SEP-94 -- TABLE OF CONTENTS

Reservoir Characterization and Monitoring

Characterization of a hydrate reservoir (ps 3464K) (src 20032K)
Ecker C.
I use marine seismic data from the Blake Outer Ridge region to characterize the lateral distribution of a methane hydrate reservoir. Detailed amplitude versus offset (AVO) analysis of the data, combined with velocity analysis and seismic impedance inversion is used to explore the extend and characteristic of the bottom simulating reflector (BSR) associated with the base of the hydrate stability field. The results suggest a strong correlation between strong BSR reflections and the presence of a low velocity zone. This is indicative of the presence of free gas beneath the hydrate. Weaker BSR amplitudes occur in areas of decreased velocity contrasts and ``fractured'' appearance of the BSR. The P-impedance inversion results in strong contrasts both at the seafloor and the BSR due to the significant velocity contrasts there. The S-wave impedance contrast, on the other hand, shows a stronger contrast in the vicinity of the BSR than at the seafloor. The BSR contrast has a rather irregular appearance and seems to be dominated by events cutting in from underneath. These events are significantly weaker in the P-impedance contrast and are partly not even visible. AVO analysis of the amplitudes at three different locations along the BSR resulted in decreasing P-wave velocities across the BSR and S-wave velocities that are either increasing, decreasing or unchanged. Possible reasons for these local variations of the shear wave velocity might be either patchy hydrate saturation or patchy gas saturation beneath the hydrate. Furthermore, the hydrate might be fractured, thus causing the properties of the hydrate to change locally.
Detecting S-wave velocity contrasts across BSRs using AVO (ps 571K) (src 1213K)
Ecker C.
Reservoir characterization of methane hydrate data requires a reasonable estimate of their S-wave velocity behavior. AVO analysis at the bottom simulating reflector (BSR), which is the reflection off the bottom of a stable hydrate structure, is a tool to extract this information from surface P-wave data. Using sonic and density logging data from a recent ODP (Ocean Drilling Program) cruise at the Blake Outer Ridge, offshore Florida and Georgia, I evaluate the effect of different S-wave velocity structures on the seismic BSR amplitudes by forward seismic modeling. I show that even for an angle coverage of 30 degrees, S-wave velocity contrasts of about $$ 17 m/s across the BSR can still be uniquely distinguished in the case of ideal amplitudes. Introduction of random Gaussian noise of a S/N ratio of about 2:1 strongly reduces the ability to differentiate between such small contrasts. In that case, errors between $$ 50 m/s and $$ 100 m/s can be introduced into the estimation of the shear wave velocity contrast. A subsequent 2-D inversion of a model, that varies laterally only in S-wave velocity, results in a more stable resolution in the presence of noise.
Bandwidth-equalization and phase-matching of time-lapse seismic datasets (ps 535K) (src 2278K)
Rickett J.
Time and frequency domain methods for equalizing bandwidth and making phase corrections are tested on synthetic time-lapse data. Cross-equalization filters are primarily designed to equalize amplitude spectra by decreasing the bandwidth of the higher frequency survey to that of the lower frequency survey. I also tested approaches that tried to maximize the bandwidth of the difference between surveys, in such a way to avoid amplifying noise.
Cross-equalization of a very shallow seismic experiment (ps 511K) (src 1139K)
Rickett J. and Bachrach R.
Cross-equalization is catch-all term for the removal of systematic non-fluid related differences between reservoir monitoring surveys. It is an important step in the processing of time-lapse seismic data as it allows interpretation of difference images in terms of fluid parameters. In this paper we apply a post-stack cross-equalization algorithm to detect differences between two high-resolution seismic surveys caused by the addition of 20 liters of honey in the near-surface. It follows the cross-equalization work on synthetic data presented in the previous paper ...
Systematic AVO response with depth (ps 107K) (src 249K)
Boyd D.
Well logs reaching several kilometers into the surface of the earth show that the P-wave reflection coefficient changes behavior with depth. I examined one particular well log, reaching approximately 5 km below the surface of the earth and assumed to be saturated with water. Simple two-layer models consisting of an upper layer of saturated sand and a lower layer of shale were constructed for three different depths to study the AVO response. Studying how the amplitude variation with offset (AVO) response changes with depth for brine and gas saturated sand shows that the well log exhibits all three classes of reflectivity. The shallow portions of the well display typical Class III reflections. The deepest sections exhibit Class I behavior. Near 2.4 km, the well log comes to a crossover point and displays Class II reflection behavior.

