Gravity exploration
DATY 21 DATY 2 − DATY 1 Saad and Bishop, 1989
3.5 Idealized source gravity modeling The gravity effects of idealized 3D and 2D bodies are
modeled by setting the appropriate limits and executing the relevant integrals in Equations 3.22–3.33 and 3.35–
3.41, respectively. Tables 3.2 and 3.3 list some of the more useful and widely used idealized source equations in Cartesian coordinates from the gravity modeling literature (e.g.Shaw, 1932;Nettleton, 1942;Telfordet al., 1990). These equations are remarkably robust for making gravity anomaly calculations with errors due to the lim- iting assumptions generally small over a large range of dimensions (e.g.Hammer, 1974).
Figures 3.8 and 3.9 depict the geometric parameters of the sources in Tables 3.2 and 3.3, respectively, beneath the principal profile along thex-axis. For these effects, the principal profile is centered on the central anomaly value CA[g]# and trends orthogonally across the strike of the source. In addition, the equations implement the density contrast (σ) because the relative sign of the anomaly always corresponds to the sign of the density contrast of the source.
The gravity effect of the spherical source, for exam- ple, isg#2=gz=g×cosθ=g(z/r)=Gmz/r3so that converting the mass into the density contrast over the volume [=(4/3)π R#33] of the spherical body gives the result in Table 3.2.Nettleton (1942) recommended dividing the gravity effect byz3so that it may be expressed as the product
g#2=
4π R3 3
Gσ
z2 =CA[g]#2
×
&
fg
x z #2=
1+
x2 z2
−32'
. (3.69)
3.5 Idealized source gravity modeling 51 TABLE 3.2 Gravity effects of the point, spherical, line, and cylindrical masses.
Source Gravity effect in spatial domain
(1) Point mass g#1=G×(m=σ×volume)×(z/r3)
(2) Sphere g#2=(4π Gσ R3)/(3z2[1+(x2/z2)]3/2) (3)∞H-line mass g#3=2G×(m=σ×area)×(z/r2) (4)∞H-cylinder g#4=(2π Gσ R2)/[x2+z2]
(A) 2.5D g#4A=g#4× {[1+(r/Y)2]−12}
(B) 2.75D g#4B=g#4× {0.5[1+(r/ya)2]−12 +[1+(r/yb)2]−12} (5)∞V-line mass g#5=(π Gσ R2)/[x2+z2]1/2
(6)∞V-cylinder g#6=(KA)ga#6A+(KB)g#6B+(KC)g#6C
(A)KA=0 & 1 ga#6A=2π Gσ[(z2+R2)1/2−z]
∀(x=0) and
(B)KB=0 & 1 g#6B=2π Gσ R(1
2
R
√
x2+z2
− 161 R
√
x2+z2
32z2−x2 x2+z2
∀(0< x≤R) +1281 R
√x2+z2
5 35z4
(x2+z2)2−x30z2+z22+3 + · · ·)
and
(C)KC=0 & 1 g#6C=2π Gσ R(
1−z
R
+142z2−x2 R2
∀(x > R) −188z4−24z2x2+3x4
R4
+ · · ·) (7) Finite V-cylinder g#7=g#6(@z1) − g#6(@z2)
(A)∀(z > x=0) ga#7A=(2π Gσ)[r1+δz−r2]
∀ri=[R2+z2i]1/2&i=1,2
(B)∀(z=x=0) ga#7B=(2π Gσ)[δz+R−(δz2+R2)1/2]
(C) for (R −→ ∞) ga#7C=2π Gσ δz
(8) V-cylinder segment ga#8=θ Gσ[r1−r2+r3−r4]
∀ri=[Rj2+zi2] & i=1,2,3,4 & j=1,2
Abbreviations include H for horizontal and V for vertical. Here,gis the vertical gravity anomaly in gal (m/s2= 105mGal) calculated at the orgin (0,0,0) due to a density contrastσin g/cm3at distancesxandz;gais the vertical gravity anomaly on the axis of the source;Gis the gravitational constant (=6.674×10−11m3kg−1s−2),ris the distance from the calculation point to the center, centerline, axis, or edge of the source;yis the uniform distance to each end of the cylinder from the bisecting profile;ya andyb are the two not necessarily equal distances from the profile to each end of the cylinder;z1andz2are respectively the vertical distances from the calculation point to the upper and lower surfaces of the source so thatδz=(z2−z1);Ris radius of the sphere or cylinder; andφis the angle from the horizontal to the appropriate point of the source in radians. All distances are in centimeters (meters) with parentheses containing SIu. The geometric parameters of these sources are illustrated in Figures 3.8 and 3.9.
