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Integral Representations in Forward Modeling of Gravity and Magnetic Fields
Chapter 6 INTEGRAL REPRESENTATIONS IN FORWARD MODELING OF GRAVITY AND MAGNETIC FIELDS 6.1
Basic equations for gravity and magnetic fields
6.1.1 Gravity and magnetic fields in three dimensions The basic formulae for gravity and magnetic fields were introduced in Chapter 1. According to these formulae the gravity field, g, satisfies the equations V - g = -47r7p, V X g = 0,
(6.1)
where 7 is the universal gravitational constant, and p is the density distribution within a domain D. The solution of these equations is given by the formula
g(r") = A»(p) = y^'
lll/'''\r-r'\'''
In particular, the commonly measured vertical component of the gravity field, g^^ can be found from the expression
In a similar way, we can represent a magnetic field. Note that in geophysical applications we usually consider the magnetic anomaly caused by the induced magnetization in the rock formations only. Assume that the volume D is filled by magnetic masses with the intensity of magnetization I(r), which is given as a product of magnetic susceptibility, x (r), and the inducing geomagnetic field H^: I(r) = x(r)HO.
(6.4)
It is well known that the corresponding magnetic field H(r'), generated by the magnetic masses within a domain D, can be represented as follows (Zhdanov, 1988): H(rO = V ' / / / I(r) • V'-^-dv.
(6.5)
Integral representations in forward modeling of gravity and magnetic fields
170
Figure 6-1 sources.
Ilustration of the concept of the plane field. F is a domain occupied by the
Substituting (6.4) into (6.5), we obtain (6.6)
where H^ = |H°| is the magnitude of the inducing field, and d/dl' denotes a directional derivative in the direction of magnetization H^. Formulae (6.2) and (6.6) provide the basis for the solution of forward and inverse geopotential (gravity and magnetic) field problems. 6.1.2 Two-dimensional models of gravity and magnetic fields In some practical situations we can assume that the density, p(r), or magnetization, I(r), of the rocks does not change in one horizontal direction, say in the y direction. In this case p(r) and I(r) are the functions of two variables, x and z: p(r) = p{x,z),
I(r) = I ( x , z ) .
(6.7a)
We assume, also, that the domain D is a cylindrical volume extended infinitely in the y direction with cross-section F in the vertical plane xz (Figure 6-1). Under these assumptions, obviously, gravity and magnetic fields vary only in the x and z directions, and formulae (6.2) and (6.5) take the form
l{x\z')=^ J JT
J-OO
H(a;', z') = V where ds is an element of area of F, i.e. ds = dxdz.
v' |3
dyds,
v' |3
dyds,
(6.8)
(6.9)
Integral representations
of potential fields based on the theory of functions
of a complex
variablel71
Note that the y - components of gravity and magnetic fields vanish, and the X- and z- components are determined by calculating an elementary integral along the y axis: 1
,
r -
1 [{x-xf
+ {y-yf
^^ + {z-z'ff'
'^
2 {x — x/) 4- (2: — z')
(6.10)
Substituting expression (6.10) back into formulae (6.8) and (6.9), we find
JJv
(.r - .X-')'+ ( s " 2')
where V (kniotc^s the oi)crator of 2-D differentiation: V ~ {D/Ox'A)/dz') . a n d d,,. and d , are tlie unit ])asis vectors of t h e Cartesian system of coorcUnates. In particular, t h e vertical component of t h e gravity field is ecjual t o <)A-
r/,s.
fn
'x ^ x'Y +
Note in conclusion t h a t two-dimensional mock^ls of geoi)otential fields ai'c wick^ly used in geophysics, because they rechice dramatically the volume of numc^rical calculations r(X|uircd for forward and inverse problems. Another advantage of 2-D models is t h a t one can use t h e |)owerful tool of complex analysis t o s t u d y 2-D ])otential fields (Zhdanov, 1988). We will consider some aspects of this approach in t h e next sections. 6.2
I n t e g r a l r e p r e s e n t a t i o n s of p o t e n t i a l functions of a c o m p l e x variable
fields
b a s e d o n t h e t h e o r y of
Most acliicvcmcmts attained in tlie theory of tw^o-dimensional potential fields involve the theory of functions of a complex variable (Strakliov, 1970a,b; Tsirulsky, 1963). This theory was exposcxl in detail in t h e book by Zhdanov (1988). We shall present here those results developed in these publications most relevant t o t h e inverse problem solution. 6.2.1
Complex
intensity
of a plane potential
field
Let us consider a 2-D potential field F satisfying t h e equations V - F = g, V x F = 0,
(6.14)
172
Integral representations in forwetrd modeling of gravity and magnetic fields
where V denotes the operator of 2-D differentiation in the vertical plane xz, and q is a source of the potential field F , concentrated within a local domain V. According to (6.14), we can write the equations for the scalar components of the vector field F : OF, dF, — + — = , ( . , z), (6.15)
In particular, outside the sources we have dx
dz
We will introduce a complex variable C, = x ^ iz and designate by F(C) the following complex function: F{0 = - F , ( x , z) + iF,{x, z).
