General source gravity modeling In contrast to idealized body effects with closed analyt-

Gravity exploration

DATY 21 DATY 2 − DATY 1 Saad and Bishop, 1989

3.6 General source gravity modeling In contrast to idealized body effects with closed analyt-

3.6 General source gravity modeling 55

FIGURE 3.11 Gravity modeling of the Minden salt dome, Louisiana, with an equivalent stack of disks of varying density contrasts constrained by available seismic and drillhole data.

Adapted fromN e t t l e t o n(1943).

presented the infinite vertical cylinder effect in terms of Legendre polynomialsPni(cosθ=z/r) of degreeni, which offers a less tedious modeling approach. For effi- cient digital computation, Equationg#6gives these effects with the Legendre polynomials expanded through degree ni=4 by the recurrence formula

*

(ni+1)P(ni+1)=(2ni+1) z

r

Pni−(ni)P(ni−1)

+

∀ni≥0 &P0≡1. (3.70) These results can provide the gravity effect of a disk with top and bottom surfaces atz1andz2, respectively, by sub- tracting the effect of a cylinder with top atz₂ from the effect of a cylinder with top atz1, as described in Equation g#7of Table 3.2.

3.6 General source gravity modeling

56 Gravity potential theory

oz’

z

x

g

Δe Δe x’

(a)

φ_i θ

(b)

P Q

a

e d

c b(x₁,z₁)

R(x₁,z)

a₁ x

Δz Δz

FIGURE 3.12 (a) Template for evaluating the gravity effect of 2D bodies of irregular cross-section. Adapted fromH u b b e r t(1948).

(b) Polygon approximation of the irregular cross-section for a 2D body with line segment endpointsa, b, c, d, andein the (x, z)-plane. Adapted fromT a l w a n iet al.(1959).

correction in Equation 3.42, and variations in density of the conceptual model can be considered as well by apply- ing the appropriate density to the area or portion thereof on the template.

Digital computing methods have essentially displaced the graphical procedure, which is effective and straightfor- ward, but lengthy and tedious to implement. The widely used digital modeling approach due toTalwani et al.

(1959) approximates the boundary of the cross-section by straight line segments as shown in Figure 3.12(b). Here, the line integral effect used inHubbert (1948) is approx- imated by the summed gravity effects of connected line segments at the observation point where all geometric ele- ments of the integral are expressed completely in terms of the 2D coordinates (x, z) of the end points or vertices of the line segments.

3.6.2 Generic 3D modeling procedures

The line integration method was also adapted by TalwaniandEwing(1960) to model the gravity effects of an irregular 3D source represented by a vertical stack of niinfinitely thin, but disconnected horizontal laminas. As

P’(0,0,0)

P’(0,0,z) Body M

dzF E D

H A G

C B y-axis

x-axis

z-axis

ztop

zbottom

x

y (a)

(b)

rn+ A

B

C E D

F G rn H

Contour at depth Z

1 1

FIGURE 3.13 (a) Determining gravity at pointPfrom a 3D body of irregular shape that can be represented by a series of horizontal laminas each bounded by a connected sequence of straight line segments. (b) Polygon approximation of the irregular cross-section for a 3D body with line segment endpointsA, B, C, D, E, F, G, and Hin the (x, y)-plane. Adapted fromT a l w a n iandE w i n g(1960).

illustrated in Figure 3.13, theith lamina is approximated by a polygon with straight line segments. The gravity effect is G(σ)S(z_i) with the line integral around the polygonS(z_i) evaluated at the observation point completely in terms of the 3D coordinates (x, y, z_i) of the end points of the line segments.

In contrast to the analytical or closed-form gravity effect of the polygon, vertical integration of the 3D source must be carried out numerically by summing at the obser- vation point the effects of thenipolygons that approximate the elevation contours of the arbitrarily shaped body. The integration can be effectively implemented using a rela- tively large number of polygons for the top third of the body, and two-thirds and one-third of this number of poly- gons for the middle third and bottom third portions of the body, respectively. The thicknesses of the laminas usually are restricted to less than one-tenth its depth. Another com- mon approach uses a conventional scheme like Simpson’s rule that assumes equal spacing of the laminas over thez- limits of integration. However, the integrand is analytical, and thus the numerical integration can be carried out more efficiently and with least squares accuracy by the Gaussian quadrature method (e.g.Stroud and Secrest, 1966;

Ku, 1977;von Freseet al., 1981b).

