Computational Mathematics and Modeling, Vol. 29, No. 3, July, 2018
CALCULATION OF THE CHARACTERISTICS OF TRAVELING WAVES
IN LAYERED MEDIA
A new method is considered for the calculation of traveling-wave characteristics in a layered medium.
It is mainly based on the transfer-matrix method [2–10] and the generalized reflection-transmission
coefficient (GRTC) method [11–14] for the calculation of characteristics. Introducing an impedance
tensor of rank 2,
, in the boundary-value stress problem and utilizing in this way the boundary
conditions and the Lame equation, we obtain a system of differential equations which is solved by the
fourth-order Runge-Kutta method. Specifying a model of the layered medium in terms of known con-
stants, we find the traveling-wave characteristic
as a function of frequency
, we easily
calculate the dispersion curves, which closely fit the GRTC method. Compared with the GRTC method,
the impedance-tensor method fully solves the dispersion curve of the normal-mode traveling waves, and
it is applicable to both a homogeneous planar Earth model and a model with a slow layer.
Keywords: traveling wave characteristics, impedance-tensor method, dispersion curve, phase velocity,
Focused studies of the structure of the Earth’s interior have produced some important results in recent
years [1–8]. In particular, the velocity structure of transverse waves is now mainly investigated using surface-
wave dispersion formulas [9–12]. Solving the elastic-wave equation is one of the most significant scientific di-
rections in seismology (for instance, investigation of the characteristics of seismic phases, calculation of seismic
wave paths, calculation of synthetic seismograms, and so on.). Different waves have different characteristics.
The determination of these characteristics plays a major role in the solution of the wave equation and the calcu-
lation of wave dispersion.
Most studies of the forward seismic-wave problem compute the plane-wave dispersion. The classical meth-
od for a piecewise-constant plane-layered medium is the Thomson–Haskell transfer-matrix method . Howev-
er, the numerical computations involve loss of high-frequency accuracy. A number of new methods have been
developed to overcome this difficulty [3–10]. X. F. Chen  has proposed the generalized reflection-
transmission coefficient (GRTC) method . The impedance tensor method is often used in magnetotelluric
sounding. Dmitriev  applies this method in seismology to solve for the spectral characteristics in an isotropic
layered medium. In the present article, we consider a new impedance-tensor method to calculate the traveling-
wave characteristics from seismic data.
Calculation of Traveling-Wave Characteristics
Consider a layered medium with the boundary
z = z
n ∈[0, n]
is the Earth’s surface.
Inside each layer,
n ∈[1, N ]
. The seismic parameters
are assumed constant.
Translated from Prikladnaya Matematika i Informatika, No. 56, 2017, pp. 43–53.
1046–283X/18/2903–0287 © 2018 Springer Science+Business Media, LLC 287