Gravity exploration
3.4 Gravity effects of an extended body An extended mass body made up of many point poles has
gravity effects that can be evaluated at an observation point from the superimposed effects of the differential point masses. However, the above equations are for the gravity effects of the point mass only, whereas in geophysical practice the operative physical property is the mass per unit volume (in kg/m3) or its densityσ of the source so that
m=
v
∂m=σ×
v
∂x∂y∂z=σ×v, (3.21) where∂m=σ×∂x∂y∂z andvis the source’s volume (in m3). Thus, taking density into account and integrating Equation 3.16 over the lowera- and upper b-limits in each spatial dimension of the 3D source gives its gravity potential (in joules)
U=U(x, y, z)
= zb
za
yb ya
xb xa
U=G
1 r
σ
∂x∂y∂z, (3.22)
where the gravitational constant G = 6.67 × 10−11m3kg−1s−2. Note that the kernelU is the poten- tial per unit volume or point mass effect in Equation 3.16.
The kernel can also be generalized in terms of the prod- uct of the density functional and the inverse displacement distance functional for the generalized gravity potential U =U(x, y, z)
=G zb
za
yb ya
xb xa
σ 1
r
∂x∂y∂z. (3.23)
Indeed, all gravity forward modeling equations in all coor- dinate systems have kernels that can be generalized as the product of a physical property functional times a geometry functional or Green’s function.
The gravity vector components of the 3D body from integrating Equation 3.18 over the body’s finite volume limits are
Fx = ∂U
∂x
= zb
za
yb ya
xb xa
Fx = −G x
r3
σ
∂x∂y∂z,
(3.24) Fy = ∂U
∂y
= zb
za
yb ya
xb xa
Fy = −G y
r3
σ
∂x∂y∂z,
(3.25) and
Fz= ∂U
∂z
= zb
za
yb ya
xb xa
Fz= −G z
r3
σ
∂x∂y∂z
≡g. (3.26)
These vector components make up the generalized gravity vector of the 3D body given by
Fg =Fg(x, y, z)
= −G zb
za
yb ya
xb xa
σ
∇ 1
r
∂x∂y∂z. (3.27) In addition, integrating Equation 3.19 over the body’s 3D volume limits results in its three diagonal tensors that
44 Gravity potential theory
-10 -10
-10
-10 -10
-1,500 -1,000 -500 0 500 1,000 1,500
-1,000
-1,500 -500 0 500 1,000 1,500
North (m)
East (m)
-1,500 -1,000 -500 0 500 1,000 1,500
-1,000
- 1,500 -500 0 500 1,000 1,500
North (m)
East (m)
-1,500 -1,000 -500 0 500 1,000 1,500
-1,000
-1,500 -500 0 500 1,000 1,500
North (m)
East (m)
-1,500 -1,000 -500 0 500 1,000 1,500
-1,000
-1,500 -500 0 500 1,000 1,500
North (m)
East (m) -1,500
-1,000 -500 0 500 1,000 1,500
-1,000
-1,500 -500 0 500 1,000 1,500
North (m)
East (m) -1,500
-1,000 -500 0 500 1,000 1,500
-1,000
-1,500 -500 0 500 1,000 1,500
North (m)
East (m)
-1,500 -1,000 -500 0 500 1,000 1,500
-1,000
-1,500 -500 0 500 1,000 1,500
North (m)
East (m)
-1,500 -1,000 -500 0 500 1,000 1,500
-1,000
-1,500 -500 0 500 1,000 1,500
North (m)
East (m)
-1,500 -1,000 -500 0 500 1,000 1,500
-1,000
-1,500 -500 0 500 1,000 1,500
North (m)
East (m)
-1,500 -1,000 -500 0 500 1,000 1,500
-1,000
-1,500 -500 0 500 1,000 1,500
North (m)
East (m) -1,500
-1,000 -500 0 500 1,000 1,500
-1,000
-1,500 -500 0 500 1,000 1,500
North (m)
East (m) -1,500
-1,000 -500 0 500 1,000 1,500
-1,000
-1,500 -500 0 500 1,000 1,500
North (m)
East (m)
(a) (b) (c)
(d) (e) (f )
(g) (h) (i)
(j) F (k) (l)
F F
Fxx (E)
Fyx (E)
Fzx (E) Fzy (E) Fzz (E)
Fyy (E) Fyz (E)
Fz (mGall) Fy (mGal)
Fx (mGal)
Fxy (E) Fxz (E)
FIGURE 3.3 Examples of the gravity vector componentsFx,Fy, andFz(Equation 3.18) and gradient tensors (Equations 3.19 and 3.20) in E ¨otv ¨os units (1 E=0.1 mGal/km) for the point mass of Figure 3.2.
