# SEP-113 (2003)

Book (pdf)

## Multiples

### A comparison of three multiple-attenuation methods for a Gulf of Mexico dataset

(ps.gz 11869K) (pdf 1837K) (src 28662K)

**Guitton A.**

Three multiple attenuation techniques are tested on a Gulf of Mexico dataset. These methods are (1) a hyperbolic Radon transform followed by a mute (2), the Delft approach and (3), a pattern-based technique. The Radon transform separates multiples and primaries according to their moveout. The Delft approach models the multiples and subtracts them by estimating adaptive filters. The pattern-based method uses the multiple model from the Delft approach to extract and separate multiples from the primaries according to their multivariate spectra. Because of the complex geology and the modeling uncertainties introduced by 3-D effects and the acquisition geometry, the Radon transform and the Delft approach do not perform as well as the pattern-based method. In addition, the pattern-based method works significantly better when higher dimension filters are utilized: diffracted multiples are well attenuated while preserving the primaries.

### Least-squares joint imaging of primaries and pegleg multiples: 2-D field data test

(ps.gz 2373K) (pdf 745K) (src 15221K)

**Brown M.**

In this paper I present an improved least-squares joint imaging algorithm for primary reflections and pegleg multiples (LSJIMP) which utilizes improved amplitude modeling and imaging operators presented in two companion papers in this report Brown (2003a,b). This algorithm is applied to the entire Mississippi Canyon 2-D multiples test dataset and demonstrates a good capability to separate pegleg multiples from the data.

### Multiple attenuation in the image space

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**Sava P. and Guitton A.**

Multiples can be suppressed in the angle-domain image space after migration. For a given velocity model, primaries and multiples have different angle-domain moveout and therefore can be separated using techniques similar to the ones employed in the data space prior to migration. We use Radon transforms in the image space to discriminate between primaries and multiples. This method has the advantage of working with 3-D data and complex geology. Therefore, it offers an alternative to the more expensive Delft approach.

### Multiple suppression in the angle domain with non-stationary prediction-error filters

(ps.gz 6236K) (pdf 2164K) (src 27948K)

**Haines S., Guitton A., and Sava P.**

Non-stationary Prediction Error Filters (PEF's) present an effective approach for separating multiples from primaries in the angle domain. The choice of models to be used for estimation of the PEF's has a substantial impact on the final result, but is not an obvious decision. Muting in the parabolic radon transform (PRT) domain produces an effective multiple model, but the corresponding primary model must be massaged in order to minimize remaining multiple energy and achieve a satisfactory result.

### Multiple attenuation with multidimensional prediction-error filters

(ps.gz 7555K) (pdf 1770K) (src 16128K)

**Guitton A.**

Multiple attenuation in complex geology remains a very intensive research area. The proposed technique aims to use the spatial predictability of both the signal (primaries) and noise (multiples) in order to perform the multiple attenuation in the time domain. The spatial predictability is estimated with multidimensional prediction-error filters. These filters are time-variant in order to handle the non-stationarity of the seismic data. Attenuation of surface-related multiples is illustrated with field data from the Gulf of Mexico with 2-D and 3-D filters. The 3-D filters allow the best attenuation result. In particular, 3-D filters seem to cope better with inaccuracies present in the multiple model for short offset and diffracted multiples.\

### Source-receiver migration of multiple reflections

(ps.gz 1962K) (pdf 565K) (src 5791K)

**Shan G.**

Multiple reflections are usually considered to be noise and many methods have been developed to attenuate them. However, similarly to primary reflections, multiple reflections are created by subsurface reflectors and contain their reflectivity information. We can image surface related multiples, regarding the corresponding primaries as the sources. Traditional source-receiver migration assumes that the source is an impulse function. I generalize the source-receiver migration for arbitrary sources, and apply it to the migration of multiple reflections. A complex synthetic dataset is used to test the theory. Results show that my multiple migration algorithm is effective for imaging the multiple-contaminated data.

