Exploration Geophysics (1999) 30, 161-166

161Exploration Geophysics (1999) Vol. 30, Nos. 3 and 4

Correcting aerial gamma-ray survey data for aircraft altitude

Maurice Craig1 Bruce Dickson2 Stewart Rodrigues3

Key Words: Deconvolution, inverse filtering, gamma-radiometry, Wiener filter

ABSTRACT

In their ascent to the aircraft detector system, the gamma-raysrecorded in airborne radiometric surveys are attenuated, first by thesurface materials wherein they originate, then by the interveningatmosphere. Increased ground-clearance thus entails reducedcount-rates. It also implies diminished spatial resolution, becausethe same cone of incident radiation derives from progressivelylarger ‘footprints’ that, for a given sampling rate, increasinglyoverlap. But suitable post-processing of the gridded, two-dimensional imagery can be used to correct these types of height-dependent degradation and hence produce sharper, quantitativelyuseful maps of radioactive isotope distribution. An essentialprerequisite for such inverse filtering is noise-suppression,achieved here through maximum-noise-fraction (MNF)transformation of multi-channel data. High-frequency noiseamplified by the deconvolution step is brought under control by avariant of Wiener filtering. The combined de-noising anddeconvolution process is illustrated by application to an airbornegamma-ray survey from the Marble Bar area, Western Australia.

INTRODUCTION

Airborne radiometric surveys attempt to map the distributions,at or just below the ground surface, of 40K, 214Bi and 209Tl - isotopesthat effectively map the surface distribution of K, U and Th (Minty,1997). The data are contaminated by the cosmic-ray backgroundand, in the case of uranium, by atmospheric radon. However, evenafter correction for these effects there remain both an inevitableblurring of surface detail, due to overlapping successive fields-of-view (around 170 by 120 m at 60 m ground clearance), and a fall-off in count rates, the result of the aircraft’s remoteness from thesources of emanation.

Although conceptually distinct, blurring and attenuation areboth height-dependent. Blurring limits interpretability much aswith photographs taken by an unfocussed camera, whereasattenuation corresponds to underexposure. To minimise blur andmaximise count-rates, detailed surveys are generally flown as lowas safely permissible (typically 30-100 m).

Gunn (1978) offered an incisive treatment of the gamma-raymeasurement process. His analysis leads to an explicit form for theassociated transfer function, i.e., the loss of amplitude as a functionof spatial frequency in (gridded) survey data. Unfortunately theexpression obtained is an integral whose oscillating integrandfrustrates easy numerical estimation. Craig (1993) reduced theintegral to a convergent series, allowing effective computation ofthe transfer function appropriate to a given flying height andgamma-ray energy. A Postscript below references significant recentextensions and refinements of this theory.

Gunn’s transfer function is radially symmetric about the zero-frequency. As usual with surfaces of revolution, the graph of thisbivariate function is fully described by just one-half its central,vertical cross-section. Thus Figure 1, calculated for 60 m altitudeand 2614 keV radiation of 208Tl, shows a curve somewhat Gaussianin general appearance, though with slower asymptotic descent tozero.

Granted knowledge of the proper transfer function, and with thedata gridded and thus transformable to the frequency domain, wehave all the requirements for restoration of airborne radiometricimagery by ‘downward continuation’. The process is wholly

1 BHP Research and Technical DevelopmentGPO Box 86A, MelbourneVictoria, Australia 3001Phone: 61 3 9609 4477Fax: 61 3 9609 4489 E-mail: craig.maurice.md@bhp.com.au

2 CSIRO Exploration and MiningP.O. Box 136, North RydeN.S.W., Australia 2113E-mail: b.dickson@syd.dem.csiro.au

3 CSIRO Exploration and MiningPrivate Bag, P.O. WembleyW.A., Australia 6014E-mail: s.rodrigues@per.dem.csiro.au

Fig. 1. The attenuation function I(p) for 208Tl gamma-rays (2614 keV)and detector height 60 m.

analogous to the more familiar operation of continuing griddedaeromagnetic data. In all such high-frequency-enhancementoperations we must exercise moderation regarding the highestspatial frequencies. Low-amplitude, high-frequency componentsmay reflect noise levels in the ungridded data, defects of thegridding algorithm, and a host of other effects well known inpractical Fourier analysis. In place of ‘naive’ deconvolution -division of the transform by the unmodified transfer function - wetherefore apply a ‘parametric Wiener filter’, as further detailedbelow.