3.4 Gravity effects of an extended body 45
satisfy Laplace’s Equation 3.6 given by Fxx = ∂2U
∂x∂x
= zb
za
yb ya
xb xa
Fxx =G
3x2 r5 − 1
r3
σ
∂x∂y∂z
= −(Fyy+Fzz), (3.28)
Fyy = ∂2U
∂y∂y
= zb za
yb ya
xb xa
Fyy =G
3y2 r5 − 1
r3
σ
∂x∂y∂z
= −(Fxx+Fzz), (3.29)
and Fzz= ∂2U
∂z∂z
=zb za
yb ya
xb xa
Fzz =G
3z2 r5 − 1
r3
σ
∂x∂y∂z
= −(Fxx+Fyy). (3.30)
Further integrating Equation 3.20 yields the body’s three unique off-diagonal tensors
Fxy = ∂2U
∂x∂y
= zb za
yb ya
xb xa
Fxy =G
3xy
r5
σ
∂x∂y∂z
=Fyx, (3.31)
Fyz = ∂2U
∂y∂z
= zb za
yb ya
xb xa
Fyz =G
3yz
r5
σ
∂x∂y∂z
=Fzy, (3.32)
and Fxz= ∂2U
∂x∂z
=zb za
yb ya
xb xa
Fxz =G
3xz
r5
σ
∂x∂y∂z
=Fzx. (3.33)
These nine gradient tensors make up the generalized gra- dient tensor of the 3D body given by
∇Fg = ∇Fg(x, y, z)
=G zb
za
yb ya
xb xa
σ
∇
∇ 1
r ∂x∂y∂z. (3.34)
For 2D modeling of a profile on thex-axis across an elliptical or narrow elongated anomaly with essentially infinite strike length along the y-axis and the vertical z-axis positive downwards, the gravitational potential is U(x, z)=
zb za
xb xa
∂x∂z ∞
−∞
G 1
r
σ ∂y
= zb
za
xb xa
U=2Glog 1
r
σ
∂x∂z. (3.35) For the above logarithmic potential, the displacement vec- tor is 2D in the (x, z)-plane with magnitude
r=
(x−x)2+(z−z)2=
x2+z2, (3.36) and the operative physical property is the body’s mass per unit length or surface density (in kg/m2)
m y =
S
∂m y =σ×
S
∂x∂z=σ×S, (3.37) where (∂m/y)=σ×∂x∂zandSis the source’s cross- sectional area or surface (in m2). Thus, the gravitational field components of the 2D source are
Fx = ∂U
∂x
=zb za
xb xa
Fx = ∂U
∂x = −2G
x
r2
σ
∂x∂z,
(3.38) and
Fz = ∂U
∂z
=zb za
xb xa
Fz =∂U
∂z = −2G
z
r2
σ
∂x∂z≡G.
