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8/3/2019 Interpretacion Fourier_Oldenburg

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GEOP HYSICS, VOL. 39, NO. 4 (AUGU ST 1974), P. 526-536, 6 FIGS., 1 TABLE

THE INVERSION AND INTERPRETATION OF GRAVITY ANOMALIES

DOUGLAS W. OLDENBURG*

A rearrangem ent of the formula used for therapid calculation of the gravitational anom alycaused by a two-dimen sional uneven layer of ma-terial (P arker, 197 2) leads to an iterative pro-cedure for calculating the shape of the perturbingbody given the anomaly. The method readilyhandles large numbers of model points, and it isfound em pirically that convergence of the itera-tion can be assured by ap plication of a low-pass

INTRODUCTIONGravity profiles are often characterized by a

smooth regional trend superimposed on higherfrequency information which is usua lly referredto as the gravitational anomaly. It is of interestto invert these data to determine the nature ofthe mass distribution that would give rise to suchan anom alous field. Unfortunately, as with otherproblems in potential theory, the solution is non-unique, i.e., many different distributions of masscan produc e the same gravitational field, and tventhe veiy strong assumption that the gravityanomaly is caused by a single two-dimensionaldisturbing mass of uniform density d oes not leadto an unamb iguous interpretation (Skeels, 1947).Nevertheless, a mean ingful interpretation may beobtained if seismic data, and perhaps geologicdata from boreholes or wells, are used to constrainsufficiently the range of possible models.

The problem we shall study is the following:Given a single profile of a gravitational anom aly,what is the shape of a two-dimensiona l mass ofconstant density which will produce this anom-aly? Original inversion attempts were based ontrial-and-error methods where the shape of aninitial starting structure wa s perturbed until itsgravitational attraction, calculated by means of

filter. The nonu niqueness of the inversion c an becharacterized by two free parameters: the as-sumed density contrast between the two media,and the level at which the inverted topograp hy iscalculated. Add itional geoph ysical knowled ge isrequired to reduce this amb iguity. The inversionof a gravity profile perpend icular to a continentalmargin to find the location of the Moh o is offeredas a practical examp le of this method.graticules, matched the observed field (Skeels,1947).

More recent attempts have centered about au-tomating the perturbation scheme (Bott, 1960;Corbat6, 1965; Tanner, 1967; Scgi and Garde,1969). The perturbing body is usually approxi-mated by a set of rectangular prisms of constantdensity, and its gravitational field is then calcu-lated. Th e residu als between the calculated an dobserved fields are used to adjust the heights ofthese rectangles; the amount of the adjustmentsis calculated by solving a linearized set of normalequations (Corbat6, 1965).

Disadvantages to this technique are numerous.The final m odel consists of a num ber of rectangu-lar prisms rather than a smooth curve whichwould be more reasonable geologically; and twotypes of iteration schemes, depending u ponwhether the upper or lower surface of the per-turbing body is assumed known and fixed, arerequired to insure probab le stability and conver-gence (Tanner, 1967). Published examples usingthis method are characterized by slow conver-gence and instability of the iteration scheme whenthe number of parameters, i.e., the number ofrectangles, is increased. For buried bodies convcr-gence of the iteration method also requires that

Man uscript receivedby the Editor August 22, 1973; evisedmanuscript eceived February 7, 1974.* University of California, San Diego, La Jolla, Calif. 92037.@ 1974 Society of Exploration Geophysicists. All rights reserved.

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Inversion of Gravity Anomalies 527

the depth of burial of the rectangular prism begreater than its width (Tanner, 1967).

Dyrelius and Vogel (1 972 ) increased stabilityand rate of convergence by using an alternatemethod of calculating some of the linearizedquantities in the normal equations and preventingthe iteration scheme from m aking large adjust-ments to the heights of the rectangles by requiringthat the center of mass and the total mass of themodel be the sam e as those calculated from theobserved anomaly by Gauss formula. Theirmethod , however, w as applied only wh en theupper surface of the rectangular prisms coincidedwith the observational plane, and the effective-ness of their m ethod is not know n when the per-turbing body is buried at depth.

