Evolution of the propagator matrix method and its implementation in seismology

In this paper, we review development of an algorithm that is referred to in seismology as the Haskell matrix method, the Thomson-Haskell matrix method, or the propagator matrix method. The roots of this algorithm and main developments are examined to offer a better understanding of its essential features. The underlying theory is highlighted by removing specific expressions and manipulations that often shroud the common method involved. Also, I discuss implementations of the algorithm in Python, with a reference to source code. These implementations calculate dispersion curves for guided waves.

[1]  Frederick A. Leckie,et al.  Matrix Methods in Elasto Mechanics , 1963 .

[2]  W. Thomson,et al.  Transmission of Elastic Waves through a Stratified Solid Medium , 1950 .

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

[4]  L. ANDEKSONf.  LOVE WAVE DISPERSION IN HETEROGENEOUS ANISOTROPIC MEDIA , 2001 .

[5]  Ronen Ben-Hador,et al.  Free-mode surface-wave computations , 1996 .

[6]  Katsutada Sezawa,et al.  Dispersion of Elastic Waves propagated on the Surface of Stratified Bodies and on Curved Surfaces , 1927 .

[7]  Stephen P. Timoshenko,et al.  Vibration problems in engineering , 1928 .

[8]  Michael A. Slawinski,et al.  Sensitivity of Love and quasi-Rayleigh waves to model parameters , 2017 .

[9]  W. Thomson.  Vibration Theory and Applications , 1966 .

[10]  T. Matsuoka,et al.  Computation of Rayleigh Waves on Transversely Isotropic Media by the Reduced Delta Matrix Method , 2013 .

[11]  C. H. Chapman,et al.  Yet another elastic plane-wave, layer-matrix algorithm , 2003 .

[12]  M. Ewing,et al.  Study of shear-velocity distribution in the upper mantle by mantle Rayleigh waves , 1960 .

[13]  J. W. Dunkin,et al.  Computation of modal solutions in layered, elastic media at high frequencies , 1965 .

[14]  Agustín Udías,et al.  Principles of Seismology , 2000 .

[15]  M. Thompson,et al.  Modified Thomson–Haskell Matrix Methods for Surface-Wave Dispersion-Curve Calculation and Their Accelerated Root-Searching Schemes , 2011 .

[16]  Freeman Gilbert,et al.  Propagator matrices in elastic wave and vibration problems , 1966 .

[17]  N. A. Haskell.  The Dispersion of Surface Waves on Multilayered Media , 1953 .

[18]  D. L. Anderson,et al.  Computation of surface wave dispersion for multilayered anisotropic media , 1962 .

[19]  David G. Harkrider,et al.  Surface waves in multilayered elastic media. I. Rayleigh and Love waves from buried sources in a multilayered elastic half-space , 1964 .

[20]  Lisandro Dalcin,et al.  Parallel distributed computing using Python , 2011 .

[21]  Frank Press,et al.  A fast, convenient program for computation of surface-wave dispersion curves in multilayered media , 1961 .

[22]  W. T. Thomson.  Matrix Solution of the N-Section Column , 1949 .

[23]  N. O. Myklestad,et al.  Fundamentals of Vibration Analysis , 1956 .

[24]  N. Frazer.  Seismic Wave Propagation in Stratified Media , 1987 .

[25]  Don L. Anderson,et al.  Elastic wave propagation in layered anisotropic media , 1961 .

[26]  N. A. Haskell.  Radiation pattern of surface waves from point sources in a multi-layered medium , 1964 .

[27]  J. Hudson.  A Quantitative Evaluation of Seismic Signals at Teleseismic Distances—I Radiation from Point Sources , 1969 .