(3.39) In addition, the two unique gravity tensors of the 2D source are
Fzz= ∂2U
∂z∂z
= zb
za
xb xa
Fzz =2G
2z2−r2 r4
σ
∂x∂z
= −Fxx, (3.40)
and Fxz= ∂2U
∂x∂z
= zb
za
xb xa
Fxz =4G xz
r4
σ
∂x∂z
=Fzx. (3.41)
46 Gravity potential theory
y’a
y’b
Offset
Offset Offset
Strike-oblique
Strike-oblique x
+y –y
FIGURE 3.4 Geometric attributes of elongated bodies with polygonal cross-sections in the (x, y)-plane used in GAMMA for the symmetric 2D and 2.5D and asymmetric 2.75D cases relative to the principal profile along thex-axis. Adapted fromS a a dandB i s h o p(1989).
For an elongated source that is finite with ends at dis- tancesya andyb from the profile, the 2D effects can be corrected based on their ratios to the related 3D effects.
For example, using Equations 3.26 and 3.39, the 3D grav- ity effectgof an elongated finite mass can be expressed as the mass’s infinitely elongated 2D effectGmultiplied by the factor
g G = 1
2 yb
ya
1 r
∂y. (3.42)
Telford et al.(1990) note that this correction factor for an elongated source with circular cross-section (i.e., a horizontal cylinder, HC) may be approximated by
g(HC) G(HC) 1
2
⎡
⎣ ya
rc2+ya212 + yb rc2+yb212
⎤
⎦, (3.43)
whererc is the distance from the station to the central axis of the source,g(HC) is the 3D effect of the finite y-length horizontal cylinder, andG(HC) is the 2D effect of the infinitey-length horizontal cylinder. The so-called 2.5D effect is obtained when|ya| = |yb|<∞, whereas for
|ya| = |yb|, the corrected result is called the 2.75D effect.
Note that the terms 2.5D and 2.75D are misnomers because they refer in reality to 3D bodies with 3D gravity poten- tials, and vector and gradient tensor components given by Equations 3.23, 3.27, and 3.34, respectively. In the explo-
ration literature, however, they commonly refer to end cor- rections developed for elongated sources with arbitarily shaped cross-sections that can be represented by a poly- gons (e.g.Rasmussen and Pedersen, 1979;Cady, 1980).
In general, the end-corrected, 2D polygonal source finds extensive use for interactive, forward modeling inter- pretation of both gravity and magnetic anomalies. A par- ticularly comprehensive suite of algorithms for 2D, 2.5D, and 2.75D Gravity and Magnetic Modeling Applications (GAMMA) developed bySaad(1991, 1992, 1993) facil- itates efficient interactive modeling of combined gravity and magnetic anomalies. The GAMMA algorithms for gravity and gravity gradients are outlined below, whereas the corresponding magnetic algorithms are described in Section 9.7.1. The magnetic equations are the same as those describing the gravity gradient tensor components after re-scaling.
The GAMMA algorithms are based on the geomet- ric attributes of an elongated source with a polygonal cross-section perpendicular to the strike that is invariant along the strike direction (Figures 3.4 and 3.5). The strike length along they-axis can be either infinite (the 2D case), finite and symmetrical with respect to the profile orx-axis (the 2.5D case), or finite and asymmetrical (the 2.75D case). The most general 2.75D case includes not only bod- ies that are crossed by the profile, but also bodies that are completely offset from the profile, as well as bodies
3.4 Gravity effects of an extended body 47
(X1, Z1) ; (X2, Z2)
Orig. coords. of end points (vertices) of edge 1
P (Xo, yo , zo)
(x1, z1) (x2, z2) φ1
Translated
X ( ⊥ to strike)
X1 X2
Rotat ed X–axis Parallel to strik
e
(Rotat ed
z–axis)
W
Z Y
Z X
FIELD POINT P (O, O, O)
Rotation of (x, z)-axes about y-axis
Rotation of body about y-axis (U1, W1) ; (U2, W2)
Rotated coords. of end points (vertices) of edge 1
U1 X
W1
V or Y V or Y
Z1 Z2 (O, O, O)
U2 U
U
(U1, W1) (U2, W2) W2 = W1
or vertices Vertix (N + 1) = Vertix (1) N = Number of edges
W
Z C = cos φi ; S = sin φi
(i = 1, 2, …, N) Ui
Wi
Xi Zi C
= –S S
C
FIGURE 3.5 Coordinate transformation used in GAMMA to shift the origin to the observation point and rotate the (x, z)-axes or body about they-axis. FromS a a dandB i s h o p(1989).
striking obliquely at any arbitrary angle from the profile (Figures 3.4 and 3.5).