Imaging

Azimuth moveout + common-azimuth migration: Cost-effective prestack depth imaging of marine data (ps 1668K) (src 4370K)
Biondi B.
Common-azimuth imaging can significantly reduce the cost of full-volume 3-D prestack depth imaging of marine data sets. The common-azimuth imaging procedure comprises of two steps: first, transformation of the prestack data to common azimuth data by azimuth moveout (AMO); second, imaging the transformed data by common-azimuth migration. Both these steps are computationally efficient. AMO is a partial-migration operator and thus it has narrow spatial aperture. Common-azimuth migration is based on downward-continuation of the wavefield; therefore, its computational cost increases only as the square of the image depth. In contrast, the cost of conventional Kirchhoff migration is proportional to the cube of the image depth. Because it is a wavefield-continuation method, common-azimuth migration does not require the computation of asymptotic Green functions. Therefore, common-azimuth imaging is likely to overcome some of the accuracy problems encountered by Kirchhoff migration in the presence of complex wave-propagation phenomena. The proposed common-azimuth imaging procedure successfully depth imaged a marine data set recorded in the North Sea. These positive results suggest the application of common-azimuth imaging to velocity estimation based on wavefield focusing.
Iterative estimation of depth migration operators (ps 143K) (src 252K)
Clapp R. G. and Biondi B.
We divide the velocity estimation problem into two steps: focusing operator estimation and back projection to the velocity field. We propose estimating perturbations in the focusing operators by analyzing homothetic scans of the travel-time field. In the case of systematic bias in the velocity model, the correct perturbations to the focusing operator are indicated by the homothetic scans.
3-D prestack datuming in midpoint and offset coordinates (ps 204K) (src 352K)
Crawley S.
3-D seismic surveys are designed to have even sampling in midpoints, but often have irregular sampling in offset. For instance, marine surveys tend towards poor sampling and small aperture in the crossline offset direction, owing to the necessity of towing streamers. Integral operators which work on 3-D prestack data must deal with the problems that arise from 3-D prestack geometry in order to be useful; a straightforward generalization of a 2-D operator is likely to consume a great many CPU cycles to produce suboptimal output. Prestack Kirchhoff datuming can be made more effective in 3-D by reformulating it in midpoint and offset coordinates, and choosing an antialiasing strategy to take advantage of the resulting limitation of the time dip of events.
Two-step equalization of irregular data (ps 164K) (src 385K)
Chemingui N. and Biondi B.
Sampling irregularities in seismic data may introduce noise, cause amplitude distortions and even structural distortions when wave equation processes such as dip moveout, azimuth moveout, and prestack migration are applied. Data regularization before imaging becomes a processing requirement to preserve amplitude information and produce a good quality final image. We propose a new technique to invert for reflectivity models while correcting for the effects of irregular sampling. The final reflectivity model is a two-step solution where the data is equalized in a first stage with an inverse filter and an imaging operator is then applied to the equalized data to invert for a model. Based on least-squares theory, the solution estimates an equalization filter that corrects the imaging operator for the interdependencies between data elements. Each element of the filter is a mapping between two data elements. It reconstructs a data trace with given input geometry at the geometry of the other data element. This mapping represents an AMO transformation. The filter is therefore a symmetric AMO matrix with diagonal elements being the identity and the off-diagonal elements being the trace-to-trace AMO transforms. We explore the effectivness of the method in the 2D case for the application of partial stacking by offset continuation. The equalization step followed by imaging has proved to correct and equalize the processing for the effects of fold variations.
Depth focusing analysis for 3-D migration velocity estimation (ps 234K) (src 203K)
Malcotti H. and Biondi B.
We present partial results of the implementation of 3D depth focusing velocity analysis, based on 2-D and 3-D prestack downward continuation algorithm in the CMP domain. We discuss the focusing analysis methodology for point diffractors in a constant velocity media. In addition, we discuss the 2-D and 3-D kinematic bases of 3D focusing analysis and the implicit approximations of depth focusing analysis technique. We show the depth error panels resulting from using diffractors points and discuss the differences between the depth error gather obtained by a 2-D and 3-D downward continued operator.
Traveltime computation with the linearized eikonal equation (ps 90K) (src 4061K)
Fomel S.
Traveltime computation is an important part of seismic imaging algorithms. Conventional implementations of Kirchhoff migration require precomputing traveltime tables or include traveltime calculation in the innermost computational loop . The cost of traveltime computations is especially noticeable in the case of 3-D prestack imaging where the input data size increases the level of nesting in computational loops. The eikonal differential equation is the basic mathematical model, ...

