An explicit procedure is presented for computing both model and data resolution matrices within a Paige-Saunders LSQR algorithm for iterative inversion in seismic tomography. These methods are designed to avoid the need for an additional singular value decomposition of the ray-path matrix. The techniques discussed are completely general since they are based on the multiplicity of equivalent exact formulas that may be used to define the resolution matrix. Thus, resolution matrices may also be computed for a wide variety of iterative inversion algorithms using the same ideas.

Transmission caustics cause velocity estimation techniques based on straightforward analysis of image volume moveout to fail. For example, the differential semblance objective function is smooth and quasi-convex when the incident wavefield is free of caustics, but loses these properties when caustics are present. The reason for this phenomenon is closely related to that identified by Geoltrain and Brac for the inadequacy of Kirchhoff migration based on first arrival times for complex velocity structure. controlled illumination of the subsurface, introduced by Rietvold and Berkhout to improve imaging, given an adequate macromodel, also suggests a new approach to image volume moveout analysis for macromodel estimation. In particular, the differential semblance objective function again becomes, through controlled illumination, a smooth quasi-convex function of the macromodel, hence suitable for optimization using variants of Newton's method. The transmission inversion problem provides a simple framework within which to illustrate the controlled illumination reformulation of differential semblance and its effectiveness.

In modern high frequency inversion of acoustic scattering data, standard ray theoretic methods play an important role, since the inversion operation is usually formulated in terms of traveltimes and amplitude functions computed along the rays. In this approach one formulates the inversion procedure in terms of a Fourier-integral operator acting on the scattering data. It is well known that this Fourier-integral operator reproduces the most singular part of the scattering potential, provided that there are no multiple ray paths between any two points in the medium. This rules out many practical situations in which caustics in the ray field are present. We show that under a relatively mild condition the inversion technique mentioned above can be extended to allow for caustics. We also present an explicit inversion formula.

We apply the elastic wave equation traveltime and waveform inversion (elastic WTW) method to the McElroy crosshole data. These data are characterized by a well offset of 184 ft and a wide-band source wavelet (250 - 2000 Hz). Numerical tests indicate that the following processing steps are necessary, but not sufficient, for successful waveform inversion: FK filtering of the upgoing and downgoing waves, FK-fan filter extraction of the PP and SS reflections, and balancing the amplitudes of the upgoing and downgoing reflections. To reduce the complexity of the data we follow the divide and conquer strategy, i.e., extract the PP reflections and SS reflections, then invert each wave mode separately. Numerical results show that the vertical spatial resolution of the WTW P-wave and S-wave tomograms are approximately 7 - 10 feet and 4 feet, respectively. This compares favorably to the 40 - 50 feet vertical resolution of the P-velocity tomogram obtained from the first arrival traveltime data. There is good to very good agreement between the sonic logs and the velocity from the WTW tomogram. These preliminary results demonstrate that high resolution Poisson ratio, S- velocity, and P-velocity tomograms can be extracted from crosshole data, and therefore can be used for lithological interpretation.

Data produced by a reproducible source contains redundant information which allows seismic inversion to simultaneously determine the high-frequency fluctuation in the p-wave velocity (or reflectivity) as well as the input energy source. The seismogram model is the plane-wave convolutional model derived from the constant density, variable sound velocity acoustic wave equation. The first step is to analyze this linearized model when the background velocity is constant. Then perturbations in the seismic data stably determine corresponding perturbations in the source and reflectivity. The stability of this determination improves as the slowness aperture over which the data is defined increases. Further, the normal operator for the convolutional seismogram model is continuous with respect to velocity. Thus the stability result for constant background velocities may be extended to more realistic background velocity models which vary slowly and smoothly with depth. The theory above is illustrated with four synthetic numerical examples derived from marine data. The examples indicate that for a wide slowness aperture, inversion is very effective in establishing the true shape of the reflectivity and the shape and location of the compactly supported energy source. As this aperture window narrows, the corresponding inversion-estimated model still describes the data quite accurately, but the inversion is not able to recover the original two distinct parameters.

Problems of great economical interest as improving recovery rate of hydrocarbon reservoirs, optimizing gas storage fields or controlling underground pollution, require a very detailed knowledge of porous media, and, in particular, of the spatial variations of their hydraulic properties. To describe the hydrocarbon reservoirs, geological models using statistical concepts are used more and more. They bring a new horizon for reservoir engineering studies. This article discusses some of the questions raised by the introduction of these geological models, and a methodology is proposed to account for the heterogeneities in the reservoir production. A link between the detailed reservoir images generated by the probabilistic geological models, and the flow simulators is established through the selection of these images, and the averaging of the petrophysical data. This approach is illustrated by an example using a specific software.