### Prestack time imaging operator for 2-D and 3-D pegleg multiples over nonflat geology

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**Brown M.**

My Least-squares Joint Imaging of Multiples and Primaries (LSJIMP) algorithm Brown (2003b) separates pegleg multiples and primaries. LSJIMP computes separate images of the peglegs and primaries, and then uses the mutual consistency of the images to discriminate against unwanted noise types in each image. The images must be consistent in two respects: kinematics and amplitudes. A companion paper Brown (2003a) paper addresses the amplitude issue. In this paper, I address the kinematics. Kinematically, the events must be correctly positioned in time and flat with offset. To this end, ...

### Amplitude modeling of pegleg multiple reflections

(ps.gz 896K) (pdf 407K) (src 13402K)

**Brown M.**

My Least-Squares Joint Imaging of Multiples and Primaries (LSJIMP) algorithm Brown (2003a) separates pegleg multiples and primaries. LSJIMP computes separate images of the peglegs and primaries, and then uses the mutual consistency of the images to discriminate against unwanted noise types in each image. The images must be consistent in two respects. Kinematically, the events must be correctly positioned in time and flat with offset. This is accomplished by an improved normal moveout operator (HEMNO) introduced in a companion paper in this report Brown (2003b). This paper addresses the second aspect ...

## Migration

### Narrow-azimuth migration of marine streamer data

(ps.gz 335K) (pdf 482K) (src 1227K)

**Biondi B.**

I introduce a new migration method that overcomes the limitations of common-azimuth migration while retaining its computational efficiency for imaging marine streamer data. The method is based on source-receiver downward-continuation of prestack data with a narrow range of cross-line offsets. To minimize the width of the cross-line offset range, while assuring that all the recorded events are correctly propagated, I define an ``optimal'' range of cross-line offset dips. To remove the effects of the boundary artifacts I apply a coplanarity condition on the prestack image. This process removes from the image cube the events that are not correctly focused at zero offset. Tests of the proposed method with the SEG-EAGE salt dataset show substantial image improvements in particularly difficult areas of the model and thus confirm the capability of the new method to overcome the limitations of common-azimuth migration in complex areas.

### Equivalence between shot-profile and source-receiver migration

(ps.gz 24K) (pdf 38K) (src 5K)

**Shan G. and Zhang G.**

Shot-profile migration and source-receiver migration seem different, but the image and Common Image Gather they obtain is the same. In this paper, we prove that shot-profile migration and source-receiver migration are equivalent, assuming that the imaging condition is cross-correlation and the method for propagating the source and receiver wavefields is a one-way wave equation. This is achieved after generalizing source-receiver migration to arbitrary sources.

### Multichannel deconvolution imaging condition for shot-profile migration

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**Valenciano A. A. and Biondi B.**

A significant improvement of seismic image resolution is obtained by framing the shot-profile migration imaging condition as a 2-D deconvolution in the shot position/time (*x*_{s},*t*) domain. This imaging condition gives a better image resolution than the crosscorrelation imaging condition and is more stable than the ``more conventional'' 1-D deconvolution imaging condition. A resolution increment is also observed in common image gathers (CIGs) computed with the 2-D deconvolution imaging condition, thus allowing a more accurate velocity analysis.

### Operator aliasing in wavefield continuation migration

(ps.gz 152K) (pdf 98K) (src 360K)

**Artman B., Shragge J., and Biondi B.**

With the widespread adoption of wavefield continuation methods for prestack migration, the concept of operator aliasing warrants revisiting. While zero-offset migration is unaffected, prestack migrations reintroduce the issue of operator aliasing. Some situations where this problem arises include subsampling the shot-axes to save shot-profile migration costs and limited cross-line shot locations due to acquisition strategies. These problems are overcome in this treatment with the use of an appropriate source function or band-limiting the energy contributing to the image. We detail a synthetic experiment that shows the ramifications of subsampling the shot axis and the efficacy of addressing the problems introduced with our two approaches. Further, we explain how these methods can be tailored in some situations to include useful energy residing outside of the Nyquist limits.