Optimization and Filtering

Diagonal weighting: An elementary challenge to mathematicians (ps 38K) (src 3K)
Claerbout J.
Geophysical mapping and imaging are applications where we seek an approximate pseudo inverse of a matrix of very high order. Say, $<B>bold</B>d = <B>bold</B>F<B>bold</B>m$ constructs theoretical data $<B>bold</B>d$from model parameters $<B>bold</B>m$ using a linear operator $<B>bold</B>F$.Experience shows that the transpose of the simulation operator ...
Preconditioning and scaling (ps 38K) (src 3K)
Claerbout J.
In geophysical mapping and imaging applications we set up linear equations of high order. We face subjective issues like how to scale components of an operator and how much damping (regularization) to use. Here I summarize a few scaling tricks. Please keep in mind that we do not have matrices (data structures) but operators, i.e., function pairs for applying an operator $<B>bold</B>F$ and its adjoint (transpose). ...
On model-space and data-space regularization: A tutorial (ps 775K) (src 16852K)
Fomel S.
Constraining ill-posed inverse problems often requires regularized optimization. I describe two alternative approaches to regularization. The first approach involves a column operator and an extension of the data space. The second approach constructs a row operator and expands the model space. In large-scale problems, when the optimization is incomplete, the two methods of regularization behave differently. I illustrate this fact with simple examples and discuss its implications for geophysical problems.
On the general theory of data interpolation (ps 98K) (src 58K)
Fomel S.
Data interpolation is one of the most important tasks in geophysical data processing. Its importance is increasing with the development of 3-D seismics, since most of the modern 3-D acquisition geometries carry non-uniform spatial distribution of seismic records. Without a careful interpolation, acquisition irregularities may lead to unwanted artifacts at the imaging step Chemingui and Biondi (1996); Gardner and Canning (1994). ...
Cross product operator detects plane reflectors (ps 126K) (src 1068K)
Schwab M.
I propose an estimation of the dip of a plane layer volume by minimizing the output of a cross product differential expression. This rather peculiar expression allows reliable estimates in small expectation volumes and it is easily extended to Prediction Error (PE) filters. I believe the method yields reasonable dip estimates even for cases that only approximate a plane layer volume (e.g., after the addition of noise). However, the approach does not yield a straightforward technique for subtracting the dominant plane-layered contribution from the original image. The cross product expression yields a vectorial, not a scalar output. The back-projection of the vectorial output by the adjoint operation results in a scalar function which does not have a simple, meaningful interpretation. The usefulness of the cross product expression is uncertain.
Pre-whitening and coherency filtering (ps 407K) (src 640K)
Schwab M.
Ultimately, a seismic image serves interpreters as a means of building a geological model of the subsurface. Automatic edge detection schemes can help to produce images that emphasize critical geological features such as faults and river channels. Showing some striking coherency images of complex geological structures, Bahorich and Farmer 1995 brought the coherency attribute to the attention of the geophysical ...

