Free-mode surface-wave computations

Abstract

The theory of Love-and Rayleigh-wave dispersion for plane-layered earth models has undergone a number of developments since the initial work of Thomson and Haskell. Most of these were concerned with computational difficulties associated with numerical overflow and loss of precision at high frequencies in the original Thomson-Haskell formalism. Several seemingly distinct approaches have been followed, including the delta matrix, reduced delta matrix, Schwab-Knopoff, fast SchwabKnopoff, Kennett's Reflection-Transmission Matrix and Abo-Zena methods. This paper analyses all these methods in detail and finds explicit transformations connecting them. It is shown that they are essentially equivalent and, contrary to some claims made, each solves the loss of precision problem equally well. This is demonstrated both theoretically and computationally. By extracting the best computational features of the various methods, we develop a new algorithm (see Appendix A5), called the fast delta matrix algorithm. To date, this is the simplest and most efficient algorithm for surface-wave dispersion computations (see Fig. 4).The theory given in this paper provides a complete review of the principal methods developed for Love-and Rayleigh-wave dispersion of free modes in plane-layered perfectly elastic, isotropic earth models and puts to rest controversies that have arisen with regard to computational stability.

A note on the equivalence of three major propagator algorithms for computational stability and efficiency

“…(16) Panza 1985;Ivansson 1993;Buchen & Ben-Hador 1996) which prevents overflow in a single layer and equals to {E u }, can be dispatched into different matrices, i.e.…”

“…Dunkin 1965;Thrower 1965;Kind & Odom 1983;Chapman & Orcutt 1985;Buchen & Ben-Hador 1996;Wang 1999). Here we summarize the two key problems.…”

“…However it is helpful to keep their phases to ensure the final determinant in eq. (38) remains real-valued (Buchen & Ben-Hador 1996), i.e.,…”

Tool for analysis of multichannel analysis of surface waves (MASW) field data and evaluation of shear wave velocity profiles of soils

“…Many of these were formulated to resolve numerical overflow and loss-ofprecision problems that can occur at high frequencies when the original Thomson-Haskell method is applied (Schwab 1970). Available methods include the propagator-matrix approach described by Knopoff (1964) and Schwab (1970) with later improvements of, e.g., Abo-Zena (1979), Menke (1979), and Buchen and Ben-Hador (1996); the stiffness matrix formulation of Kausel and Roësset (1981); and the reflection-transmission matrix method developed by Kennett (1974) and Kennett and Kerry (1979). In this work, the stiffness matrix method was used for computations of theoretical dispersion curves.…”

The Modification of the Matrix Method for the Modelling of Propagation of the Body Waves

“…The matrix method was developed in the works [7][8][9][10]24]. The stable algorithms of seismograms calculation for all angles of seismic waves propagation is obtained.…”