MATHICSE Technical Report : Analysis and computation of the elastic wave equation with random coefficients

We analyze the stochastic initial-boundary value problem for the elastic wave equation with random coefficients and deterministic data. We propose a stochastic collocation method for computing statistical moments of the solution or statistics of some given quantities of interest. We study the convergence rate of the error in the stochastic collocation method. In particular, we show that, the rate of convergence depends on the regularity of the solution or the quantity of interest in the stochastic space, which is in turn related to the regularity of the deterministic data in the physical space and the type of the quantity of interest. We demonstrate that a fast rate of convergence is possible in two cases: for the elastic wave solutions with high regular data; and for some high regular quantities of interest even in the presence of low regular data. We perform numerical examples, including a simplified earthquake, which confirm the analysis and show that the collocation method is a valid alternative to the more traditionalMonte Carlo sampling method for problems with high stochastic regularity

Seismic Ray Impedance Inversion

This thesis investigates a prestack seismic inversion scheme implemented in the ray parameter domain. Conventionally, most prestack seismic inversion methods are performed in the incidence angle domain. However, inversion using the concept of ray impedance, as it honours ray path variation following the elastic parameter variation according to Snell’s law, shows the capacity to discriminate different lithologies if compared to conventional elastic impedance inversion. The procedure starts with data transformation into the ray-parameter domain and then implements the ray impedance inversion along constant ray-parameter profiles. With different constant-ray-parameter profiles, mixed-phase wavelets are initially estimated based on the high-order statistics of the data and further refined after a proper well-to-seismic tie. With the estimated wavelets ready, a Cauchy inversion method is used to invert for seismic reflectivity sequences, aiming at recovering seismic reflectivity sequences for blocky impedance inversion. The impedance inversion from reflectivity sequences adopts a standard generalised linear inversion scheme, whose results are utilised to identify rock properties and facilitate quantitative interpretation. It has also been demonstrated that we can further invert elastic parameters from ray impedance values, without eliminating an extra density term or introducing a Gardner’s relation to absorb this term. Ray impedance inversion is extended to P-S converted waves by introducing the definition of converted-wave ray impedance. This quantity shows some advantages in connecting prestack converted wave data with well logs, if compared with the shearwave elastic impedance derived from the Aki and Richards approximation to the Zoeppritz equations. An analysis of P-P and P-S wave data under the framework of ray impedance is conducted through a real multicomponent dataset, which can reduce the uncertainty in lithology identification.Inversion is the key method in generating those examples throughout the entire thesis as we believe it can render robust solutions to geophysical problems. Apart from the reflectivity sequence, ray impedance and elastic parameter inversion mentioned above, inversion methods are also adopted in transforming the prestack data from the offset domain to the ray-parameter domain, mixed-phase wavelet estimation, as well as the registration of P-P and P-S waves for the joint analysis. The ray impedance inversion methods are successfully applied to different types of datasets. In each individual step to achieving the ray impedance inversion, advantages, disadvantages as well as limitations of the algorithms adopted are detailed. As a conclusion, the ray impedance related analyses demonstrated in this thesis are highly competent compared with the classical elastic impedance methods and the author would like to recommend it for a wider application

Stochastic band structure for waves propagating in periodic media or along waveguides

We introduce the stochastic band structure, a method giving the dispersion relation for waves propagating in periodic media or along waveguides, and subject to material loss or radiation damping. Instead of considering an explicit or implicit functional relation between frequency $\omega$ and wavenumber $k$, as is usually done, we consider a mapping of the resolvent set in the dispersion space $(\omega, k)$. Bands appear as as the trace of Lorentzian responses containing local information on propagation loss both in time and space domains. For illustration purposes, the method is applied to a lossy sonic crystal, a radiating surface phononic crystal, and a radiating optical waveguide. The stochastic band structure can be obtained for any system described by a time-harmonic wave equation

Solving seismic wave propagation in elastic media using the matrix exponential approach

Three numerical algorithms are proposed to solve the time-dependent elastodynamic equations in elastic solids. All algorithms are based on approximating the solution of the equations, which can be written as a matrix exponential. By approximating the matrix exponential with a product formula, an unconditionally stable algorithm is derived that conserves the total elastic energy density. By expanding the matrix exponential in Chebyshev polynomials for a specific time instance, a so-called ``one-step'' algorithm is constructed that is very accurate with respect to the time integration. By formulating the conventional velocity-stress finite-difference time-domain algorithm (VS-FDTD) in matrix exponential form, the staggered-in-time nature can be removed by a small modification, and higher order in time algorithms can be easily derived. For two different seismic events the accuracy of the algorithms is studied and compared with the result obtained by using the conventional VS-FDTD algorithm.Comment: 13 pages revtex, 6 figures, 2 table

The Commonality of Earthquake and Wind Analysis

Earthquakes and wind loadings constitute dynamic effects that often must be considered in the design of buildings and structures. The primary purpose of this research study was to investigate the common features of general dynamic analysis procedures employed for evaluating the effects of wind and earthquake excitation. Another major goal was to investigate and develop a basis for generating response spectra for wind loading, which in turn would permit the use of modal analysis techniques for wind analysis in a manner similar to that employed for earthquake engineering. In order to generate wind response spectra, the wind loading is divided into two parts, a mean load treated as a static component and a fluctuating load treated as a dynamic component. The spectral representation of the wind loading constitutes a simple procedure for estimating the forces associated with the dynamic component of the gusting wind. Several illustrative examples are presented demonstrating the commonality.National Science Foundation Grants ENV 75-08456 and ENV 77-0719