(6.18)
This function, F{(), is called a complex intensity of a plane field. It is an analytical function outside the sources, which vanishes at infinity because, according to equation (6.17), its real and imaginary parts, R e F = —F^, I m F = F^, satisfy the Cauchy-Riemann conditions: dReF dx
dlmF dz
dReF dlmF dz dx Let us proceed from the real variables x, 2 to the complex ones C = x + iz, C =x-iz,
(6.19)
(6.20)
and employ differentiation operators: d__l,d__ dC ~ 2^dx
.d_ ^dz''
where the asterisk * means complex conjugate. Multiplying equation (6.15) by (—1) and equation (6.16) by (—z) and summing them up, we obtain — ( - F , + zF,) + i-^i-F.
+ iF,) = - g ( x , z),
Integral representations of potential fields based on the theory of functions of a complex variablel73
Figure 6-2
Solution of the equation for the complex intensity of the potential field using
the Pompei formula. F is the domain occupied by the field sources.
-|.F,o^-i,(»
(6.22)
It is iiotcHvorthy t h a t for an arbitrary function vHC) fli^if '^^ analytical in a cloniain 6'. t-h(^ Cauchy-Hicniaini relations imply t h a t (G.23)
^ V ' ( C ) - 0.
T h e solution of the differential equation (6.22) can be obtained using the Pornpei formula (Zhdanov, 1988):
FiO = ~ I y^FiOdC ~- ff 2711 Jos C-C
T^ JJsC
-(,
^^FiOds,
(G.24)
oC
where S is a domain in tlic complex plane, dS is its boundary, and ("' is a fixed point within 5'. Note t h a t for any function ip t h a t is analytical in S, property (6.23) reduces (6.24) to the well known Cauchy integral formula:
^iC)
1
1
7V^(C)^C-
(6.25)
We will use the Pompei formula to examine the complex intensity of a plane field. In accordance with the definition, tlic complex intensity satisfies equation (6.22) everywdiere in a complex plane and vanishes in infinity. Let us take an a r b i t r a r y point
174
Integral representations in forward modeling of gravity and magnetic fields
(^ and draw therefrom a circle LR (Figure 6-2). The domain bounded by Lji will be SR. We assume that the radius R is big enough that the domain F is completely inside 5 ^ . Applying Pompei formula (6.24), we write
The integral taken over the circle LR can be written, upon substitution of the variable C = C -^R-e'^, in the form
Being analytical outside F, the function F(C' + /^ • e^^) tends uniformly over 6 to zero at infinity, R —^ oo. Hence the limit of the integral (6.27) as i? ^ oo is zero. Thus we write finally:
where we have substituted the domain of the source concentration F for the domain of integration SR, assuming that q{(^) = 0, if C ^ F. 6.2.2 Complex intensity of a gravity field The gravity field g of a two-dimensional distribution of masses concentrated with a density p(x, z) within the domain F satisfies the equations V • g = -ATTJP,
V X g - 0,
(6.29)
where 7 is the universal constant of gravitation. Let us define a complex intensity: ^(C) = -9x{x, z) + ig,{x, z).
(6.30)
In accordance with (6.22), the function g{C,) satisfies the equation ^ g ( C ) = 27r7p,
(6.31)
whose solution is governed by a formula following from (6.28),
ff(C') = -27 jj^ ^p{Ods,
(6.32)
where p{Q = p{x,z). In particular, the complex intensity of the gravity field generated by masses with a constant density p{() = PQ is defined by ge(C') = - 2 7 P o / ( ^ -
(6.33)
Integral representations
6.2.3
Complex
of potential fields based on the theory of functions of a complex
intensity
and potential
of a magnetic
variablel75
field
Let domain T be filled with magnetized masses with an intensity of magnetization I(x, z) = {Ix{x, z),Iz{x, z)). Let us define the complex intensity of mugnetization as I{0 T h e complex intensity (6.12), as
= L{x,z)+ih{x,z).
of a plane magnetic
(6.34) field
can be defined, according to
H{C) = - i / . ( x ^ z') + ^7/,(x^ z') = - 2 JJ^ ^-±-^^I{(;)ds.