However, while precautions against amplifying noise in mistakefor signal are important, only by enhancing signal can we restorepicture quality. Thus, ensuring a signal level above the noise-flooris as essential to success as the core process itself. To this end, the‘Maximum Noise Fraction (MNF)’ process of Green et al. (1988)is brought to bear on the 256-channel data, so as to reduce noiseprior to attempting deconvolution. In default of some equivalentpre-cleaning step, little improvement can be expected from thealtitude-compensation process described below.

162 Exploration Geophysics (1999) Vol. 30, Nos. 3 and 4

The next section briefly summarises the passage from Gunn’sintegral to effectively computable quantities, as well as themodification of the resulting (continuous) transfer function topermit use with discrete data. We discuss then construction of theparametric Wiener filter, including choice of the parameter, andillustrate the whole procedure with reference to a survey flownsome 35 km west of Marble Bar, Western Australia. Thedescription of pre-cleaning accompanies this case study.

DOWNWARD CONTINUATION PROCEDURE

In the similar but simpler aeromagnetic situation, the theory ofdownward continuation can be derived wholly within theframework of the discrete Fourier transform, the setting proper alsoto its practical implementation (Appendix A). For the radiometriccase we seem obliged instead to deduce our formulae, for discretelysampled data, as approximations to their continuous-variableanalogues. The immediate task here is to make this transition, fromthe continuous results (Craig, 1993) to a form that we can apply toactual surveys.

The Discrete Transfer Function

To begin, let f(x, y) denote a (suitably well behaved) function ofthe point (x, y) in the Cartesian plane, and

its (continuous) Fourier transform. Further, let g(x, y) and G(u, v)be a second transform-pair. Suppose also that g represents theground-level radiometric count (for 40K, say) and that f is thedegraded version of g observable at flying height h. How, we mayask, is f related to g?

Gunn (1978) gives the answer as a multiplicative relationbetween F and G. Thus

F = PG (2)

where

As regards the new symbols in Equation (3), let z denote aspatial coordinate measured vertically downward from the plane ofobservation. Then

r = (x2 + y2 + z2)1/2 , s = [ah + b(z - h)]/z (4)

where a and b are the respective coefficients of attenuation(exponential with distance) for passage through air, and throughearth (whether ‘source’ or ‘host’ rock).

In naive deconvolution we seek to recover g from f as the inverseFourier transform of G = F/P (Equation (2)). Of course, errors in Fwill destabilise this ratio at places (u, v), in the frequency domain,where P approaches zero. However, notwithstanding theimpracticality of naive deconvolution per se, Equation (2) formsthe basis for constrained deconvolution, a modification capable ofaverting numerical instability.

To evaluate the function P we exploit its radial symmetry,expressing it in terms of the sole independent variable p = (u2 +v2)1/2. Introducing the abbreviations

I(p) = P(u/2ππ, v/2ππ)

A = (a2 +p2 )1/2 , B = (b2 + p2)1/2

and, for µ > 1,

we have then the formulae (Craig, 1993)

(b/2)I(0) = exp (-ah) - ahE(ah) (5a)

(5b)

In Equation (5a) the symbol E refers to the exponential integral,defined as

Further progress depends on numerical evaluation of the definiteintegral K, for the various arguments µ that arise. The problem isone of simple quadrature involving a positive, monotonicallydecreasing integrand, and might be approached in various ways. Aseries expansion well suited to automatic computation of K(µ) isdescribed in Craig (1993).

Figure 1 shows the graph of I(p) appropriate to the 2614 keVradiation for a survey conducted at 60 m. It was computed asindicated in Craig (1993), with the aid of physical parameters takenfrom Grasty (1979, Table 2). To explain the practical application ofsuch plots or, rather, the arrays of numerical values that they depict,it is enough to indicate the path from continuous to discrete Fouriertransforms. Thus, imagine now that f(x, y) is a function discretelysampled at integers x, y with 1<x<M , 1<y<N. In place ofEquation (1), and summing over all pairs (x, y), we have

where m, n take integer values with -M 1 < m < M 2 ,- N 1 < n < N 2 , say. (Here M1 = M2 = (M-1)/2 if M is odd,whereas M1 = M/2, M2 = M/2-1 if M is even; and similarly for N1,N2.)