In this more simplified format, CA[g]#2 is the central anomaly maximum or minimum that reflects the sign of the density contrast, and varies with depthz(nc=2)where the decay ratenis one power lower than the number of effec- tive volume dimensions of the source. Figure 3.10 illus- trates the dimensionless curvefg[x/z]#2that describes the decay of CA[g]#2with offsetx.
Table 3.4 lists the more easily applied formats for sev- eral of the idealized body equations thatNettleton (1942) developed, and Figure 3.10 illustrates the related
dimensionless curves that modulate the CA[g]#with off- set from the central value. These results are useful start- ing points for the interpretation of simple anomalies and for developing preliminary conceptual models. Further- more, the plot of the normalized distance function, which gives the shape of the anomaly from sources irrespective of amplitude, is helpful in comparing anomalies to differ- entiate sources.
In contrast to the 3D effect of the sphere, the infi- nite horizontal cylinder equation (#4) exemplifies the 2D
52 Gravity potential theory
TABLE 3.3 Gravity effects of tabular and sheet masses.
Source Gravity effect in spatial domain
(9) Bouguer H-slab g#9=2π Gσ δz=ga#7C
(10) Thinα-inclined sheet (2D) g#10=2Gσ t
sinαlnr2
r1
−(θ2+θ1) cosα (11) Thin V-sheet (2D) g#11=(2Gσ t) ln
z22+x2 /
z21+x2 (12) Thin H-Sheet (2D) g#12=2Gσ δzφ
(13) Thin S-∞H-slab (fault) g#13=2Gσ δz[(π/2)−tan−1(x/z)]
(A) ∀(x=0) ga#13A=π Gσ δz
(14) Thick S-∞H-slab (fault) g#14=2Gσ[xln(r2/r1)+π t−z2φ2+z1φ1] (15) Oblique faced S-∞H-slab g#15=2Gσ
z2φ2−z1φ1+x2z1−x1z2
x2+δz2
δzlnr2
r1
+xφ
∀x=(x2−x1) &φ=(φ1−φ2) Abbreviations and variables are the same as listed for Table 3.2, but also includes S for semi.
The geometric parameters of these sources are illustrated in Figure 3.9.
P (x, 0) P (x, 0)
r z
V
z
#2
#1
x
P (x, 0)
R z
#5 & #6
x
P (x, 0) P (x, 0)
r z
S
R R
z
#4
#3
#7
#8
x
P (0,0)
R
Y Z1
Z1 Z2
r1
r1r2
r2 R1 R2 r3 r4
x
#9 #10
P (x, y, z)
P (0,0,0)
P (x, y, z)
Z2 θ
Z
ΔZ ΔZ
ΔX
FIGURE 3.8 Pictorial representations of the geometric parameters for modeling the gravity effects of the point mass, sphere, line mass, and cylinder identified in Table 3.2 and the horizontal and inclined slabs identified in Table 3.3.
3.5 Idealized source gravity modeling 53
P
)
P
#11 #12
#15 P
t
Z2 Z1
Z2 Z1
2 1
x2, z2
x1, z1
t
x x
P P
#13
#14
r
r1
r2
r2
ϕ
ϕ2
ϕ1
ϕ1
ΔZ ΔZ
ΔZ
ΔZ Δϕ
FIGURE 3.9 Pictorial representations of the geometric parameters for modeling the gravity effects of tabular and sheet bodies identified in Table 3.3.
effect. It is applicable for a body with horizontal length (L#4) that is ten or more times greater than its radius (R#4), which in turn is less than or not much greater than half the depth (zc) to the central axis of the source (i.e.