### Phase-shift migration of approximate zero-offset teleseismic data

(ps.gz 220K) (pdf 194K) (src 2759K)

**Shragge J.**

A hybrid of traditional survey-sinking migration is derived that is applicable to teleseismic wavefields. To reconfigure teleseismic data to an approximate equivalent of zero-offset, an adjoint linear moveout shift is applied. This transformation enables the straightforward development of phase-shift operators to downward continue the modified teleseismic data. This method also affords an opportunity for imaging earth structure with a variety of forward- and backscattered modes through appropriate choices of wavefield velocities. This method is applied to a synthetic teleseismic data set, and several migration results are presented to demonstrate its effectiveness.

### Imaging with buried sources

(ps.gz 64K) (pdf 71K) (src 2748K)

**Shragge J. and Artman B.**

Because the shot-profile migration algorithm largely mimics the data acquisition process, simple thought experiments may extend its utility to image the subsurface with less conventional geometries and/or sources. Imaging with the forward- and backward-scattered wavefields in an elastically modeled earth from buried sources is easily implemented without the development of new tools. With this potential in mind, we identify several novel applications of this wave-equation imaging technique, detail the requirements and processing required for its success, and give an example of the process and results by applying these concepts to a crustal-scale imaging experiment using emergent teleseismic plane-waves as sources.

### Improving the amplitude accuracy of downward continuation operators

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**Vlad I., Tisserant T., and Biondi B.**

While wave-equation downward continuation correctly accounts for traveltimes, the amplitude and phase of the image can be improved. We show concrete ways of implementing a previously proposed improvement using both mixed-domain and finite-difference extrapolators. We apply the corrections to constant velocity, constant vertical velocity gradient and *v*(*x*,*z*) cases and show that the correction brings amplitudes closer to the theoretical values.

## Angle gathers

### Angle-domain common-image gathers for migration velocity analysis by wavefield-continuation imaging

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**Biondi B. and Symes W.**

We analyze the kinematic properties of offset-domain Common Image Gathers (CIGs) and Angle-Domain CIGs (ADCIGs) computed by wavefield-continuation migration. Our results are valid regardless of whether the CIGs were obtained by using the correct migration velocity. They thus can be used as a theoretical basis for developing Migration Velocity Analysis (MVA) methods that exploit the velocity information contained in ADCIGs. We demonstrate that in an ADCIG cube the image point lies on the normal to the apparent reflector dip, passing through the point where the source ray intersects the receiver ray. Starting from this geometric result, we derive an analytical expression for the expected movements of the image points in ADCIGs as functions of the traveltime perturbation caused by velocity errors. By applying this analytical result and assuming stationary raypaths, we then derive two expressions for the Residual Moveout (RMO) function in ADCIGs. We verify our theoretical results and test the accuracy of the proposed RMO functions by analyzing the migration results of a synthetic data set with a wide range of reflector dips. Our kinematic analysis leads also to the development of a new method for computing ADCIGs when significant geological dips cause strong artifacts in the ADCIGs computed by conventional methods. The proposed method is based on the computation of offset-domain CIGs along the vertical-offset axis (VOCIGs) and on the ``optimal'' combination of these new CIGs with conventional CIGs. We demonstrate the need for and the advantages of the proposed method on a real data set acquired in the North Sea.

### Wavefield-continuation angle-domain common-image gathers in 3-D

(ps.gz 194K) (pdf 140K) (src 800K)

**Tisserant T. and Biondi B.**

Angle-Domain Common-Image Gathers (ADCIGs) are often used for velocity analysis or Amplitude Versus Angle studies. Wavefield-continuation methods can easily generate angle gathers before Prucha et al. (1999) or after imaging Sava and Fomel (2000). The method proposed by Sava and Fomel (2000) assumes that the source and ...

### Angle decompositions of images migrated by wavefield extrapolation

(ps.gz 356K) (pdf 248K) (src 604K)

**Sava P.**

I present an extension to the angle-domain decomposition of images migrated using wavefield extrapolation. Traditionally, reflectivity is described by a 1-D function of scattering angle. I show that we can further decompose the image function of other angles related to the structure and acquisition. In the 2-D case, the reflectivity is described function of two angles, while in the 3-D case the reflectivity is described function of four angles. Applications for such a multi-angle decomposition include amplitude and illumination compensation due to limited acquisition.