The purpose of this paper is to present an in-version technique which can accommodate easilya large number of model points (more than ahund red) and still maintain the desired num ericalproperties of stability and relatively rapid con-vergence. The inversion is accomplished by aniterative scheme based on the rearrangem ent ofthe forward algorithm given by Parker (1973).The scheme calculates the Fourier transform ofthe gravitational anom aly as the sum of theFourier transforms of powers of the perturbingtopography. Because Fourier transforms can becomputed rapidly (IEEE, 1967), this method iscom putationally muc h more efficient than calcu-lating the gravitational field by breaking u p themodel into a set of prisms whose contributionsare calculated separately an d summed. Indeed, itis this speed with which the forward algorithm isexecuted that allows us to present the followinginversion m ethod as a practical one.

THEORY

We con sider Parkers method of calculating thegravitational attraction of a two-dimensiona l, u n-even layer of material of constant density. In anX-Z Cartesian coordinate system the gravita-tional an om aly is given by Ag(x), and the lowerand upper boundaries of the perturbing layer aredenoted by z=O and z= h(x), respectively. Theentire m ass of the perturbing layer m ust lie belowthe horizontal line on which the observations arespecified. Since our profile is of only finite lengthand in order to avoid problems of convergence, weassum e that the layer vanishes outside some finitedomain D, i.e., h(x)=0 if xaD. In practice h(r)

is measu red relative to som e reference level a dis-tance zo below the su rface.

We define the one-dimensional Fourier trans-form of a function h(x) by

F[h(x)] = J mh(z)t+dx, (1 )--mwhere k is the wavenumber of the transformedfunction. The Fourier transform of the gravita-tional anom aly is obtained by reducing Parkerstwo-dimensional formu la to the one-dimension alform required here. Hence,Fbdd

= - 2&pe-lklzo cm F[h(x)], (2)n-1 n.

where p is the density contrast between the twomedia and G is Newton s gravitational constant.Transpo sing the n= 1 term from the infinite sumand rearranging, we obtain

F[h(x)] = - FIAg(x)]eikizo__2aGp_ 2 I : F[h(. )]. (3)x

n=2 ?t.!

When p and z. are known (or assumed), this equa-tion m ay be used iteratively to calculate h(x) inthe following manner: The most recent determi-nation of h(x) [for the first iteration a guessedsolution or h(x)=0 is satisfactory] is used toevaluate the right-hand side of equation (3); theinverse Fou rier transform of this quan tity thengives an updated value for the topography. Theiterative procedu re is continued until some con-vergence criterion is met or a maximum numberof iterations has been completed. It is impo rtantto note that the calculation of h(r) by equation (3)involves approximately the same number of com-putations as solving the forward algorithm; hence,each iteration can be done quickly, and totalcomputation time for the inversion is thereforegoverned by the number of iterations required be-fore the convergence criterion is satisfied.

CONVERGENCE AND UNIQUENESS

Tests for convergence must be applied in evalu-ating the infinite sum of Fou rier transforms inequation (3) and also in the iteration technique



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528 Oldenburg

used to find the topography h(r). Before conver-gence properties can be considered, reaso nable re-strictions mu st be imposed upon h(r). In a man nersimilar to that given by Parker (1972), we requirethat h(z) be bounded and integrable and that itvanishes outside some finite dom ain D, i.e.,h(x) = 0 for x~D. Then,

F[h(x)] = J h(x)ekaxDn

J,\ h(x)\ dx< LH,where L is the length of D and H= maxi h(x) 1.The sum

=L(Ik l elkiH - 1 - ) k 1 H).

