R-3 Repository :: Browsing by Author "Symes, William W."

Browsing by Author "Symes, William W."

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2D and 2.5D Kirchhoff Inversion Using Upwind Finite Difference Amplitudes
(1996-07) Araya, Kidane; Symes, William W.
Finite difference solution of the transport equation provides an efficient and accurate method for computation of 2.5D geometric acoustics amplitudes. These amplitudes can be used in simulation, migration and inversion formulas. Remodeled data based on high frequency asymptotic inversion using these amplitudes shows excellent agreement with both synthetic and field input data.
A C++ class supporting adjoint-state methods
(2008) Enriquez, Marco U.; Symes, William W.
The adjoint-state method is widely used for computing gradients in simulation-driven optimization problems. The adjoint-state evolution equation requires access to the entire history of the system states. There are instances, however, where the required state for the adjoint-state evolution is not readily accessible; consider large-scale problems, for example, where the entire simulation history is not saved to conserve memory. This thesis introduces a C++ state-access class, StateHistory , to support a myriad of solutions to this problem. Derived StateHistory classes implement a (simulation) time-altering function and data-access functions, which can be used in tandem to access the entire state history. This thesis also presents a derived StateHistory class, GriewankStateHistory , which uses Griewank's optimal checkpointing scheme. While only storing a small fraction of simulation states, GriewankStateHistory objects can reconstitute unsaved states for a small computational cost. These ideas were implemented in the context of TSOpt, a time-stepping library for simulation-driven optimization algorithms.
A comparison of finite difference stencils on two forms of the acoustic wave equation
(2000) Hill, Regina Shaylean; Symes, William W.
In practice, two forms of the acoustic wave equation, the velocity-stress and pressure forms, are used to simulate seismic experiments. These equations in their discrete forms lead to two families of finite difference schemes, the staggered-grid and centered difference schemes. These two difference schemes are widely used to numerically generate seismograms. Although these two difference schemes are widely used, there has been no distinction whether one is better than the other. The goal of this research is to formulate a heuristic based on computational cost and storage to determine which scheme is better than the other.
A Consortium Proposal: "The Rice Inversion Project"
(1992-11) Symes, William W.
This document details a proposal for an industrially sponsored consortium for research in seismic inversion at Rice University. This consortium project will be directed by Professor William W. Symes in the Department of Computational and Applied Mathematics (formerly called The Department of Mathematical Sciences), George Brown School of Engineering. This project will develop novel approaches pioneered by Professor Symes to velocity and reflectivity estimation from waveform data, and will offer its sponsors both pilot software for state-of-the-art vector and parallel computing platforms, and a database of experience in waveform inversion. Participants will include graduate student assistants, postdoctoral research associates, and (whenever possible) short-and long-term visitors from the sponsoring organizations.
A discrepancy-based penalty method for extended waveform inversion
(Society of Exploration Geophysicists, 2017) Fu, Lei; Symes, William W.; The Rice Inversion Project
Extended waveform inversion globalizes the convergence of seismic waveform inversion by adding nonphysical degrees of freedom to the model, thus permitting it to fit the data well throughout the inversion process. These extra degrees of freedom must be curtailed at the solution, for example, by penalizing them as part of an optimization formulation. For separable (partly linear) models, a natural objective function combines a mean square data residual and a quadratic regularization term penalizing the nonphysical (linear) degrees of freedom. The linear variables are eliminated in an inner optimization step, leaving a function of the outer (nonlinear) variables to be optimized. This variable projection method is convenient for computation, but it requires that the penalty weight be increased as the estimated model tends to the (physical) solution. We describe an algorithm based on discrepancy, that is, maintaining the data residual at the inner optimum within a prescribed range, to control the penalty weight during the outer optimization. We evaluate this algorithm in the context of constant density acoustic waveform inversion, by recovering background model and perturbation fitting bandlimited waveform data in the Born approximation.
A Nonlinear Differential Semblance Algorithm for Waveform Inversion
(2013-07-24) Sun, Dong; Symes, William W.; Heinkenschloss, Matthias; Zhang, Yin; Zelt, Colin A.
