Ž . In solving Eq. 4 , the value of l must be selected
so that an acceptable misfit is achieved. This requires some form of trial and error. At each iteration, Eq.
Ž . 4 is solved for several values of l, and for each
updated model the forward modeling is carried out to evaluate the misfit S. By approximating the behavior
of S as a function of l with a polynomial, a new value of l that will minimize the misfit is estimated.
3. Forward modeling
In the inversion, the earth is divided into a num- ber of blocks of constant conductivity and so the
forward modeling method needs to have the ability to handle a wide variety of conductivity distribu-
tions. The 3-D EM fields can be obtained by numeri- cally solving either Maxwell’s differential or integral
equations. For the purpose of inversion, the differen- tial equation approach is superior to the integral
equation approach, because the former is more effi-
Ž cient and flexible to model complex structures New-
. man, 1995 .
If displacement currents are ignored and an e
i v t
time dependence is assumed, then the secondary electric field E
in the frequency domain is de-
s
scribed by a second-order equation,
= = = = E q jvms E s yjvm s y s E . 6
Ž .
Ž .
s s
p p
Here the conductivity and the magnetic permeability are denoted by s and m, respectively, and subscript
p designates a background or primary value. It is assumed that m s m s 4p = 10
y7
Hrm every- where. The total field is given by
E s E q E . 7
Ž .
p s
In a finite-difference scheme on a staggered grid, the Ž
. solution region
including air is discretized into
rectangular cells. The secondary electric field is de- fined at the center of the cell edges as shown in Fig.
1. The conductivity at each sampling point is repre- sented by a weighted average of conductivities of the
four adjoining cells; the weights are proportional to the cross-sectional area of each cell that is normal to
the sampled field component. As boundary condi-
Fig. 1. Sampling positions for the electric field components on a staggered grid.
tions, the tangential component of E is set equal to
s
zero on the boundaries of the model. Approximating Ž .
Eq. 6
with finite differences results in a linear system of equations,
K f s s, 8
Ž .
where K is a symmetric complex matrix, f is the unknown vector for the secondary electric field, and
s is a vector containing source terms. All elements of
Ž .
K are real except for the diagonal elements. Eq. 8
Ž .
can be solved using the biconjugate gradient BCG method, preconditioned with an incomplete Cholesky
decomposition. The preconditioning schemes using standard incomplete Cholesky decompositions break
down because K is not positive-definite. However,
Ž .
Mackie et al. 1994 shows that this difficulty can be avoided by applying the decomposition only to the
diagonal subblocks that are positive-definite, when K is grouped into subblocks depending on the related
components of the electric fields. Although reason- able convergence rates can be obtained with the
Ž .
incomplete Cholesky biconjugate gradient ICBCG method in many instances, they tend to be degraded
as the frequency falls. This is because conservation of current is not guaranteed in a discretized version
Ž . of Eq. 6 when the second term of the left-hand side
Ž .
becomes small. Smith 1996 derived a correction procedure that enforces divergence-free conditions
on the current density and electric field. The conver- gence rate at low frequencies can be improved sig-
nificantly by alternating ICBCG iterations with this correction procedure.
4. Sensitivity
