Extension of the Thomson--Haskell method to non-homogeneous spherical layers

Abstract

Amplitude spectra of Rayleigh and Love waves in a layered nongravitating spherical earth have been obtained using as a source, displacement and stress discontinuities. In each layer elastic parameters and density follow specified functions of radial distance and the solutions of the equations of motion are obtained in terms of exponential functions. The Thomson-Haskell method is extended to this case. The problem reduces to simple calculations as in a plane-layered medium. Numerical results of phase and group velocities up to periods of 300 s in various earth models when compared with earlier results (obtained by numerical integration) show that the present method can be used with sufficient accuracy. The differences in phase velocity, group velocity and amplitude (also surface ellipticity in the case of Rayleigh waves) between spherical-and flat-earth models have been investigated in the range 20-300-s period and expressed in polynomials in the period. * Present address: India Meteorological Department, Mohanbari Airfield, Dibrugarh, Assam 78601 2, India. = P;(COS e ) [cos 14, sin l41. = a In (r/a). = %(a2 + 2P2)/(a2 -202).

Rayleigh-wave dispersion function for a transversely isotropic layered spherical earth using the Thomson-Haskell matrix method

“…Similarly, results for the first higher mode, M , , , of the Rayleigh wave are given in Table 2. Our results for an isotropic medium agree with those of Bhattacharya (1976) to four decimal places. In the transversely isotropic GBA model, the value of T increases for a given value of n. The increase in T is a maximum at n = 75.…”

“…The present values of T completely agree to four or five significant figures with those of Takeuchi et al (1964) and Bhattacharya (1976). Abe (1970) obtained results for this model considering gravitational perturbations.…”

“…However, evaluation of amplitude response, group velocity and other properties of Rayleigh waves with Bessel functions is not as convenient as with exponential functions. Bhattacharya (1976) andArora et al (1996) developed a useful technique by considering a specific inhomogeneity within each layer, and obtained solutions in terms of exponential functions.…”

“…To obtain solutions, we do not resort to the conventional method of representing the displacement field by potential functions and looking for a set of differential equations, each containing a single potential function as in Bhattacharya (1976), but instead we follow Anderson (1965). With a simple transformation, it is seen that the two radial functions involved in the displacement components satisfy a fourth-order homogeneous linear differential equation which is a quadratic in the squared differential operator and has solutions in terms of exponential functions.…”

Extended formulation for Rayleigh-wave computations at high frequencies in a spherical layered earth

“…The above developments are related to a flat layered earth. In an earlier paper (Bhattacharya 1976, hereafter referred to as paper l), for a non-gravitating layered earth a suitable inhomogeneity has been chosen in each shell so that solutions of the equations of motion are obtained in terms of exponential functions. With these solutions the calculation of surface wave parameters in a spherical layered earth can be obtained as simply as in a flat layered earth.…”