Preface 1. Introduction 2. The elastodynamics and its simple solutions 3. Seismic rays and travel times 4. Dynamic ray tracing. Paraxial ray methods 5. Ray amplitudes 6. Ray synthetic seismograms Appendix. Fourier transform, Hilbert transform and analytical signals References Index.

Seismic Ray Theory

Preface 1. Introduction to rock physics 2. Rock physics interpretation of texture, lithology and compaction 3. Statistical rock physics: combining rock physics, information theory, and statistics to reduce uncertainty 4. Common techniques for quantitative seismic interpretation 5. Case studies: lithology and pore-fluid prediction from seismic data 6. Workflows and guide lines 7. Hands-on References Index.

Quantitative Seismic Interpretation: Applying Rock Physics Tools to Reduce Interpretation Risk

Just as synthetic seismic data can be created by expressing the wave field radiating from a seismic source as a set of Gaussian beams, recorded data can be downward continued by expressing the recorded wave field as a set of Gaussian beams emerging at the earth’s surface. In both cases, the Gaussian beam description of the seismic‐wave propagation can be advantageous when there are lateral variations in the seismic velocities. Gaussian‐beam downward continuation enables wave‐equation calculation of seismic propagation, while it retains the interpretive raypath description of this propagation. This paper describes a zero‐offset depth migration method that employs Gaussian beam downward continuation of the recorded wave field. The Gaussian‐beam migration method has advantages for imaging complex structures. Like finite‐difference migration, it is especially compatible with lateral variations in velocity, but Gaussian beam migration can image steeply dipping reflectors and will not produce unwanted reflectio...

Gaussian beam migration

Prestack depth migration of shot profiles by downward continuation is a practical imaging algorithm that is especially cost-effective for sparse-shot wide-azimuth geometries. The interpretation of offset as the displacement between the downward-propagating (shot) wavefield and upward-propagating (receiver) wavefield enables us to extract offset-domain common image-point (CIP) gathers during shot-profile migration. The offset-domain gathers can then be transformed to the angle domain with a radial-trace mapping originally introduced for shot-geophone migration. The computational implications of this procedure include both the additional cost of multioffset imaging and an implicit transformation from shot-geophone to midpoint-offset coordinates. Although this algorithm provides a mechanism for imaging angle-dependent reflectivity via shot-profile migration, for sparse-shot geometries the fundamental problem of shot-aliasing may severely impact the quality of CIP gathers.

Offset and angle-domain common image-point gathers for shot-profile migration

Historically, seismic migration has been the practice (science, technology, and craft) of collapsing diffraction events on unmigrated records to points, thereby moving (“migrating”) reflection events to their proper locations, creating a true image of structures within the earth. Over the years, the scope of migration has broadened. What began as a structural imaging tool is evolving into a tool for velocity estimation and attribute analysis, making detailed use of the amplitude and phase information in the migrated image. With its expanded scope, migration has moved from the final step of the seismic acquisition and processing flow to a more central one, with links to both the processes preceding and following it. In this paper, we describe the mechanics of migration (the algorithms) as well as some of the problems related to it, such as algorithmic accuracy and efficiency, and velocity estimation. We also describe its relationship with other processes, such as seismic modeling. Our approach is tutorial; we avoid presenting the finest details of either the migration algorithms themselves or the problems to which migration is applied. Rather, we focus on presenting the problems themselves, in the hope that most geophysicists will be able to gain an appreciation of where this imaging method fits in the larger problem of searching for hydrocarbons.

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Seismic migration problems and solutions

A rapid nonrecursive prestack Kirchhoff migration is implemented (for 2-D or 2.5-D media) by computing the Green’s functions (both traveltimes and amplitudes) in variable velocity media with the paraxial ray method. Since the paraxial ray method allows the Green’s functions to be determined at points which do not lie on the ray, two‐point ray tracing is not required. The Green’s functions between a source or receiver location and a dense grid of thousands of image points can be estimated to a desired accuracy by shooting a sufficiently dense fan of rays. For a given grid of image points, the paraxial ray method reduces computation time by one order of magnitude compared with interpolation schemes. The method is illustrated using synthetic data generated by acoustic ray tracing. Application to VSP data collected in a borehole adjacent to a reef in Michigan produces an image that clearly shows the location of the reef.

