A Rapid Location Independent Full Tensor Gravity Algorithm

V. Jayaram, K.D. Crain, G.R. KellerMewbourne College of Earth and Energy

The University of OklahomaNorman OK, U.S.A

Email: {vjayaram,kevin.crain,grkeller}@ou.edu

Mark BakerGeomedia Research & Development

El Paso TX, U.S.AEmail: bakergrd@cs.com

Abstract—We present an algorithm to rapidly calculatethe vertical gravitational attraction and full tensor gravitygradient (FTG) values due to a 3D geologic model. Ourtechnique is based on the vertical line source (VLS) elementapproximation with a constant density within each grid cell.This type of parameterization is well suited for high-resolutionelevation datasets with grid size typically in the range of 1m to 30 m. Our approach can perform rapid computationson large topographies including crustal-scale models derivedfrom complex geologic interpretations. Most importantly theproposed model is location independent i.e. we can computeFTG anywhere in the geologic volume of interest (VOI) and isnot limited to performing computations outside the VOI.

Keywords-3D Geology & Interpretation, full tensor gravity,GPU, line-element, crustal-scale models.

I. INTRODUCTION

A modern geophysical interpretation requires an inte-grated approach in which a variety of geological and geo-physical data are employed in and 3-D analysis. However,many computational challenges exist, particularly when con-sidering the available ’state of the art’ computing resources.Gravity data are widely available and relatively inexpensiveto obtain and are a good starting point for an integratedanalysis. Forward modeling of mass distributions is a pow-erful tool to model and visualize gravity anomalies thatresult from different geologic settings.Thanks to databasedevelopment efforts around the world and the emergence ofhigh-quality gravity field models based on satellite measure-ments, gravity data are available globally as are high qualitydigital elevation models (DEM) thanks to NASA’s ShuttleRadar Topography Mission and national and local efforts toprovide DEM data. Thus, development of an efficient andflexible approach to 3-D modeling and inversion of gravityanomaly data is very timely. Historically, a classic techniqueused to model gravity data in 2D was developed by Talwani(1959)[6]. The gravity anomaly resulting from either a 2Dor 3D model is computed as the sum of contributions ofindividual bodies, each with a given density (ρ) and volume(V ) which is a mass m directly proportional to ρ× V .

In this paper, we demonstrate our new 3D gravity mod-eling approach utilizing the derived VLS algorithm [4].However, the computational requirements are substantial,and below, we describe our approach to perform large scale

(in terms of areas and elevation changes) location indepen-dent calculations can be executed utilizing high throughputprocessing hardware.

THEORY

Unlike traditional gravity computational algorithms, theVLS approach [5], [1] can calculate gravity effects at lo-cations interior or exterior to the model.The only conditionthat must be met is the observation point cannot be locateddirectly above the line element. Therefore, we perform alocation test and then apply appropriate formulation to thosedata points. We will present and compare the computationalperformance of the traditional prism method versus the lineelement approach on different CPU-GPU system configura-tions.

The algorithm calculates the expected gravity at stationlocations where the observed gravity and FTG data wereacquired. This algorithm can be used for all fast forwardmodel calculations of 3D geologic interpretations for datafrom airborne, space and submarine gravity, and FTG in-strumentation.

A. Equations

Now, consider a vertical gravitational line source shownin Figure 1. The vertical line source is located at (xo, yo)and extends from ztop to zbot (positive z direction). The linesource have a linear mass density of λ ( kg/m). The elementof mass dmo = λ dzo. The total mass is given by∫ zo= zbot

zo= ztop

dmo = λ

∫ zo= zbot

zo= ztop

dzo = λ ( zbot − ztop ) (1)

Since, λ = ρ Area where Area is the cross-sectionalarea of a vertical rectangular prism ( ∆x ∆y ) then

λ = ρ Area

= ρ Area zBOT − zTOP

zBOT − zTOP= mass

zBOT − zTOP

Since we know the anomalous gravitational field andgradients of the point source, we merely integrate to findthe gravity tensor components.

2931978-1-4799-1114-1/13/$31.00 ©2013 IEEE IGARSS 2013

As we derive these components, there are several integralsthat must be evaluated. We will set these up first.

