## Imaging Hessian

### Wave-equation Hessian by phase encoding

[pdf 962K] [src 13Mb] Yaxun Tang
I present a method for computing wave-equation Hessian operators, also known as resolution functions or point-spread functions, under the Born approximation. The proposed method modifies the original explicit Hessian formula, enabling efficient computation of the operator. A particular advantage of this method is that it reduces or eliminates on-disk storage of Green's functions. The modifications, however, also introduce undesired cross-talk artifacts. I examine two different phase-encoding schemes, namely, plane-wave phase encoding and random phase encoding, to suppress the cross-talk. I apply the Hessian operator obtained by using random phase encoding to the Sigsbee2A synthetic data set, where a better subsalt image with higher resolution is obtained.

### Reconciling processing and inversion: Multiple attenuation prior to wave-equation inversion

[pdf 732K] [src 3.1Mb] Claudio Guerra and Alejandro Valenciano
Seismic inversion is very sensitive to the presence of noise. In an inversion scheme, noise is any event in the data not predicted by the forward modeling because of either inaccurate physics or inadequate model parameterization. Therefore, noise exists in the residual space and, if coherent, slows convergenceto an acceptable result, or in the worst case, dominates the whole process, inhibiting the efficacy of inversion. The ideal solution is to incorporate the modeling of the noise into the forward modeling. A more practical approach is to make the data agree with the physical assumptions of the inversion scheme. In this context, noise attenuation is a pre-processing step before inversion. Here, we illustrate the problem by applying one-way wave-equation inversion to a portion of the well known Sigsbee2b data. Inthe present example, noise takes the form of multiples, not modeled by the one-way wave-equation. After characterizing the noise in the migrated data, we use a dip filter to estimate it and a non-stationary adaptive filter technique to subtract it from the migrated data.

### Time-lapse wave-equation inversion

[pdf 2.8M][src 37Mb] Gboyega Ayeni and Biondo Biondi
A regularized least-squares inversion scheme is proposed as a method for correcting poor and uneven subsurface illumination under complex overburden and for attenuating image differences resulting from differences in acquisition geometries. This approach involves a joint inversion of migrated images from different vintages with filters that are explicitly derived from the Hessian of the least-squares solution to the linearized wave equation. By using such a formulation, both the imaging and monitoring challenges are solved as a single problem that yields a more accurate image of the subsurface and its time-evolution. Numerical tests show that this joint inversion technique yields more accurate time-lapse images than those obtained by differencing independently migrated or inverted images.

## Velocity Analysis

### Automatic wave-equation migration velocity analysis

[pdf 335K] [src 1.6Mb] Biondo Biondi
I present a general framework to formalize a wide class of wave-equation velocity-estimation methods that search for the velocity function that optimally focuses the migrated image. Two well-known methods, the Differential Semblance Optimization (DSO) method and the Wave-Equation Migration Velocity Analysis (WEMVA), are particular instances of velocity-estimation methods that can be formalized in this framework. I define a WEMVA-like algorithm that exploits the potential of residual migration for velocity estimation but does not require the picking of residual-migration parameters. This result enables the derivation of a closed-form expression for the gradient of the objective function, making possible to use quasi-Newton methods to solve the optimization problem. I use a simple synthetic example to illustrate some of the characteristics of the method, and to evaluate its potential as a velocity estimation method.

### Prestack exploding reflector modeling: The crosstalk problem

[pdf 348K] [src 6.3Mb] Claudio Guerra and Biondo Biondi
The recently introduced prestack exploding reflector modeling aims to model a small dataset comprised by areal shots, while keeping the correct kinematics to be used in iterations of migration velocity analysis. To achieve this goal, the modeled areal data must be combined into sets. This procedure generates data which is subjected to crosstalk during migration. Here, using a simple constant velocity model, we describe two different origins for the crosstalks and we show that by applying the concept of random phase-encoding during the modeling of the areal shots we achieve an almost complete elimination of the crosstalks.

### Ray tracing modeling and inversion of light intensity under a water surface

[pdf 741K] [src 1.7Mb] Abdullah Al~Theyab
Light propagating in a water-filled pool is perturbed by the water surface, creating patterns on the pool floor. In this report, I use ray tracing to compute an approximation of the light intensity field on the pool floor using point source and exploding surface models. The ultimate goal is to infer the water surface from the intensity field. In a seismic imaging, it is similar to imaging using amplitudes rather than travel-times. I present a geometric approachto invert for a discretized surface from a ray-count representation of the intensity field. With this formulation, the inversion becomes a combinatorial problem, which can be solved using non-deterministic search techniques. The formulation has a large inherent null space. The low cost of the technique allows a large number of iterations to be applied.

