Understanding Gravity Gradients-A Tutorial

7/23/2019 Understanding Gravity Gradients-A Tutorial

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The use of gravity gradient (GG) data in exploration isbecoming more common. However, interpretation of grav-ity gradient data is not as easy as the familiar vertical grav-ity data. For a given source, regardless of its simplicity,gravity gradients often produce a complex pattern of anom-alies (single, doublet, triplet, or quadruplet) as comparedto the simple single (monopolar) gravity anomalies. Thispaper is a minitutorial on gravity gradients and is designedto provide a simple explanation of the complex pattern ofGG anomalies and suggest some guidelines for the inter-pretation of measured surface GG data.

To demonstrate the complex pattern of anomalies asso-ciated with gravity gradients, I will compute the gravity gra-dient components of the full gradient tensor starting withthe basic building block, the gravitational potential. This will

be followed by computing and examining:

the first derivatives of the potential in x,y, andz direc-tions (i.e., the horizontal and vertical components of thegravity field vector)

the second derivatives of the potential (x-, y-, and z-derivatives of each gravity vector component) whichconstitute the nine components of the full GG tensor (ofwhich only five are independent).

Figure 1 shows the model, constructed with GOCAD,used for these computationsa diapiric salt body in a sed-imentary section whose density increases with depth, in ageologic setting typical of the U.S. Gulf coast. The upper part

of the salt is above the nil zone and, thus, has positive den-sity contrasts with the surrounding sediments; the lowerpart of the salt body has negative density contrasts. The nilzone, at depth of about 1 km in this example, is the area wherethe density of the surrounding sediments is identical to thatof salt; hence, its gravity effect is nil (Figure 2).

This model is very realistic and useful because it wasdigitized from a real case history and is really two modelsin onea shallow one with positive density contrasts, anda deeper one with negative density contrasts. Hence, it isuseful for testing the resolving capabilities of gravity gra-dients from shallow to deep sources.

The gravitational potential and its first derivatives. Figure3 shows color contour maps of the gravitational potential

(P) and its first derivatives in the x,y, andz directions (P,x;P,y; and P,z). These derivatives are the horizontal (P,x; P,y)and vertical (P,z) gravity components of the gravity field vec-tor. The salt model depth contours (0.2, 0.5, 1, 2, 3, 4, 5, and6 km) are projected on all the maps for reference and to aidin interpretation.

The potential (P) shows mainly a broad bell-shaped neg-ative anomaly due to the main salt body; the effect of theshallow part of the salt is not obvious although, on closerexamination, there is a subtle change in the contour spac-ing in the northeast, suggesting a small positive anomaly.It is interesting to note that, in spite of the apparent sim-plicity of the potential anomaly, it contains all the informa-

tion that produces the enhanced details and complex anom-alies of the gravity and gravity gradient components shownlater.

The first horizontal derivatives of the potential in x andy or E and N directions produce doublet anomalies, a neg-ativepositive pair along the x and y axes, respectively(Figure 3, top row). These are equivalent to the horizontalgravity componentsgx andgy that would be measured by

Understanding gravity gradientsa tutorial

AFIFH. SAAD, Saad GeoConsulting, Richmond, Texas, USA

THE METER READER

Coordinated by Bob Van Nieuwenhuise

942 T HELEADINGEDGE AUGUST2006

Figure 1. GOCAD salt model and Cartesian coordinates system used.

Figure 2. Density-depth curves for salt and sediments typical ofGulf of Mexico geologic setting.


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a horizontal gravimeter. The pattern of doublet anomaliesis coordinate-dependent as suggested by the rotated patternin Figure 4 for the NE directional horizontal derivative. We

should expect this pattern of gravity anomalies if we con-sider the characteristic properties of the horizontal deriva-tives. The horizontal derivative operator is a phase filter (leftpanel in Figure 5) which will shift the location of anomaliesor, in this case, split the negative Panomaly into a negative-positive pair along the x- or y-axis, respectively. The fre-quency response of /x, for example, is ikx where i is theimaginary number, and kx is the wavenumber in the x direc-tion. Hence, the x-derivative involves a phase transforma-

AUGUST2006 THELEADINGEDGE 94

Figure 3. Gravitational potential P and its first derivatives P,x, P,y, and P,z (x-, y-, and z-gravity field components of the gravity vector gdue to thesalt model shown).

Figure 4. First horizontal derivative of P in the NE direction.

Figure 5. Frequency responses and characteristics of first derivative fil-ters: horizontal derivatives (left), vertical derivative (right).