(6.35)
We can introduce a complex magnetic potential ^(C) as H{C) = ^U{C).
(6.36)
From (6.35) and (6.36), we have at once
f/(C') = - 2 11^ ^ / ( C ) r f . s .
(6.37)
It is useful to compare cc|uation (6.37) with the corresponding formula for a gravity iicld:
fjiO = - 2 7 11 ^^piOd-^.
(6.38)
One can see t h a t these expressions are similar, with one very i m p o r t a n t dificrence: the density p{Q is a real function, while the complex intensity of magnetization, /(C), is a complex function. T h e complex magnetic potential generated by a uniformly magnetized body, /(C) = /(), is defined by
U^{C) = ^2IoJI^^.
(6.39)
Comparing formulae (6.39) and (6.33), we see
IPo Substituting the last expression into (6.36), we arrive at the famous Poisson
theorem
which provides a simple connection between the magnetic field, H^ and t h e gravity field, gc, of t h e same b o d y with uniform distribution of the density and magnetization.
176
Integral representations in forward modeling of gravity and magnetic fields
The analytical representations derived above for anomalous gravity and magnetic fields provide a useful tool for the solution of the inverse problems.
References to Chapter 6 Strakhov, V. N., 1970a, Some aspects of the plane inverse problem of magnetic potential (in Russian): Izvestia AN SSSR, Fizika Zemli, No. 9, 31-41. Strakhov, V. N., 1970b, Some aspects of the plane gravitational problem (in Russian): Izvestia AN SSSR, Fizika Zemh, No. 12, 32-44. Tsirulsky, A. V., 1963, Some properties of the complex logarithmic potential of a homogeneous domain (in Russian): Izvestia AN SSSR, Fizika Zemli, No. 7, 1072-1075. Zhdanov, M. S., 1988, Integral transforms in geophysics: Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 367 pp.
Basic equations for gravity and magnetic fields
6.1.1 Gravity and magnetic fields in three dimensions The basic formulae for gravity and magnetic fields were introduced in Chapter 1. According to these formulae the gravity field, g, satisfies the equations V - g = -47r7p, V X g = 0,
(6.1)
where 7 is the universal gravitational constant, and p is the density distribution within a domain D. The solution of these equations is given by the formula
g(r") = A»(p) = y^'
lll/'''\r-r'\'''
In particular, the commonly measured vertical component of the gravity field, g^^ can be found from the expression
In a similar way, we can represent a magnetic field. Note that in geophysical applications we usually consider the magnetic anomaly caused by the induced magnetization in the rock formations only. Assume that the volume D is filled by magnetic masses with the intensity of magnetization I(r), which is given as a product of magnetic susceptibility, x (r), and the inducing geomagnetic field H^: I(r) = x(r)HO.
(6.4)
It is well known that the corresponding magnetic field H(r'), generated by the magnetic masses within a domain D, can be represented as follows (Zhdanov, 1988): H(rO = V ' / / / I(r) • V'-^-dv.
(6.5)
Integral representations in forward modeling of gravity and magnetic fields
170
Figure 6-1 sources.
Ilustration of the concept of the plane field. F is a domain occupied by the
Substituting (6.4) into (6.5), we obtain (6.6)
where H^ = |H°| is the magnitude of the inducing field, and d/dl' denotes a directional derivative in the direction of magnetization H^. Formulae (6.2) and (6.6) provide the basis for the solution of forward and inverse geopotential (gravity and magnetic) field problems. 6.1.2 Two-dimensional models of gravity and magnetic fields In some practical situations we can assume that the density, p(r), or magnetization, I(r), of the rocks does not change in one horizontal direction, say in the y direction. In this case p(r) and I(r) are the functions of two variables, x and z: p(r) = p{x,z),
I(r) = I ( x , z ) .