### Stable simulations of illumination patterns caused by focusing of sunlight by water waves

[pdf 213K] [src 0.8Mb]Sjoerd~de Ridder
Illumination patterns of underwater sunlight have fascinated various researchers in the past. I derive a set of equations that models these patterns at arbitrary depths from arbitrary surface topology functions. The rays are approximated by a statistical distribution function that is shifted to different lateral positions depending on the ray's refraction angle at the water surface. I perform simulations using either delta or Gaussian distributions. Using delta distributions proves unstable because of the Gibbs phenomenon, which can be suppressed using a low-pass filter. The simulations using a Gaussian distribution are stable and, apart from minor smoothing, do not suffer from additional artifacts.

## Imaging with non-standard coordinates and sources

### Angle-domain common-image gathers in generalized coordinates

[pdf 2.0M] [src 57Mb]Jeff Shragge
The theory of angle-domain common-image gathers (ADCIGs) is extended to migrations performed in generalized coordinate systems and subsurface offset axes generated by nonlinear wavefield shifts. I develop an expression linking the definition of reflection opening angle to various geometric and nonlinear shifting factors. I demonstrate that, under certain circumstances, generalized coordinate ADCIGs can be calculated directly using Fourier-based offset-to-angle approaches. Cartesian and elliptic coordinate examples are given to validate the theory. A method for eliminating geometric factors from the ADCIG expression using judicious wavefield shifts is derived; however, this approach is not likely computationally advantageous in practice.

### 3D shot-profile migration in ellipsoidal coordinates

[pdf 256K] [src 3.9Mb]Jeff Shragge and Guojian Shan
We present an approach for performing 3D shot-profile migration in ellipsoidal coordinate systems. Wavefields are extrapolated on confocal ellipsoidal shells that are well suited for accurately propagating steeply dipping and turning waves in all azimuthal directions. Numerical implementation of the corresponding dispersion relationship, though, is somewhat problematic due to first-order, complex-valued wavenumbers. We show that an integral transform recasts the problem in a way that eliminates first-order wavenumbers. The corresponding dispersion relationship is similar to that in elliptically anisotropic media. This similarity allows us to use existing implementations of wavefield extrapolation in elliptically anisotropic media to propagate wavefields on ellipsoidal meshes. Impulse response tests demonstrate the stability and accuracy of the approach.

### Toward 3D conical-wave migration in tilted elliptic cylindrical coordinates

[pdf 269K] [src 4.6Mb]Jeff Shragge and Guojian Shan
We extend conical-wave migration to tilted elliptic cylindrical (TEC) coordinate systems. When inline coordinate tilt angles are well-matched to the inline plane-wave ray parameters, the TEC coordinate extension affords accurate propagation of steep-dip and turning-wave components of conical wavefields in both the in- and crossline directions. We show that wavefield extrapolation in TEC coordinates is no more complicated than propagation in elliptically anisotropic media. Impulse response tests illustrate the accuracy of the the approach. Future work will apply the conical-wave migration approach to field data sets.

### Spectral analysis of the non-proliferation experiment

[pdf 2.6M] [src 23Mb]Sjoerd~de Ridder
To monitor a UN Test Ban Treaty, the US Department of Energy conducted a 1.5 kiloton chemical explosion at the Nevada Test Site, named the Non-Proliferation Experiment (NPE). I study a rarely known recording of the seismic waves of the NPE on a dense, transversely oriented array; the recording contains an extraordinary coda. The spectral information in this wavefield is analyzed to infer information about the subsurface. To first order, the energy arrives isotropically at the array. This allows for interferometric reconstruction of the direct wave between the receivers in the array.

## Computational Interpretation

### Image segmentation for velocity model construction and updating

[pdf 1.3M] [src 83Mb]Adam Halpert and Robert G. Clapp
Image segmentation can automatically delineate salt bodies in seismic data, an otherwise human-intensive and time-consuming task. In many instances, current segmentation algorithms successfully pick salt boundaries; a logical extension of such work is to apply these methods to the task of building and updatingseismic velocity models. We apply image segmentation tools in conjunction with sediment- and salt-flood velocity estimation techniques to identify the top and base of a salt body. Furthermore, previously existing velocity models may be updated based on the results of segmentation and automated boundary picking. By using the existing model as a priori information for the picking algorithm in areas where the segmentation is ambiguous, we calculate an optimized boundary path across a seismic image. For both synthetic and real seismic data, migrations with velocity models derived from this method produce greatly improved images.