Further, let g and G be the discrete analogues of the functionspreviously so denoted. To obtain G(m, n) from F(m, n), the propervalue by which to divide is I(2πq/w). The symbol w (in the sameunits as h) represents the sample-spacing, assumed the same in bothx and y directions, while q , the discrete analogue of p , is given by

q = [(m/M)2 + (n/N)2]1/2 (7)

For our implementation, an M-by-N image of the function q isfirst constructed, along with a list of (say, 100) equispaced valuesfor I(p). We convert the q-image to an image P(m, n) by use of thelist as a look-up table, supplemented by linear interpolationbetween table entries. The range of values for p must clearly extendfrom zero to beyond the largest value for 2πq/w that will berequired. Of course, this maximum value occurs at a corner of theimage and is easily predicted.

To conclude this section we note that, in Gunn’s theory,intensities do not follow the widely accepted rule: ‘decayexponential with flying height’. Indeed, by Equation (6) thefunction f has mean value F(0, 0). Similarly, G(0, 0) is the meanvalue for g. But the ratio of these ‘average count rates’, namely I(0)in Equation (5a), is not strictly exponential in form.

Constrained Deconvolution

In ‘parametric’ Wiener deconvolution (Hunt, 1972) onemodifies Equation (2), replacing the factor 1/P by

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163Exploration Geophysics (1999) Vol. 30, Nos. 3 and 4

P/(P2 + γγ L/S) (8)

Here L, S are power spectra for the ‘noise’ and ‘signal’,respectively, while symbol γ is the parameter. (Such formulaeusually incorporate also P*, the asterisk denoting complexconjugation, but our P is real-valued. There seems to be no agreedname for γ.) When γ = 0 we revert to naive deconvolution. Theusual Wiener filter has γ = 1. However, at least for the choice of Lindicated in the following paragraph, the effect is then often tosmooth rather than sharpen the image.

Of course, we have no true means for separating signal fromnoise. As an estimate for S the most obvious choice is the powerspectrum FF* of f, admittedly a mixture of both signal and noisecomponents. Following Hunt (1973, Equation (10)), as a surrogatefor L we may take the power spectrum of the image obtained byLaplacian filtering of f. This operation (Huang et al., 1971,Equation (l14); Rosenfeld and Kak, 1982, p. 241-244) is widelyused for isotropic edge-detection. Accordingly, the result offiltering will generally include ‘signal’ (edge locations beingcoherent information) as well as ‘noise’, whence our choice for Lis no nearer to ideal than is that for S. However, since L is nowlinearly related to S, the expression (8) has the necessary scale-independence.

The usual facts regarding Wiener filtering apply (Rosenfeld andKak, 1982, p. 285). For frequencies (m,n) where L/S is negligible,expression (8) approximates 1/P. At high frequencies, where Papproaches zero, the generally higher noise levels bound thedenominator in (8) away from zero; and so on.

CASE STUDY: LYNAS FIND AREA, MARBLE BAR, WA

The application of downward continuation is illustrated by itsuse on a low-level, fixed-wing survey conducted in 1997 in theLynas Find area, near Marble Bar and 90 km south-south-east ofPort Hedland in Western Australia. The parameters of the surveyare described in Table 1. The Lynas Find area is located in thenorthern section of the Pilgangoora Syncline and encompasses abelt of Archaean ultramafic-through-basaltic rocks with lesserfelsic volcanics and cherts flanked by Archaean granites (Neumayret al., 1998). Data were obtained as located, 256-channel spectra,along with calibration and stripping coefficients, and background-correction parameters.

Data Processing

To provide comparative illustrations, the survey was processedtwice: first conventionally, then with the altitude correctiondescribed above. Some steps are common to both processes. In theconventional treatment, channel data within the standard windowsfor K, U and Th (Table 2) were extracted and corrected for cosmicand aircraft background, Compton scattering and flying height. Theheight-correction involved here is a minor one that convertsground-clearances of individual readings to the nominal surveylevel. It is needed also before downward continuation to groundlevel (next paragraph), because that process assumes measurementsto be on a horizontal plane. The located, corrected data were thengridded at a cell size of 15 × 15 m using the algorithm of Briggs(1974). Figure 2a shows the Th image, for a small part of thesurvey. Low count-rates help explain its moderately noisyappearance.