Multiple Removal

Multiple suppression using prediction-error filter (ps 187K) (src 1351K)
Sun Y.
I present an approach to multiple suppression, that is based on the moveout between primary and multiple events in the CMP gather. After normal moveout correction, primary events will be horizontal, whereas multiple events will not be. For each NMOed CMP gather, I reorder the offset in random order. Ideally, this process has little influence on the primaries, but it destroys the shape of the multiples. In other words, after randomization of the offset order, the multiples appear as random noise. This ``man-made'' random noise can be removed using prediction-error filter (PEF). The randomization of the offset order can be regarded as a random process, so we can apply it to the CMP gather many times and get many different samples. All the samples can be arranged into a 3-D cube, which is further divided into many small subcubes. A 3-D PEF can then be estimated from each subcube and re-applied to it to remove the multiple energy. After that, all the samples are averaged back into one CMP gather, which is supposed to be free of multiple events. In order to improve the efficiency of the algorithm of estimating the PEF for each subcube, except for the first subcube which starts with a zero-valued initial guess, all the subsequent subcubes take the last estimated PEF as an initial guess. Therefore, the iteration count can be reduced to one step for all the subsequent subcubes with little loss of accuracy. Three examples demonstrate the performance of this new approach, especially in removing the near-offset multiples.
Inverse NMO stack in depth-variable velocity (ps 237K) (src 1434K)
Sun Y.
Inverse NMO stack is a procedure which combines conventional NMO and stacking into one step. By solving a set of simultaneous equations using optimization methods, such as conjugate gradient, inverse NMO stack tries to find the most ``reasonable'' stack trace for a CMP gather. Claerbout 1994 discusses inverse NMO stack in constant velocity to illustrate how back projection can be upgraded towards inversion. In this note, I extend his idea to the case of depth-variable velocity. ...
Attenuation of long period multiples (ps 813K) (src 1409K)
Holden T. C.
I present a multiple suppression method that utilizes inverse beam stacking to model and remove long period multiples from CMP gathers. The method is applied to synthetic data and is compared with the Parabolic Radon Transform (PRT) method. The results are very good except at the nearest offsets.

Deep Reflections

The changing face of deep reflection seismic profiling (ps 412K) (src 442K)
Long A.
The often complex and poor-quality seismic events resolved by deep seismic profiling are generally not dissimiliar to those found on conventional exploration data. Historically however, the processing technology used to image these data has been simplistic, by comparison to exploration studies. This gap in processing technology is generally not viewed as an urgent issue by the crustal geophysics community, who are typically more concerned with broader geological issues. Nevertheless, it would appear that at least on one front, things are going to change. The future of deep seismic profiling is clearly taking two separate paths. On the gross scale, academic researchers are beginning to take time out from their fervent acquisition to review their techniques, their Earth geological models and their perceived future challenges. On the other hand, efforts using deep seismic profiling to place natural resources in a crustal-scale context are rapidly growing in strength, with many such surveys being acquired. These latter studies will become the technical leaders for deep seismic studies, incorporating more advanced processing technologies better known in the exploration industry. The results of these commercially-driven studies should then be of great use to the academic community, providing many previously unattainable insights into the Earth.