(4 )

The right-hand side of (4) is boun ded for anyfinite value of k; hence, from properties of theexponential function (Whittaker and Watson,1962, p. 581), the sum of Fourier transforms isabsolutely and uniformly convergent in anybounded domain of the k-plane. In principle then,no problem exists with the convergence of theinfinite s um althoug h, in practice, large relief mayrequire many terms (say 20 or 30) of the sum tobe evaluated before an approp riate convergencecriterion is met. Let

S, = max 1kin-lOYer ll I- F[k"(x)] 1n.The convergence criterion chosen requires thatsuccessive terms in the su m are comp uted untilS,,/S2

and hence that the series is uniformly convergent,independent of the value of k, when H/zo< 1.Therate of convergence of ~(H/.z~ )~ is maxim izedwhen H/z0 is a minimum, thus the obvious re-quirement that the topography be measured rela-tive to a level zo which is the median of the largestand smallest values of k(x). Num erical experi-ments carried out by Parker have shown that thischoice of zo falls very close to the optimu m one.Unfortunately, if the material giving rise tothe gravitational anomaly comes into contactwith the ob servation p lane, then Z Z/zo= 1 and~(H~zo) is not convergent. However, the firstinequality in equation (5 ) remains valid therebyensuring the convergen ce of the sum for finite k,but in practice this rate of convergence is slowedas the perturbing layer ap proaches the observa-tional surface. Convergence of the forward algo-rithm is monitored by

A sufficient num ber of terms h as been evaluatedwhen R,/R1


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Inversion o f Gravity Anom alies 529ally associated with noise in the data or trunca-tion errors in the Fourier transforms, are multi-plied by large exponential factors.

As a result of these complications we have notbeen able to mak e analytical statements concern-ing the convergence of the iterative procedurebut, instead, have been forced to resort to em-pirical resu lts. We have used as our convergencecriterion the requirement that the root-mean-square difference between two successive approxi-mations of h(x) be less than some arbitrarilychosen value. It has been found that the straight-forward application of the iterative procedure usu-ally resulted in a divergent solution characterizedby high wavenumber oscillations. Only in specialcases, when the observed gravitational anom alycould be accounted for by an k(x) with sufficientlysmall relief, was convergence obtained.However, problems involving downward con-tinuation of potential field data usually providemeaningful results only when the downward con-tinued data are smoothed (Bullard and Cooper,1948). In gravitational problems such smoothingmay be physically realistic since short wave lengthanomalies are more easily generated by structuresnear the surface than those at dep th. If it is re-quired to invert a gravitational anom aly the bulkof whose source is believed to b e at co nsiderabledepth, then we may justifiably filter ou t muc h ofthe short wavelength information which is prob-ably cau sed by near-surface structure. Thesehigh-frequency oscillations can be eliminated bymultiplying the right-hand side of equation (3) bya su itable low pass filter B(k) which passes all fre-quencies up to WH and passes none above a cut-off frequency SH.

If the gravitational anom aly to be inverted isgiven at N equa lly spaced points, then the model(i.e., the shap e of the topograph ic surface causingthe anom aly) is determined by specifying the Ncomplex am plitudes of the equally spaced fre-quency components between -,& and Lv, whereJo is the Nyqu ist frequency. The effect of thefilter B(k) is to reduce the number of free parame-ters of the model by approximately the factor ofSH/j*v, since for SH WH to reduce the effect of with lower density, and the choice of a large p soGibbs phenomenon in the untransformed domain. diminishes the magnitude of k(x) that only the

1Editors note: Throughout this report the author first term in the infinite sum in equation (2) isuses he term frequency or the quantity k/2n. important. This term in fact calculates the gravi-



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530 Oldenburgtational attraction of a surface mass distributionobtained by condensing the matter in the verticalcolumn from z = 0 to e = h(x) down into the z = 0plane. Although there is no upper bound to thevalue of the density which can give rise to theanomaly, there is a minimum value.

Maximum depth rules (Grant and West, 1965;Parker, 1974) indicate that for an assumed densitycontrast there is a maximum depth above whichsome portion of the perturbing body m ust lie inorder to cause the anom aly; alternatively, if theanomaly is caused by a density contrast at somespecified depth, then the magn itude of this con-trast m ust be greater than some Pmin.Therefore,if, in the inversion scheme, the assumed value of pis too sm all and/or the as sumed value of zo is toolarge, no filter exists which will force the iterationprocedure to converge to a topography satisfyingthe observations.