3.6 General source gravity modeling 57

3.6.3 Least squares 3D modeling

In the generic modeling application of the previous section, interpolation polynomials approximate the integrand by a summation formula like

z_b z_a

S(z)∂z ni

i=1

AiS(z_i), (3.71)

where thenivaluesAiare the weights to be given to theni functional valuesS(z_i) evaluated at the interpolation coor- dinates,z_i. However, because the integrand is analytical, Gaussian quadrature formulae can be developed to yield selected values of the interpolationz_iand coefficientsAi

so that the sum in Equation 3.71 gives the integral exactly when S(z) is a polynomial of degree 2ni or less (e.g.

StroudandSecrest, 1966;Carnahanet al., 1969).

The Gaussian coefficientsAiare obtained from a poly- nomial of orderniwhich is orthogonal over the interval of integration such that thenipoints of interpolationz_i are the zeros of the polynomial. Orthogonal polynomials commonly used to develop Gaussian quadrature formu- lae include Legendre, Laguerre, Chebyshev, and Hermite polynomials. Here, however, only the prototype Gaussian method involving Legendre polynomials is considered.

Legendre polynomialsPni(¯z) of ordernithat are orthog- onal over the interval (−1≤z¯≤1) are given by Pni(¯z)=

1 2ⁿⁱni!

dⁿⁱ

d¯zⁿⁱ[¯z²−1]ⁿⁱ

∀ P0(¯z)≡1, (3.72) and abide by the recurrence relations expressed in Equa- tion 3.70. Thus, the standard Gauss–Legendre quadrature (GLQ) integration over the interval (−1,1) is

1

−1S(¯z)∂z¯ ni

i=1

AiS(¯zi), (3.73)

where the interpolation points ¯ziat which the integrand is evaluated are the zeros of Equation 3.72 and the Gaussian coefficients are

Ai= 2(1−z¯²_i)

ni²[Pni−1(¯zi)]². (3.74) Now, for arbitrary limits of integration such as in Equation 3.71 it is necessary to map the standard inter- val (−1≤z¯i≤1) into the interval of integration (z_a≤ z_i≤z_b) by the transformation

z_i= z¯i(z_b−z_a)+(z_b+z_a)

2 . (3.75)

Thus, the integral in Equation 3.71 can be approximated as

z_b z_a

[S(z)]∂z= 1

−1

S

z¯i(z_b−z_a)+(z_b+z_a)

2 ∂z

(z_b −z_a) 2

ni i=1

AiS(z_i), (3.76)

so that the – GLQ expression for the 3D gravity effect of an irregularly shaped body is

g=Gσ z_b

za

S_i(z)∂zGσ

(z_b−z_a) 2

ni i=1

A_iS(z_i).

(3.77) A large value of the order ni insures a very accurate solution, but meaningful improvements in solution accu- racy are difficult to achieve in practice once the nodal spacing (i.e. the spacing between the roots or zeros of Equation 3.72) is smaller than the depth to the source (e.g.

Ku, 1977; von Frese et al., 1981b). Thus, choosing the lowest orderniso that (z_i< z_b) facilitates efficient implementation of Equation 3.77.

The Gaussian coefficientsA_iand corresponding nodes

¯

ziare tabulated to 30 significant figures for Legendre poly- nomials of ordersni=2 to 512 inStroudandSecrest (1966). However, algorithms for digitally computing these values are also available in books on numerical methods (e.g.Carnahanet al., 1969;Presset al., 2007). Expe- rience suggests that these values to 10 digit accuracy for orders up to ni=16 are sufficient for most geological applications.

Equation 3.77 models the anomalous gravity effect of an irregular 3D body by summing at each observation point the gravity effects ofnipolygons located at source coordinates given by Equation 3.75 and weighted by GLQ coefficientsAi. Relative to more conventional numerical integrations, GLQ integration minimizes the number ni of laminas for accurate least squares modeling because the integrand (i.e. the gravity effect of the polygon) is analytical, and thus can be taken at the nodes or roots z_i(¯zi) of the Legendre polynomial selected to span the integration interval (z_a, z_b).