The derivations of the GAMMA algorithms for grav- ity are developed from the basic equations of the New- tonian gravitational potential for 3D sources (Equation 3.23) and the logarithmic potential for 2D sources (Equa- tion 3.35) and consider the 2.5D source as a special case of the 3D source. The volume integrals for the 3D sources, and surface integrals for the 2D sources are reduced to surface and line integrals, respectively, using Gauss’ divergence theorem (e.g. Equation 3.99).
To evaluate the surface integrals for 2.5D bodies and line integrals for 2D bodies, a method similar to that described byOkabe(1979) orRasmussen andPed- ersen (1979) is used. The method involves a coordi- nate transformation to shift the origin to the observation point followed by rotating the (x, y, z)-coordinate sys- tem about they-axis to the (u, v, w)-coordinate system (Figure 3.5). In the rotated system, theu-axis is parallel to the polygonal edge or facet for which the integrals are to be evaluated, thew-axis is perpendicular to the facet,
φ1 θ1
θ2 φ1
u’1
u’2 z’1
z’2
x’1 x’2
(x’2, z’2) (x’1, z’1)
(u’2, w’1) R1
Definitions of parameters used in gamma
Observations
Field point
0
0
2
z u
1
W
X
x Z
DX
DL DZ 1
2 Topography
DX = (x’2 - x’1) and DZ = (z’2 - z’1) so that the EDGE or SIDE LENGTH is DL = [(DX)2 + (DZ)2]1/2 = |u’2 - u’1| = |DU| where
C = cos φ1 = DX / DL and S = sin φ1 = DZ / DL and the transformed coordinates are u’i = C x’i + S z’i and w’i = -S x’i + C z’i and (w’i = w’i+1)
with Ri2 = x’i2 + z’i2 = u’i2 + w’i2 for (i = 1, 2, …, N) and θi = tan-1(z’i / x’i) = tan-1(w’i / u’i) + φi
and φi = tan-1(DZi / DXi) .
(u’1, w1’) w’1 =
w’2
R2
FIGURE 3.6 Definitions of edge parameters used by the GAMMA algorithms in the (x, z)- or rotated (u, w)-planes. Adapted from S a a dandB i s h o p(1989).
and thev-axis coincides with they-axis. The (x, z)-axes are rotated clockwise about they-axis by the dip angle ϕof the polygonal edge. Mathematically, the rotation is achieved according to the matrix equation at the bottom of Figure 3.5 or the equations in Figure 3.6.
The geometric parameters used in GAMMA for each polygonal edge are illustrated in Figures 3.6 and 3.7.
Notice that all the parameters and functions used by the GAMMA algorithms are expressed fully in terms of the (x, z)-coordinates of the polygonal vertices and the lower a- and upperb-limits of they y-coordinates of the end faces in the 2.5D and 2.75D cases.
The geometric functions described in Table 3.1 for the 2D, 2.5D and 2.75D cases apply to both gravity and mag- netic algorithms. These functions are expressed for a given polygonal edge as differential natural logarithmic (ln) and arctangent (tan−1) functions – i.e. the difference in values at two successive vertices (1,2) which define the edge. For the 2D case, the functions are designated DLR (differen- tial logarithm of radial distances), and DAT (differential arctangent). Both of these functions are fully defined in
48 Gravity potential theory
Coordinates & parameters used in 2D / 2.5D / 2.75D
GAMMA algorithms
(X1, Y1, Z1) (X2, Y1, Z2)
(X1, Y2, Z1)
(X2, O, Z2) (X1, O, Z1)
z Y2
Y1
v R1
R2
ŵ
û
û
φ1 x R12
R22
R11
R21
O
y
(X2, Y2, Z2)
x
FOR SYMMETRIC BODY (Y1 = –Y ; Y2 = +Y ; Y > 0) Ri02 = Ri2 + Y2(i = 1, 2, …, N)
FOR ASYMMETRIC BODY
(i = 1, 2, …, N ; j = 1, 2) Rij2 = Ri2 + Yj2
= Xi2 + Zi2 + Yj2
= Ui2 + Wi2 + Yj2
FIGURE 3.7 Isometric illustrations of the coordinates and parameters used in the 2.5D and 2.75D GAMMA algorithms. Adapted from S a a dandB i s h o p(1989).