The data were then reprocessed. The first step is to reduce noise.Only 211 of the 256 channels were used, the top 16 channels beingrejected as comprising mostly zero readings, the bottom 29 becauseof their coincidence with a steep Compton edge. To decrease noisewe used the Maximum Noise Fraction (MNF) method for orderingmultispectral data by image quality. Application of this process to

Correcting gamma-ray data for altitude

Flight Direction east-west

Line spacing 50 m

Survey height 30 m

Sample interval 1 s ~60 m

Crystal volume 33.6 litre

Table 1. Lynas Find Survey Parameters.

Element Potassium Uranium Thorium

Isotope 40K 214Bi 208Tl

Peak (keV) 1460 1760 2614

Energy Range(keV)

1370 - 1570 1660 - 1860 2400 - 2800

0.0068 0.0062 0.00505a (m-1)

b (m-1) 13.2 12.05 9.9

Table 2. Parameters used for data preparation and downwardcontinuation. Windows set as recommended by IAEA (1991).Attenuation coefficients of air and rock (Equation. (4) coefficients a, b)derived from Grasty (1979).

Fig. 2. A section of the thorium image for the Lynas Find survey,gridded at 15 × 15 m. Image (a) was derived from conventionalprocessing, (b) using MNF-cleaning on the spectral data prior toprocessing and (c) after downward continuation on the cleaned data.

164 Exploration Geophysics (1999) Vol. 30, Nos. 3 and 4

aerial gamma-ray survey data has been described by Dickson andTaylor (1998). It comprises essentially two Principal Component(PC) transformations, separated by a rescaling. The first transformdecorrelates the noise covariance. After the rescaling to equalisenoise-variance in all channels, the second PC transformationdecorrelates the resulting, noise-whitened data. See Green et al.(1988) where, however, the simultaneous diagonalisation of thesignal and noise covariance matrices operates with their rolesreversed. The advantage of MNF over the simpler PC (or SVD)transform, which merely maximises variance, comes from thelatter’s inability to discriminate sufficiently ‘loud’ noise fromsignal.

To obtain a suitable measure of multivariate noise, theconventionally processed images were searched visually for asubscene of minimal geological signal. The raw spectra in this areawere used to generate a ‘noise-covariance’ matrix. Here ‘noise’ isdefined by a shift-difference image, such as results from theLaplacian filtering discussed above. For the ‘signal’ covariancematrix, the full scene was used. The two covariance matricesdetermine forward and reverse MNF transforms, as described inGreen et al. (1988). The forward transform orders the data into newcomponents with signal-to-noise ratio graded monotonically. Noiseremoval is effected by retaining only the better components, theothers (comprising mostly noise) being reset to zero before reversetransformation.

In our case 42 of 230 components were retained, although justthe first dozen components comprise nearly the total variance (traceof signal covariance). The back-transform was applied to restorethe gamma-ray spectra, which were then processed and gridded inthe conventional manner. Figure 2b shows the ‘cleaned’ results forTh. While insisting on the need for pre-cleaning, we recognise thepossibility of variations on our approach so have kept this accountbrief.

The pre-cleaning improved image quality sufficiently to permitdownward continuation. To minimise FFT artefacts, the original526 × 436 image was extended to 1024 × 1024 by use of a cosineramp to generate synthetic borders. Figure 3 shows in perspectivethe ratio L/S of power spectra, computed as described above butplotted with logarithmic vertical scale (i.e., log(L/S +10-6)is shown).The tornado-like funnel overlies the origin (zero-frequency point)of the frequency plane, within whose neighbourhood signalgenerally dominates noise.

As regards methods for choosing the parameter, Hunt (1973),and Dines and Kak (1977) solve iteratively for γ in terms of anotherconstant that must itself be estimated. However, as such anapproach seems to risk getting the right answer to the wrongquestion, we have preferred trial-and-error evaluation of γ itself,based on subjective assessment of output image quality. Hunt(1972) quotes Twomey (1965) as follows: ‘the solution changesquite slowly as γ is altered, so that, in practice, the selection of γposes no problems’. Such slow variation suggests ‘bracketing’ thecorrect range for γ by a sequence of (negative) powers of ten, aprocess well suited to covering a wide range of possibilitiesquickly.