The nonu niqueness of the gravity inversionproblem results from the existence of two freeparameters, p and za. Skeels (1947) appears tohave given one of the first published examples ofhow the change in level is responsible for an am-biguity in interpretation. After fixing the densitycontrast between the basement rocks and theoverlying sediments, he varied the depth to theundisturbed basement and obtained different con-figurations of the basement and overburden whoseresultant gravitational attraction fitted the orig-inal observations to within 0.1 mgal. Sk eels alsopointed ou t that g ravity data assum ed to arisefrom a density contrast of a single layer at depthcould be inverted uniquely if the value of thedensity contrast and either the depth to the sur-face of density contrast at one point or the maxi-mum relief of the structure were known.

Parker (1973) has shown that a formula anal-ogous to equation (2) can be used for the rapidcalculation of a magnetic anomaly arising from atwo-dimensional magn etized layer. If the thick-ness and location of this layer are known, thenthe rearrangem ent of this formula (Parker andHuestis, 1974) has been shown to lead to aniterative procedu re for calculating a linear distri-bution of magnetization m(x) which could causethe anomaly. It is interesting to note the differ-ence in the roles that the ch oice of zg (i.e., thelevel relative to which the topography is mea-sured) plays in the magn etic and gravitational in-version procedures. For the magnetic problem the

value of zo affects the rate of convergence of theiterative procedu re but n ot the resulting nz(x). Incontrast, for the gravitational problem a differenth(l) is obtained for each value of ZO,hence thisparam eter characterizes an entire family of possi-ble topog raphies which can cause the observedanomaly.

The inversion of any gravitational profile there-fore involves a set of three parameters, p, ZO, ndB(k). To provide some insight a s to the effects ofthese parame ters, the nonuniqu eness of gravita-tional interpretation, and the rapidity of conver-gence of the iterative process, a num erical exam-ple is given.

NUMERICAL EXAMPLEWe consider the gravitational anomaly caused

by the bump on the subsurface stratum inFigure 1. The bump is 20 km wide at the base; ithas a height of 4 km and an assumed density con-trast of 1.0 gm/cm 3. The gravitational attractionof this bump has been calculated at 128 equi-distant points (1 km apart) by using equation (2).Evalu ation of only the first nine terms of the in-finite sum resulted in Rg/Rl< 10e6.

This grav ity anom aly has been inverted for dif-ferent assumed values of p and ZO. The results,shown in Table 1, typify the interrelation betweenZO,p, and the low-pass filter for such an inversion.Three different low-pass filters were applied, andthe iterative procedu re was terminated when therms difference between successive values of h(x)was less than 0.5 m or when a maximum of 10iterations had been completed. T he infinite su m.which had to be evaluated at each iteration, wassaid to have converged when S,/& < 5 X 10p3.

For each inversion, Tab le 1 shows the rms dif-ference in h(r) at the last iteration (ERR ), thetotal num ber of iterations taken (ITER), and theabsolute value of the maximum error (in mgals)between the observed gravity and the gravitycalculated from the inverted top ography (M.E.).Also indicated is whether the iterative process haddiverged (D IV) or was still converging (S.C.)There is some difficulty in concisely presenting theeffects of the three different parameters on theinversion procedure. Nevertheless, from Table 1we can draw the following general conclusionswhich h ave been empirically shown to be validfor the inversion of many different profiles.We shall first consider the effects of varying p



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Inversion of Gravity Anomalies 531and za for a pa rticular low-pass filter. If z0 is fixed, verges. As examples, the filter (WH , SH) = (.lj,then a decrease in the assum ed value of p requires 20) enabled an acceptable II(X) to be found forthat m ore iterations be performed before conver- z0=3 .0 km but not for zO= 3.0 km or below.gence is obtained. Indeed, there existsp,,i,, a mini- For the filter (.075, ,150) convergence was ob-mu m value of p, such that for p SH,,,, the iterative procedu re no longer con- models satisfying the observations. The speed of

6

7

h(x)=Acor(Znx/W)

~W=ZOkmc(FIG. 1. Gravitational anomaly from a bump on a flat substratum.



