Robert Burridge

Poroelasticity equations derived from microstructure

Equations are derived which govern the linear macroscopic mechanical behavior of a porous elastic solid saturated with a compressible viscous fluid. The derivation is based on the equations of linear elasticity in the solid, the linearized Navier–Stokes equations in the fluid, and appropriate conditions at the solid–fluid boundary. The scale of the pores is assumed to be small compared to the macroscopic scale, so that the two‐space method of homogenization can be used to deduce the macroscopic equations. When the dimensionless viscosity of the fluid is small, the resulting equations are those of Biot, who obtained them by hypothesizing the form of the macroscopic constitutive relations. The present derivation verifies those relations, and shows how the coefficients in them can be calculated, in principle, from the microstructure. When the dimensionless viscosity is of order one, a different equation is obtained, which is that of a viscoelastic solid.

Linearized inverse scattering problems in acoustics and elasticity

Abstract Using the single-scattering approximation we invert for the material parameters of an acoustic two-parameter medium and then for a three-parameter isotropic elastic medium. Our procedure is related to various methods of depth migration in seismics, i.e. methods for locating major discontinuities in the subsurface material without specifying which quantities are discontinuous or by how much they jump. Our asymptotic multiparameter inversion makes use of amplitude information to reconstruct the size of the jumps in the parameters describing the medium. We allow spatially varying background parameters (both vertically and laterally) and an almost arbitrary source-receiver configuration. The computation is performed in the time domain and we use all available data even if it is redundant. This ability to incorporate the redundant information in a natural way is based upon a formula for double integrals over spheres. We solve for perturbations in different parameters treating separately P -to- P , P -to- S , S -to- P , and S -to- S data. It turns out that one may invert using subsets of the data, or all of it together. We also describe modifications to our scheme which allow us to use the Kirchhoff instead of the Born approximation for the forward problem when the scatterers are smooth surfaces of discontinuity.

The Gelfand-Levitan, the Marchenko, and the Gopinath-Sondhi integral equations of inverse scattering theory, regarded in the context of inverse impulse-response problems

In this paper, we are concerned with interpreting some classical inverse-scattering theories so that they are relevant to the one-dimensional inverse impulse-response problem which arises in reflection seismology First the Gelfand-Levitan integral equations (which arise in the inverse scattering theory for the Schrodinger equation) are derived strictly in the time domain. Originally these celebrated equations were derived as a means of solving an inverse spectral problem, which is naturally posed in the frequency domain. We show not only that these equations have a time-dependent formulation, but that their derivation is actually simpler here than in its original context. Next we give a similar derivation for the Marchenko integral equation. We then obtain, by an independent method, the integral equation of Gopinath and Sondhi, which is not unlike the Gelfand-Levitan linear equation. It was used by them as a means of solving a time-dependent inverse problem arising in speech synthesis. A new integral equation, similar to the Marchenko equation is also derived. Finally the integral equations are related to each other by direct transformation independent of the corresponding differential equations. The present paper opens with a section in which the equation of one-dimensional elastic waves and the corresponding seismic inverse impulse-response problem are transformed into a form to which the Gelfand-Levitan theory applies, and then into the equations which arose in Gopinath and Sondhis work. We regard the integral equation of Gopinath and Sondhi as being directly applicable to the interpretation of seismic reflection data. The Gelfand-Levitan theory is also applicable but only after considerable transformation. Our calculations are entirely in the time domain. The resulting equations have the merit that in order to recover the unknown coefficient on a finite interval (0, L) we need to use the boundary data also only on a finite segment of the reflection time series. The length of the record is just the two-way travel time associated with length L. By contrast, frequency-domain approaches require that long records be Fourier transformed. We distinguish results which are true only when the unknown coefficient is continuous from those true when it is bounded but only piecewise continuous. In this work we have not addressed the important problem of deconvolution.