Figure 4 illustrates a sequence of extracts from the Th-surveythat demonstrate the effect of varying γ from 10-5 to 10-1, fromwhich range the value 0.03 was chosen. (Being noise-specific,values for K and U must be chosen individually.) With a littleexperience, one can predict the quality of results by displayinggraphs of the filter in formula (8). A perspective view of such asurface (not illustrated) shows a bowl-shape surrounded by athicket of vertical spikes, the exact form variable with γ.Proliferation of useless results can then be avoided by withholdingapplication of the filter until it appears satisfactory. Figure 2cshows the effectiveness of downward continuation on the ‘MNF-cleaned’ image for Th.

DISCUSSION

What does downward continuation of gamma-ray survey dataachieve? The method sharpens edges and can reveal subtle linearfeatures. It is therefore a tool to inspect surveys for lineaments andstructural features. The drawback is noise enhancement, as is wellillustrated in Figure 2. From its nature this noise seems related tothe gridding procedure used on the original located data. Minimumcurvature gridding (Briggs, 1974) appears to produce ‘circular’noise features whereas spline fitting gives ‘worm-like’ features (G.Taylor, personal communication). Our experience of regridding theprocessed data from just the flight-line locations, with a view tooptimising image quality, is that it effects little improvement.

In the course of choosing data for this work we encounteredsurveys that seemed to be of good quality, with little radon noise.However, displays of power spectra uncovered a serious deficiency,their appearance strongly suggesting that ‘notch filtering’ in thefrequency domain had been used to suppress striping sub-parallelto flight lines. Notch filters operate by resetting to zero theamplitudes of wave components with frequencies within chosen

Craig, Dickson and Rodrigues

Fig. 3. Ratio L/S between the power spectra, of the cleaned thoriumdata and its plane Laplacian. This quantity, pictured with a logarithmicstretch, permits stable inverse filtering in the frequency-domain.

Fig. 4. Examples of downward-continued thorium data with variousvalues parameter of γ.

165Exploration Geophysics (1999) Vol. 30, Nos. 3 and 4

limits. Although the notches were not sharply defined, the locateddata that we imaged may have derived from a previous gridding, ata different scale, used to perform such filtering. If so, regridding atour scale might fail to reproduce edges sharply.

Once amplitudes have been set to zero, downward continuationbecomes impossible; we cannot amplify a lost signal. Thus, thenotched data had to be rejected as unusable. However, non-leveldata will impede height deconvolution (as well as ‘stripping’corrections and similar refinements) as intractably as amplitudesset deliberately to zero. If the processes described in this paper areto hold practical interest, it behoves us therefore to suggest some‘safe’, alternative levelling procedure. In fact, such methods arewell known in related contexts, the problem of mismatch betweenflight lines being quite analogous to the line-striping seen inairborne and satellite reflectance imagery. Levelling ofaeromagnetic data, to account for diurnal variations in the earth’sfield over the course of a large survey, is another familiar instance.For brevity we limit our remarks here to the following suggestion.In the interests of those, like ourselves, wanting to maximise thepotential of airborne radiometry, along with the ‘corrected’ data letlocated data files retain channels of raw values.

CONCLUSIONS

Downward continuation of gridded airborne gamma-rayspectrometric data can enhance the resolution of such images. Theprocess suffers from limitations similar to those besetting thedownward continuation of gridded potential-field data, a process towhich it is closely analogous. But their seriousness diminishes asdata quality improves, and they can be managed systematically ina way that benefits from user interaction without, moreover,requiring inordinate skill.

Granted the possibility of thus recovering the height-dependentloss of signal strength and spatial resolution, the chief determinantsof success in mapping surface radioelement distribution areperhaps: (a) data quality, and (b) effectiveness of the griddingalgorithm. Successful restoration at a particular frequency willdepend on the signal-to-noise ratio, itself a frequency dependentquantity. As regards (b), we can only note with regret that this areahas seen no definitive improvement for over 20 years.

ACKNOWLEDGEMENTS

Permission to use the Lynas Find data was generously given byLynas Gold NL. These data (free from faulty prior processing)were obtained with the assistance of Mark Deuter and Geoff Taylorof Pitt Research, Adelaide, S.A. Their continuing interest andassistance in developing the downward continuation method isgratefully acknowledged.