A further simplification extends the GLQ integration to evaluate the line integral around the polygonS(z_i) using the versatile analytical point pole or mass gravity expres- sions at the nodes of Legendre polynomials of ordersnj andnkthat respectively span the polygon’sxandylim- its. This extension transforms the 3D Equation 3.77 into Equation 3.26 with kernelF_zand least squares numerical

58 Gravity potential theory

solution

g≡Fz

y_kb −y_ka 2

× nk

k=1

⎧⎨

⎩

x_{j b}−x_{j a} 2

× nj j=1

&

z_ib−z_ia 2

ni i=1

[F_z]Ai

' Aj

⎫⎬

⎭Ak. (3.78)

Here theni×nj×nkpoint pole gravity effects per unit volumeF_zare evaluated at the coordinates within the body given by

z_i=0.5[¯zi(z_ib −z_ia)+z_ib+z_ia], x_j =0.5[ ¯xj(x_{j b} −x_{j a})+x_{j b} +x_{j a} ],

and y_k =0.5[ ¯yk(y_kb −y_ka)+y_kb +y_ka], (3.79)

where ¯z_i, ¯x_j, and ¯y_k are the respectiveni, nj, and nk Gaussian nodes in the standard (−1,1) interval.

For a uniformly dimensioned body like the prism, the integration limits for evaluating Equation 3.78 are easy to specify. In this case, for example, (x_{j a} , x_{j b})=(x_a, x_b), (y_ka , y_kb ) = (y_a, y_b), and (z_ia, z_ib)=(z_a, z_b). However, for the irregular body, the differential integration limits in each of the three dimensions must be interpolated at every node from a set of body point coordinates that provide an approximation of the surface envelope of the body (e.g.Ku, 1977;von Frese et al., 1981b). Typically, the body is considered for its longest dimension, which for the sake of argument might be thexdimension. Then thea-lower andb-upperx-limits of the body (x_{j a} , x_{j b} ) are established from which thenjGauss–Legendre nodes x_j are determined. Next, interpolations of the body point coordinates are performed at eachx_j to determine the (y_ka , y_kb )-limits of the body for thenknodesy_k. Similarly, the vertical coordinates of the body points are interpo- lated at each horizontal coordinate (x_j, y_k) to appropriate vertical (z_ia, z_ib)-limits of integration for theninodesz_i. In practice, maximum accuracy effectively results when the spacing between the equivalent point source nodes is smaller than the depth to the top of the source (e.g.Ku, 1977;von Freseet al., 1981b).

The 3D body accordingly is represented by a distribu- tion of ni×nj×nk point poles from which all grav-

FIGURE 3.14 Details of the2×2×2Gauss–Legendre quadrature formula for estimating the vertical gravity anomalyGZ3=Fz≡g of a uniform density, 3D prism. Adapted fromK u(1977). Here, the Gaussian coefficientsV1andV2in all spatial dimensions are equal in magnitude, but opposite in sign because the heightH, widthW, and lengthLof the cube are equal. The GLQ formula for the prism’s magnetic total field effect is illustrated in Figure 9.8.

ity effects of the body can be least squares estimated by

[U;Fx;Fy;Fz≡g;Fxx;Fyy;Fzz;Fxy;Fxz;Fyz]

y_kb −y_ka 2

nk k=1

x_{j b}−x_{j a} 2

× nj

j=1

&

z_ib−z_ia 2

ni i=1

[U;F_x;F_y;F_z;F_xx ;

F_yy ;F_zz;F_xy ;F_xz ;F_yz ]Ai

Aj

Ak. (3.80) Here, the unprimed variables in the left portion of the equa- tion are the complete gravity effects of the extended 3D body in Equations 3.22–3.33 as derived from the weighted triple sum of the primed variables in the right portion which are the corresponding integrands of the equations. How- ever, for the integration to hold, these kernels must be evaluated at the strategic (x_j, y_k, z_i)-coordinates within the body given by Equation 3.79. A 3D example giv- ing the details of estimating the gravity effect (Fz≡g) of a rectangular prism by GLQ integration is shown in Figure 3.14.