Horizontal ray theory for ocean acoustics

The solution of the reduced scalar wave equation for an almost stratified medium is written in the form of an asymptotic power series. The vertical structure of the solution is expressed as a linear combination of the normal‐mode eigenfunctions whose coefficients satisfy two‐dimensional eikonal and transport equations. Although smooth caustics may be treated for the two‐dimensional scalar wave equation, the problem of a uniform approximation near a point source has not yet been resolved. Finally, the theory is applied to acoustic propagation in a realistic model ocean and the results are compared with measurements.The solution of the reduced scalar wave equation for an almost stratified medium is written in the form of an asymptotic power series. The vertical structure of the solution is expressed as a linear combination of the normal‐mode eigenfunctions whose coefficients satisfy two‐dimensional eikonal and transport equations. Although smooth caustics may be treated for the two‐dimensional scalar wave equation, the problem of a uniform approximation near a point source has not yet been resolved. Finally, the theory is applied to acoustic propagation in a realistic model ocean and the results are compared with measurements.

Eulerian Gaussian beams for high-frequency wave propagation

We design an Eulerian Gaussian beam summation method for solving Helmholtz equations in the high-frequency regime. The traditional Gaussian beam summation method is based on Lagrangian ray tracing and local ray-centered coordinates. We propose a new Eulerian formulation of Gaussian beam theory which adopts global Cartesian coordinates, level sets, and Liouville equations, yielding uniformly distributed Eulerian traveltimes and amplitudes in phase space simultaneously for multiple sources. The time harmonic wavefield can be constructed by summing up Gaussian beams with ingredients provided by the new Eulerian formulation. The conventional Gaussian beam summation method can be derived from the proposed method. There are three advantages of the new method: (1) We have uniform resolution of ray distribution. (2) We can obtain wavefields excited at different sources by varying only source locations in the summation formula. (3) We can obtain wavefields excited at different frequencies by varying only frequencies in the summation formula. Numerical experiments indicate that the Gaussian beam summation method yields accurate asymptotic wavefields even at caustics. The new method may be used for seismic modeling and migration.

A method to detect and characterize ellipses using the Hough transform

We describe a new technique for detecting and characterizing ellipsoidal shapes automatically from any type of image. This technique is a single pass algorithm which can extract any group of ellipse parameters or characteristics which can be computed from those parameters without having to detect all five parameters for each ellipsoidal shape. Moreover, the method can explicitly incorporate any a priori knowledge the user may have concerning ellipse parameters. The method is based on techniques from projective geometry and on the Hough transform. This technique can significantly reduce interpretation and computation time by automatically extracting only those features or geometric parameters of interest from images and making exact use of a priori information.

The resolving power of seismic amplitude data : An anisotropic inversion/migration approach

A description of the theory and numerical implementation of a 3-D linearized asymptotic anisotropic inversion method based on the generalized Radon transform is given. We discuss implementation aspects, including (1) the use of various coordinate systems, (2) regularization by both spectral and Bayesian statistical techniques, and (3) the effects of limited acquisition apertures on inversion. We give applications of the theory in which well-resolved parameter combinations are determined for particular experimental geometries and illustrate the interdependence of parameter and spatial resolutions. Procedures for evaluating uncertainties in the parameter estimates that result from the inversion are derived and demonstrated.

Probing a random medium with a pulse

This paper studies the reflection of pulses from a randomly layered half space. It characterizes the statistical properties of the reflected signals at the surface in a suitable asymptotic limit in...

The Sitar String, a Vibrating String with a One-Sided Inelastic Constraint

The small transverse motion of a stretched string vibrating against a rigid, inelastic curved obstacle is calculated. This system models the vibration of the strings of certain Indian musical instruments, the sitar, the tanpura and the viva, where the bridge does not have a sharp edge but is smooth and forms a curved impenetrable “obstacle” around which the string wraps and unwraps during its vibration.The complete motion of the string plucked at its midpoint is calculated in closed form from the instant it is released to its asymptotic approach to equilibrium. A similarity is pointed out between the solution obtained here and the solution obtained by Helmholtz for the bowed violin string. The solution is illustrated graphically.

Statistics for pulse reflection from a randomly layered medium

We consider reflection of a pulse incident on a layered halfspace whose density and bulk modulus vary randomly. We show that when the pulse width is long compared to the average time it takes to travel over one correlation length, the reflected signal is approximately a Gaussian random process. The parameters of this process change slowly on a scale long compared to the pulse width. We give a full characterization of the power spectrum of the Gaussian process in terms of a universal function that does not depend on the medium.