SUMMARY We present a new massively parallel method for computation of first arrival times in arbitrary velocity models. An implementation on conventional sequential computers is also proposed. This method relies on a systematic application of Huygens’ principle in the finite difference approximation. Such an approach explicitly takes into account the existence of different propagation modes (transmitted and diffracted body waves, head waves). Local discontinuities of the time gradient in the first arrival time field (e.g., caustics) are built as intersections of locally independent wavefronts. As a consequence, the proposed method provides accurate first traveltimes in the presence of extremely severe, arbitrarily shaped velocity contrasts. Associated with a simple procedure which accurately traces rays in the obtained time field, this method provides a very fast tool for a large spectrum of seismic and seismological problems. We show moreover that this method may also be used to obtain several arrivals at a given receiver, when the model contains reflectors. This possibility significantly extends the domain of potential geophysical applications.

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Finite difference computation of traveltimes in very contrasted velocity models: a massively parallel approach and its associated tools

A few important issues for performing nonlinear refraction traveltime tomography have been identified. They include the accuracy of the traveltime and raypath calculations for refraction, the physical information in the refraction traveltime curves, and the characteristics of the refraction traveltime errors. Consequently, we develop a shortest path ray‐tracing method with an optimized node distribution that can calculate refraction traveltimes and raypaths accurately in any velocity model. We find that structure ambiguity caused by short and long rays in the seismic refraction method may influence the inversion solution significantly. Therefore, we pose a nonlinear inverse problem that explicitly minimizes the misfits of the average slownesses (ratios of traveltimes to the corresponding ray lengths) and the apparent slownesses (derivatives of traveltimes with respect to distance). As a result, we enhance the resolution as well as the convergence speed. To regularize our inverse problem, we use the Tikhon...

/pdf/nonlinear-refraction-traveltime-tomography-27qf2xwa2b.pdf

Nonlinear refraction traveltime tomography

SUMMARY A method has been developed to compute seismic reflection traveltimes in complex 3-D velocity models with complex 3-D reflector geometry. An existing finitedifference algorithm for calculating first-arrival traveltimes was modified to handle large, sharp velocity contrasts properly. The modified algorithm is faster and more accurate than several alternative schemes, and was incorporated in a procedure to compute reflection traveltimes. Snell's law for reflections is used in the vicinity of the reflecting interface. The reflector model is allowed to vary smoothly in depth, increasing the accuracy compared with a discretized reflector model. Reflection traveltimes are computed simultaneously for all revivers, requiring only two applications of the finite-difference algorithm for each shot. This results in a significant saving in computation time in comparison with other algorithms which require one pass of the finite-difference algorithm for each shot and each receiver. The reflection traveltime procedure is well suited for incorporation in inversion schemes for 3-D velocity and reflector structure.

3-D finite-difference reflection travel times

SUMMARY The fast marching method (FMM) is a grid based numerical scheme for tracking the evolution of monotonically advancing interfaces via finite-difference solution of the eikonal equation. Like many other grid based techniques, FMM is only capable of finding the first-arriving phase in continuous media; however, it distinguishes itself by combining both unconditional stability and rapid computation, making it a truly practical scheme for velocity fields of arbitrary complexity. The aim of this paper is to investigate the potential of FMM for finding later arriving phases in layered media. In particular, we focus on reflections from smooth subhorizontal interfaces that separate regions of continuous velocity variation. The method we adopt for calculating reflected phases involves two stages: the first stage initializes FMM at the source and tracks the incident wave front to all points on the reflector surface; the second stage tracks the reflected wave front by reinitializing FMM from the interface point with minimum traveltime. Layer velocities are described by a regular grid of velocity nodes and layer boundaries are described by a set of interface nodes that may be irregularly distributed. A triangulation routine is used to locally suture interface nodes to neighbouring velocity nodes in order to facilitate the tracking of wave fronts to and from the reflector. A number of synthetic tests are carried out to assess the accuracy, speed and robustness of the new scheme. These include comparisons with analytic solutions and with solutions obtained from a shooting method of ray tracing. The convergence of traveltimes as grid spacing is reduced is also examined. Results from these tests indicate that wave fronts can be accurately tracked with minimal computational effort, even in the presence of complex velocity fields and layer boundaries with high curvature. Incident wave fronts containing gradient discontinuities or shocks also pose no difficulty. Further development of the wave front reinitialization scheme should allow other later arrivals such as multiples to be successfully located.