Anisotropy

Analysis of Thomsen parameters for finely-layered VTI media (ps 83K) (src 212K)
Berryman J. G., Grechka V., and Berge P. A.
Since the work of Postma (1955) and Backus (1962), much has been learned about elastic constants in vertical transversely isotropic (VTI) media when the anisotropy is due to fine layering of isotropic elastic materials. Nevertheless, there has continued to be a degree of uncertainty about the possible range of Thomsen's anisotropy parameters $$ and $$ for such media. We show that $$ lies in the range $-3/8 {12}[<v_p^2 <v_p^{-2}- 1]$, for finely layered media having constant density; smaller positive and all negative values of $$occur for media with large fluctuations in the Lamé parameter $$.We show that $$ can also be either positive or negative, and that for constant density media ${sign}() = {sign}(<v_p^{-2}- <v_s^{-2} <v_s^2/v_p^2)$.Among all theoretically possible random media, positive and negative $$ are equally likely in finely layered media limited to two types of constituent layers. Layered media having large fluctuations in Lamé $$are the ones most likely to have positive $$.Since Gassmann's results for fluid-saturated porous media show that the effects of fluids influence only the $$ Lamé constant, not the shear modulus $$, these results suggest that positive $$ occurring together with positive but small $$ may be indicative of changing fluid content in layered earth.
Prestack time migration for anisotropic media (ps 2200K) (src 2304K)
Alkhalifah T.
Prestack phase-shift migration is implemented by evaluating the offset-wavenumber (k_h) integral using the stationary-phase method. Thus, the stationary point along k_h must be calculated prior to applying the phase shift. This type of implementation allows for migration of separate offsets, as opposed to migration of the whole prestack data when using the original formulas. For zero-offset data, the stationary point (k_h=0) is known in advance, and, therefore, the phase-shift migration can be implemented directly. For non-zero-offset data, we first evaluate k_h that corresponds to the stationary point solution either numerically or through analytical approximations. The insensitivity of the phase to k_h around the stationary point solution (its maximum) implies that even an imperfect k_h obtained analytically can go a long way to getting an accurate image. In transversely isotropic (TI) media, the analytical solutions of the stationary point (k_h) include more approximations than those corresponding to isotropic media (i.e., approximations corresponding to weaker anisotropy). Nevertheless, the resultant equations, obtained using Shanks transforms, produce accurate migration signatures for strong anisotropy ($$0.3) and even large offset-to-depth ratios (>2). The analytical solutions are particularly accurate in predicting the non-hyperbolic moveout behavior associated with anisotropic media, a key ingredient to performing an accurate non-hyperbolic moveout inversion for strongly anisotropic media. Although the prestack correction achieved using the phase-shift method can also be obtained using a cascade of normal-moveout correction, dip-moveout (DMO) correction, and zero-offset time migration, the prestack approach can handle sharper velocity models more efficiently. In addition, the resulting operator is cleaner than that obtained from the DMO method. Synthetic and field data applications of the proposed prestack migration demonstrate its usefulness.
Seismic anisotropy in Trinidad: Processing and Interpretation (ps 7518K) (src 7543K)
Alkhalifah T. and Rampton D.
The lithology of offshore Trinidad is formed of alternating sequences of sand and shale dominated layers. Average (effective) anisotropy is much lower in Trinidad compared to the prevoiusly studied area of offshore Angola due to the large amount of sand in the subsurface. Nevertheless, accounting for anisotropy in seismic processing results in improved imaging of structural and stratigraphic features. The imaging improvement is shown for two different lines from that region. Inversion for an interval value of the anisotropy parameter ($$), suggests that low values are correlated with sands (or any other isotropic material), while high interval $$ values are correlated with shales. Correlation between separate independent measurements for $$ across common midpoints (CMPs) enhances the credibility of such estimates as a representation of real geologic parameters. Finally, the $$ curve agrees well with gamma-ray well-log measurements used as a shale estimate. This result confirms the hypothesis that anisotropy is due to shales in the subsurface, and the inversion for interval $$ can subsequently be used to predict lithology.
Residual migration in VTI media using anisotropy continuation (ps 117K) (src 67K)
Alkhalifah T. and Fomel S.
We introduce anisotropy continuation as a process which relates changes in seismic images to perturbations in the anisotropic medium parameters. This process is constrained by two kinematic equations, one for perturbations in the normal-moveout (NMO) velocity and the other for perturbations in the dimensionless anisotropy parameter $$. We consider separately the case of post-stack migration and show that the kinematic equations in this case can be solved explicitly by converting them to ordinary differential equations by the method of characteristics. Comparing the results of kinematic computations with synthetic numerical experiments confirms the theoretical accuracy of the method.

TEX and Java

Empowering SEP's documents (ps 227K) (src 759K)
Fomel S., Schwab M., and Schroeder J.
The arrival of L^ATEX2_e at SEP enhanced our L^ATEX typesetting system and led us to overhaul SEP's customized macros. The revised system enables us to use the latex2html script Drakos (1996) to publish our documents routinely on the Internet. Additionally, we improved the communication between a document's makefile and its corresponding L^ATEX file. Finally, we replaced a gigantic c-shell script (texpr) that governed SEP's entire document processing, by a set of small Perl scripts. These Perl ...
A seismic inversion library in Java (ps 186K) (src 246K)
Schwab M. and Schroeder J.
Jag is a Java library for numerical optimization of geophysical problems. It shares the fundamental class hierarchy with HCL, a C++ library. We found writing Java easier than writing C++. Java freed us from garbage collection and pointer arithmetic and gave us multiple inheritance of interfaces. During the development we guided our design decisions on a small set of research scenarios. We are confident Jag will excel in prototyping solutions to geophysical inversion problems. Furthermore, we are at the verge of delivering Jag results wrapped in reproducible documents on the World Wide Web. Unfortunately, Java's current performance is inferior to even C++, which might restrict Jag to small- and medium-sized research projects.