The radiation modes of a vertically varying half-space: a new representation of the complete Green's function in terms of modes

SUMMARY We prove that the complete Green’s function for an isotropic vertically varying halfspace can be written in a dyadic form as a sum on the usual normal modes plus an integral over radiation modes. The radiation modes form the basis on which to represent the body waves, which are not trapped in the structure. The expressions for their eigenfunctions, the so-called improper eigenfunctions, are found for both the SH and the P-SV wave operators. We prove that they are orthogonal to each other and to the normal-mode eigenfunctions. In a numerical example, we present improper eigenfunctions, and compare complete wavefield seismograms synthesized by the mode method and by the discrete wavenumber integration method.

[1]  J. Woodhouse.  Surface Waves in a Laterally Varying Layered Structure , 1974 .

[2]  Spectral representation of the Love wave operator , 1976 .

[3]  H. Takeuchi,et al.  Seismic Surface Waves , 1972 .

[4]  B. Kennett,et al.  2‐D reflectivity method and synthetic seismograms for irregularly layered structures—II. Invariant embedding approach , 1991 .

[5]  L. E. Alsop.  The leaky-mode period equation—a plane-wave approach , 1970, Bulletin of the Seismological Society of America.

[6]  F. Gilbert.  Propagation of transient leaking modes in a stratified elastic waveguide , 1964 .

[7]  I. Herrera.  On a method to obtain a Green's function for a multi-layered half space , 1964 .

[8]  V. Maupin.  Modelling of laterally trapped surface waves with application to Rayleigh waves in the Hawaiian swell , 1992 .

[9]  B. Kennett.  Guided wave attenuation in laterally varying media , 1990 .

[10]  R. Odom.  A coupled mode examination of irregular waveguides including the continuum spectrum , 1986 .

[11]  V. Maupin.  Surface waves across 2-D structures: a method based on coupled local modes , 1988 .

[12]  R. Snieder.  3-D linearized scattering of surface waves and a formalism for surface wave holography , 1986 .

[13]  L. Felsen,et al.  Coupled mode theory of intrinsic modes in a wedge , 1986 .

[14]  R. Snieder,et al.  Surface‐wave mode coupling for efficient forward modelling and inversion of body‐wave phases , 1995 .

[15]  F. A. Dahlen,et al.  The Effect of A General Aspherical Perturbation on the Free Oscillations of the Earth , 1978 .

[16]  R. Haddon.  Exact evaluation of the response of a layered elastic medium to an explosive point source using leaking modes , 1986 .

[17]  M. Campillo,et al.  Wave diffraction in multilayered media with the Indirect Boundary Element Method: application to 3-D diffraction of long-period surface waves by 2-D lithospheric structures , 1996 .

[18]  A. McGarr,et al.  Transmission and reflection of Rayleigh waves at vertical boundaries , 1967 .

[19]  Calvin H. Wilcox.  Sound Propagation in Stratified Fluids , 1984 .

[20]  K. Kohketsu,et al.  2-Dreflectivity method and synthetic seismograms for irregularly layered structures -I. SH-wave generation , 1987 .

[21]  M. Bouchon.  A simple method to calculate Green's functions for elastic layered media , 1981 .

[22]  Robert B. Herrmann,et al.  A numerical study of P-, SV-, and SH-wave generation in a plane layered medium , 1980 .

[23]  R. Haddon.  Response of an oceanic wave guide to an explosive point source using leaking modes , 1987 .

[24]  R. Kind,et al.  The reflectivity method for a buried source , 1977 .

[25]  K. Fuchs,et al.  Computation of Synthetic Seismograms with the Reflectivity Method and Comparison with Observations , 1971 .

[26]  Paul G. Richards,et al.  Quantitative Seismology: Theory and Methods , 1980 .

[27]  R. Haddon.  A simple exact method for Green's functions for SH motion in a layered elastic medium using leaking modes , 1987 .

[28]  M. Saito.  Excitation of free oscillations and surface waves by a point source in a vertically heterogeneous Earth , 1967 .

[29]  F. Gilbert.  Excitation of the Normal Modes of the Earth by Earthquake Sources , 1971 .

[30]  Danny J. Harvey,et al.  Seismogram synthesis using normal mode superposition: the locked mode approximation , 1981 .

[31]  M. Bostock.  Reflection and transmission of surface waves in laterally varying media , 1992 .

[32]  Thomas H. Watson,et al.  A real frequency, complex wave-number analysis of leaking modes , 1972, Bulletin of the Seismological Society of America.

[33]  Donald H. Menzel,et al.  Fundamental Formulas of Physics , 1955 .

[34]  Dietrich Marcuse,et al.  Surface Waves and discontinuities , 1988 .

[35]  B. Kennett.  Guided wave propagation in laterally varying media — I. Theoretical development , 1984 .

[36]  D. Marcuse.  Theory of dielectric optical waveguides , 1974 .

[37]  C. Chapman,et al.  Symmetry of the wave equation and excitation of body waves , 1981 .

[38]  Martin Tygel,et al.  Transient Waves in Layered Media , 1987 .

[39]  R. Haddon.  Computation of synthetic seismograms in layered earth models using leaking modes , 1984 .