For microhomogeneous media with inclusions the problem of construction of macroscopic equations is solved using diagram technique. Distribution of inclusions in a medium is considered as an arbitrary stochastic tensor field of macroscopic elastic modules, which are defined by correlation functions of an arbitrary order. The exact expression of an effective tensor of elastic modules is deduced using mass operator. The calculation of Green functions, mass operator and effective elastic tensor for media with inclusions is carried out in a frame of correlation approximation. The existence of elastic anisotropy due to an order in location of pores in a space is shown. Numerical estimations of influence of space distribution of inclusions on a coefficient of effective elastic anisotropy are obtained.

We introduce a theoretical model describing wave propagation in layered media. This theory relates medium properties (i.e. P- and S-wave velocities and density) to travel time delay and pulse attenuation. Approximation schemes are based on the assumption that the change of medium parameters is not large. If changes are large, then higher order approximations can be obtained by successive iterations. Several numerical examples show the effects of target-zone scattering. The main purpose of this work is to provide an accurate understanding of seismic amplitudes in realistic geological models. It is hoped that these results will lead to more robust seismic inversion analysis.

We review recent work on paraxial equation based migration methods for 3D heterogeneous media. Two different methods are presented: one deals directly with the classical paraxial equations, by solving a linear system at each step in depth. The other method derives new paraxial equations that lend themselves to splitting in the lateral variables, without losing either accuracy or isotropy. We also show how to incorporate Berenger's perfectly matched layers in this framework. We detail the discretization schemes, both for the full paraxial equations, and for the newly derived equations.

We introduce a migration algorithm based on paraxial wave equation that does not use any splitting in the lateral variables. The discretization is first derived in the constant coefficient case by higher order finite differences, then generalized to arbitrarily varying velocities via finite elements. We present a detailed plane wave analysis in a homogeneous medium, and give evidence that numerical dispersion and anisotropy can be controlled. Propagation along depth is done with a higher order method based on a conservative Runge Kutta method. At each step in depth we have to solve a large linear system. This is the most time consuming part of the method. The key to obtaining good performance lies in the use of a Conjugate Gradient like iterative solver. We show the performance of the method with a model example.

The quasi-random migration (QM) method is tested on 3D seismic data from West Texas. Results show that the QM images, at 1/4 subsampling of the entire data set, are about the same quality as regularly migrated (RM) images obtained from a regular 1/4 subsampling of the original data. However, it appears that there are fewer artifacts in the QM images than in the regular images. At 1/8 subsampling the QM images have moderately better quality than the corresponding RM images. Previous studies suggested that the QM images should be of much higher quality than the regularly sampled images. Our field test results did not show this to be true because the source and receiver lines were spaced more than 1300 feet apart, and so did not allow for a good quasi-random sampling of the data.

We consider a family of inverse problems, which include well-known problems of diffraction tomography. The aim of this paper is to present the method of sequential projections on image taking into account the specific character of these problems and to present the uniqueness criterion.

In this paper, we present a new approach for travel-time tomography in which ray tracing is not required. Analytical expressions for the travel-time between any two points and the slowness in a 2D inhomogeneous medium are obtained by minimizing the difference between the analytical travel-time function and observed travel-times. The analytical functions satisfy the Eikonal equation everywhere in the medium, and hence provide a unique solution for the slowness. The computation of travel-times using the analytical function can be much faster than that using ray tracing for pre-stack migration. Furthermore, the computation can be carried out during the migration which will avoid storing of the travel-time field and increase the efficiency of pre-stack migration.

I derive the formulas for the seismic response of a line scatterer for the crosswell migration and traveltime tomography operators. These formulas are used to estimate the limits of spatial resolution in reflectivity images obtained from migration and slowness images reconstructed from traveltime tomography. In particular, for a crosswell geometry with borehole length L, well offset 2x_o, source wavelength (lambda) , and a centered line scatterer I show that: (1) The vertical resolution (Delta) _z^mig of the migration image is equal to 2(lambda) x_o/L under the far-field approximation. (2) The horizontal resolution (Delta) _x^mig of the migration image is equal to 16(lambda) x²_o/L². The lateral resolution of the migrated image is worse than the vertical resolution by the factor 8x_o/L (where x_o/L > 1 under the far-field approximation). (3) For inverting traveltimes associated with a localized slowness perturbation midway between the wells, the vertical resolution (Delta) _x^tomo of the slowness tomogram is proportional to (root)(lambda) x_o. This estimate agrees with that of a previous study. (4) The horizontal resolution of the slowness image in a traveltime tomogram is equal to [4x_o/L](root)3x_o(lambda) /4, a factor 4(root)3x_o/L worse than the vertical resolution. (5) For N_s and N_g geophones, the dynamic range of the migrated image is proportional to N_sN_g. The dynamic range of the slowness tomogram is proportional to (root)N_gN_s.