Figure 1 shows the sphere model used in our experiments.The diameter of this sphere was set to 1kms. The sphereis modeled using vertical line elements at different grid sizespacings ranging from 10 m to 100 m.

Figure 1. Figure 1: Sphere model constituting vertical line sources witheach line source being placed at the center of the grid.

We define as beforeA2 = (x− xo)

2+ (y − yo)

2

and r =∣∣∣→r − →

ro

∣∣∣=

√( x− xo)

2+ ( y − yo)

2+ ( z − zo)

2.

These integrals are:

I1 =∫

1r3 dz0 =

A2 ̸= 0

z0 − zA2 r

A2 = 0− 1

2( zo − z )| zo − z |

I2 =∫

z0r3 dz0 =

A2 ̸= 0

− A2 + z (z − z0)A2 r

A2 = 0z − 2 z0

2( z0 − z )| z0 − z |

I4 =∫

1r5 dz0 =

A2 ̸= 0( z0 − z )⟨ 3A2 + 2(z0 − z )2⟩

3 A4 r3

A2 = 0

− | z0 − z |4( zo − z )5

I5 =∫

z0r5 dz0 =

A2 ̸= 0

−A4 − 2 z (z0 − z)3 + 3 A2 z ( z − z0)3 A4 r3

A2 = 0z − 4 z0

12(z0 − z)3| z0 − z |Each integral will be evaluated from zTOP to zBOT so,

for example, the I1 integral will become (if A2 ̸= 0)

∆I1 =∫ zbotztop

1r3 dz0 = − zo−z

A2r

∣∣∣z0 =zBOTz0 = zTOP

= − zBOT zA2rBOT

+ zTOP−zA2rTOP

and the same notation will be used for the other fiveintegral evaluations.

Then the gravitational field at (x,y,z) due to the verticalline source is given by

• lineTz = linegz(x, y, z) =∫ zbotztop

dgz

• lineTxx = linegxx(x, y, z) =∫ zbotztop

dgxx

• lineTxy = linegxy(x, y, z) =∫ zbotztop

dgxy

• lineTxz =line gxz(x, y, z) =∫ zbotztop

dgxz

• lineTyy = linegyy(x, y, z) =∫ zbotztop

dgyy

• lineTyz = linegyz(x, y, z) =∫ zbotztop

dgyz

• lineTzz = linegzz(x, y, z) =∫ zbotztop

dgzz

In most gradiometry applications, the vertical derivative Tz

is the most meaningful component as it locates the target [9].The Txx and Tyy components identify N-S and E-W edgesof the target. In interpretations, the horizontal derivativesof the vertical component Tzx and Tzy , and horizontalcomponent derivatives Txx and Tyy provide the centralaxes of target mass, highs and lows defining fault trends.Similarly, Txy shows anomalies associated with corners ofthe target. Finally, Tzz identifies vertical changes in gravityand also represents the difference between the near and farresponse. It highlights all edges and is the easiest gradientto interpret directly. Geologic structure is usually evident inthe data when large mass anomalies, such as salt dome, arepresent. Notice from the equations above, the Tzz gradientdata is a summation of Txx and Tyy gradients. It highlightsall edges and is useful for understanding the approximateshape of the dominant mass anomaly.Figures 3-6 show thecontour plots of FTG computed at different levels - above,below and inside the sphere. Now, in order to perform abenchmark test the accuracy of our VLS algorithm, we are

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compare it to the calculated analytical solution of a buriedsphere model as shown below.

Figure 2. A buried sphere model for analytical gz calculations. Figuredepicted is a courtesy of [7].

The buried sphere model (Telford et. al. 1990)[8]illustrated in Figure 2 depicts the fundamental proper-ties of gravity anomalies. Here we describe the analyticalformulation of the buried sphere model and compare the gzcalculations to our derived VLS sphere models at varyinggrid size. Using G = 6.67× 10−11Nm2/kg2 [9]

δgz =4π

3G(δρ)R3 z

(x2 + z2)3/2

where the variables and units are:• δgz = vertical component of gravitational attraction

measured by a gravimeter (mGal)• δρ = difference in density between the sphere and the

surrounding material (g/cm3)• R = radius of the sphere (m)• x = horizontal distance from the observation point to

a point directly above the center of the sphere (m)• z = vertical distance from the surface to the center of

the sphere (m)In Table I below we show the various gz model errors

based on maximum absolute error (MAE) that is associatedwith the prism model [3] and the proposed VLS sphere mod-els with varying grid sizes. The closed expression solution ofa right angular prism is derived by Nagy 1966 [3]. We havecompared the closed form expression of the right angularprism to the VLS at different prism resolutions. Through thisresult we have tried to demonstrate that the proposed VLStechnique very closely approximates the analytical closedform solution of a right angular prism. In this section wealso show the computations speeds achieved while usingdifferent compute architectures.