This thesis proposes a nonlinear differential semblance approach to full waveform inversion as an alternative to standard least squares inversion, which cannot guarantee a reliable solution, because of the existence of many spurious local minima of the objective function for typical data that lacks low-frequency energy. Nonlinear differential semblance optimization combines the ability of full waveform inversion to account for nonlinear physical effects, such as multiple reflections, with the tendency of differential semblance migration velocity analysis to avoid local minima. It borrows the gather-flattening concept from migration velocity analysis, and updates the velocity by flattening primaries-only gathers obtained via nonlinear inversion. I describe a general formulation of this algorithm, its main components and implementation. Numerical experiments show for simple layered models, standard least squares inversion fails, whereas nonlinear differential semblance succeeds in constructing a kinematically correct model and fitting the data rather precisely.
A Nonlinear Differential Semblance Strategy for Waveform Inversion: Experiments in Layered Media
(2009-04) Sun, Dong; Symes, William W.
This paper proposes an alternative approach to the output least-squares (OLS) seismic inversion for layered-media. The latter cannot guarantee a reliable solution for either synthetic or field data, because of the existence of many spurious local minima of the objective function for typical data, which lack low-frequency energy. To recover the low-frequency lacuna of typical data, we formulate waveform inversion as a differential semblance optimization (DSO) problem with artificial low-frequency data as control variables. This version of differential semblance with nonlinear modeling properly accounts for nonlinear effects of wave propagation, such as multiple reflections. Numerical experiments with synthetic data indicate the smoothness and convexity of the proposed objective function. These results suggest that gradient-related algorithms may successfully approximate a global minimizer from a crude initial guess for typical band-limited data.
A Software Framework for the Abstract Expression of Coordinate-Free Linear Algebra and Optimization Algorithms
(2005-10) Symes, William W.; Padula, Anthony D.; Scott, Shannon D.
Object oriented design solves a fundamental programming problem arising in scientific and engineering applications of linear algebra and optimization: the separation in code of multiple levels of abstraction naturally appearing in solution algorithms for such problems. The Rice Vector Library provides C++ classes expressing core concepts (vector, function,...) of calculus in Hilbert space with minimal implementation dependence, and standardized interfaces behind which to hide application-dependent implementation details (data containers, function objects). A variety of coordinate free algorithms from linear algebra and optimization, including Krylov subspace methods and various relatives of Newton's method for nonlinear equations and constrained and unconstrained optimization, may be expressed purely in terms of this system of classes. The resulting code may be used {\em without alteration} in a wide range of control, design, and parameter esti mation applications, in serial and parallel computing environments.
A Sparse, Bound-Respecting Parametrization of Velocity Models
(2005-04) Dussaud, Eric; Symes, William W.
We present a parsimonious representation of velocity models which allows for user-defined placement of nodes. Mild restrictions are imposed on the data structure so that a computationally efficient algorithm can be used to smoothly approximate nodal values on finely sampled regular grids. The building block of the algorithm operates on one-dimensional arrays, allows for user-defined control of smoothness and guarantees that bounds are preserved. The specific data structure allows to carry out this process recursively to obtain multi-dimension smooth and regularly gridded velocity models.
A study of viscous effects in seismic modeling, imaging, and inversion: Methodology, computational aspects, and sensitivity
(1996) Blanch, Joakim Oscar; Symes, William W.
Real Earth media are anelastic, which affects both the kinematics and dynamics of propagating waves: Waves are attenuated and dispersed. If anelastic effects are neglected, inversion and migration can yield erroneous results. The anelastic effects in real rocks can be well described by a viscoelastic model. Hence, viscoelastic wave propagation simulation is a well suited base for realistic seismic inversion algorithms derived through the adjoint state technique. The thesis develops a finite-difference simulator to model wave propagation in viscoelastic media. The viscoelastic scheme, which is dispersion and stability analyzed, is only slightly more expensive than analogous elastic schemes. The thesis also presents a method for modeling of constant Q as a function of frequency based on an explicit closed formula for calculation of the parameter fields. The $\tau$-p (intercept time-slowness) domain permits economical modeling and inversion of 3-D wave propagation in attenuative (viscoacoustic) layered media. A recomputation scheme for adjoint calculations permits efficient inversion in multidimensional attenuative (viscoacoustic) physically consistent media. The inversion method is proved to be feasible by successfully being applied to real field data. Synthetic data inversion shows that neglect of attenuation can lead to interpretation errors. Analysis in thesis indicate the necessary precision in attenuation (Q) to reliably estimate Earth parameters, such as velocity and density. Attenuation has a large effect on the magnitude of inversion estimates of Earth parameters, however ratios (relative amplitudes) between parameters are not as sensitive to the amount of attenuation in the medium. This is a positive result since the amount of attenuation in a medium is rather difficult to determine accurately from (seismic) data. Hence, a reasonably well estimated amount of attenuation would allow for reliable estimation of Earth parameter ratios, such as the ratio between normalized velocity and density fluctuations. Ratios between estimated Earth parameters are generally having an extreme point for the correct amount of attenuation. If this extremum does not exist, the ratios are well determined. The existence of an extreme point could be the base for estimation algorithms for attenuation, through search for the extreme point in Earth parameter ratios as a function of attenuation.