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tion as well as enhancement of high fre-quencies (or high wavenumbers) relativeto low frequencies. The phase transfor-mation generally produces anomalypeaks (or troughs) approximately overthe source edges in the case of wide bod-ies (width w is large relative to depth d,w > d). The enhancement of highwavenumbers sharpens these peaks toincrease the definition of body edges inaddition to emphasizing the effects ofshallow sources. Another explanation,from elementary calculus, is that in thespace domain the horizontal derivativeis defined as the rate of change of Pwithrespect to x ory. Hence, the horizontalderivative is a measure of the slope orgradient of the anomalies in the x or

y direction (Figure 5, bottom left). If weconsider the P surface as topography,

the potential P (Figure 3) has a negativeslope on the west and south sides of theminimum (going downhill), zero slopeat the minimum, and positive slope onthe east and north sides (goinguphill)thus producing the negative-positive pairs of gravity anomalies P,xand P,y. Notice that we can obtain the P,ypattern of anomalies by a simple 90counterclockwise rotation of the P,x pat-tern, in the same manner as one rotatesthe x-axis to they-axis. In fact, if we rotatethe x- andy- axes 45 counterclockwise,or if we take the directional horizontalderivative of the potential P in the NEdirection, the negative-positive patternof anomalies obtained is rotated in thesame direction as shown in Figure 4,emphasizing the fact that these anom-alies are coordinate-dependent.


Figure 6. Gravity gradients (second derivatives of the potential).

Figure 7. Frequency responses of secondderivative filters.


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The first vertical derivative, on the other hand, is a zero-phase filter (right panel of Figure 5); hence, it will not affectthe location of anomaly peaks, but it will sharpen the poten-tial anomalies and will emphasize the high-frequency com-

ponents due to shallow sources relative to the deeper effects,as seen in the P,z map of Figure 3 (lower right). The verticalderivative of P is, by definition, the rate of change of P withdepth; hence, its effect will be similar to downward continu-ation, making the anomalies sharper and emphasizing shal-lower effects. Notice that the P,z data are the vertical gravitycomponentgz measured by modern-day gravimeters.

The frequency response of all three first derivative fil-ters (Figure 5) is proportional to the wavenumber; hence,we expect these derivatives to enhance the short wave-lengths or high frequencies due to the shallow part of thesalt with positive density contrast as suggested by the bend-ing or embayment of the contours at that location in Figure3. Notice the reverse polarity of the shallow anomalies in

response to the positive density contrast of the salt as com-pared to the deeper salt effect.

The second derivatives of the potential. The various grav-

ity gradient components are computed by taking the hori-zontal x- andy-derivatives and verticalz-derivative of eachof the three gravity components of Figure 3. Figure 6 showsthe five independent components of the gravity gradient ten-sor (second derivatives of the potential P): P,xx; P,xy; P,xz;P,yy; and P,yz along with the dependent second verticalderivative P,zz (P,zz = P,xx P,yy by Laplaces equation).Again, we can expect the single, double, triple, and quadru-ple pattern of anomalies produced, if we keep in mind theproperties and effect of the derivative operators explainedabove, or the frequency responses of the second derivativefilters shown in Figure 7.

The gravity gradient component P,xx is computed by tak-ing the x-derivative of P,x. This results in a second phase


Figure 8. The full gravity gradient tensor.


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Figure 9. Combined products of gravity gradient components: Horizontal gradient and total gradient of gz.

Figure 10. Combinedproducts of gravity gradi-ent components:Differential curvaturemagnitude.


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transformation and further enhancement of the high fre-quencies of the anomalies of P,x. Thus, the negative anom-aly of the doublet of P,x splits into a negative-positive pair,and the positive anomaly splits into a positive-negative pairwest-to-east along the x-axis, resulting into a negative-

strong positivenegative triplet (P,xx of Figure 6). We canalso explain this pattern by examining the slopes of theanomalies of P,x as we proceed from left to right along thex-axis. Notice that the steepest slope is at the center of themap of P,x (Figure 3) and it is positive; the zero slopes areat the trough and peak of P,x, and the gentle negative slopesare to the left and right of the trough and peak, respectively.In a similar manner, we can explain the triplet pattern ofthe component P,yy (center panel of Figure 6) which is sim-ply a 90 counterclockwise rotation of the P,xx pattern.

The gravity gradient component P,xy is computed by tak-ing the derivative of P,x in they (or N) direction or by tak-ing the derivative of P,y in the x (or E) direction. This resultsin a second phase transformation and further enhancementof high-frequencies of the anomalies of P,x or P,y.Considering P,x, the negative anomaly of the doublet of P,xsplits into a negative-positive pair, along they-direction orsouth-to-north and the positive anomaly splits into a posi-tive-negative pair along the y-direction or south-to-north,resulting in a negative-positivenegativepositivequadruplet (P,xy of Figure 6, top center panel). We can alsoexplain this pattern by examining the slopes of the anom-alies of P,x in Figure 3 as we proceed from south-to-northin the y-direction, or the slopes of the anomalies of P,y aswe proceed from west-to-east in the x-direction.