Time-lapse 3D seismic monitoring of subsurface rock property changes incurred during reservoir fluid-flow processes is an emerging new diagnostic technology for optimizing hydrocarbon production. I discuss the physical theory relevant for three-phase fluid flow in a producing oil reservoir, and rock physics transformations of fluid-flow pressure, temperature and pore-fluid saturation values to seismic P-wave and S-wave velocity. I link fluid-flow physical parameters to seismic reflection data amplitudes and traveltimes through elastic wave equation modeling and imaging theory. I demonstrate in a simulated data example that changes in fluid-flow can be monitored and imaged from repeated seismic surveys acquired at varying production calendar times.

Born and Kirchhoff approximations are frequently used in seismic imaging and have different assumptions. A Born and a Kirchhoff prestack depth migration/inversion techniques formulated in the space-time domain are briefly reviewed. These are used and compared on a field data collected above a hydrocarbon reservoir. Synthetic tests although interesting could not provide the answer to the question: can the physics (measurements) distinguish between these two approximations? Multiparameter images (relative changes in P-wave and S-wave impedances) produced by these two m/i techniques are presented along with their differences. Forward modeling with a calibrated ray-Born modeling technique is performed on resulting images from the Born and Kirchhoff m/i. Synthetics are compared to the field data. A residual energy function computed as a function of shot point number appraises which approximation represents best the real data. Although differences are small, results show that for this real data case the Kirchhoff approximation is more physical than the Born approximation.

We present a new moment method of solving the acoustical scattering equation, and then apply this method to non-linear wave equation tomography. We describe the formulation, the implementation, and numerical testing of the method. The main characteristic of this method is that a bilinear basis function is used instead of a pulse basis function to evaluate the Green function, total field, and scattering potential at any arbitrary point of the image region. In this way, the integral equation may be discretized to arbitrary fineness in order to increase the accuracy of the computations. From simulation tests, we find that this method is accurate and the number of unknowns can be greatly reduced. Finally, we utilize this method in solving a non-linear wave equation inverse problem. The simulation results show that this method is effective and very useful for both forward and inverse problems.

Reflection data consists of primary and multiply reflected events. Since most methods for estimating target properties from reflection data assume a linear model, these multiply reflected events often masquerade as, or interfere with, primaries. although there are many circumstances when current methods for removing multiples can be effective, there remain many important cases where state-of-the-art techniques are either suboptimal or fail. We present a multidimensional multiple suppression method for surface and internal multiples that doesn't depend on periodicity, differential moveout, or a model of the reflectors that generate the multiples. The method derives from separating an inverse scattering series into subseries that carry out specific tasks in the inversion process. Examples will be presented to illustrate the technique.

This presentation will demonstrate the use of image processing technology applied to seismic interpretation. Two sets of tools are available for this purpose: (1) Processes to improve the perception of geological attributes of the seismic image. (2) Processes to extract, analyze and quantify information to construct geological models. These tools reproduce the intuitive work performed by the interpreter during the elaboration of a geological model (visualization, analysis, recognition and extraction of information). Our objective is to facilitate routine tasks, and not to replace the interpreter in his specific functions. It must be noted however that the distinction between routine and special tasks is constantly evolving. In an operational environment, three criteria must be met for these tools to be accepted: reliability, speed of execution and cost effectiveness.

We investigate the inversion of seismic data for the recovery of combinations of elastic parameters. The theory is based on a single scattering and a high-frequency (stationary phase) approximation and maps the most singular part of the wave field onto the most singular part of the medium within this approximation. In this framework, it is assumed that the smooth constituents of the medium are known. Three asymptotic parameters play a role, one for propagation in the background medium, one for scattering in the medium perturbation, and one describing the smoothness of the medium perturbation itself. A procedure concerning what combinations of parameters can be determined given an acquisition geometry is discussed. The zero-offset case in particular simplifies the expressions but allows only one parameter to be reconstructed. We find that in anisotropic media we are forced to follow a `generalized' inverse approach, which we implement through a singular-value decomposition. Finally, we mention a simplification which is based on a micro-local analysis. Assuming that the medium jumps in a single direction only, and assuming that the associated dip can be estimated separately, the theory becomes essentially 1D and the inversion procedure reduces to an AVA analysis.

Basic principle of 2D refraction analysis first illustrated by Hagedoom is reformulated in terms of migration theory. Reciprocity relationships are utilized to derive imaging principles which relate to the location of the refractor and velocity of the refractor. In this theory, two sets of images are simultaneously developed via migration of the refraction data. The first image called `the real image' locates the refractor in depth. The second image called `the virtual image' provides information regarding true subsurface velocity of the refractor. Methodology is independent of the geometry of the refractor or velocity variations along the refractor. Since the technique utilizes both kinematic and dynamic characteristics of the wave field, no travel time picking is required.

