Three-dimensional frequency-domain modeling of ... - CiteSeerX

Exploration Geophysics (1998) 29, 111-119

Three-dimensional frequency-domain modeling of airborne electromagnetic responses Dmitry B. Avdeev Alexei V. Kuvshinov Oleg V. Pankratov

Gregory A. Newman

Geoelectromagnetic

Albuquerque,

Research Institute,

Russian Academy of Sciences, 142092 Troitsk,

Sandia National Laboratories, P.O. Box 5800, NM 87185-0750,

Org.6116,

USA.

Moscow region, Russia.

ABSTRACT A novel three-dimensional (3D) frequency-domain modeling code has been implemented for airborne electromagnetic (AEM) simulations. The code is based on the volume integral equation (IE) solution to Maxwell's equations. To verify the code we consider three earth models of practical interest: (1) a 3D elongated body (dyke) residing in a layered earth; (2) faulted half-space (vertical contact) without, and, (3) with topography. Comparison of the code with a staggered-grid finite-difference solution for these models produces results in excellent agreement. In addition we have simulated responses that were previously numerically intractable, such as anisotropic and highcontrast 3D targets. We believe that our code is an efficient tool for AEM applications; as an example, for a model of a 1 Qm body in a 100 Qm half-space the code takes 8 minutes on 100MI:Iz Pentium PC per AEM system position when the body is discretised into 1500 cells. INTRODUCTION Advances in computers and numerical methods now make it possible to model the airborne electromagnetic (EM) responses over three-dimensional (3D) structures, which have previously been very difficult to simulate numerically. Encouraging examples of the 3D airborne EM (AEM) simulations with a finite-difference (FD) solution have been recently demonstrated in Newman and Alumbaugh (1995). Here we efficiently simulate the AEM responses to 3D structures using an integral equation (IE) approach. The approach is based on properties of the Green's function of Maxwell's equations that follow from the energy conservation law. For a quasi-static field these properties were presented in Singer and Fainberg (1985) and Singer (1995), as the mathematical foundation of the iterative dissipative method. The concept was later extended to media with polarisation and displacement currents (Pankratov et aI., 1995) and to anisotropic media (Pankratov et aI., 1997). Independently, similar results were obtained in Singer and Fainberg (1995; 1997). In this paper we first present, following Pankratov et al. (1997), the IE approach governing equations as applied to an anisotropic 3D earth model, and discuss computational requirements of our numerical solution. The solution is a modified from Avdeev et al. (1997) to simulate the responses for models excited by airborne EM systems. Then, for a model of a 3D target residing in a half-space, and models of a two-dimensional (2D) faulted half-space with and without topography, we check our solution against FD solution by Newman and Alumbaugh (1995). Finally we simulate AEM responses over 3D anisotropic targets and over 3D targets with resistivity contrasts up to 104.

GOVERNING EQUATIONS Assuming a time harmonic dependence of e-iOJt, Maxwell's equations for electric ~(r,w) and magnetic JJCr,w) fields are given by V'xH

= SE+

jP,

(1a)

V'xE=iwflH+mP.

(1b)

In these expressions the source vectors, jP and f!lP are current densities for impressed electric and magnetic sources. Hereinafter dependencies on rand W are omitted, where they are not essential. The magnetic permeability within the earth, fl, is set to flo = 4nlO-7 Him. The generalised conductivity tensor, ~, is defined via the conductivity

~

and dielectric

permittivlty

~=

§, as

~(r,w)-iw§,(r,w).

We assume in this paper that ~ and ~ are 3 x 3 complexvalued diagonal matrices ~= diag-(O"xx'O"yy,O"zz) and §,= diag(Exx' Gyy,Gzz). In accordance with the IE approach, we introduce some reference formation. Here we assume a reference formation of generalised conductivity fa, which is a 3x 3 diagonal matrix given by ~o= diag(S;'S!"SII)' where Sr and SII are complex-valued fiinctions of depth z and frequency w; this specific choice of diagonal elements is driven by the desire to efficiently solve thefcattered field versions of Maxwell's equations, where

- -

",,-

(2a)

= iWflHS,

(2b)

V'xHS =CEs+j', V'x ES for the scattered fields

~s=~-~o

and .ijs=.ij-.ijo.

Here

the reference formation fields, ~o and .ijo, satisfy Maxwell's equations Yx.ijo = ~o~o+jp and Yx ~o= iWfl.ijo+f!lP. In Equation (2a) an unknown equivalent source, js, is

-

defined by jS

= (S-C

)~.

(3)

Ultimately one finds js on the scattering volume Vs C IR3, . where (~ - fo) differs trom zero, by

j' =2~~(~+~~O),

(4)

Avdeev, Kuvshinov, Pankratov, Newman

112

where 1:: = diag (--JRe ST' ..JRe S r, --JRe S/l) here Re S is the real part of-So In this expression the vector field is a solution

~

of an integral equation for electromagnetic scattering, where .

~(~)

=X3+ Ivs ~(~':)~(~) ~(~)dv'.

