Blocky models and Sparsity-promoting solvers
Blocky models via the L1/L2 hybrid norm
This paper seeks to define robust, efficient solvers of regressions of FigFig nature with two goals: (1) straightforward parameterization, and (2) ``blocky'' solutions. It uses an L1 hybrid norm characterized by a residual L1/L2 of transition between FigFig and Rd for data fitting and another L2 for model styling. Both the steepest descent and conjugate direction methods are included. The 1-D blind deconvolution problem is formulated in a manner intended to lead to both a blocky impedance function and a source waveform. No results are given.
Generalized-norm conjugate direction solver
In optimization problems, the L1 norm outperforms the L2 norm in presence of noise and when a blocky or sparse solution is appropriate. These applications call for a solver that can redefine the optimum criteria for a particular problem. We have implemented a generalized norm solver that is useful for a wide range of problems. Our solver modularizes the norm function so that it can easily be interchanged to experiment with different schemes on any particular geophysical problem. We implement L1, L2 , and two additional norms: Huber and Hybrid FigFig. These are useful for problems that seek the benefits of both the L1 and L2 norms.
Dix inversion constrained by L1-norm optimization
To accurately invert for velocity in a model with a blocky interval velocity inversion using Dix inversion, we set up our optimization objective function using L1 criterion. In this study, we analyze and test an improved version of the Iterative Reweighted Least Squares (IRLS) solver, a hybrid FigFig solver and a conjugate direction L1 solver. We use a 1-D synthetic velocity data set and a 1-D field RMS velocity data set as test cases. The results of the inversion are promising for applications on realistic geophysical problems.
Applications of the generalized norm solver
The application of a L1/L2 regression solver, termed the generalized norm solver, to two test cases, shows that it is potentially an efficient method for L1 inversion and is easy to parameterize. The generalized norm solver iterates with conjugate direction. Our first test case, the line fitting problem, shows that the generalized solver is capable of removing outliers in data. Our second test case, the 1D Galilee problem, shows that the generalized solver can produce a satisfactory ``blocky" solution. In terms of parameters, a low threshold value, if giving convergent solution, gives the best result. Experience shows the optimal number of inner loop iterations is one.
Alternatives to conjugate direction optimization for sparse solutions to geophysical problems
Throughout much of this summer, we experimented with extensions to the conjugate direction method to find optimal solutions to sparse geophysical problems. However, this category of techniques is not unique in its ability to optimize L1-styled fitting goals. We also investigated a variety of other techniques, including a pure L1 solution via the weighted median; a steepest-descent algorithm using the signum-function as a gradient of the true L1 norm; and a totally different approach using the Simplex Algorithm, by mapping our objective function into a linear programming form. Categorically, the approaches that relied on the true L1 method failed due to what we believe is a theoretical shortcoming of the direct application of the pure L1 norm to geophysical optimization problems. The use of linear programming turned out to be quite successful. This could be an interesting option for future research in geophysical optimization.
Velocity model building
Measuring velocity from zero-offset data by image focusing analysis
Migration velocity can be estimated from zero-offset data by analyzing focusing and defocusing of residual-migrated images. The accuracy of these velocity estimates is limited by the inherent ambiguity between velocity and reflector curvature. However, velocity resolution improves when reflectors with different curvatures are present, as demonstrated by simple synthetic examples. The application of the proposed method to zero-offset field data recorded in the New York harbor yields a velocity function that is consistent with available geologic information and clearly improves the focusing of the reflectors.
Gradient of image-space wave-equation tomography by the adjoint-state method
Optimization with gradient-descent techniques requires computing the gradient of the objective function. The gradient can be determined by using the Frech FigFigt derivatives, but, for practical problems, this can be very expensive. The gradient can be more efficiently computed by the adjoint-state method, which does not require the use of the Frech FigFigt derivatives. Here, I derive the gradient of the image-space wave-equation tomography using the adjoint-state method. I also show its application with a numerical example using image-space phase-encoded gathers.
Geophysical data integration and its application to seismic tomography
For oil exploration and reservoir monitoring purposes, we probe the earth's subsurface with a variety of geophysical methods, generating data with different natures, scales and frequency content. This diversity represents a large problem when trying to integrate all the gathered information. The concept of a shared earth by all these geophysical surveys suggests the presence of structural similarities in different data sets. For that purpose, it is necessary to work with geophysical properties that are scale-independent and not physical properties in individual layers. In this paper, I overview two methods for extracting structural information from data and using it as a constraint to the seismic tomography problem to compare different techniques and their effectiveness.
