . . . . . . . 208 5.3 Extending Weight transformation to splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 6 Effect of FBA parameters on the deconvolution of radiometric data . . . . . . . . . . . . . . . . . . 210 7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 NOISSUCSI D 8RETPAHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 siseh tf oyrammu S2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 langi syar-amma geh tgnilledo M1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 2.2 Spatial resolution, assay precision and Wiener filtering . . . . . . . . . . . . . . . . . . . . . . . 217 2.3 The FBA approach to interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 2.4 Fourier transforms, convolution and deconvolution of FBA surfaces . . . . . . . . . . . . . 220 sdohte mf onoitargetn I5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222