L#4≥10×(R#4≤0.5zc)). The effective mass here, how- ever, is the mass per unit length of the source given by me(#4)=σ×[2π R2#4=A(#4)], where now the effec- tive volume is the cross-sectional area of the sourceA(#4).
In other words, the effective mass for the 2D source is the mass per unit length given by the product of its density contrast and cross-sectional area so that CA[g]#4varies withz(nc=1).
The effect of finite horizontal cylinder length can be accommodated by relatively simple end corrections. For example, Equationg#4.A in Table 3.2 gives the so-called 2.5D effect for the finite cylinder where the principal pro- file is an equal distanceY from both ends of the cylinder.
Equationg#4.Bgives the 2.75D effect where the distances ya andybto the ends of the finite horizontal cylinder from the profile are not equal.
For a 1D source like the Bouguer slab, the effective mass is simply m(#9)=2π σ, where 2π is the solid angle that the infinite slab subtends at any observation point (e.g.Nettleton, 1942). Thus, any tabular source with horizontal dimensions that are large relative to the elevation of the observation point has a gravity effect that varies withz(n=0).
A disk with large radius (i.e. R−→ ∞) compared with its depth zt, for example, has the same effect (Equationg#7.C) as the Bouguer slab. Hence, a disk with a smaller radius will have an effect ω/2π as large, or g=ωGσ δz, whereωis the solid angle that the disk sub- tends at the observation point. This approximation holds to within about 5% or less for disk thicknessesδz≤0.5zt
54 Gravity potential theory
TABLE 3.4 Examples of generalized dimensionless equations for calculating the gravity effect of idealized geometric sources.
Source CA[g]# fg[x/z]#
(2) Sphere 27.979×10−6σv(R3/z2) [1+(x/z)2]−32 (4)∞H-cylinder 11.936×10−6σ(R2/z) [1+(x/z)2]−1 (5)∞V-line mass 20.984×10−6σ(R2/z) [1+(x/z)2]−12 (11) Thin V-sheet (2D) 30.738×10−6σ t log[1+(z/x)2]−12 (14) Thin S-∞H-slab (fault) 41.936×10−6σ δz (1/2)+(1/π) tan−1(x/z) CA[g]#is the center anomaly value or constant andfg[x/z]#is the normalized distance function for the numbered sources in Tables 3.2 and 3.3. All parameters are in SIu, but the CA[g]#are in mGal.
Modified fromNettleton(1942).
1.0
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
0.0 0 10 20 30 40
0.8 0.6 0.4 0.2 0
0 1 2 3 4
1.4
1.0
0.6
0.2 0.0 0.4 0.8 1.2
0.1 0.5 5.0 10.0
0.05 0.10 0.20
1.0 -0.2
0.15 0.3 1.5 3.0
fg [x/z]# x/z
x/z x/z
#2
#4
#13
#5
#11 (a)
(b) (c) #11-continued
fg [x/z]#
fg [x/z]#
Dimensionless offset curves for some
idealized gravity sources
FIGURE 3.10 Distance functions from Table 3.4 used in calculating the gravity anomaly of (a) the sphere (source#2), infinite horizontal cylinder (source#4), and infinite vertical line mass (source#5); (b) the thin, horizontal semi-infinite slab (fault) (source#13); and (c) the thin vertical sheet (source#11).
(e.g.DobrinandSavit, 1988), and is especially use- ful for numerically modeling the effect of an arbitrarily shaped geological body. The calculation involves filling in the volume of the body with disks and summing or integrating the effects of the disks at the observation point.
In computing these gravity effects, effectiveω esti- mates typically are visually selected from charts because of the difficulty in calculating solid angles, even for sim- ple geometric bodies (e.g.Nettleton, 1942;Dobrin and Savit, 1988). However, Telford et al. (1990)
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