A Time-Stepping Library For Simulation-Driven Optimization
(2007-03) Symes, William W.
The Timestepping Simulation for Optimization ("TSOpt") library provides an interface for time-stepping simulation. It packages a simulator togehter with its derivatives ("sensitivities") and adjoint derivaties with respect to simulation parameters, and uses the aggregate to define a Rice Vector Library Operator subclass.
A Trace Theorem for Solutions of Linear Partial Differential Equations
(1989-10) Bao, Gang; Symes, William W.
The main goal of this work is to determine circumstances under which the trace of the solution of a linear partial differential equation is as smooth as the solution itself. Clearly, if the linear partial differential equation is strictly hyperbolic with smooth coefficients, standard energy estimates will yield the fact that the solution along any spacelike trace is as smooth as itself locally, provided a sufficiently smooth right-hand side. Unfortunately, for more general equations or even a strictly hyperbolic differential equation but this time along a nonspacelike trace, the same idea will not work, essentially because one does not know how to apply energy estimates to a nonhyperbolic problem directly. In this paper, we shall investigate the trace regularities of solutions to linear P.D.E. Our result shows that the difficulties discussed above may be cured by imposing some additional microlocal smoothness.
All Stationary Points of Differential Semblance Are Asymptotic Global Minimizers: Layered Acoustics
(1999) Symes, William W.
Differential semblance velocity estimators have well-defined and smooth high frequency asymptotics. A version appropriate for analysis of CMP gathers and layered acoustic models has no secondary minima. Its structure suggests an approach to optimal parametrization of velocity models.
An abstract analysis of differential semblance optimization
(1994) Gockenbach, Mark Steven; Symes, William W.; Tapia, Richard A.
Differential Semblance Optimization (DSO) is a novel way of approaching a class of inverse problems arising in exploration seismology. The promising feature of the DSO method is that it replaces a nonsmooth, highly nonconvex cost function (the Output Least-Squares (OLS) objective function) with a smooth cost function that is amenable to standard (local) optimization algorithms. The OLS problem can be written abstractly as a partially linear least-squares problem with linear constraints. The DSO objective function is derived from the associated quadratic penalty function. It is shown that one way to view the DSO objective function is as a regularization of a function that is dual (in a certain sense) to the OLS objective function. By viewing the DSO problem as a perturbation of this dual problem, this method can be shown to be effective. In particular, it is demonstrated that, under suitable assumptions, the DSO method defines a parameterized path of minimizers converging to the desired solution, and that for certain values of the parameter, standard optimization techniques can be used to find a point on the path. The predictions of the theory are motivated and illustrated on two simple model problems for seismic velocity inversion, the plane wave detection problem and the "layer-over-half-space" problem. It is shown that the theory presented in this thesis extends the existing theory for the plane wave detection problem.
An Adaptive Finite Difference Method for Traveltime and Amplitude
(1999) Qian, Jianliang; Symes, William W.
The eikonal equation with point source is difficult to solve with high order accuracy because of the singularity of the solution at the source. All the formally high order schemes turn out to be first order accurate without special treatment of this singularity. Adaptive upwind finite difference methods based on high order ENO (Essentially NonOscillatory) Runge-Kutta difference schemes for the paraxial eikonal equation overcome this difficulty. The method controls error by automatic grid refinement and coarsening based on an a posteriori error estimation. It achieves prescribed accuracy at far lower cost than fixed grid methods. Reliable auxiliary quantities, such as take-off angle and geometrical spreading factor, are by-products.