## Inversion and filtering

### Multiple realizations and data variance: Successes and failures

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**Clapp R. G.**

Geophysical inversion usually produces a single solution to a given problem. Often it is desirable to have some error bounds on our estimate. We can produce a range of models by first realizing that the single solution approach produces the minimum energy, minimum variance solution. By adding appropriately scaled random noise to our residual vector we change the minimum energy solution. Multiple random vectors produce multiple new estimates for our model. These various solutions can be used to assess error in our model parameters. This methodology strongly relies on having a decorrelated residual vector and, previously, was used primarily on the model styling portion of our inversion problem because it came closer to honoring the decorrelation requirement. With an appropriate description of the noise covariance, multiple realizations can be estimated. Examples of perturbing the data fitting portion of the standard inversion are shown on a 2-D deconvolution and 1-D velocity estimation problems. Results indicate that methodology has potential but is not well enough understood to be generally applied.

### Flattening 3-D data cubes in complex geology

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**Lomask J.**

3-D volumes of data can be efficiently flattened with Fourier domain methods as long as the reflections are continuous and depth invariant. To handle faults (discontinuous reflections), I pose the flattening problem in the time-space (T-X) domain to apply a weight. This ignores fitting equations that are affected by the faults. This approach successfully flattens a synthetic faulted model. Also, the flattening method is applied repeatedly in the T-X domain to flatten a synthetic model that has pinch-outs and structure that varies with depth. There are two possible schemes for handling unconformities. One scheme requires that the unconformity be picked, the data separated into different volumes, flattened individually, and then recombined. Another scheme is to apply the flattening method which picks travel-times for all horizons at once without being restricted to time-slices. It is expected that this method will be much more computationally intensive but will require less initial picking. Both of these methods need more development and testing.

### Amplitude balanced PEF estimation

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**Guitton A.**

Inverse theory teaches us that the residual, or misfit function, should be weighted by the inverse covariance matrix of the noise. Because the covariance operator is often difficult to estimate, we can approximate it with a diagonal weight that can be more easily computed. This paper investigates the possible choices of weighting functions for the data residual when prediction-error filters are estimated. Examples with 2-D and 3-D field data prove that it is better to weight the residual than to weight the data before starting the inversion.

### Coherent noise suppression in electroseismic data with non-stationary prediction-error filters

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**Haines S. and Guitton A.**

Non-stationary prediction-error filters (PEF's) provide an effective means for separating signal from coherent noise in electroseismic data. The electroseismic signal is much weaker than the noise, so we can not rely on windowing, transforms, or other alterations of the original data to create models for the PEF estimation. Instead, we design signal PEF's using the physical predictability of the phenomena, and estimate noise PEF's using portions of the original data. This technique is effective on synthetic and real data.

### Subtraction versus filtering for signal/noise separation

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**Guitton A.**

In Guitton (2001) I presented an efficient algorithm that attenuates coherent noise based on the spatial predictability of both the noise and the signal. I called this algorithm the subtraction method. This algorithm was first presented by Nemeth (1996) and works generally better than the standard projection filtering technique Abma (1995); Soubaras (1994). The main motivation in writing this paper is to better understand why the subtraction method attenuates the noise better than the filtering approach Guitton and Claerbout (2003). This is difficult to answer...

### Enhanced random noise attenuation

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**Vlad I.**

Spatial prediction filtering attenuates random noise uncorrelated from trace to trace, while preserving linear, predictable events. The prediction is formulated as a least-squares problem in either the * t-x* or the *f-x* domain. The methods are casually known as ``* f-x* decon'' and ``*t-x* decon.'' Although established by common practice, the name ``decon'' is not appropriate in these cases, because it suggests a similarity with the much better-known deconvolution of the signal along the time axis. However, deconvolution removes the predictable information (wavelet + multiples) and keeps the unpredictable (the reflectivity function), ...