The gravity gradient components P,xz and P,yz and P,zz(right column of Figure 6) are com-puted by taking thez-derivative of P,xand P,y and P,z, respectively. This only

causes further sharpening of the anom-alies and enhancements of the high fre-quencies of P,x and P,y and P,z withoutany changes in the location or shapesof the anomalies, thez-derivative beinga zero-phase filter (Figure 7).

The full gradient tensor can be con-structed by noting that P,yx = P,xy andP,zy = P,yz and P,zx = P,xz (Figure 8).The tensor is symmetric about its diag-onal and its trace, the sum of the diag-onal components (P,xx + P,yy + P,zz), isidentically equal to zero in source-freeregions, according to Laplaces equa-

tion. Thus, the tensor has only five independent components.It is interesting to note from Figure 8 that the first (top) rowof the tensor is identical with the first (left) column and itscomponents are the x-, y- and z-derivatives of the gravityfield horizontal component gx of the gravity vector g (Figure

3). Similarly, the second (center) row of the tensor is iden-tical with the second (center) column and its componentsare the x-,y- andz-derivatives of the horizontal gravity fieldcomponent gy of the gravity vector g; the third (bottom) rowof the tensor is identical with the third (right) column andits components are the x-,y- andz-derivatives of the grav-ity field vertical component gz of the gravity vector g.

Notice the greater enhancement and better definition ofthe shallow anomaly pattern associated with the upper partof the salt in all gravity gradient maps (Figure 6). This is

because the frequency response of all second derivative fil-ters is proportional to the square of the wave number (Figure7). Notice also the reverse polarity of the high-frequencyanomaly pattern in all components as expected from the pos-itive density contrast of the shallow salt. Thus, for example,the triplet of P,xx is positive-negative-positive for shallowsalt as compared to the main negative-positive-negativepattern for the deep salt.

One should emphasize that the pattern of anomaliesproduced is coordinate-dependent. However, one can usethese patterns and shapes of gravity gradient anomalieswith the projected outline of the causative salt body in thisexample to develop interpretation techniques for locatingthe main salt body, its edges, and its shallow part. For exam-ple, the zero contours of P,xx and P,yy closely define the west-east edges and south-north edges of the main salt body,


Figure 11. Gravity gradient invariants (after Pedersen and Rasmussen,1990).

Figure 12. Other gravity gradient combinations: Euler deconvolutionusing GG tensor components.


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respectively (Figure 6, top left and center panels). Also, thepeaks and troughs of the quadruplet pattern of P,xy anom-alies are located roughly around the perimeter of the salt

body (Figure 6, top center) and can be used to delineate thesalt boundary. The negative-positive pairs of the P,xz andP,yz anomalies are near or on the west-east and southnorthedges of the body, respectively (Figure 6, top-right and cen-ter-right panels). These relations depend on the width/depthratio of the source and are generally valid only for wide bod-ies, i.e., bodies whose width is greater than their depth (w>d). It should be emphasized that narrow sources (wd),including point masses, will produce similar geometric pat-tern of complex anomalies as in Figure 6; however, the rela-

tions discussed above do not hold inthis case; the locations of the zero con-tours, the lows and highs, and size ofthe anomalies in general will dependmainly on the depth to the source,rather than the width/depth ratio.Finally, the P,zz anomalies (Figure 6,

bottom-right panel) can be used tolocate the center of the anomaloussource mass.

Combinations of GG components(invariants). Various combinations ofthe gravity gradient components can

be used to simplify their complex pat-tern and to further enhance and aid inthe interpretation of the data. Figures9 and 10 show three examples: ampli-tude of the horizontal gradient of ver-tical gravity (gz); amplitude of the totalgradient or analytic signal of gz; andthe differential curvature which is alsoknown from the early torsion balanceliterature as the horizontal directive

tendency or HDT. The horizontal andtotal gradients ofgz (Figure 9) are com-puted from combinations of the ele-ments of the third column (or thirdrow) of the gravity gradient tensorP,xz and P,yz and P,zz (Figure 6). Thelatter are the x,y, andz derivatives ofP,z (orgz). The horizontal gradient ofgz can be used as an edge-detector orto map body outlines. The analyticsignal can be used for depth interpre-tation. The differential curvature(Figure 10) is computed by a combi-nation of the other components of thetensor: P,xx and P,xy and P,yy. The

magnitude of the differential curva-ture emphasizes greatly the effects ofthe shallower sources. Several inter-pretation techniques for the differen-tial curvature are available in the earlyliterature of the torsion balance.