(5)

Here

ST

R = diaa Su - Sr Syy '"( Su +Sr'

=0;

-

Xo(~) =

Syy +Sr'

S::

-

S

11

(6)

S:: +S/l ] '

f ~(~,~)~(~)

~(z')E:(~)dv',

(7)

Vs

S is the complex conjugation of S, r' = (x',y',z'), and dv' = dx'dy'dz'. In addition,

- -=0;- - -

K(r,r') = 8(r - r')! + 2 ~ ge (r,r'):::

(8)

where 8(r) is Dirac's delta function, ! is the identity matrix, and ge is 3 x 3 dyadic electric Green's function of the reference formation. Note that similar definition of 3 x 3 dyadic function K was used in (Singer, 1995) for the quasistatic field. The solution, ~, of Equation (5) is obtained from an always convergent Neumann (Bom) series expansion

-

= X 0+ f ~(~,~) ~(~) X °(~)dv'

-

+

(9)

-

Vs

f ~(~,~)~(~)f ~(~,r~)~(r~)Xo(rJdv"dv' -

Vs

Vs

+..', as ~(r) = ~~~!,t~(N),where N-th partial sum ~(N) is determined iteratively by

(~) = X_a +

~(N+l)

M

°0xNyNJN.

N:'+(N +NJlog2

=0;

5

where the

X(~)

demonstrated in Equation (11). The reader is referred to (Avdeev et aI., 1997) for details of the numerical volume integrations and the overall numerical solution scheme. Although that paper dealt with an isotropic earth, the estimates of the number of operations, M, and memory sizes, 5, needed to obtain a numerical solution remains the same for an anisotropic earth considered in this paper. In particular,

f ~(~,~)

~(~) ~(N)(~' )dv',

Vs

(0)

=0;

Nx log2 Ny))

O(NxNy N'1 }ytes.

(13) (14)

Here N is the number of terms of the Neumann series expansion (9) needed to get the solution ~ with a given relative accuracy E, and N" N v and Nz are the number of nodal points describing the vo!ume, Vs. Our discretisation implies the 3D mesh is equidistant along each horizontal axis, but is variable along verticalaxis. From Equation (13) it can be seen that M depends on horizontal sizes tV., and Nv as O(NT10gNr), and depends quadratically on the vertical size Nz. Meanwhile, the existing 3D IE solvers (Newman et aI., 1986; Xiong and Tripp, 1995) give a cubic dependence, O(N} N? Ni), for each size N" Ny or Nz. Moreover, the number of terms N can be dramatically diminished by a proper choice of the reference formation. As an example, for a case of isotropic earth and for frequency range when the displacement currents can be ignored, the number N is directly proportional to square root of the resistivity contrast (Singer, 1995). Thus, the expression for M in Equation (13) points to the high-performance nature of our solver. The expression in Equation (14) shows that on workstation-type platforms, one can perform all calculations exclusively in RAM without access to disk, which will accelerate the solver performance strongly. As an example if the scattering volume is divided into 80 x 80 x 5 = 32,000 or 40 x 40 x 10 = 16,000 or 20 x 20 x 20 = 8,000 cells, the solver takes about 32 Mbytes of memory. Note that expressions in Equations (13) and (14) also demonstrate an increase in the performance of our solver when N" and Nv

exceed Nz.

.

VERIFICATION OF SOLUTION

{ x(l)(r)

= Xo,

N = 1,2,...

With j s now known from Equation (4), one determines the scattered electric ~s and scattered magnetic tls fields for

any r E IR3by

In order to verify the accuracy of our solution for airbome simulations, we compared our results with those of a FD solution (Newman and Alumbaugh, 1995) for a number of models. 3D target in a half-space

~s (~) = fvs £' (~,~' ) ~s(~)dv',

r

~s (~) = Jvs

h

£ (~, ~

,

, ) ~s(~ )dv'.

(11)

(12)

The explicit form of the 3 x 3 dyadic Green's functions ge and flh of the reference formation can be found in (Avdeev et aI.~ 1997). The total fields are obtained by adding the appropriate reference formation fields and scattered fields. COMPUTATIONAL REQUIREMENTS It is seen from Equations (11) and (12) that the calculation of electric ~s and magnetic tls fields involves volume integration. Moreover, from the iteration rule in Equation (10), one can see that the Neumann series summation is again nothing but the repeated volume integrations of type

The first model shown in Figure I consists of 1 Qm body of dimensions 20 x 200 x 75 m buried at a depth of 50 m in a 100 Qm half-space. A helicopter AEM system is simulated using a vertical magnetic dipole (VMD) and a horizontal magnetic dipole (HMD) at 900 Hz, deployed 20 m above the earth's surface. The transmitter and receiver separation is 10 m with the receiver leading the transmitter to the right. In this and all the following examples, the results have been plotted mid-way between the transmitter and the receiver along the flight line where this line bisects the body. . The scattering volume VS is confined to the 3D body. We discretised the body by 4 x 40 x 9, or 1,440 rectangular cells with dimensions of 5 x 5 x 8.33 m3. The code required N = 195 terms of Neumann series expansion (9) to compute the solution with relative accuracy I %. On a Pentium100-MHz PC the run times for the VMD and HMD sources each required about 8.5 minutes per source position. The

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Figure 5. Comparisons of the 900-Hz responses over vertical fault contact with and without topography. The middle panels illustrate Hz and Hx responses for VMD source, while the lower panels illustrate those for HMD source.

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Midpoint (m)

Figure 6. Comparisons of the 900-Hz VMD source responses over vertical fault contact with and without topography. The middle and lower panels show the responses for the models with resistivities of 300 Qm and 3000 Qm right of the fault contact, respectively.

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