(6.7a)
We assume, also, that the domain D is a cylindrical volume extended infinitely in the y direction with cross-section F in the vertical plane xz (Figure 6-1). Under these assumptions, obviously, gravity and magnetic fields vary only in the x and z directions, and formulae (6.2) and (6.5) take the form
l{x\z')=^ J JT
J-OO
H(a;', z') = V where ds is an element of area of F, i.e. ds = dxdz.
v' |3
dyds,
v' |3
dyds,
(6.8)
(6.9)
Integral representations
of potential fields based on the theory of functions
of a complex
variablel71
Note that the y - components of gravity and magnetic fields vanish, and the X- and z- components are determined by calculating an elementary integral along the y axis: 1
,
r -
1 [{x-xf
+ {y-yf
^^ + {z-z'ff'
'^
2 {x — x/) 4- (2: — z')
(6.10)
Substituting expression (6.10) back into formulae (6.8) and (6.9), we find
JJv
(.r - .X-')'+ ( s " 2')
where V (kniotc^s the oi)crator of 2-D differentiation: V ~ {D/Ox'A)/dz') . a n d d,,. and d , are tlie unit ])asis vectors of t h e Cartesian system of coorcUnates. In particular, t h e vertical component of t h e gravity field is ecjual t o <)A-
r/,s.
fn
'x ^ x'Y +
Note in conclusion t h a t two-dimensional mock^ls of geoi)otential fields ai'c wick^ly used in geophysics, because they rechice dramatically the volume of numc^rical calculations r(X|uircd for forward and inverse problems. Another advantage of 2-D models is t h a t one can use t h e |)owerful tool of complex analysis t o s t u d y 2-D ])otential fields (Zhdanov, 1988). We will consider some aspects of this approach in t h e next sections. 6.2
I n t e g r a l r e p r e s e n t a t i o n s of p o t e n t i a l functions of a c o m p l e x variable
fields
b a s e d o n t h e t h e o r y of
Most acliicvcmcmts attained in tlie theory of tw^o-dimensional potential fields involve the theory of functions of a complex variable (Strakliov, 1970a,b; Tsirulsky, 1963). This theory was exposcxl in detail in t h e book by Zhdanov (1988). We shall present here those results developed in these publications most relevant t o t h e inverse problem solution. 6.2.1
Complex
intensity
of a plane potential
field
Let us consider a 2-D potential field F satisfying t h e equations V - F = g, V x F = 0,
(6.14)
172
Integral representations in forwetrd modeling of gravity and magnetic fields
where V denotes the operator of 2-D differentiation in the vertical plane xz, and q is a source of the potential field F , concentrated within a local domain V. According to (6.14), we can write the equations for the scalar components of the vector field F : OF, dF, — + — = , ( . , z), (6.15)
In particular, outside the sources we have dx
dz
We will introduce a complex variable C, = x ^ iz and designate by F(C) the following complex function: F{0 = - F , ( x , z) + iF,{x, z).
(6.18)
This function, F{(), is called a complex intensity of a plane field. It is an analytical function outside the sources, which vanishes at infinity because, according to equation (6.17), its real and imaginary parts, R e F = —F^, I m F = F^, satisfy the Cauchy-Riemann conditions: dReF dx
dlmF dz
dReF dlmF dz dx Let us proceed from the real variables x, 2 to the complex ones C = x + iz, C =x-iz,
(6.19)
(6.20)
and employ differentiation operators: d__l,d__ dC ~ 2^dx
.d_ ^dz''
where the asterisk * means complex conjugate. Multiplying equation (6.15) by (—1) and equation (6.16) by (—z) and summing them up, we obtain — ( - F , + zF,) + i-^i-F.
+ iF,) = - g ( x , z),
Integral representations of potential fields based on the theory of functions of a complex variablel73
Figure 6-2
Solution of the equation for the complex intensity of the potential field using
the Pompei formula. F is the domain occupied by the field sources.
-|.F,o^-i,(»
(6.22)
It is iiotcHvorthy t h a t for an arbitrary function vHC) fli^if '^^ analytical in a cloniain 6'. t-h(^ Cauchy-Hicniaini relations imply t h a t (G.23)
^ V ' ( C ) - 0.