### Lloyd and Viterbi for QC and auto-picking

[pdf 892K][src 5.8Mb] Robert G. Clapp
Automatic picking and the QCing of these picks are crucial step in the velocity analysis loop. In this paper I show that a modified version of Viterbi's algorithm can be an effective auto-picker when used interactively. In addition I show that Lloyd's algorithm can reduce densely auto-picked information to a representative subset that simplifies the QCing process.

### Hypercube viewer

[pdf 6.1M] [src 7.9Mb]Robert G. Clapp and David M. Chen and Simon Luo
Efficient viewing and interacting with multi-dimensional data volumes is an essential part of many scientific fields. This interaction ranges from simple visualization to steering computationally demanding tasks. The mixing of computation and interpretation requires a library that allows user inputs and generated results to easily be transferred We wrote {\tt Hyperview} in C++ using the QT library to facilitate this interaction. We describe the graphical user interface to the library and the basic design principles. We demonstrate the flexibility of the underlying libraries through a simple semblance picking application.

## Interpolation

### Interpolation of near offsets using multiples and prediction-error filters

[pdf 11.4M] [src 57Mb]William Curry and Guojian Shan
Most conventional marine reflection seismic data lack sampling of near offsets. We address this problem by interpolating data with a nonstationary prediction-error filter (PEF) that is first estimated from fully-sampled training data and then is used to interpolate missing data to produce an interpolated output. These training data need not be perfect, and may differ in amplitude and phase but should contain the local multi-dimensional amplitude spectra of the data we wish to recreate. We generate pseudo-primary data by crosscorrelating multiples and primaries in the recorded data. These pseudo-primary data can be generated at missing near offsets, but contain many artifacts, so it is undesirable simply to replace the missing data with the pseudo-primaries. Fortunately, many of the problems with the pseudo-primaries do not influence PEF estimation, so a desirable PEF can be obtained from these data, and then used to interpolate the missing near inline offsets to produce a result that is superior to direct substitution of the pseudo-primaries into the missing offsets.

### An algorithm for interpolation using Ronen's pyramid

[pdf 103k] [src 0.7Mb]Jon Claerbout and Antoine Guitton
Shuki Ronen has shown that a dip spectrum in 2-D may be characterized by a 1-D Prediction Error Filter (PEF) after his pyramid transform'' where $(t,x)$-space is transformed to $(\omega,u=\omega x)$-space. The transform to $(\omega, u)$-space creates empty locations (missing data). Thus we have the question of unknown PEF along with missing data. Here we propose to find both simultaneously by iterative linear least squares.

### Data interpolation in pyramid domain

[pdf 112K] [src 1.3Mb]Xukai Shen
Pyramid domain is defined as a frequency-space domain with different spatial grids for different frequencies. Data interpolation in pyramid domain is preferable since for stationary events only one prediction error filter (PEF) is needed for estimating all offsets and frequencies. However, when it is necessary to estimate both missing data and the PEF, it becomes a nonlinear problem. By solving both iteratively, we can linearize the problem and frame it as a least-squares problem. Initial results show that the iterative method will recover the missing data and PEF quite well.

## Amplitudes and anisotropy

### Biot-Gassmann analysis of partial and patchy saturated reservoirs for both reflected and transmitted seismic waves

[pdf 456K] [src 3.6Mb]James G. Berryman
This paper is a tutorial on the relationships between Biot-Gassmann theory for AVO-AVA analysis of reflection seismic amplitudes and also for the related analysis of propagation speeds for transmitted seismic waves in applications such as VSP or crosswell seismic. The main observation of special note is that the so-called fluid-line'' in AVO analysis appears to be related to patchy saturation of fluids when the Biot-Gassmann analysis of laboratory data is considered.

### Maximum entropy spectral analysis

[pdf 933K] [src 3.9Mb]James G. Berryman
A review of the maximum entropy spectral analysis (MESA) method for time series is presented. Then, empirical evidence based on maximum entropy spectra of real seismic data is shown to suggest that $M = 2N/\ln 2N$ is a reasonable {\em a priori} choice of the operator length $M$ for discrete time series of length $N$. Various data examples support this conclusion.

### Seismic anisotropy for polar media and an extended Thomsen formulation for longer offsets

[pdf 438K] [src 1.7Mb]James G. Berryman
Crack-influence parameters of \cite{Sayers1991} have been shown to be directly related to Thomsen weak-anisotropy parameters for seismic wave speeds. These results are applied to the problem of seismic propagation in reservoirs having polar (HTI) symmetry due to aligned vertical fractures. To take full advantage of these relationships, it is also helpful to obtain more accurate expressions for seismic wave speeds in polar media at longer offsets than those originally intended for Thomsen's weak anisotropy formulation.