ISSN 0278-6419, Moscow University Computational Mathematics and Cybernetics, 2018, Vol. 42, No. 2, pp. 55–62.
Allerton Press, Inc., 2018.
Original Russian Text
A.V. Baev, S.V. Gavrilov, 2018, published in Vestnik Moskovskogo Universiteta, Seriya 15: Vychislitel’naya Matematika i Kibernetika, 2018,
No. 2, pp. 7–13.
An Iterative Way of Solving the Inverse Scattering Problem
for an Acoustic System of Equations
in an Absorptive Layered Nonhomogeneous Medium
A. V. Baev
Faculty of Computational Mathematics and Cybernetics,
Moscow State University, Moscow, 119991 Russia
Received December 18, 2017
Abstract—An algorithm is considered for solving the inverse scattering problem of seismic waves
in a layered medium. The algorithm is based on solving a nonclassical ordinary diﬀerential equation
with respect to an acoustic impedance, which also contains an unknown function characterizing
the dissipative properties of the medium. The uniqueness of determining of these functions and
the functional dependence associating them is established by solving the inverse problem of ground
seismics. Results are presented from a computing experiment on applying the proposed algorithm.
Keywords: acoustic impedance, dissipative medium, generalized function.
When solving diﬀerent applied problems [1–7], including those of inverse scattering, we use a
diﬀerential equation of the form
+ ϕ(y(x)) = f (x),x∈ [0,a], (1)
where y(x) is the desired solution, which is generally not continuously diﬀerentiable; ϕ(y) is a function
continuous on (−∞, ∞);andf(x) is a given right-hand side. Such equations are typical when
processing seismic survey data, where function y(x) describes the elastic properties of a layered medium,
and f(x) is a seismogram recorded by instruments. Independent variable x is in this case the eikonal,
i.e., the time needed for the propagation of a seismic signal from its current point to the Earth’s surface.
Meanwhile, the functional term in the left-hand side of (1) characterizes the dissipative properties of the
medium. In most mathematical models, it is speciﬁed a priori.
It is clear that in the general formulation, the problem of simultaneously determining the elastic and
dissipative properties of a layered medium, i.e., functions y(x) and ϕ(y), does not have a unique solution
in the Cauchy problem for (1). However, the very nature of the geological origin of the upper strata of the
Earth’s crust allows us to formulate a reasonable mathematical statement of such a problem. The depths
of the Earth’s crust consist of homogeneous layers of diﬀerent rocks whose thickness and stiﬀness diﬀer
greatly for seismic observations. This means the problem is normally posed of ﬁnding layer boundaries
and the coeﬃcients of reﬂection at them that correspond to identifying discontinuities of function y(x).
There is another reason why we should abandon the classical understanding of (1) as an equation
that holds at all points of [0,a]. A discrete approach is used in the vast majority of cases when recording
and processing observations, and when constructing numerical algorithms. At the same time, the
interval of discretization is identiﬁed with a thin layer, and the diﬀerence derivative determines the