Wavelet transforms have a simple representation in the frequency domain (Daubchies, 1992; Veterlli and Herley, 1992; Mosher and Foster, 1995). Since wave propagation also has a simple representation in the frequency domain, frequency domain wavelet transforms provide a useful framework for studying the nature of wave propagation in the wavelet domain. In this paper, we study phase shift extrapolators for 2-dimensional wavefields that have been Fourier transformed over time and wavelet transformed over space. The wavelet transform over the space axis is implemented in the wavenumber-frequency domain by complex multiplication of low and high pass wavenumber filter functions to form wave packet trees. To differentiate this operation from time-frequency wavelet transforms, we refer to the space-wavenumber-frequency transform as the 'beamlet transform.' The interaction of beamlet transform filter banks and phase shift wavefield extrapolators are simple complex multiplications. Wavefield propagation in the beamlet domain is complicated, however, by the digital implementation of decimation and upsampling operators used in orthogonal wavelet transforms. Unlike the filter functions, which can be viewed as diagonal matrix operators, the decimation and upsampling operators have significant off-diagonal terms. Since these operators do not commute with the filter and phase shift operators, the effects of the non-diagonal operators must be accounted for in the application of wave propagation operators. A simple (but unsatisfying) solution would be to apply forward-inverse transforms at each extrapolation step. Beamlet transforms with compact support in the wavenumber domain (Mosher and Foster, 1995) provide an alternate solution. Analysis of phase shift migration in the beamlet domain yields a simple matrix representation defining the interaction of filters, phase operators, and decimation/upsampling. The effects of decimation/upsampling are represented by simple folding operations. Use of filters designed for simple shape and compact support in the wavenumber domain reduces the domain of the interactions, resulting in efficient implementations of phase shift extrapolators that compare favorably with traditional Fourier approaches. Coupled with data compression, implementations of phase shift migration with multi-dimensional wavelet/beamlet transforms that exceed traditional implementations in computational efficiency may be possible.

A new Kirchhoff-type true-amplitude migration to zero-offset (MZO) algorithm is proposed for 2.5-D common-offset reflections in 2-D laterally inhomogeneous layered isotropic earth models. It provides a transformation of a common- offset seismic section to a simulated zero-offset section and is thus closely related to a dip-moveout correction (DMO). The simulated primary zero-offset reflections, even from curved interfaces, have the best possible signal character, i.e., the geometrical-spreading factor of an original primary common-offset reflection is replaced by that of a correct zero-offset reflection. A single weighted stacking procedure needs to be performed only similar to the familiar Kirchhoff or diffraction-stack migration. Moreover, in analogy to true-amplitude Kirchhoff migration, the weight function can be computed by dynamic ray tracing in the macro-velocity model which is supposed to be available. As the simulated zero-offset reflection amplitudes are controlled by the zero-offset geometrical-spreading factor and the (angle-dependent) offset reflection coefficients, one can thus perform a post-MZO, but pre-migration AVO analysis. If compared to correct zero-offset reflections, the simulated ones turn out to be stretched (frequency shifted) by the cosine of the reflection angle.

Can a developed oil field image itself? That is, can the oil pumps be used as seismic sources to image the subsurface geology in real time? This assumes that seismic data are recorded by a semi-permanent array of receivers. Some simple algebra and numerical test suggest such a possibility.

The acoustic pseudo-screen propagator is a kind of one-way wave propagator implemented in the wavenumber-space domain (dual-domain). It takes into account lateral velocity and density variations. We have recently used it for modeling primary reflected waves. In this paper, we use it as recursive downward wave extrapolators to develop a 3-D prestack depth migration method for common-shot data. In the method, the transversal Laplacian operators in the acoustic wave equation for heterogeneous media are applied exactly. The method has fast computational speed due to the use of fast Fourier transform algorithm and capability of handling wide-angle downward wave propagation/backpropagation in laterally heterogeneous media. Another main advantage of the method is huge memory saving relative to finite-difference or ray tracing based methods. Therefore, the 3-D acoustic pseudo-screen prestack depth migration method may enable 3-D prestack migration for large real 3-D data sets to be implemented within a reasonable CPU time on a supercomputer. Numerical examples of migrating synthetic reflected data generated by a finite-difference algorithm for 2-D and 3-D models are presented to demonstrate the feasibility of the method.

