Body waves as normal and leaking modes, 3. Pseudo modes and partial derivatives on the (+ −) sheet

Abstract

In a multiple elastic wave guide, the phase velocity curves of the Π and Σ modes on the (+ −) sheet combine to form two families of almost continuous curves. The curves of the first family, called Π pseudo modes, are formed by the plateaus of the dispersion curves; they have a high group velocity, a low attenuation, and a low excitation; they depend solely on the P velocities. The curves of the second family, called Σ pseudo modes, are formed by the slopes of the dispersion curves and have the opposite characteristics. The Π pseudo modes make up the oscillatory part of the seismogram between P and S. The study is based on a simplified CIT 11 GB model, which consists of 6 layers, is 653 km thick, and contains a 54‐km low‐velocity P channel. The frequency range studied is from 0 to 0.146 Hz. The influence of the P channel is very restricted. At wavelengths smaller than ⅓ of its thickness, the layer below the channel (α = 8.19 km/sec) supports what is essentially a head wave.

Multiple Leaking Mode Dispersion Observations and Applications From Ambient Noise Cross‐Correlation in Oklahoma

“…According to the sensitivity of the S‐wave velocity (Vs) and P‐wave velocity (Vp), the leaking mode can be divided into $${\Sigma}$$ and $${\Pi}$$ modes (Cochran et al., 1970). The higher‐mode leaking modes (Π modes) extracted from earthquake records were found to be most sensitive to the Vp structure (Li, Shi et al., 2021).…”

Theoretical investigation of the excitation of leaky modes for multi-layered models

“…The direct strategy for searching roots is the grid search in the 2-D complex 𝑘-plane, but it turns out time-consuming and yields less accurate results (Roth et al, 1998;Gao et al, 2014). If we resort to some iterative methods such as the Newton-Raphson method, with a series of initial values the roots may be located very efficiently (Gilbert, 1964;Cochran et al, 1970;Watson, 1972;Radovich and De Bremaecker, 1974). Similar to the approach of Watson (1972), we search the roots from high frequency to low frequency, extrapolate the trial value of the root of the same order at the next frequency point, and employ the Newton-Raphson method to achieve the roots to the desired accuracy.…”

“…Leaky modes have long been suggested be used to constrain P-wave velocity structure and been thought to have unique advantages where traditional surface waves and classic seismic reflection and refraction methods may fall short (Oliver and Major, 1960;Su and Dorman, 1965;Roth et al, 1998). On the way to putting leaky modes into practical inversion, earlier studies had been devoted to the effects of the crustal structure and P-wave velocity structure on the leaky modes (Haskell, 1966;De Bremaecker, 1967;Cochran et al, 1970;Stalmach and De Bremaecker, 1973;Su and Dorman, 1965) and to the attenuation and the excitation of the leaky modes (Laster et al, 1965;Haskell, 1966;Dainty, 1971) and found that a deeper event in the crust could more efficiently excite the fundamental leaky mode (the most commonly seen mode) than a near-surface source (Dainty, 1971). Notwithstanding these progresses after 1960, leaky 5 modes have found limited applications in imaging underground structures (Su and Dorman, 1965;Ibrahim, 1969;Fujita and Nishimura, 1991) and have been approximately computed in the case of high Poisson's ratios (Roth et al, 1998;Roth and Holliger, 1999;Boiero et al, 2013;Gao et al, 2014).…”

Scattering versus intrinsic attenuation in periodically layered media

“…It has no real roots for ω, while the complex roots ω = α + iβ ∈ C define the frequencies and attenuation of the self-induced oscillations of a periodically layered medium. Gilbert (1964) and Cochran et al (1970) studied the dependence of these complex roots on the horizontal slowness (or horizontal wave number) in a horizontally layered medium. This dependence is given by the dispersion equation and is algebraically very complicated even for the four-layer medium.…”