T h e solution of the differential equation (6.22) can be obtained using the Pornpei formula (Zhdanov, 1988):
FiO = ~ I y^FiOdC ~- ff 2711 Jos C-C
T^ JJsC
-(,
^^FiOds,
(G.24)
oC
where S is a domain in tlic complex plane, dS is its boundary, and ("' is a fixed point within 5'. Note t h a t for any function ip t h a t is analytical in S, property (6.23) reduces (6.24) to the well known Cauchy integral formula:
^iC)
1
1
7V^(C)^C-
(6.25)
We will use the Pompei formula to examine the complex intensity of a plane field. In accordance with the definition, tlic complex intensity satisfies equation (6.22) everywdiere in a complex plane and vanishes in infinity. Let us take an a r b i t r a r y point
174
Integral representations in forward modeling of gravity and magnetic fields
(^ and draw therefrom a circle LR (Figure 6-2). The domain bounded by Lji will be SR. We assume that the radius R is big enough that the domain F is completely inside 5 ^ . Applying Pompei formula (6.24), we write
The integral taken over the circle LR can be written, upon substitution of the variable C = C -^R-e'^, in the form
Being analytical outside F, the function F(C' + /^ • e^^) tends uniformly over 6 to zero at infinity, R —^ oo. Hence the limit of the integral (6.27) as i? ^ oo is zero. Thus we write finally:
where we have substituted the domain of the source concentration F for the domain of integration SR, assuming that q{(^) = 0, if C ^ F. 6.2.2 Complex intensity of a gravity field The gravity field g of a two-dimensional distribution of masses concentrated with a density p(x, z) within the domain F satisfies the equations V • g = -ATTJP,
V X g - 0,
(6.29)
where 7 is the universal constant of gravitation. Let us define a complex intensity: ^(C) = -9x{x, z) + ig,{x, z).
(6.30)
In accordance with (6.22), the function g{C,) satisfies the equation ^ g ( C ) = 27r7p,
(6.31)
whose solution is governed by a formula following from (6.28),
ff(C') = -27 jj^ ^p{Ods,
(6.32)
where p{Q = p{x,z). In particular, the complex intensity of the gravity field generated by masses with a constant density p{() = PQ is defined by ge(C') = - 2 7 P o / ( ^ -
(6.33)
Integral representations
6.2.3
Complex
of potential fields based on the theory of functions of a complex
intensity
and potential
of a magnetic
variablel75
field
Let domain T be filled with magnetized masses with an intensity of magnetization I(x, z) = {Ix{x, z),Iz{x, z)). Let us define the complex intensity of mugnetization as I{0 T h e complex intensity (6.12), as
= L{x,z)+ih{x,z).
of a plane magnetic
(6.34) field
can be defined, according to
H{C) = - i / . ( x ^ z') + ^7/,(x^ z') = - 2 JJ^ ^-±-^^I{(;)ds.
(6.35)
We can introduce a complex magnetic potential ^(C) as H{C) = ^U{C).
(6.36)
From (6.35) and (6.36), we have at once
f/(C') = - 2 11^ ^ / ( C ) r f . s .
(6.37)
It is useful to compare cc|uation (6.37) with the corresponding formula for a gravity iicld:
fjiO = - 2 7 11 ^^piOd-^.
(6.38)
One can see t h a t these expressions are similar, with one very i m p o r t a n t dificrence: the density p{Q is a real function, while the complex intensity of magnetization, /(C), is a complex function. T h e complex magnetic potential generated by a uniformly magnetized body, /(C) = /(), is defined by
U^{C) = ^2IoJI^^.
(6.39)
Comparing formulae (6.39) and (6.33), we see
IPo Substituting the last expression into (6.36), we arrive at the famous Poisson
theorem
which provides a simple connection between the magnetic field, H^ and t h e gravity field, gc, of t h e same b o d y with uniform distribution of the density and magnetization.
176
Integral representations in forward modeling of gravity and magnetic fields
The analytical representations derived above for anomalous gravity and magnetic fields provide a useful tool for the solution of the inverse problems.
References to Chapter 6 Strakhov, V. N., 1970a, Some aspects of the plane inverse problem of magnetic potential (in Russian): Izvestia AN SSSR, Fizika Zemli, No. 9, 31-41. Strakhov, V. N., 1970b, Some aspects of the plane gravitational problem (in Russian): Izvestia AN SSSR, Fizika Zemh, No. 12, 32-44. Tsirulsky, A. V., 1963, Some properties of the complex logarithmic potential of a homogeneous domain (in Russian): Izvestia AN SSSR, Fizika Zemli, No. 7, 1072-1075. Zhdanov, M. S., 1988, Integral transforms in geophysics: Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 367 pp.