Migration is commonly used as a wavefield focussing tool in the study of the variation of reflection amplitude with offset (AVO), or with angle of incidence at reflectors. Migrations are typically applied to common offset or common incident angle sections. In many processing systems, common angle sections are formed by simple 1-d transformations from offset to angle of common midpoint (CMP) gathers based on ray tracing. In this paper, we provide a wave-equation framework for migrating common incidence angle sections that have been formed from Radon transforms over offset in CMP gathers. Radon transformation of the scalar wave equation results in an independent wave equation for each offset plane wave. The offset plane wave equation is nearly equivalent to the zero offset wave equation, except for an additional term related to dip in the mid-point direction, and to offset ray parameter (angle of incidence at the surface). Within this framework, finite difference, pseudo- spectral, and Kirchhoff migrations for common angle sections can be easily adapted from existing algorithms. The availability of a wave equation for common angle sections allows rigorous and efficient application of wave equation techniques for AVO studies and complex structural imaging problems.

We present algorithms based on ray + Born approximation for 2.5 D and 3D prestack preserved amplitude migration (PPAM) of seismic reflection data. The Green's functions are estimated by dynamic ray tracing in 2D or 3D heterogeneous smooth velocity fields with a wavefront construction method. The algorithms were implemented on a Sparc 20 workstation and special attention was paid to CPU efficiency and low RAM storage. We give some basic idea for optimization and present some applications on synthetic and real datasets. For the 2D version about 10 Mbytes RAM and 10 minutes CPU time are needed for the migration of a marine line, using 300 shots with 120 receivers. For the 3D algorithm, about 48 Mbytes RAM and a weak CPU time are needed for migrating a 3D marine survey using 29 lines consisting of ca. 500 shots with 240 receivers (13 Gbytes).

Conventional finite difference eikonal solvers produce only the first arrival time. However suitable solvers (of sufficiently high order of accuracy) may be extended via Fermat's principle to yield a simple algorithm which computes all travel times to each subsurface point, with cost on the same order as that of a first arrival solver.

Differential semblance measures are unique amongst velocity inversion objectives in having well-defined and smooth high frequency asymptotics. A version appropriate for analysis of CMP gathers and layered models is particularly easy to analyze and economical for numerical experimentation. For model-consistent data, the DS objective measures the discrepancy in takeoff slowness of rays, weighted by the energy in the data -- without of course requiring that events be identified or rays traced. Numerical experiments with synthetic data illustrate the theoretical properties.

The general problem of 2D image reconstruction based on a tomographic approach is here explored in a particular case. Exploiting the relationship between microwave attenuation and rainfall intensity in a microwave tomography approach was demonstrated to be a valid possibility for the reconstruction of rainfall fields in limited areas. At each time step, path-integrated attenuation measurements are provided by a set of microwave transmitter-receiver pairs, and two-dimensional Gaussian basis functions are utilized to reconstruct the space-time distribution of the rainfall intensity field. The inversion problem to be solved is highly ill-conditioned, due to the practical (and economical) impossibility to set up an adequate network for performing classical tomography, therefore a global optimization stochastic technique has been developed to obtain valid results in quasi real time, the implemented multiresolution algorithm is explored. Some significant examples are briefly shown in order to demonstrate the speed, precision and flexibility of the technique. Therefore a significant analysis of the typical algorithm output is presented to show how the problem of recognizing the best solution could be faced.

The purpose of the authors is to present a comparison of the merits and demerits of three different approximations in acoustic scattering theory, viz. the (first) Born approximation, the eikonal, and the Rytov approximation. For the most part, we limit our attention to far-field scattering in a spherically symmetric potential, and we are mainly interested in the case where the product of the wave number and the radius of the scattering potential is well in excess of unity.