## Velocity

### Wave-equation migration velocity analysis by inversion of differential image perturbations

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**Sava P. and Biondi B.**

Wave-equation migration velocity analysis is based on the linear relation that can be established between a perturbation in the migrated image and the corresponding perturbation in the slowness function. Our method formulates an objective function in the image space, in contrast with other wave-equation tomography techniques which formulate objective functions in the data space. We iteratively update the slowness function to account for improvements in the focusing quality of the migrated image. Wave-equation migration velocity analysis (WEMVA) produces wrong results if it starts from an image perturbation which is not compliant with the assumed Born approximation. Other attempts to correct this problem lead to either unreliable or hard to implement solutions. We overcome the major limitation of the Born approximation by creating image perturbations consistent with this approximation. Our image perturbation operator is computed as a derivative of prestack Stolt residual migration, thus our method directly exploits the power of prestack residual migration in migration velocity analysis.

### Steering filters in 3-D: Tubes rather than planes

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**Clapp R. G. and Clapp M. L.**

An operator composed of non-stationary plane-wave destruction filters, called a steering filter, has practical applications to many problems. Clapp et al. (1997) demonstrated their use for the missing data problem. Fomel (2000) showed how they could be used for signal-noise separation. In several papers Clapp and Biondi (1998, 2000); Clapp (2001), they are used for regularizing tomography. Prucha and Biondi (2002) used them to better handle the null-space in wave equation migration. Several different methods have been suggested for constructing the 2-D representation of these filters...

### Semblance picking

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**Clapp R. G.**

Obtaining reasonable moveout measurements is an essential step in migration velocity updating. The problem is at some level non-linear. A viable solution is to solve the problem iteratively by relinearizing the problem several times. Preliminary results are encouraging.

### Realization and analysis of an integration tomography scheme

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**Chen W.**

I realize and analyze a model-based joint tomography scheme in this paper. Surface reflection seismic data and VSP traveltimes are used simultaneously to invert the velocity by using an integrated inversion scheme. Since more data are used, the integrated tomography can obtain more accurate inversion results with lower uncertainty. Using identity operator as integration operator, I apply this method to a synthetic anticline model. Compared to the surface reflection tomography, the integrated inversion result is better in areas where the VSP ray coverage is good and conversely, the integrated inversion result is poorer in the areas where VSP ray coverage is sparse or inexistent. The results suggest we can obtain improved velocity field by this integration tomography scheme if using a well-designed integration operator.

## Migration and illumination

### The oddities of regularized inversion: Regularization parameter and iterations

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**Clapp M. L.**

Proper imaging in areas with complex overburdens can not be done effectively with an adjoint operator such as migration. To image in complex areas, we really want to apply an inverse operator, but most imaging problems can be represented by very large matrices that are difficult to invert directly. Therefore, many schemes to approximate an inverse operator have been developed. Regularized least-squares inversion implemented in an iterative scheme can be very effective in dealing with illumination problems when the imaging and regularization operators are well chosen. However, those aren't the only decisions that need to be made. Both the choice of regularization parameter () which balances the data and model residuals and number of iterations (*n*_{iter}) can have a significant effect on the quality of the final image. In this paper, I describe some of the issues that must be taken into account when choosing and *n*_{iter} for an imaging problem with poor illumination. I also examine their effects on a simple synthetic data example. These experiments show that the effects of and *n*_{iter} are related and must be considered when performing an inversion for imaging.

### Amplitude and kinematic corrections of migrated images for non-unitary imaging operators

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**Guitton A.**

Obtaining true-amplitude migrated images remains a challenging problem. One possible solution to address it is iterative inversion. However, inversion is an expensive process that can be rather difficult and expensive to apply, especially with 3-D data. In this paper, I propose computing an image that is close to the least-squares inverse image by approximating the Hessian, thus avoiding the need for iterative inversion. The Hessian is approximated with non-stationary matching filters. These filters are estimated from two images: one is the migration result () and the second is the migration result of the remodeled data computed from . Tests on the Marmousi dataset show that this filtering approach gives results similar to iterative least-squares inversion at a lower cost. In addition, because no regularization was used in the inversion process, the filtering method produces an image with fewer artifacts. Applying this method in the angle domain yields similar conclusions.

