Computational Mathematics and Modeling, Vol. 29, No. 3, July, 2018
I. MATHEMATICAL MODELING
TWO-DIMENSIONAL INVERSE PROBLEM OF MAGNETOTELLURIC SOUNDING
IN A NONHOMOGENEOUS MEDIUM
V. I. Dmitriev
Unique solvability theorems are proved for the inverse two-dimensional sounding problem in a nonho-
mogeneous conducting half-space with different primary-field polarizations.
Keywords: inverse problems, differential equations, solution uniqueness.
Magnetotelluric sounding (MTS) studies the structure of the Earth by using its natural electromagnetic field.
It is assumed that the field source is at a large distance from the Earth’s surface, where the electromagnetic field
is measured. Under these assumptions, we may treat the primary field as constant in the observation region and
assume that its variations are associated with the nonhomogeneous distribution of the subsurface conductivity.
The strength of the source creating the Earth’s natural electromagnetic field is unknown, and we observe the
relationship between the electric and magnetic fields:
The linear coefficients in (1) constitute the impedance tensor, which is independent of the strength of the
distant field source and is determined only by the field frequency and the distribution of the electrical conductiv-
ity below the Earth’s surface (
z > 0
The forward MTS problem determines the impedance tensor
z = 0
when the conductivity distribu-
is known for
z > 0
. The field source is a plane wave normally incident on the Earth’s surface from
z < 0
The inverse MTS problem determines
z > 0
given the impedance tensor on the Earth’s surface
z = 0
as a function of the observation point and the frequency
Z(x, y, ω)
In the one-dimensional case, when the conductivity
z > 0
, is a function of depth only (a layered me-
dium), the inverse problem has a unique solution for piecewise-analytical functions
z > H
. The uniqueness of the inverse-problem solution has been proved for a layered medium
containing a thin layer with longitudinally varying conductivity . We apply the method of  to investigate
Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, Russia; e-mail: email@example.com.
Translated from Prikladnaya Matematika i Informatika, No. 56, 2017, pp. 5–17.
1046–283X/18/2903–0253 © 2018 Springer Science+Business Media, LLC 253