Reverse Time Migration
Selecting the right hardware for Reverse Time Migration
The optimal computational platform for Reverse Time Migration (RTM) has recently become a topic of significant debate, with proponents of the Central Processing Unit (CPU), General Purpose Graphics Processing Unit (GPGPU), and Field Programmable Gate Arrays (FPGA) all claiming superiority. The difficulty of comparing these three platforms for RTM performance is that the underlying architecture leads to significantly different algorithmic approaches. The flexibility of the CPU allows for significant algorithmic changes, which can lead to more than an order of magnitude improvement in performance. The GPGPU's large number of computational threads and overall memory bandwidth provide a significant uplift but require a simpler algorithmic approach, requiring more computation for the same size problem. The FPGA's streaming programming model results in an attractive but different cost metric. The current lack of a standardized high-level language is problematic.
Reverse Time Migration of up and down going signal for ocean bottom data
We present the results of reverse time migration (RTM) on ocean-bottom data as a precursor to applying reverse time migration and inversion of multi-component ocean-bottom data using the two-way acoustic wave-equation. We propose a joint-inversion scheme that constructively combines up- and down-going migration results and removes spurious artifacts in the final image. Reverse time migration of up-going data gives stronger reflector amplitude and weaker artifacts than migration of down-going data; however, due to mirror-imaging, the area of sub-surface illumination is narrower when using up-going energy. In addition, we observe that a new class of artifacts is present due to RTM injection of receiver wavefields with the ocean bottom geometry.
Theory and practice of interpolation in the pyramid domain
With the pyramid transform, 2-D dip spectra can be characterized by 1-D prediction-error filters (pefs) and 3-D dip spectra by 2-D pefs. This transform takes data from (w,x) -space to data in (w, u = w*x)-space using a simple mapping procedure that leaves empty locations in the pyramid domain. Missing data in (w,x)-space create even more empty bins in (w,x)-space. We propose a multi-stage least-squares approach where both unknown pefs and missing data are estimated. This approach is tested on synthetic and field data examples where aliasing and irregular spacing are present.
Schoenberg's angle on fractures and anisotropy: A study in orthotropy
For vertical-fracture sets at arbitrary orientation angles to each other - but not perfectly randomly oriented, I present a detailed model in which the resulting anisotropic fractured medium generally has orthorhombic symmetry overall. Analysis methods of Schoenberg are emphasized, together with their connections to other similarly motivated and conceptually related methods by Sayers and Kachanov, among others. Examples show how parallel vertical fracture sets having HTI symmetry turn into orthotropic fractured media if some subsets of the vertical fractures are misaligned with the others, and then the fractured system can have VTI symmetry if all the fractures are aligned either randomly, or half parallel and half perpendicular to a given vertical plane. Another orthotropic case of vertical fractures in an otherwise VTI earth system treated previously by Schoenberg and Helbig is compared to, and contrasted with, other examples treated here.
Poroelastic measurements resulting in complete data sets for granular and other anisotropic porous media
Poroelastic analysis usually progresses from assumed knowledge of dry or drained porous media to the predicted behavior of fluid-saturated and undrained porous media. Unfortunately, the experimental situation is often incompatible with these assumptions, especially when field data (from hydrological or oil/gas reservoirs) are involved. The present work considers several different experimental scenarios typified by one in which a set of undrained poroelastic (stiffness) constants has been measured using either ultrasound or seismic wave analysis, while some or all of the dry or drained constants are normally unknown. Drained constants for such a poroelastic system can be deduced for isotropic systems from available data if a complete set of undrained compliance data for the principal stresses is available, together with a few other commonly measured quantities such as porosity, fluid bulk modulus, and grain bulk modulus. Similar results are also developed here for anisotropic systems having up to orthotropic symmetry if the system is granular (i.e., composed of solid grains assembled into a solid matrix, either by a cementation process or by applied stress) and the grains are known to be elastically homogeneous. Finally, the analysis is also fully developed for anisotropic systems with nonhomogeneous (more than one mineral type), but still isotropic, grains - as well as for uniform collections of anisotropic grains as long as their axes of symmetry are either perfectly aligned or perfectly random.