An adaptive multiscale algorithm for efficient extended waveform inversion
(Society of Exploration Geophysicists, 2017) Fu, Lei; Symes, William W.; The Rice Inversion Project
Subsurface-offset extended full-waveform inversion (FWI) may converge to kinematically accurate velocity models without the low-frequency data accuracy required for standard data-domain FWI. However, this robust alternative approach to waveform inversion suffers from a very high computational cost resulting from its use of nonlocal wave physics: The computation of strain from stress involves an integral over the subsurface offset axis, which must be performed at every space-time grid point. We found that a combination of data-fit driven offset limits, grid coarsening, and low-pass data filtering can reduce the cost of extended inversion by one to two orders of magnitude.
An alternative approach to differential semblance velocity analysis via normal moveout correction
(2010) Wang, Chao; Symes, William W.
This thesis develops a new computation of the objective function and gradient for normal moveout-based differential semblance (DS). The DS principle underlies a class of algorithms for seismic velocity analysis. The simplest variant of DS is based on a drastic approximation to the scattering of waves, called "normal moveout" (NMO) in the seismic literature. This simple NMO-driven DS algorithm is very fast relative to other variants based on more faithful approximations to wave physics, but nonetheless accurate enough to be used to process field data. A recent implementation of NMO-based DS demonstrated these capabilities, but it also exhibited numerical irregularity which may have affected the stability of its velocity estimates. My alternative approach avoids interpolation noise that existed in previous work and so results in more stable numerical optimization.
An alternative formula for approximate extended Born inversion
(Society of Exploration Geophysicists, 2017) Hou, Jie; Symes, William W.
Various modifications of reverse time migration (RTM) provide asymptotic inverses to the subsurface offset extended Born modeling operator for constant-density acoustics. These approximate inverses have the same quality (asymptotic accuracy) as do generalized Radon transform pseudoinverses, but they can be computed without any ray tracing whatsoever. We have developed an approximate inverse of this type whose additional computational cost, above that of subsurface offset extended RTM, is negligible.
An approximate inverse to the extended Born modeling operator
(Society of Exploration Geophysicists, 2015) Hou, Jie; Symes, William W.; The Rice Inversion Project
Given a correct (data-consistent) velocity model, reverse time migration (RTM) correctly positions reflectors but generally with incorrect amplitudes and wavelets. Iterative least-squares migration (LSM) corrects the amplitude and wavelet by fitting data in the sense of Born modeling, that is, replacing migration by Born inversion. However, LSM also requires a correct velocity model, and it may require many migration/demigration cycles. We modified RTM in the subsurface offset domain to create an asymptotic (high-frequency) approximation to extended LSM. This extended Born inversion operator outputs extended reflectors (depending on the subsurface offset and position in the earth) with correct amplitude and phase, in the sense that similarly extended Born modeling reproduces the data to good accuracy. Although the theoretical justification of the inversion property relies on ray tracing and stationary phase, application of the weight operators does not require any computational ray tracing. The computational expense of the extended Born inversion operator is roughly the same as that of extended RTM, and the inversion (data-fit) property holds even when the velocity is substantially incorrect. The approximate inverse operator differes from extended RTM only in the application of data- and model-domain weight operators, and takes the form of an adjoint in the sense of weighted inner products in data and model space. Because the Born modeling operator is approximately unitary with respect to the weighted inner products, a weighted version of conjugate gradient iteration dramatically accelerates the convergence of extended LSM. An approximate LSM may be extracted from the approximate extended LSM by averaging over subsurface offset.
An Infeasible Point Method for Minimizing the Lennard-Jones Potential
(1993-10) Gockenbach, Mark S.; Kearsley, Anthony J.; Symes, William W.
Minimizing the Lennard-Jones potential, the most-studied problem for molecular conformation, is an unconstrained global optimization problem. In this paper, the problem is reformulated as an equality constrained nonlinear program in such a way that the likelihood of finding a global minimizer is increased. Implementation of an algorithm for solving this nonlinear program is discussed, and results of numerical tests are presented.

Browsing by Author "Symes, William W."

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