In classical travel time tomography, seismic signals are approximated in the high frequency limit. In wave equation tomography methods, the problem is reformulated in the finite bandwidth sense. Although this inverse problem is nonlinear, it is usually approached in the linearized sense, with the help of either Rytov or Born approximations. In this study, we introduce the asymptotic ray + Born and ray + Rytov formalisms. Notwithstanding its ability to provide computationally efficient methods to model seismic scattering, the introduction of wave asymptotics allows simple geometrical interpretations of Born and Rytov integrals. It also allows qualitative discussion of their respective validity domains, and to identify the principal sources of errors and artefacts. It results that validity conditions for Born and Rytov linearizations are very different. In particular, forward scattering due to spatially extended slowness perturbations is correctly accounted for by Rytov summation (in terms of both wave kinematics and amplitudes), whereas Born approximation clearly fails. Numerical tests with diverse perturbations provide insights on the effects of nonlinearity, and illustrate the role of Rytov summation as a 'slowness smoothing operator' accounting for the actual source bandwidth.

Strongly heterogeneous media such as those with fine layers and/or aligned heterogeneities and/or empty pores can pose problems for classical finite-difference methods to simulate wave propagation. This is due to the difficulties of handling sharp interfaces in these media and total reflections from boundaries of empty pores in a porous medium. The phononic lattice solid by interpolation (PLSI) is a microscopic approach to P wave propagation in strongly heterogeneous media. The method is capable of handling sharp interfaces and, therefore, provides a powerful tool to simulate wave propagation in such media. Numerical simulations by the PLSI to simulate P wave propagation in these media are presented. Anisotropy induced by fine layers and aligned heterogeneities is observed. Numerical results demonstrate that the scattering effect of empty pores is much stronger than non-empty heterogeneities. Ultimately, the approach could enable numerical experiments to be conducted to study the microscopic mechanisms responsible for anisotropy and attenuation of seismic waves. This would require the approach to be extended to the elastic case.

Chevron recently developed and deployed a seismic data compression algorithm based on multidimensional wavelet transforms. Development was motivated by the large volumes of data acquired in modern 3D marine surveys. We demonstrate an algorithm that can compress seismic data at ratios between 50 and 100 to 1 without losing geophysically significant information. The algorithm was successfully field tested on a vessel in the North Sea in July 1995, demonstrating the feasibility of on board real-time compression and satellite transmission of the data to a land based processing center. Compressed and decompressed data from the field test were processed into a final image. Differences between this image and the image based on the original data are geophysically insignificant, demonstrating that all geophysical information in the original was retained

The method involves a system of computer-assisted algorithms. This system enables us to unite all refractions rays which outcame or reflected from it into beams of parallel rays. For each beam, in the near area of the crystal, the amplitudes of electromagnetic field components and the coordinates of the vertexes of its polygonal cross- section are determined. Then, the electromagnetic fields of refraction beams are recalculated into a far area by means of method of physical optics. The algorithm system is developed in assumption that the crystal under investigation is a convex polyhedron. In the algorithms, the crystal form is defined by the coordinates of its vertexes and can be arbitrary.

One-way wave extrapolators, such as the split-step Fourier method, generalized screen methods including the phase- screen, complex screen and wide-angle pseudo-screen methods, have been proposed recently as part of the solution to lessen the CPU and memory size requirement of wave equation based 3D subsurface imaging methods. The phase space path- integral formulation and the vertical slowness symbol analysis provide a general and convenient background for accuracy estimation and improvement of screen propagators. In this paper we review and elucidate the relevant theory and formulation in paper I (de Hoop and Wu, 1996), and present the results of accuracy analysis and numerical tests for screen propagators. By comparing the dispersion relations of screen propagators in the high-frequency limit with the leading term high-frequency asymptotics to the vertical slowness symbol, and through numerical experiments of thin-slab transmission, it is seen that for weak perturbations both the phase-screen and the wide-angle pseudo-screen propagators perform well; while for the case of strong medium contrasts, the modified wide-angle pseudo- screen propagator has much better accuracy for large-angle waves than the phase screen propagator. Unlike the traditional phase screen propagator which is purely a space domain operator, the wide-angle pseudo-screen propagator is a dual-damain operator. The screen propagators have great potential in the application to 3D prestack depth migration/inversion as extrapolators.

We discuss several aspects of raytracing solutions in smooth velocity models compared to hard interface models. The emphasis of this paper is to identify a fundamental open problem of describing wave propagation in a medium with singularities.