Theoretical Investigation of Modal Seismograms for a Layer Over a Half‐space

Abstract

The use of normal modes to represent the propagation of elastic waves at large horizontal offset is well known. By addition of the relevant leaky modes, the mode theory is shown to be useful for representation of the seismograms at shorter range. A theoretical model consisting of a 2‐cm brass layer over a steel half‐space is considered. Dispersion curves and excitation functions are computed for the first four normal modes and the first three PL modes. Attenuation as a function of frequency also is computed for the PL modes. A suite of seismograms is computed for the distance range 50–70 cm, showing each mode individually and their sum (the total seismogram). It is found that, for the distances used, the individual modes do not approximate transients with a definite “arrival” time. Only their sum is required to exhibit this physical behavior. In addition, at short distances, the dispersion of a single mode is not visually obvious although the dispersion curve may be recovered by use of Fourier transform methods. Determination of the dispersion curves from the total seismogram is more difficult and requires some seperation of the modes, as they overlap in frequency and velocity. This work shows the preponderance of the leaky modes in the early part of the seismogram and indicates their importance in the later part of the seismogram for short horizontal offset.

Synthesis of Regional Ground Motion from Western U. S. Earthquakes.

“…It is possible that these contributions are also provided by leaking modes (Laster , et al, 1965). At regional distances , the main consequence of omitting branch-line contributions is a slight acausality in the S-wave motion ( Swanger and Boore , 1978) and perhaps a misrepresentation of the Sn phase , which is a small phase anyway Herrmann ( 1977) presents informative examples of the deficiencies of Love-mode solutions and shows they are not serious at epicentral digtances much larger than the source depth.…”

Slowly-attenuating P-SV leaky waves in a layered elastic halfspace. Effects on the coherences of diffuse wavefields

“…2), where their numerical computation is simpler. The branch-cut discontinuities of both As was noticed by Laster et al [19], a weakness of this choice resides in the difficulties to evaluate some oscillatory events by means of the branch-cut integrals. These instabilities are related with the existence of complex-k roots of Anomalous poles of this type have been described in more detail by Surkov and Reshetnikov [20].…”

“…Table 1. Example models presenting a pole in the P-SV integrand at To investigate the origin of these real poles, the scheme of branch-cut rerouting used by Laster et al [19], Ben-Menahem and Singh [13] or Watson [18] is followed. Conjugation of the integrands is required to meet the definition of Fourier transform used in the first and second references.…”

A Comparison of Leaking Modes and Generalized Ray Theory

“…3 shows the solution s(k) of d(s, k) = 0 as an s-k diagram (Gilbert 1964 f (s, k)/(ad/8s),=,(,, and g(s, k)/(&f/8s),,,~,, for the same mode. Note that Figs 3-5 cannot be compared directly with Laster et al (1965), since wavenumber has been used as the independent variable instead of frequency. Fig.…”

“…Model used Table 1 summarizes the elastic constants of the model, which is taken from Laster, Foreman & Linville (1965). The model is two-dimensional and there is no transverse motion.…”