In this paper I apply two processing methods to reverse VSP data: iterative conjugate gradient migration and coherency stacking. Iterative conjugate gradient migration helps to eliminate the migration artifacts and produce a reflectivity distribution which is consistent with the observed seismogram. The second method, coherent stacking, helps to coherently stack the prestack migrated sections to produce a more highly resolved composite migrated section. Numerical results show that these procedures are capable of eliminating a fault-like artifact from the conventionally processed VSP migrated section.

Scalable parallel algorithms for seismic imaging remain a significant challenge for the oil and gas industry. Scalability must address both the computational and the input/output portions of the algorithm in question. These issues are addressed by the ARCO Seismic Benchmark Suite, a public domain software system that provides an environment for development and performance analysis of parallel seismic processing algorithm. We illustrate some of the issues in the design of scalable parallel imaging algorithms with an example process, 3D post-stack depth migration. The algorithm used is based on an implicit finite difference formulation described by Zhiming Li. Scalability is obtained by designing computation, communication between processors, and input/output as parallel operations. The resulting application runs efficiently on both distributed memory and shared memory hardware platforms with processor counts from 1 - 128 nodes.

No abstract available.

Wozniakowski recently achieved a mathematical breakthrough in understanding the tractability of multidimensional integration using nearly-optimal quasi-Monte Carlo methods. Inspired by the new mathematical insights, we have studied the feasibility of applying quasi-Monte Carlo methods to seismic imaging by 3D pre-stack Kirchhoff migration. This earth imaging technique involves computing a large (10⁹) number of 3- or 4-dimensional integrals. Our numerical studies show that nearly-optimal quasi-Monte Carlo migration can produce the same or better quality earth images using only a small fraction (one fifth or less) of the data required by a conventional Kirchhoff migration.

In this paper, I present a derivation of a 3D one-pass post-stack depth migration algorithm which is based on the use of the linearly transformed wave equation (LITWEQ). This 3D migration operator is able to properly migrate steeply dipping events in a 3D heterogeneous media. Additionally, I propose an explicit finite-difference scheme formulation for LITWEQ 3D, which is eighth order in space and second order in time. This formulation leads to dispersion free seismograms at Nyquist with higher degree of accuracy than those derived from conventional schemes of second order in time and space. The Von Neumann stability analysis shows that the finite-difference scheme is conditionally stable, then, a proper discretization of the medium is required. Examples with synthetic models show how the wavefield is properly extrapolated by the finite-difference formulation of LITWEQ 3D. The impulse response of the 3D migration fits very well that calculated analytically for a homogeneous medium. Impulse responses are also checked in an heterogeneous medium composed of two materials separated by a 90 degrees corner interface. Finally, a LITWEQ 3D migration is performed on a 3D model which is built in a linearly varying velocity in all three spatial coordinates.

As a condition for further generalization of the migration to zero-offset in variable velocity media, I develop the theory for 2D migration to zero offset (MZO) in constant velocity media, starting from prestack migration in midpoint-offset coordinates. In arrive at an integral formulation for the MZO operator, analytically derived from the double square root (DSR) prestack migration equation. The integral formulation for the MZO is similar in form to the DSR equation, suggesting a generalization to variable velocity media using a phase-shift algorithm.

Due the huge amount of computation and internal memory required, wave backpropagation becomes the bottleneck of prestack migration or other 3D imaging/inversion procedures. We propose to use the multi-screen backpropagator for 3D prestack migration in laterally inhomogeneous background (depth migration). Multi-screen (phase-screen for scalar waves, elastic complex-screen for elastic waves) backpropagator shuttles between space-domain and wavenumber-domain using FFT and therefore avoids the time-demanding matrix multiplication. The time saving is tremendous for large-size elastic wave problems. Because it needs to store the medium parameters only one grid-plane for each step, the enormous computer memory saving makes it capable of handling large 3D problem prohibitive to other methods. The method of elastic complex screen (ECS) is a one-way propagation algorithm by neglecting the backscattered waves. However, all the forward multiple-scattering effect, such as the focusing/defocusing, diffraction, interference, wave conversion between P and S, interface waves, guided waves, etc., can be correctly handled. In this paper first the Love integral and Love migration integral are introduced. The formulation of elastic complex-screen as elastic wave one-way propagator is summarized. Numerical tests and comparisons with other full-wave methods (elastic wave finite difference and eigenfunction expansion method) are presented to show the validity of the propagator. Finally, two numerical examples of single-shot prestack migration using the ECS backpropagator, one for homogeneous background and the other for inhomogeneous background, are shown to demonstrate the feasibility of the proposed scheme.