Paraxial ray Kirchhoff migration

Acquisition of on‐bottom hydrophone data recording of a near‐surface source provides an opportunity to treat water column multiples as useful signal. A ray‐equation based Kirchhoff depth migration is used to image primary reflections and deep‐water multiples recorded on an Ocean Bottom Hydrophone (OBH). The image of the subbottom sediments is shown to be improved by inclusion of the deep‐water multiple in the imaging process. Field data, jointly acquired by Woods Hole Oceanographic Institute and University of Texas Institute for Geophysics at Austin and consisting of an OBH (2300 m depth) recording a 10 800 cubic inch air gun array, are used to illustrate the feasibility of the technique. Images are obtained from both the primary reflections and from energy that has undergone an additional passage through the water column. Comparison of these images reveals an excellent correlation of reflectors with the predicted polarity reversal observed in the multiple’s image. Synthetic data are used to examine the d...

Imaging with deep-water multiples

Amplitude variation with offset (AVO) analysis is often limited to areas where multidimensional propagation effects such as reflector dip and diffractions from faults can be ignored. Migration-inversion provides a framework for extending the use of seismic amplitudes to areas where structural or stratigraphic effects are important. In this procedure, sources and receivers are downward continued into the earth using uncollapsed prestack migration. Instead of stacking the data as in normal migration, the prestack migrated data are used in AVO analysis or other inversion techniques to infer local earth properties. The prestack migration can take many forms. In particular, prestack time migration of common-angle sections provides a convenient tool for improving the lateral resolution and spatial positioning of AVO anomalies. In this approach, a plane-wave decomposition is first applied in the offset direction, separating the wavefield into different propagating angles. The data are then gathered into common-angle sections and migrated one angle at a time. The common-angle migrations have a simple form and are shown to adequately preserve amplitude as a function of angle. Normal AVO analysis is then applied to the prestack migrated data. Examples using seismic lines from the Gulf of Mexico show how migration improves AVO analysis. In the first set of examples, migration is shown to improve imaging of subtle spatial variations in bright spots. Subsequent AVO analysis reveals dim spots associated with dry-hole locations that were not resolvable using traditional processing techniques, including both conventional AVO and poststack migration. A second set of examples shows improvements in AVO response after migration is used to reduce interference from coherent noise and diffractions. A final example shows the impact of migration on the spatial location of dipping AVO anomalies. In all cases, migration improves both the signal-to-noise ratio and spatial resolution of AVO anomalies.

The impact of migration on AVO

The paraxial ray method is an economical way of computing approximate Green’s functions in heterogeneous media. The method uses information from the standard dynamic ray‐tracing method to extrapolate the seismic wave field at receivers in the neighborhood of a ray so that two‐point ray tracing is not required. Applicability conditions are explicit: they define where asymptotic (high‐frequency) methods are valid, and how far away from the ray the extrapolation remains accurate. Increasing the density of the ray fan improves accuracy but increases computation time. However, since reasonable accuracy is obtained with relatively few rays, the method yields results similar to the two‐point ray‐tracing method, but at a fraction of the cost. Examples of wave‐field extrapolation from a ray to neighboring receivers show that traveltime extrapolation is more accurate than amplitude extrapolation. Accuracy, robustness, and efficiency tests, comparing paraxial ray synthetic seismograms with acoustic finite‐difference...

The paraxial ray method

An improved method and apparatus for conducting seismic prospecting is provided wherein more accurate estimates of the source wavelet can be achieved. Receivers (202) detect pulses at vertically offset locations. The pulses are converted to digital signals in a signal processor (212) and stored on a recorder (210). A processor (108) operates on the stored signals to produce an output wavelet A'(107 ,θ) in accord with the equation ∫(P.G n -G.P n )dx g .

Timothy H. Keho

Papers

Paraxial ray Kirchhoff migration

Imaging with deep-water multiples

The impact of migration on AVO

The paraxial ray method

Method and apparatus for improved seismic prospecting