POSTSCRIPT

This paper was first prepared in 1995, and results from themethod were presented to the ASEG workshop on Processing ofAirborne Gamma-ray Spectrometric Data held at Manly, NSW inFebruary 1998. While unfortunate circumstances delayed writtenpublication, several related papers have appeared. Thus, Gunn andAlmond (1997) have described an equivalent-layer technique forboth deconvolution, and downward continuation from irregularsurfaces.

Again, Billings and Hovgaard (1999) have modified theresponse function to account for both aircraft movement, and therectangular shape of actual detectors. The treatment of Gunn

(1978) refers to a point-detector (not to be confused with a sphereof positive radius). While this alternative function also deconvolvesaerial gamma-ray data (Billings, 1998), so far as we know themethod presented here remains the only one availablecommercially.

REFERENCES

Billings, S., 1998, Geophysical aspects of soil mapping using airborne gamma-rayspectrometry: PhD thesis, The University of Sydney (unpublished).

Billings, S. and Hovgaard, J., 1999, Modeling detector response in airborne gamma-ray spectrometry: Geophysics, 64, (in press).

Briggs, I. C., 1974, Machine contouring using minimum curvature: Geophysics, 39,39-48.

Craig, M., 1993, The point-spread function for airborne radiometry: Math. Geol., 25,1003-1013.

Dickson, B.L. and Taylor, G., 1998, Noise reduction of aerial gamma-ray surveys:Explor. Geophys. 29, 324-329.

Dines, K. A. and Kak, A. C., 1977, Constrained least squares filtering: IEEE Trans. onAcoustics, Speech and Signal Processing, 25, 346-350.

Grasty, R. L., 1979, Gamma ray spectroscopic methods in uranium exploration -theory and operational procedures: in J. Hood (Ed.), Geol. Survey of CanadaEconomic Geology Report 31: Ottawa, Canada, 147-161.

Green A. A., Berman, M., Switzer, P. and Craig, M. D., 1988, A transformation forordering multispectral data in terms of image quality with implications for noiseremoval: IEEE Trans. Geosci. and Remote Sensing, 26, 65-74.

Gunn, P. J., 1975, Linear transformations of gravity and magnetic fields:Geophys. Prosp., 23, 300-312.

Gunn, P. J., 1978, Inversion of airborne radiometric data: Geophysics, 43, 133-142.

Gunn, P.J. and Almond, R., 1997, A method for calculating equivalent layerscorresponding to large aeromagnetic and radiometric grids: Explor. Geophys., 28,72-79.

Huang, T. S., Schreiber, W. F. and Tretiak, O. J., 1971, Image processing: Proc. IEEE,59, 1586-1609.

Hunt, B. R., 1972, Deconvolution of linear systems by constrained regression and itsrelationship to the Wiener theory: IEEE Trans. Automat. Control, 17, 703-705.

Hunt, B. R., 1973, An application of constrained least squares estimation to imagerestoration by digital computer: IEEE Trans. Comput., 22, 805-812.

IAEA, 1991, Airborne gamma-ray spectrometer surveying: Internat. Atomic EnergyAgency, Technical Report Series, No 323.

Minty, B. R. S., 1997, Fundamentals of airborne gamma-ray spectrometry: AGSOJournal of Australian Geology and Geophysics, 17, 39-50.

Neumayr, P., Ridley, J.R., McNaughton, N.J., Kinny, P.D., Barley, M.E., and Groves,D.I., 1998, Timing of gold mineralization in the Mt York district, PilgangooraGreenstone Belt, and implications for the tectonic and metamorphic evolution ofan area linking the western and eastern Pilbara Craton: Precambrian Research 88,249-265.

Rosenfeld, A. and Kak, A. C., Digital Picture Processing, 2nd Ed., vol. 1. AcademicPress, Orlando, 1982, 435 p.

Twomey, S., 1965, The application of numerical filtering to the solution of integralequations encountered in indirect sensing measurements: J. Franklin Inst., 279, 95-109.

APPENDIX - A

The functions exp{2π[qz + i(mx/M + ny/N)]} (see Equation (6)-(7)) are all trivariate harmonic, whence a linear combination ofthem, say with coefficients amn, is likewise harmonic. Putting z = 0we see that, for agreement, the amn must be chosen as the Fouriercoefficients of the gridded data. The formula thus obtained isadequate for deducing the frequency-domain filters that effectvertical continuation, directional derivative, and conversion toanother field component. These comprise all the usual operations,except reduction-to-pole (Gunn, 1975).

Correcting gamma-ray data for altitude