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532 OldenburgTable 1. The effects of the density contrast p, thelevel Z, and the low-pas filter (WH, SH), on theinversion of a gravitational pr0tile.lfilter: c.15, ,201 EIWH. SH)

ERR=.3 m ERR=.3 mITER=6 ITER = IO (SC.)M.E.=.09 M.E. = .87ERR=70m ERR=200 m

ITER=2 (DIV) ITER=Z(DIV) ITER=Z(DIVIM.E.=7.9

All inversions diverged

filter: c.075, ,150)P\I 1.25 1.0 .7 52

ERR=.13 m ERR= .27 m ERR = .38 m3. 0 ITER=~ ITER =4 ITER = 5

M.E.=.05 M.E. =.04 M.E.=.IOERR=.44m ERR = I.2 m ERR=37 m

5. 0 ITER=7 lTER= IO (SC.1 ITER =3 (DIV)M.E.=.07 M.E.=.29 M.E. =9.95

7. 0 All inversions ivergedfiller: t.025, .075)

PL\ I .25 I .o .7 s~.

ERR=.29 m ERR= .12 m ERR= .33 m3.0 ITER = 3 ITER=4 ITER=4

M&=7.2 M.E.=6.9 M.E.=6.4ERR=.38 m ERR=.17 m ERR= .47 m

5.0 ITER=3 ITER =4 ITER=IM.E.=7.1 M.E.=6.8 M.E.=6.3ERR=.l2m ERR = .26 m ERR=.22 m

7.0 ITER =4 ITER=4 ITER=SM.E.=7.0 M.E.=6.7 M.E. = 6.0

TER is the number of interations completed; ERR isthe rms difference between the last two iterations ofh(x); and M.E. is the absolute value of the maximumerror (in mgals) between the observed gravitationalanomaly and that calculated from the inverted topog-raphy. S.C. beside ITER means the iteration procedurewas still converging, while DIV means it was diverging,i.e., the rms error between successive iterations hadbegun to increase.

the inversion scheme is illustrated b>, the fact thatthe six inverted profiles in Figure 2 were calcu-lated in approximately 1.2 set on a CDC 7600.

GEOPHYSICAL EXAMPLEUnfortunately, the assum ptions that an ob-

served gravitational anom aly is caused by a singlesurface between two co nstant density m edia andthat the perturbing body is two-dimensional are

sufficiently strong that they arc never exactly metin nature. In many cases though, enough is knownabout the near-surface structure from seismologyor from boreholes that its effect can be strippedoff leaving a reduced gravitational anom aly w hichmay ap proximately fulfill ou r assum ptions. Anexample of geophysical interest is the determina-tion of the depth to the interface between theearths crust and mantle, i.e., the MohoroviEii:discontinuity or Moho.

Seismological work indicates that the Moho isat a depth of 11 km beneath m ost ocean surfaces(Hart, 1969, p. 228) and that it plunges underthe thick c ontinental crust. WC applied this in-version technique to see if gravitational observa-tions can be used to delineate this discontinuit)along a traverse perpendicu lar to a continentalmargin . Since continental margins are approxi-mately linear features over lengths of a few hun-dred kilometers, the error introduced by the as-sumption of two-dimcnsionalit~. of the variouslayers is probab ly small.

We used as gravity observations the 22 pendu-lum m easuremen ts for the Mt. 1)escrt Sectionprofile given by Worzel (1965) and redrawn inFigure 3. We interpo lated these points with aspline routine and cxtendcti the lengths of thewater and seditnent profiles in order to produce atotal p rofile length of 10 24 km.The reasons for this bordering are tcvo-fold.The com putation of a finite transform assume sthat the data series is infinitely replicated.Achievement of this replication with n o disconti-nuities requires that the beginning and end pointsof the series should have appro ximately the samevalue. It is also desirable to extrapolate the ob-served layering of water and sedimen ts so thatthe calculated gravitational attraction from theselayers will be approxim ately correct even close tothe edges of the unbordered profile. In the presentcase the w ater and sedim ents were extrapolatedas horizontal layers for a distance of 150 km, thenboth rose gently to zero in the next 150 km andremained at zero until 1024 km. All structureswere then digitized at intervals of 8 km yieldingseries 128 points long. In spite of this bordering,it must b e rememb ered that we are interested ina model fitting the observations only in the first600 km, where real data exist.