The three examples of combinedGG products discussed above are use-ful in simplifying and focusing thecomplex pattern of anomalies overtheir source, providing more enhance-ments to the high-frequency part ofanomalies due to shallow sources, andproducing coordinate-independent or

invariant anomalies. These are per-haps easier to interpret than the original gradient compo-nents. Other coordinates-independent invariants can becomputed and used as well for interpreting the data usingdifferent combinations of the GG components. For exam-ple, one can compute the horizontal and total gradients of

gx andgy from the elements of the first row and second rowof the GG tensor, respectively. Figure 11 defines other grav-ity gradient invariants, I0, I1, and I2 suggested by Pedersenand Rasmussen (1990) and used for interpretation of GGdata. Gravity gradient components can also be combinedto form three different Euler equations for gx, gy, and gz thatcan be used to solve for source depth (Figure 12), as sug-gested by Zhang et al. (2000).


Figure 13. Similarity between surface horizontal gravity (in the X-Y plane) and subsurface verticalgravity (in the X-Z plane).

Figure 14. Similarity between surface horizontal gravity gradient difference (in the X-Y plane) andsubsurface vertical gravity gradient (in the X-Z plane).


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Similarities between surface and subsurface gravity andgravity gradients. It is interesting to note that there are simi-larities between surface variations of the horizontal gravityand GG components and subsurface variations of verticalgravity and vertical GG (or anomalous apparent density) suchas those observed in a borehole. Figures 13 and 14 show exam-ples illustrating these similarities. Figure 13 compares surfacevariations in the x-y plane of the horizontal gravity compo-nent P,y (Figure 3) with subsurface variations in the x-z planeof vertical gravity due to a spherical source. Figure 14 showsa similar comparison between surface gravity gradient dif-ference (P,xx P,yy) and subsurface vertical gravity gradientor apparent density anomaly, as used in borehole gravitywork, due to the same spherical mass. Vertical profiles in the

z direction extracted from the maps on the right-hand sidesof Figures 13 and 14 show the anomalous responses expectedin boreholes and measured in borehole gravity surveys (Figure15). In this example, the boreholes are located at a remote dis-

tance X=2R from the center of the sphere of radius R. Theapparent gravity doublet and GG triplet patterns encoun-tered in the borehole are similar to the patterns of gravity gra-dient profiles that would be observed on the surface. Thus,interpretation techniques developed and used for boreholegravity and gravity gradient data can be extended and usedfor surface gravity gradient data interpretation. Overall, expe-rience with interpretation of borehole gravity data can bevaluable for the interpretation of surface gravity gradient pro-file and map data.

Conclusions.Gravity gradients (GG) often produce a patternof complex anomalies that is coordinate-dependent, not nec-essarily reflecting the shape of the underlying sources.Understanding GG anomalies is important in the interpreta-tion of measured data. It is easy to understand the complexpattern of gravity gradients if one considers the fact that theyare derivable from the simple gravitational potential, beingthe directional second derivatives of the potential. In general,for 3D sources producing single bell-shaped potential and ver-tical gravity anomalies, the P,zz gravity gradient componentconsists of a single anomaly; the P,xz and P,yz componentsconsist of doublet anomalies; the P,xx and P,yy componentsconsist of triplet anomalies; and the P,xy component consists

of quadruplet anomalies. Various combinations of GG com-ponents can be used to produce coordinate-independentinvariants that are simple, easy to interpret, more localized,and more related to the size and shape of the sources. Thereare also similarities between surface and subsurface (or bore-hole) variations of certain gravity and gravity gradient com-ponents. Hence, interpretation methods developed and usedfor borehole gravity data may be applicable or can be extendedto surface GG data interpretation. Certainly past experiencewith borehole gravity can be valuable in interpreting surfacegravity gradient data.

Suggested reading. Gravity gradiometry resurfaces by Bell etal. (TLE, 1997). Gravity gradiometry in resource exploration

by Pawlowski (TLE, 1998). The gradient tensor of potential field

anomalies: Some implications on data collection and data pro-cessing of maps by Pedersen and Rasmussen (GEOPHYSICS, 1990).Euler deconvolution of gravity tensor gradient data by Zhanget al. (GEOPHYSICS, 2000). TLE

Acknowledgments: Parts of this work were conducted while the author wasemployed by Gulf Research and Development, Chevron, and Unocal com-

panies. This paper was presented at the SEG75 Annual Meeting in Houston,Texas.

Corresponding author: [email protected]


Figure 15. Borehole vertical gravity and gravity gradient (apparent den-sity) profiles due to a sphere of radius R, density contrast . Boreholedistance X = 2R from the center of the sphere.

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Understanding Gravity Gradients-A Tutorial