3.6 General source gravity modeling 59

3.6.4 Least squares 2D modeling

For the 2D body, the gravity effects are similarly obtained by fitting thex- andz-limits of the cross-section with Legendre polynomials of ordersnj and ni, respectively (e.g.Ku, 1977). The body accordingly is represented by a 2D distribution ofnj×nipoint poles that estimate the source’s least squares gravity effects by

[U;Fx;Fz≡G;Fzz= −Fxx;Fxy]

x_{j b}−x_{j a} 2

nj j=1

&

z_ib−z_ia 2

ni i=1

[U;Fx;Fz;Fzz

= −Fxx;Fxz]Ai

'

Aj. (3.81) Here again, the unprimed variables in the left portion of the equation are the complete gravity effects of the 2D body in Equations 3.35–3.41 as derived from the weighted double sum of the primed variables in the right portion which are the corresponding integrands of the equations evaluated at the strategic (x_j, z_i)-coordinates within the cross-section according to Equation 3.79.

3.6.5 Least squares modeling accuracy

In 2D and 3D cell applications, the minimum numbers of equivalent point sources required for GLQ gravity modeling arenj×ni=2×2=4 andnk×nj×ni= 2×2×2=8, respectively. As illustrated inKu(1977), the modeling assumes that the cell is subdivided into quadrants where mass is concentrated at points displaced towards the corners relative to the centers of the quadra- tures. If the estimates are made too close to the body, the gravity effect is dominated by the individual point source effects, whereas adding sources so as to make the source spacing smaller than the elevation of the estimates inte- grates the point effects to a closer least squares estimate of the body’s gravity effect.

Thus, the practical trade-off between accuracy and the speed and computational labor of GLQ integration is con- trolled by the number of point sources. The integration must be computed at a sufficiently great distance from the body that the individual point pole effects coalesce into an acceptable approximation of the analytical solution. In practice, this can be achieved either by subdividing the body into smaller blocks or by simply increasing the num- ber of point sources until the distance between them is smaller than the depth to the top of the body (e.g.Ku, 1977;von Freseet al., 1981b). For example, Gaussian nodes and weights tabulated to 30 significant figures for

Legendre polynomial orders 2 to 512 are readily available (e.g.StroudandSecrest, 1966). The tabulated values could be used to model the gravity effects of the prism by a (512×512×512=134,217,728)-point GLQ formula, but it would match the analytical solution in many more significant figures than is applicable in practice.

In general, changes in the significant figures of the GLQ estimate decrease as the number of nodes increases.

Thus, the trade-off in accuracy and computational effort is effectively optimized by the smallest number of point source nodes beyond which changes in the estimate’s least significant figure are negligible.

3.6.6 Least squares modeling in spherical coordinates

Regional gravity anomalies registered in spherical coordi- nates are becoming increasingly available for geological studies of large areas of the Earth and other planetary bodies of the solar system as the result of satellite mea- surements and continental-scale and larger compilations of terrestrial, marine, and airborne surveys. The above mod- eling results can be adapted by determining the equivalent point source (EPS) effects in spherical coordinates. This conversion requires the Cartesian-to-spherical coordinate transformations given by

x=rcos(θ) cos(ϕ), y=rcos(θ) sin(ϕ),and

z=rsin(θ), (3.82)

where the body-fixed spherical coordinates include geo- centricr-distance,θ-latitude (i.e. the co-latitude), andϕ- longitude so that

r=zcos(θ)+Ksin(θ), θ= −zsin(θ)+Kcos(θ), andϕ=ycos(ϕ)−xsin(ϕ) (3.83) withK=[xcos(ϕ)+ysin(ϕ)] (Figure 3.15).

By Equation 3.82, the Cartesian gravity point pole potential (Equation 3.16), for example, transforms into the spherical coordinate EPS expression given by U(r, θ , ϕ)= G×m

|r−r|= G×m

R , (3.84)

where G is universal gravitational constant, m is the mass (or mass contrast) of the point pole, R= r²+r²−2rrcos(δ), r is the radius vector directed from the Earth’s center to the observation point (r, θ , ϕ), ris the radius vector directed from the Earth’s center to the source point (r, θ, ϕ),δis the angle betweenrandr such that cos(δ)=cos(θ) cos(θ)+sin(θ) sin(θ) cos(ϕ− ϕ), (θ, θ) are the geocentric latitude (i.e. co-latitude)