/pdf/wave-front-evolution-in-strongly-heterogeneous-layered-media-2e8304e9mq.pdf

Wave front evolution in strongly heterogeneous layered media using the fast marching method

Prestack depth migration is the most glamorous step of seismic processing because it transforms mere data into an image, and that image is considered to be an accurate structuraldescriptionoftheearth.Thus,ourexpectationsofitsaccuracy, robustness, and reliability are high.Amazingly, seismic migration usually delivers. The past few decades have seen migration move from its heuristic roots to mathematically sound techniques that, using relatively few assumptions, render accurate pictures of the interior of the earth. Interestingly,theearthandthesubjectswewanttoimageinside itarevariedenoughthat,sofar,nosinglemigrationtechnique has dominated practical application.All techniques continually improve and borrow from each other, so one technique mayneverdominate.Despitetheprogressinstructuralimaging,wehavenotreachedthepointwhereseismicimagesprovide quantitatively accurate descriptions of rocks and fluids. Nor have we attained the goal of using migration as part of a purelycomputationalprocesstodeterminesubsurfacevelocity. In areas where images have the highest quality, we might be nearing those goals, collectively called inversion. Where data are more challenging, the goals seem elusive. We describe the progress made in depth migration to the present and the most significant barriers to attaining its inversion goalsinthefuture.Wealsoconjectureonprogresslikelytobe made in the years ahead and on challenges that migration mightnotbeabletomeet.

An overview of depth imaging in exploration geophysics

We show that a scheme to solve the 2-D eikonal equation by a finite‐difference method can violate causality for moderate to large velocity contrasts (V2/V1>2). As an alternative, we present a finite‐difference scheme in which the solution region progresses outward from an “expanding wavefront” rather than an “expanding square,” and therefore honors causality. Our method appears to be stable and reasonably accurate for a variety of velocity models with moderate to large velocity contrasts. The penalty is a large increase in computational cost and programming effort.

Finite‐difference solution of the eikonal equation along expanding wavefronts

Primary reflections, like multiples, can generate false images when reverse time migration (RTM) algorithms are used. The false images are formed by the zero-lag correlation of the source wavefields and primary reflections, which are propagated by the migration algorithm along nonphysical paths. These paths are generated by strong velocity gradients or reflection interfaces when the two-way wave equation is used. Conceptually, this type of artifact can be removed by separating up- and downgoing waves, but such separation may be impractical because it often requires storing the entire wavefields at all time steps. We have developed a de-primary RTM method in which such separation can be accomplished without saving the wavefields. The computational cost of the proposed method was only approximately 33% higher than that of conventional RTM algorithms. Using field and synthetic data sets, we have demonstrated the existence of this endemic RTM problem and verified the effectiveness of the de-primary RT...

Removing false images in reverse time migration: The concept of de-primary

A beam implementation is presented for efficient full‐volume 3-D prestack Kirchhoff depth migration of seismic data. Unlike conventional Kirchhoff migration in which the input seismic traces in time are migrated one trace at a time into the 3-D image volume for the earth’s subsurface, the beam migration processes a group of input traces (a supergather) together. The requirement for a supergather is that the source and receiver coordinates of the traces fall into two small surface patches. The patches are small enough that a single set of time maps pertaining to the centers of the patches can be used to migrate all the traces within the supergather by Taylor expansion or interpolation. The migration of a supergather consists of two major steps: stacking the traces into a τ-P beam volume, and mapping the beams into the image volume. Since the beam volume is much smaller than the image volume, the beam migration cost is roughly proportional to the number of input supergathers. The computational speedup of be...

3-D prestack Kirchhoff beam migration for depth imaging

Variable-size (dynamic) smoothing operator constraints are applied in crosswell traveltime tomography to reconstruct both the smooth- and fine-scale details of the tomogram. In mixed and underdetermined problems a large number of iterations may be necessary to introduce the slowly varying slowness features into the tomogram. To speed up convergence, the dynamic smoothing operator applies adaptive regularization to the traveltime prediction error function with the help of the model covariance matrix. By so doing, the regularization term has a larger weight at initial iterations and the prediction error term dominates the final iterations with a small regularization term weight. In addition, it is shown that adaptive regularization acts by reweighting the adjoint modeling operator (preconditioning) and by providing additional damping. Comparisons of two dynamic smoothing operators, the low-pass filter smoothing and the multigrid technique, with the fixed-size (static) smoothing operators show that the dynamic smoothing operator yields more accurate velocity distributions with greater stability for larger velocity contrasts. Consequently, it is a preferred choice for regularization.

Dynamic smoothing in crosswell traveltime tomography

The present invention incorporates the use of model-driven and data-driven methodologies to attenuate multiples in seismic data utilizing a prediction model which includes multiply-reflected, surface-related seismic waves. The present invention includes beam techniques and convolving a predicted multiples beam with a segment of a modeled pegleg beam to obtain a convolved multiples beam. The convolved multiples beam can then be deconvolved to attenuate the multiples that are present in the original input beam.

Fuhao Qin

Papers

Finite‐difference solution of the eikonal equation along expanding wavefronts

Removing false images in reverse time migration: The concept of de-primary

3-D prestack Kirchhoff beam migration for depth imaging

Dynamic smoothing in crosswell traveltime tomography

Method for identifying and removing multiples for imaging with beams