Assuming the sediment and water distributionsas given in Figure 3, we calcutatcd their gravita-



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Inversion of Gravity Anomalies,OO-

=1+

$60-Eea 40-a.

LE __

533

FIG. 2. Ambiguity in gravity interpretation caused by the choice of level.The observations are those from Figure 1, and each of the six reliefs showngives rise to a gravitational anom aly differing from the observed anom alyby less than 0. I mgal.

%I30 Computed R Observed Gravity AnomaliesE -Free-Air Anomalyu.s s. Tush, 1947

Structure Section Deduced from Seismic a Gravi?yoqyhJence

0 100 200 300 400 500 600Dirtancr in Kilometers from Mt. Desert Rock

FIG. 3. The 22 pendulum measurements for the M t. Desert section andthe locations of the water and sedimentary layers (redrawn from Worzel,196 5). Also shown is the agreem ent between the observations and the g ravita-tional attraction calculated from the stratigraphic section.



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200 -

Oldenburg

Reduced Gravitational Anomoly (water 8 sediment removed)------------Resultant Fit filtrr1.05../01

FIG. 4. The observed gravitational anomaly with water and sediments removed and the fit obtained frominversion with the filter (.05, .10).

tional contribution to obtain the reduced gravi-tational profile shown in Figure 4. The densitiesused were those given by W orzel (1965 ): p(water)= 1.03 gm/cm3 and p(sediment) = 2.30 gm/cm3.Because the calculation of an unamb iguous struc-ture requires additional information, we assum eda constant density contrast between the crust andmantle of 0.43 gm/cm3, i.e., p(crust)=2.84gm/cm3 and p(mantle) =3.27 gm/cm3. We alsoassumed that the depth of the Moho at X= 600 kmis about 11 km beneath the ocean surface. Withthese added constraints a value of ZO= 22 km wasfound. Th e results of inverting the gravity profileof Figure 4, using three different filters, are shownin Figure 5. Different amo unts of high frequencydata are present in each of the curves, but thegeneral shape of the Moho is clearly outlined.

The depth of the Moho beneath the continent(0 to 15 0 km) is approximately 33-34 km, a veryreasonable value. The Moho then rises at an over-all rate of abou t .l km/k m (slope -6 degrees) inthe region 200 to 450 km, remains flat between450 to 5 25 km, then rises once more, and finallylevels off at a depth of approximately 11 km be-neath the ocean surface. Although each of thethree topographies gives rise to nearly the samegravitational effect, one feels intuitively that theundu lations seen in the two less filtered profiles

are artificial and do not represent plausible geo-logic models. The smooth profile corresponding tothe filter (.OSO , lOO) seems more likely to repre-sent the true structure.

The agreement between the reduced gravityobservations and the anomaly calculated from thetopography obtained by inversion for filter (.050,.lOO) is shown in Figure 4. The dashed line repre-sents a smoothing of the oscillations occurringbetween x= 128 and x=350 km; with the excep-tion of the points near x= 192,230, and 260 km,the agreement along the whole profile is quitegood.

The short wavelength anomalies probably arisefrom errors in defining the water-sediment boun d-ary or the sediment-crust boundary, but the non-two-dimension ality of these surfaces, and theeffect of possible additional shallow geo logic fea-tures, may also contribute significantly to theseanom alies. Indeed, application of the results ofthe 2-body gravity problem (Parker, 1974) showsthat the depth of a disturbing body, with densitycontrast p= .43 gm/cm3, giving rise to the obser-vations at s= 192,208, and 224 km must be shal-lower than 15 km. For comparison purposes, theoriginal 22 observations and the anomaly calcu-lated from the inverted topography are shown inFigure 6.



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Inversion of Gravity Anomalies 535

Filter-

FIG. 5. Location of the Moho obtained from the inversion of the reduced gravitational anom aly inFigure 4. Three different low pass filters h ave been used to produce Moho reliefs with varying amountsof structural detail.

