2.1. Efficient Finite-Difference Simulation
In CSAMT/CSRMT, the five components of the electromagnetic filed are measured (
Ex,
Ey,
Hx,
Hy,
Hz) and, after preprocessing, converted to the components of the impedance and tipper. The off-diagonal components of the impedance are employed most commonly, that is [
2],
where subscripts 1 and 2 refer to one of the two orthogonal transmitter lines. Thus, computation of the components of the impedance or tipper requires simulation of the electric and magnetic fields
and
at several locations and frequencies.
We consider electromagnetic modeling in 3D heterogeneous isotropic media in the frequency domain. Assuming time dependence of
, the electric field,
, satisfies the following system of partial differential equations,
where
is the source angular frequency,
is the magnetic permeability of the free space,
is the conductivity model, and
is the source current density. This is a linear system of three scalar equations with respect to three scalar entries of
. We solve this partial-differential equation system in a bounded domain, completed by zero Dirichlet boundary conditions. The magnetic field
is then computed via Faraday’s law,
Since geological models predominately vary in the vertical direction, we can always split the conductivity model into background and anomalous parts,
We further assume that the following double inequality holds,
which controls the contrast of anomalous conductivity with respect to the background.
Given a non-uniform computational grid with
cells, we apply the conventional edge-based second-order FD discretization. The FD method approximates the unknown electric field
with a finite set of discrete values
[
26]. Each discrete value is attached to the respective edge of the FD grid,
Figure 1.
The actual discrete equations can be found for example in [
27]. Let us form the unknown vector
of the discrete electric fields
introduced above. Now we can write the FD discretization of (2) in a matrix form:
where
is a complex symmetric system matrix with at most 13 nonzero entries per row, corresponding to the total conductivity distribution. We denote the size of this system as
,
.
Let
be the FD system matrix, corresponding to the background conductivity model. Importantly, this matrix can be implicitly factorized using a fast direct algorithm, and the action of the inverse matrix can be efficiently computed [
23,
28]. As a result, it can be used as a preconditioner to (6):
We will refer
as the Green’s function (GF) preconditioner. The complexity of applying the GF preconditioner is
and auxiliary memory required is near
only. Applying the analysis presented in [
23] the condition number of the GF preconditioned system can be estimated as follows,
This result implies that convergence of an iterative solver applied to (7) has minor or no dependence on the grid size and cell aspect ratio, as well as the frequency, while it degrades on models with high-contrast bodies.
To minimize the impact of high-contrast bodies, another preconditioner could be constructed. Denote as
and
diagonal matrices, corresponding to full and background conductivity, respectively. Let us define the modified FD Green’s operator and diagonal matrices according to the following formulae:
Using this operator, Equation (7) can be written in an equivalent form as follows (see
Appendix A):
where
. By introducing a new operator,
, we rewrite Equation (11) as follows:
We will refer this system a contraction operator (CO) preconditioned system, since it was shown in [
23] that
. Moreover, it can be proved that this preconditioned system has a smaller or equal spectral condition number than that of the GF preconditioner, implying faster or equal convergence of the iterative solvers,
The complexity of applying the CO preconditioner is
as well. The preconditioners in Equations (7) and (12) were incorporated into the BiCGStab iterative solver [
29]. We used the complex-valued version of the solver with the standard complex-valued dot product.
Practical controlled-source modeling requires excessive gridding near the source location. To avoid this, we preferred secondary field modeling, i.e., the electric field
is represented as a sum
, where
is the response due to background conductivity model known analytically [
30,
31]. In this case, the secondary field
will be the unknown function and its FD discretization is performed. The actual source
in Equation (2) is substituted with the secondary source
.
2.2. Applicability of Quasi-Statinary Simulation
As the first step, we tested applicability of the quasi-stationary simulation in the context of controlled-source electromagnetics. The displacement current in the air must be considered when the electromagnetic field generated by a high-frequency dipole oscillator is measured at large offsets [
10,
12]. Otherwise observed components of the electromagnetic field cannot be matched to the computed quasi-stationary ones; this is known as the propagation effect. Formally, it is achieved by replacing
term in Equation (1) with
, where
is the vacuum dielectric permittivity and
is the relative dielectric permittivity. It complicates the 3D numerical simulation considerably. However, in contrast to the individual components, the surface displacement-current impedance was almost identical to the quasi-stationary one, at least in a 1D Earth. To illustrate this point, we considered an X-oriented horizontal electric dipole (HED) located at the origin of Cartesian coordinate system and two receiver stations (
Figure 2).
The first station corresponded to relatively short offset (
X = 450 m,
Y = 750 m) and the second one imitated a typical CSAMT offset (
X = 450 m,
Y = 4500 m). We computed the electromagnetic field for the two cases. In the first case, the model was the one in
Table 1. Computations were conducted in the quasi-stationary regime, where the squared wavenumber of
-th layer was given by
. The air conductivity was set to
=
S/m. In the second case, the squared wavenumbers of each
-th layer were defined as
. We set
in the air, and
in the Earth. The computations were performed by the 1D code of [
13]. Computed curves are presented in
Figure 2. At frequencies higher than 7 kHz (the close receiver) and 27 kHz (the distant receiver), the quasi-stationary electromagnetic field (dashed lines) differed considerably from the displacement-current field (solid lines). However, ratios of the horizontal component remained essentially the same.