Generalized eigenfunctions of layered elastic media and application to diffuse fields.

The spectral decomposition of the elastic wave operator in a layered isotropic half-space is derived by means of standard functional analysis methods. Particular attention is paid to the coupled P-SV waves. The problem is formulated directly in terms of displacements which leads to a 2 x 2 Sturm-Liouville system. The resolvent kernel (Green's function) is expressed in terms of simple plane-wave solutions. Application of Stone's formula leads naturally to eigenfunction expansions in terms of generalized eigenvectors with oscillatory behavior at infinity. The generalized eigenfunction expansion is employed to define a diffuse field as a white noise process in modal space. By means of a Wigner transform, we calculate vertical to horizontal kinetic energy ratios in layered media, as a function of depth and frequency. Several illustrative examples are considered including energy ratios near a free surface, in the presence of a soft layer. Numerical comparisons between the generalized eigenfunction summation and a classical locked-mode approximation demonstrate the validity of the approach. The impact of the local velocity structure on the energy partitioning of a diffuse field is illustrated.

[1]  A. Paul,et al.  Long-Range Correlations in the Diffuse Seismic Coda , 2003, Science.

[2]  B. V. van Tiggelen,et al.  Green function retrieval and time reversal in a disordered world. , 2003, Physical review letters.

[3]  J. Guillot,et al.  Les ondes élastiques dans un demi-espace isotrope. Développement en fonctions propres généralisées. Principe d'absorption limite , 1985 .

[4]  Allan M. Krall,et al.  M (λ) theory for singular Hamiltonian systems with two singular points , 1989 .

[5]  Calvin H. Wilcox.  Sound Propagation in Stratified Fluids , 1984 .

[6]  D. Pearson,et al.  Quantum scattering and spectral theory , 1988 .

[7]  Richard L. Weaver,et al.  On diffuse waves in solid media , 1982 .

[8]  B. V. van Tiggelen,et al.  Quasi-two-dimensional transfer of elastic waves. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  Richard L. Weaver,et al.  Diffuse fields in open systems and the emergence of the Green’s function (L) , 2004 .

[10]  Luis Fabian Bonilla,et al.  Effects of Surface Geology on Seismic Motion , 2011 .

[11]  V. Edwards.  Scattering Theory , 1973, Nature.

[12]  B. V. van Tiggelen,et al.  Observation of equipartition of seismic waves. , 2001, Physical review letters.

[13]  Richard L. Weaver,et al.  On the emergence of the Green's function in the correlations of a diffuse field: pulse-echo using thermal phonons. , 2001, Ultrasonics.

[14]  Valérie Maupin,et al.  The radiation modes of a vertically varying half-space: a new representation of the complete Green's function in terms of modes , 1996 .

[15]  J. Scales,et al.  Extracting the Green function from diffuse, equipartitioned waves. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  Philippe Sécher.  Étude spectrale du système différentiel 2 x 2 associé à un problème d'élasticité linéaire , 1998 .

[17]  B. Kennett,et al.  Symmetries in the reflection and transmission of elastic waves , 1978 .

[18]  B. Kennett,et al.  Synthetic reflection seismograms in three dimensions by a locked-mode approximation , 1989 .

[19]  J. Guillot,et al.  Scattering of elastic waves in a perturbed isotropic half space with a free boundary. The limiting absorption principle , 1988 .

[20]  Diffuse waves in finite plates , 1984 .

[21]  B. Kennett,et al.  Seismic Wave Propagation in Stratified Media , 1983 .

[22]  R. Weaver.  Diffuse elastic waves at a free surface , 1985 .

[23]  P. Sheng,et al.  Introduction to Wave Scattering, Localization and Mesoscopic Phenomena. Second edition , 1995 .

[24]  Yves Colin de Verdiere,et al.  Mathematical models for passive imaging I: general background , 2006 .

[25]  R. Weaver,et al.  Ultrasonics without a source: thermal fluctuation correlations at MHz frequencies. , 2001, Physical review letters.