This paper is concerned with the linearized inversion of elastic wave data using the Generalized Radon Transformation to give anisotropic medium parameters. Assumptions of linearity are examined by comparing linearized reflection coefficients calculated using the Born approximation with full plane-wave reflection coefficients. In typical sand/shale models we have found that the linearity assumption is valid only to approximately 60 degrees from the normal. Linear dependencies between the scattering patterns produced by individual moduli result in an ill-posed inverse problem. Utilizing P-wave data, we find that for the Transversely Isotropic case similarities in the C₅₅ and C₁₃ scattering directivity mean that they cannot be distinguished. C₁₁ is best observed at wide-angle and hence estimates made using limited aperture data are subject to large error. Quasi-Monte Carlo techniques are adapted to carry out the 4D inversion integral.

We present a numerical method of computing asymptotic Green's functions for applications in linear inversion of seismic reflection data. The method is based on tracing a density sampled ray field, which is locally interpolated for ray field parameters. We propose a uniformly valid asymptotic formula relating the wave field to the ray field parameters. A few test bases and an application to the linearized inversion of seismic reflection data are presented.

The objective of seismic inversion is to estimate Earth parameters, such as velocity and density, from seismic data. The inversion scheme (algorithm) depends on a physical model of wave-propagation/reflection, which often is linearized. Such a linearized inversion scheme is highly dependent on a fixed reference model, i.e. propagation velocity of and amount of attenuation in the medium. Hence, it is necessary to investigate the reliability of linearized inversion results when the reference model is incorrect. To investigate sensitivity of seismic linearized inversion to attenuation (Q), we set up a very simple example of linearized reflection from a plane in an attenuating medium. The simplicity of the model allows the study of singular (eigen) values, eigenvectors of the physical model, and error sensitivity of the inversion results with easily understandable results. The model should yield at least qualitative results, even though it naturally is a substantial simplification of a realistic seismic inverse problem. The analysis shows that exact amplitudes of parameter estimates are highly dependent on an accurate reference attenuation (Q), whereas ratios between different parameters (AVO results) can be estimated with roughly 20 percent error in attenuation (Q).

Artificial neural networks are becoming increasingly popular as a method for parameter classification and as a tool for recognizing complex relationships in a variety of data types. The power of neural networks comes from their ability to `learn' from a set of training data and then to generalized to new data sets. In addition, neural networks have the potential to assimilate data over a wide range of scales and are robust in the presence of noise. A back propagation neural network has been successfully applied in predicting paleosol sequences using well log suites from two wells in a Cenozoic basin in southwestern Montana. The training set consists of neutron porosity, bulk density, and resistivity logs and the interpreted paleosol section. Training is accomplished using well log values over a range of depths rather than discrete depths. The trained network is used to predict paleosol occurrences in a neighboring well. Network prediction results show good agreement with paleosol interpretations. Neural networks are also being applied as part of a reservoir characterization project for a north central Montana oil field. A back propagation network shows limited success in predicting sonic and porosity well logs from resistivity, gamma, and density logs. Preliminary attempts to predict spatial distribution of porosity from 3D seismic data show some promise. The use of high-pass filtered seismic data and seismic trace attributes improves prediction error over the unfiltered seismic data alone.

Determination of the long wavelength, or background, velocity structure is a critical step in prestack imaging. We have developed an efficient procedure for the solution of this non-linear inverse problem using a genetic algorithm (GA). In our approach velocities are described by splines; velocity values at spline nodes are the parameters in the inversion. We distinguish between primary nodes, affecting very long wavelength variation, and secondary nodes, describing more rapid change. The GA evolves a population of trial solutions, seeking out the globally fittest velocity model. Key to the method is the evaluation of fitness, which is carried out in three steps: (1) map migration of zero-offset traveltimes through the trial velocity model to identify the approximation locations of primary reflectors; (2) prestack Kirchhoff depth migration to generate image gathers in narrow depth windows centered on the predicted reflector locations; (3) calculation of horizontal semblance within the image gathers. For the correct velocity model, reflection events appear flat in each of the gathers; thus, by calculating horizontal semblance we obtain a measure of fitness for the GA. Since the migration is performed over a narrow depth range in the neighborhood of a given reflector, rays need only be traced from a small number of depth points for each gather; additional ray information can be efficiently obtained using paraxial approximations.