. Pendulum ObservationsGravity from lnvertrd Topography

FIG. 6. Agreem ent between the 22 original gravity o bservations and the anom aly calculated from theMoh o topograp hy as shown in Figtire 5 [filter: (.OlO , .lOO)]. Effects of the water and sediments havebeen included.



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536 Oldenburg

CONCLUSIONS ACKNOWLEDGMENTS

The formula developed by Parker (1973) forthe rapid calculation of potential anom alies hasbeen rearranged to yield an iterative procedu refor inverting bne-dimension al gravity profiles.Because computations in the inversion schemeinvolve primarily the evaluation of Fourier trans-forms which can be comp uted very quickly, thismethod is capable of handling large numbers ofmodel points without requiring much time

The nonu niqueness of the gravity inversion, asshown by e xplicit ex amples in the text, is charac-terized by two free parameters: p, the densit}contrast between the perturbing layer and thesurrounding medium, and zO, the level at whichthe inversion is mad e. Witho ut additional infor-mation constraining these two param eters, theamb iguity in the gravity interpretation cannot bereduced. 4 low-pass filter has been applied to im-prove convergence of the iteration process but,in trials where the as sumed density was too smallor zo too large, no topography could be foundwhich would give rise to an anomaly agreeingwith the initial observations.

The author is grateful to Dr. R. L. Parker forsuggesting this problem and for criticall!, readingthis man uscript. He also wishes to thank Dr.Parker and Mr. S. Huestis for giving him accessto their unpublished work.

REFERENCESRott, %I. H. D., 1960, The use of rapid digital comput-ing method s or direct gravity interpretation of sedi-mentary basins:Geophys. J., R. Astr. Sot., v. 3, p.63-67.Bullard,,E. C., and Cooper. R. 1. B., 1948, The deter-minatlon of the masses necessarv to produce a given

gravitational field: Proc. Roy. Sot. London, series A,v. 194, p. 332-347.Corbath, C. E., 1965, A least-squares procedure forgravity interpretation: Geophysics, v. 30. p, 228-233.Dyrelius, D., and Vogel, .S., 1972, Improvement of con-vergency in iterative gravity interpretation. Geophys.

J. R. Astr, Sot., v. 2i, p. 195-205.Grant. 1;. S.. and IVest. G. F.. 1965, Interpretationtheory in applied geolihysics: New York, h,IcGraw-

Hill Book Co., Inc.Hart, P. J., editor, 1969, The earths crust and uppermantle: Geophysical Monograph 13.IEEE,, 1967, Past Fourier transform and its applicationto dlgital filtering and spectral analysis: Spec. iss.,AU-IS, p. 43-117.

As a real example, a gravity profile perpendicu-lar to a continental margin was inverted to de-termine the location of the Moho. The ability ofthis inversion scheme to handle large numbers ofmode l points without greatly decreasing the nu-merical stability or greatly increasing the comp u-tation time makes it particularly attractive. Also,since the original form ula for the direct problemcalculated the gravitational effect from a three-dimensional topography, an extension of this in-version scheme to invert a two-dimensional grav-tational anomaly measured over a plane ispossible.

Negi, J. G., and Garde, S. C., 1969, Symmetric matrixmethod for aravitv interpretation: J. Geophpa. Res.,v. 74,~. 380&i-380?. . -Parker, R. L., 1973, The rapid calculation of potentialanomalies: Geophys. J. 1~. Astr. Sot., v. 31, p. 447-455.-- 1974, Best hounds on density, and depth fromgravity data: submitted to Geophyws.

Parker, R. I,., and Huestis, S., Inversion of magneticanomalies in the presence of topography: J. Geophys.Res., v. 79, p. 1587-1593.Skeels, D. C., 1947, Ambiguity in gravity interpreta-tion: Geophysics, v. 12, p. 43-56.Tanner, J. G:, 196i, An automated method of gravityF3;Jg;;tatlon: Geophys. J. R. Astr. Sot., Y. 13, p.

Whittaker; E. T. and Watson, G. N.. 1962, A course ofmodern analyiis: Cambridge, Cambridge UniversityPress.Worzel, J, I,., 1965, Pendulum gravity measurements atsea 1936-1959: New York, John Wiley and Sons.

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