Comment on ‘Free-mode surface-wave computations’ by P. Buchen and R. Ben-Hador

Abstract

Buchen & Ben-Hador (1996) give a valuable review and unification of some well-known methods for computing the Rayleigh-wave dispersion function of free modes in plane-layered perfectly elastic, isotropic earth models. (They also cover the simpler Love-wave dispersion function, but we leave that aside in this note.) However, when they claim that they provide a ‘complete’ review of the principal methods developed, this is clearly not true. For example, ‘global matrix’ methods are common in underwater acoustics (Jensen et al. 1994) and plate acoustics (Mal, Xu & Bar-Cohen 1989). These methods start from the same kind of ‘global’ matrix as in Knopoff's method, see eq. (22) in Buchen & Ben-Hador (1996), but the determinant is computed in a different way. Elementary row and column operations are carried out to bring the matrix to diagonal form, and the determinant can be obtained as the product of diagonal elements. The problem with loss of numerical precision can be circumvented by special scaling and pivoting, etc., to ensure that the global coefficient matrix becomes block-diagonally dominant (Schmidt & Jensen 1985; Chin, Hedstrom & Thigpen 1984). A particular global matrix method was developed by Karasalo (1994). By eliminating the traction variables, he arrived at a symmetric and well-conditioned global matrix with half the number of unknowns. Block-diagonal dominance for the critical evanescent cases is automatically accomplished. He called his method an exact finite-element method. A general technique to eliminate the problem with numerical loss of precision for standard shooting methods, such as the Thomson-Haskell propagator method, is to orthogonalize the matrix of base solutions at appropriate moments during the propagation (e.g. Conte 1966). This idea was adapted and applied to Rayleigh-wave problems by Evans (1986). To incorporate these methods into the formalism of Buchen & Ben-Hador (1996) and to compare them with the other methods would certainly be of interest, but it is beyond the scope of this note. Instead, we will consider the ‘fast delta matrix algorithm’ proposed by Buchen & Ben-Hador (1996) and point out a more transparent derivation, in the light of Woodhouse (1980), and some variants. In fact, a similar algorithm appears in Woodhouse (1980, p. 150). Buchen & Ben-Hador (1996) apply transformation of variables to arrive at a simpler propagator T~ than the original Ti, see their eq. (44). Using the corresponding delta matrix and taking care to optimize speed by exploiting common factors, they arrive at the ‘fast delta matrix algorithm’ as specified in their Appendix A5. However, the optimization step here is not transparent, and it seems that Buchen & Ben-Hador (1996) prefer to derive their algorithm through an adapted Abo-Zena recursion (Buchen & Ben-Hador 1996, Section 7.1). By utilizing delta-matrix factorization explicitly, as Woodhouse (1980) and Ivansson (1993) did, we can obtain the ‘fast delta matrix algorithm’, as well as some variants, in a direct and convenient way. This will be discussed in Section 2. Finally, we give some comments on the viscoelastic case. In fact, the presentation in most of our references covers the viscoelastic case without giving particular attention to the simplifications (real arithmetic in particular) that may be obtained for the purely elastic case. For the viscoelastic case, Buchen & Ben-Hador (1996) seem to advocate the Kennett RT matrix method. As shown by Ivansson (1993), however, delta-matrix factorization provides an attractive and efficient algorithm for this case too. Furthermore, it is applicable to computations of the full wavefield and not just the dispersion function (Ivansson 1997a,b).