terms of the (x, z)- or (u, w)-coordinates of the vertices and invariant under this coordinate transformation. Notice that DAT is the angular difference (θ1−θ2) shown in Figure 3.6. For the 2.5D and 2.75D cases, the logarith- mic and arctangent functions depend on the (x, z) or (u, w)-coordinates of the vertices, as well as on the half- strike lengthY =yb−ya in the symmetric 2.5D case or on they-coordinates of the end facesya and yb of the strike length in the asymmetric 2.75D cases as shown in Figure 3.7 and Table 3.1.
The GAMMA algorithms for the 2D, 2.5D, and 2.75D gravity vector components can be expressed compactly in terms of the above-defined parameters and functions. In the symmetric 2D case, for example,
Fx =2Gσ N
i=1
w1[C×DLR+S×DAT] (3.44) and
Fz≡G=2Gσ N
i=1
w1[S×DLR−C×DAT], (3.45) whereNis the total number of polygonal sides or vertices, σis the density (or density contrast of the polygon), and Gis the gravitational constant. For the symmetric 2.5D case, on the other hand, the gravity vector components
are
F(2.5)x =2Gσ N
i=1
w1[C×DLRY+S×DATY]
−S[Y ×DLRU], (3.46)
F(2.5)y =0 (because of symmetry in y), (3.47)
and
F(2.5)z≡g=2Gσ N
i=1
w1[S×DLRY−C×DATY]
+C[Y×DLRU]. (3.48)
For the asymmetric 2.75D case, the gravity vector compo- nents are
F(2.75)x =Gσ N
i=1
w1[C×DLRY21+S×DATY21]
−S[Y2×DLRU2−Y1×DLRU1], (3.49) F(2.75)y =Gσ
N i=1
w1[DLRU21]
+[y2×DATY2−y1×DATY1], (3.50)
3.4 Gravity effects of an extended body 49
TABLE 3.1 Definitions of functions applied in both gravity and magnetic GAMMA algorithms as illustrated in Figures 3.4–3.7.
2D – symmetric
DLR=ln(R1)−ln(R2)=ln(R1/R2), where ln≡loge
DAT=tan−1(z1/x1)−tan−1(z2/x2)
=tan−1(w1/u1)−tan−1(w2/u2); w2=w1
2.5D – symmetric DLRY =ln(R20+Y)
(R10+Y)×RR12
=DLR−ln(R10+Y)
(R20+Y)
DLRU=ln(R20+u2)
(R10+u1)
DATY =tan−1 u2×Y
w2×R20
−tan−1 u1×Y
w1×R10
; w2=w1
2.75D – asymmetric DLRY21=ln(R22+y2)
(R12+y2)×(R(R1121++yy11)) DLRU1=ln(R21+u2)
(R11+u1)
; DLRU2=ln(R22+u2)
(R12+u1)
DLRU21=ln(R22+u2)
(R12+u1)×(R(R1121++uu12))
=DLRU2−DLRU1 DATY2=tan−1 u2×y2
w2×R22
−tan−1u1×y2
w1×R12
; w2=w1
DATY1=tan−1 u2×y1
w2×R21
−tan−1u1×y1
w1×R11
; w2=w1
DATY21=DATY2−DATY1