### Directions in 3-D imaging - Strike, dip, both?

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**Clapp M. L.**

In an ideal world, a 3-D seismic survey would have infinite extents and dense shot and receiver grids over the entire x-y plane. This would provide the best illumination possible everywhere in the subsurface. In our world, our limited source-receiver geometries allow energy to leave the survey and the density of our shot and receiver arrays depends on the equipment available. For 3-D surveys, the geometry leads to limited azimuth ranges dependent on the direction in which the survey is shot. The illumination itself depends on the subsurface structure. For all of these reasons, shooting our surveys in different directions will result in different subsurface ...

### Illumination compensation: Model space weighting vs. regularized inversion

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**Clapp M. L.**

In areas of complex geology, finite surveys and large velocity contrasts result in images full of artifacts and amplitude variations due to illumination problems. Cheap methods such as model space weighting and expensive methods such as regularized least-squares inversion are among the schemes that have been developed to deal with these issues. Model space weighting operators can be obtained by applying a forward modeling and an adjoint migration operator to a user-specified reference model, then applied *a posteriori* to an image. Regularized least-squares inversion applied in an iterative scheme requires the selection of an imaging operator and a regularization operator that will compensate for the illumination problems during the processing itself. Applying each of these methods to the Sigsbee2A dataset, a complex synthetic, shows that model space weighting *a posteriori* can help to equalize amplitudes, but will strengthen artifacts within the image. Regularized least-squares inversion will equalize amplitudes and reduce artifacts, but can be quite expensive.

## Interpolation

### Iteratively re-weighted least-squares and PEF-based interpolation

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**Curry W.**

Interpolation methods frequently deal poorly with noise. Least-squares based interpolation methods can deal well with noise, as long as it is Gaussian and zero-mean. When this is not the case, other methods are needed. I use an iteratively-reweighted least-squares scheme to interpolate both regular and sparse data with non-stationary prediction-error filters. I show that multi-scale methods are less susceptible to erratic noise than single-scale PEF estimation methods. I also show how IRLS improves results for PEF estimation in both cases, and how IRLS can also improve the second stage of the interpolation, where the unknown data is constrained by the PEF.

### Parameter optimization for multiscale PEF estimation

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**Curry W.**

Data interpolation is a long standing and persistent problem in exploration geophysics. Methods range from those based on the known behavior of the kinematics of seismic data Chemingui (1999); Fomel (2001); Vlad and Biondi (2001), to those that are based on transformations of the data into another domain, such as the Fourier Schonewille (2000) or Radon ...

### Interpolation of bathymetry data from the Sea of Galilee: A noise attenuation problem

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**Guitton A. and Claerbout J.**

We process a bathymetry survey from the Sea of Galilee. This dataset is contaminated with non-Gaussian noise in the form of glitches and spikes inside the lake and at the track ends. Drift on the depth measurements leads to vessel tracks in the preliminary depth images. We derive an inversion scheme that produces a much reduced noise map of the Sea of Galilee. This inversion scheme includes preconditioning and Iteratively Reweighted Least-Squares with the proper weighting function to get rid of the non-Gaussian noise. We remove the ship tracks by adding a modeling operator inside the inversion that accounts for the drift in the data. We then approximate the model covariance matrix with a prediction-error filter to enhance details in the middle of the lake. Unfortunately, the prediction-error filter has the property of degrading the resolution of the depth map at the edges of the lake. Our images of the Sea of Galilee show ancient shorelines and rifting features inside the lake.

## Acquisition

### Flexible 3-D seismic survey design

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**Alvarez G.**

Using all available subsurface information in the design of a 3-D seismic survey, we can better adjust the acquisition effort to the demands of illumination of the target horizon. I present a method that poses the choice of the acquisition parameters as an integer optimization problem. Rays are shot from grid points on the target reflector at uniform opening and azimuth angles and their emergence positions at the surface are recorded. The optimization (an exhaustive search in this example) minimizes the distance between the ray emergence coordinates and the source and receiver coordinates of candidate geometries subject to appropriate geophysical and logistics constraints. I illustrate the method with a 3-D subsurface model that I created featuring a target reflector whose depth changes significantly across the survey area. I show that for this model the standard approach would lead to a design requiring 200 shots/km^{2} whereas the optimum design requires only 80 shots/km^{2} without sacrificing the illumination of the target at any depth or the logistics of acquisition.