Much of the wavelet literature is focussed on wavelets with compact support in the time domain. For many geophysical applications, compact support in the frequency domain is desirable. For these applications, simple window functions can be used to construct appropriate filter banks in the frequency domain. Convolution with filter coefficients in the time domain is replaced with a Fourier transform and multiplication by window functions in the frequency domain. Given the dual nature of the Fourier transform, the time and frequency variables can be exchanged to produce a time windowing algorithm for computing wave packet transforms. Taken together, frequency-windowed and time-windowed wave packet transforms provide a comprehensive tool set for constructing new geophysical applications that take advantage of simultaneous access to time and frequency. Depending on the application, frequency windowing or time windowing may be more desirable.

In this paper a novel method for solution of the acoustic wave equation is presented. This method achieves the computational efficiency of the explicit methods while also offering the excellent numerical properties of the implicit methods, i.e., the unconditional stability. We discuss the mathematical foundation of this method as well as various numerical aspects such as inclusion of the absorbing boundary conditions and efficient parallel implementations.

Today it is quite common to find in academic and research environments as installation of heterogeneous computers connected via ethernet. In this paper we present a very cost-efficient solution for the parallel implementation of seismic algorithms in a network of heterogeneous workstations using PVM (Parallel Virtual Machine). Two algorithms were analyzed and implemented using different implementation approaches: (1) A minimum time ray-tracer based on the Fermat's principle operating on irregular grids and (2) An algorithm that calculates numerical solutions to the eikonical equations and computes the travel times for the first arrivals of seismic waves on a 2D grid by finite difference extrapolation.

Given a 3D seismic record for an arbitrary measurement configuration and assuming a laterally inhomogeneous, isotropic macro-velocity model, a unifying approach to amplitude- preserving seismic reflection imaging is provided. It consists of (a) a Kirchhoff-type weighted diffraction stack to transform (migrate) the seismic data from the (time-domain) record space into the (depth-domain) image space, and of (b) a weighted isochrone stack to transform (demigrate) the migrated seismic image from the image space back into the record space. Both the diffraction and isochrone stacks can be applied in sequence for different measurement configurations, velocity models, or elementary waves to permit a variety of amplitude- preserving image transformations. These include, e.g., (a) the amplitude-preserving transformation of a 3D constant-offset record into a 3D zero-offset record, which is known as a migration to zero offset, (b) a dip-moveout correction, or (c) the transformation (here referred to as a remigration) of a 3D depth-migrated image directly in the image space into another one for a different macro-velocity model. By analytically chaining the two stacking integrals, each image transformation can be achieved with only one single weighted stack.

A fast modeling method based on multiple-forescattering single-backscattering (MFSB) approximation, i.e. the De Wolf approximation for calculating reflected (or backscattered) wave fields in 3D heterogeneous acoustic media is introduced. The method is much faster than full wave finite difference or finite element methods. The formulation is especially suitable for the configuration of surface reflection surveying. When discontinuities in a medium are not very sharp or parameter perturbations of heterogeneities are not very strong, reverberations between heterogeneities or resonance scattering can be neglected. However, for large volume heterogeneous media or long propagation distances the accumulated effect of multiple forward scattering becomes very important for both forward modeling and inverse problems. In such cases, the Born approximation is not valid while the De Wolf approximation can be applied. After renormalizing the multiple scattering series of the Lipmann-Schwinger equation, a MFSB approximation for acoustic waves is derived and a fast dual-domain computation scheme is presented, in which the multi-screen one-way wave propagator is used. Finally numerical examples are given to demonstrate the validity of the method.

The deformable porous media possess four distinct seismic wave processes, namely, fast- and slow-waves each associated with compression and shear deformations. The fast compressional (shear) wave is essentially motion of a porous medium as a whole such that constituent phases undergo volume (shape) change in the same manner. The linear momentum fluxes are associated with these motions and they describe transport of translational kinetic energy. The slow compressional (shear) wave is basically motions of constituent phases undergoing volume (shape) in equal but opposite manner such that the medium as a whole is at rest. This new mode of deformation amounts to existence of an intrinsic angular momentum (spin). The slow waves are basically spin fluxes and they describe transport of rotational kinetic energy.