B. Tables

Table IMAE COMPARISON OF VARIOUS SPHERE MODEL WITH VARYING

GRID-SIZE TO ITS CALCULATED ANALYTICAL SOLUTION

MAEPrism 0.004167047510 m 0.08092199825 m 0.16320696450 m 0.003493141100 m 0.003493141

Table IICOMPARISON OF VARIOUS GPU-CPU COMPUTATION TIMES FOR A

SPHERE MODEL WITH VARYING GRID-SIZE

CPU CPU GPUMAT CUDAPrism Line Line Linein sec. in sec. in sec. in sec.

10 m 4595.27 24.73 11.32 2.2325 m 13.96 3.37 1.34 0.1450 m 3.46 0.78 0.42 0.05100 m 0.86 0.19 0.023 0.02

Figure 3. FTG Computations at 100 m above the top of the Sphere model.

II. CONCLUSIONS

The results in Table II suggests that giga-scale order ofcalculations can be done in matter of milliseconds with theVLS algorithm compared to the traditional prism technique[3] utilizing GPU-CPU hardware configurations. Most im-portantly the proposed model is location independent i.e.we can compute FTG anywhere in the geologic volume ofinterest (VOI) and rather than being limited to performingcomputations outside the VOI. Figures 3-6 show the contourplots of FTG computed at different levels - above, belowand inside the sphere. Notice the flip in the coloring of thecontours when the direction are changed above and belowthe zero-plane of the sphere. We also demonstrated in Table

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Figure 4. FTG Computations at 100 m below the bottom of the Spheremodel.

Figure 5. FTG Computations at 300 m inside the upper half of the Spheremodel

Figure 6. FTG Computations at 300 m inside the lower half of the Spheremodel

I that the accuracy based on the MAE metric of our modelscomes very close to the calculated analytical solution ofthe buried sphere model. Based on these results, we havebegun to apply our software to the calculation of geolog-ically realistic models with good results. When applied tolarge complex geologic structures, our approach makes thecomputations for the application of inverse methods tractableand very efficient.

REFERENCES

[1] B.J. Drenth, G.R. Keller, and R.A. Thompson, Geophysicalstudy of the San Juan Mountains batholith complex, south-western Colorado, Geosphere, June 2012, v. 8, p. 669-684.

[2] G.R. Keller, T.G. Hildenbrand, W.J. Hinze, and X. Li, Thequest for the perfect gravity anomaly: Part 2 Mass effectsand anomaly inversion: Society of Exploration GeophysicistsTechnical Program Expanded Abstracts, v. 25, p. 864., 2006.

[3] Dezso Nagy, The Gravitational attraction of a right angularprism, Geophysics, Vol. XXXI, April 1966, pp.362-371.

[4] Z. Frankenberger Danes, On a successive approximationmethod for interpreting Gravity Anomalies, Geophysics, VolXXV, No. 6, December 1960, pp. 1215-1228.

[5] Kevin Crain, Three Dimensional gravity inversion with a prioriand statistical constraints, Ph.D. Dissertation, 2006, Universityof Texas at El Paso.

[6] M. Talwani, J.L. Worzel, and M. Landisman, Rapid gravitycomputations for two-dimensional bodies with application tothe Mendocino submarine fracture zone, J. Geophys. Res.,64(1), 4959, 1959.

[7] R.J. Lille, Whole Earth Geophysics: An Introductory Textbookfor Geologists and Geophysicists”, Prentice Hall, 1999

[8] W. M. Telford, L. P. Geldart, R. E. Sheriff, Applied Geo-physics, Second Edition, Cambridge University Press, Oct 26,1990 - Science - 792 pages.

[9] R.J. Blakely, Potential Theory in Gravity and Magnetic Appli-cations, Cambridge University Press, 441 p., 1995.

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