### Ocean-bottom seismometers in Japan

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**Vlad I.**

Ocean-bottom seismometers are well-tested, functional tools commonly used in crustal seismology. They can be deployed much deeper and are more robust than ocean-bottom cables, being the only type of instrument for 4-C surveys at depths greater than 1500m. I present the state-of-the-art of the Japanese OBS technology and the logistics associated with it to the seismic industry reader.

## Rock Physics

### Dynamic permeability in poroelasticity

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**Berryman J. G.**

The concept of dynamic permeability is reviewed. Modeling of seismic wave propagation using dynamic permeability is important for analyzing data as a function of frequency. In those systems where the intrinsic attenuation of the wave is caused in large part by viscous losses due to the presence of fluids, the dynamic permeability provides a very convenient and surprisingly universal model of this behavior.

### Poroelastic shear modulus dependence on pore-fluid properties arising in a model of thin isotropic layers

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**Berryman J. G.**

Gassmann's fluid substitution formulas for bulk and shear moduli were originally derived for the quasi-static mechanical behavior of fluid-saturated rocks. It has been shown recently that it is possible to understand deviations from Gassmann's results at higher frequencies when the rock is heterogeneous, and in particular when the rock heterogeneity anywhere is locally anisotropic. On the other hand, a well-known way of generating anisotropy in the earth is through fine (compared to wavelength) layering. Then, Backus' averaging of the mechanical behavior of the layered isotropic media at the microscopic level produces anisotropic mechanical and seismic behavior at the macroscopic level. For our present purposes, the Backus averaging concept can also be applied to fluid-saturated porous media, and thereby permits us to study how and what deviations from Gassmann's predictions could arise in an elementary fashion. We consider both closed-pore and open-pore boundary conditions between layers within this model in order to study in detail how violations of Gassmann's predictions can arise. After evaluating a number of possibilities, we determine that energy estimates show unambiguously that one of our possible choices - namely, *G*_{eff}^{(2)} = (*C _{11}* +

*C*- 2

_{33}*C*-

_{13}*C*)/3 - is the correct one for our purposes. This choice also possesses the very interesting property that it is one of two sets of choices satisfying a product formula ,where are eigenvalues of the stiffness matrix for the pertinent quasi-compressional and quasi-shear modes.

_{66}*K*

_{R}is the Reuss average for the bulk modulus, which is also the true bulk modulus

*K*for the simple layered system.

*K*

_{V}is the Voigt average. For a polycrystalline system composed at the microscale of simple layered systems randomly oriented in space,

*K*

_{V}and

*K*

_{R}are the upper and lower bounds respectively on the bulk modulus, and

*G*

_{eff}

^{(2)}and

*G*

_{eff}

^{(1)}are the upper and lower bounds respectively on the

*G*

_{eff}of interest here. We find that

*G*

_{eff}

^{(2)}exhibits the expected/desired behavior, being dependent on the fluctuations in the layer shear moduli and also being a monotonically increasing function of Skempton's coefficient

*B*of pore-pressure buildup, which is itself a measure of the pore fluid's ability to stiffen the porous material in compression.

## Computers

### SEPlib programming and irregular data

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**Clapp R. G.**

SEP3D is a powerful extension to the original SEPlib hypercube format. SEP3D attempts to build as much as possible on top of the original hypercube format. As a result, a SEP3D dataset is to some extent an amalgamation of two or three SEPlib datasets. Traditionally SEPlib files are composed of two parts, an ASCII description file and a binary data file. The ASCII file provides a description and a pointer to the binary file (by default ending in @). Below is a summary of the three ASCII/binary pairs.

### MPI in SEPlib

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**Clapp R. G.**

For many applications, a very few routines are all the MPI that is needed. The basic idea of these routines is to make a local version of global files. These files are transfered to and from the master process through some simple routines. These routines can be broken into three categories: initialization, distribution, and collection. All of the routines are written in C with a Fortran interface (e.g. floats become reals, ints become integers). ...

## References

- Chemingui, N., 1999, Imaging irregularly sampled 3D prestacked data: Ph.D. thesis, Stanford University.
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