60 Gravity potential theory

cos δ = cos θ cos θ′ + sin θ sin θ′ cos (φ − φ′)

cos ψ =r − r ′cos δ

R =∂R

∂r

EPS magnetic effects EPS gravity effects

(U , F₄, F , F_{4 r}≡F , F_tt, F_tt, Ftt;F_rr;F_rr) (V, Br, B₆;B₉;B₆₆;B₆;B₉₉;B_rr;B_rr;B_rr) q(x, r,θ,ϕ, r ,θ ϕ)∂v

GENERALIZED ANOMALY

ϕb ϕa

θb θa

γb γa

q(xr,θ,ϕ, r ,θ,ϕ)(r²sinθ)∂r∂θ ∂ϕ

φ_kb−φ_ka 2

nk k = 1

θ_{j b}−θ_{j a} 2

nj

j=1

r_ib−r_ia 2

ni

j=1

q(x, r,θ,ϕ, r ,θ,ϕ) A_i A_j A_k (r, θ, φ)

R = |r − r′| = √r² + r′² 2rr′ cos δ

R ψ

δ r

r (r ,θ , φ )

0º E ≤ φ ≤360º E

90º - 0º N ≤

º N- 900º 9θ ≤

,

FIGURE 3.15 Gravity and magnetic anomaly modeling in spherical coordinates by Gauss–Legendre quadrature integration of a body of any shape and uniform physical propertyxin density and

magnetization, respectively. Adapted fromv o n F r e s eet al.(1981b).

The equivalent point source (EPS) effects refer to potential, vector, and gradient tensor effects of the gravity point pole and magnetic point dipole that are described in full analytical detail in A s g h a r z a d e het al.(2007) andA s g h a r z a d e het al.(2008), respectively. The EPS effects generalize as a generic integrand or q-function over a differential volume∂vthat can be analytically or numerically volume integrated for the body’s effect by the respective triple integral and quadrature series expressions at the base of the equation tree. The GLQ estimate includes

ϕ=(ϕ_kb −ϕ_ka ), θ=(θ_jb −θ_ja), andr=(r_ib−r_ia).

coordinates of the observation and source points, respec- tively, and (ϕ, ϕ) are the longitude coordinates of the respective observation and source points.

Thus, the gravitational EPS force field Fg(R) at the distanceRin spherical coordinates is

Fg(R)= ∇U(R)= ∂

∂R[U(R)]∇R

=(G×m) ∂

∂R 1

R

∇R , (3.85) where the gradient ofRin observation point coordinates is

∇R= ∂R

∂r

ˆer+1 r

∂R

∂θ

ˆeθ+ 1 rsinθ

∂R

∂ϕ

ˆeϕ

(3.86) with

∂R

∂r =cosψ= r−rcosδ

R = C

R, (3.87)

1 r

∂R

∂θ

= r[sinθcosθ−cosθsinθcos(ϕ−ϕ)]

R

= E

R, (3.88)

1 rsinθ

∂R

∂ϕ

= rsinθcos(ϕ−ϕ)

R = H

R, (3.89) andˆer,ˆeθ, andˆeϕare the spherically orthogonal unit basis vectors at the observation point. With the results from the above three equations, Equation 3.85 becomes

Fg(R)=(G×m) ∂

∂R 1

R C Rˆe_r+E

Rˆe_θ+H Rˆe_ϕ ,

(3.90) where the radial scalar component is

Fr =(G×m) ∂

∂R 1

R ∂R

∂r

=

−G×m R³

C≡g, (3.91)

the horizontalθ-co-latitude scalar component is Fθ =(G×m)

∂

∂R 1

R 1 r

∂R

∂θ

=

−G×m R³

E, (3.92)

and the horizontalϕ-longitude scalar component is Fϕ =(G×m)

∂

∂R 1

R 1 rsinθ

∂R

∂ϕ

=

−G×m R³

H. (3.93)

Additional details on these spherical gravity components, the associated gradient tensors, and their implementation for modeling the gravity effects of extended bodies are described byAsgharzadehet al.(2007).

The equation tree at the bottom of Figure 3.15, for example, shows the adaption of Equation 3.80 to spher- ical coordinates where (r, r) are the respective observa- tion and source point radial distances from the Earth’s center, (θ, θ) are the respective observation and source point co-latitudes, and (ϕ, ϕ) are the longitude coordi- nates of the respective observation and source points.

However, to convert density to mass in spherical coor- dinate applications, the standard Cartesian unit volume