Changsoo Shin

An optimal 9-point, finite-difference, frequency-space, 2-D scalar wave extrapolator

In this study, a new finite-difference technique is designed to reduce the number of grid points needed in frequency-space domain modeling. The new algorithm uses optimal nine-point operators for the approximation of the Laplacian and the mass acceleration terms. The coefficients can be found by using the steepest descent method so that the best normalized phase curves can be obtained. This method reduces the number of grid points per wavelength to 4 or less, with consequent reductions of computer memory and CPU time that are factors of tens less than those involved in the conventional second-order approximation formula when a band type solver is used on a scalar machine.

Waveform inversion using a logarithmic wavefield

Although waveform inversion has been studied extensively since its beginning 20 years ago, applications to seismic field data have been limited, and most of those applications have been for global-seismology- or engineering-seismology-scale problems, not for exploration-scale data. As an alternative to classical waveform inversion, we propose the use of a new, objective function constructed by taking the logarithm of wavefields, allowing consideration of three types of objective function, namely, amplitude only, phase only, or both. In our waveform inversion, we estimate the source signature as well as the velocity structure by including functions of amplitudes and phases of the source signature in the objective function. We compute the steepest-descent directions by using a matrix formalism derived from a frequency-domain, finite-element/finite-difference modeling technique. Our numerical algorithms are similar to those of reverse-time migration and waveform inversion based on the adjoint state of the wave equation. In order to demonstrate the practical applicability of our algorithm, we use a synthetic data set from the Marmousi model and seismic data collected from the Korean continental shelf. For noisefree synthetic data, the velocity structure produced by our inversion algorithm is closer to the true velocity structure than that obtained with conventional waveform inversion. When random noise is added, the inverted velocity model is also close to the true Marmousi model, but when frequencies below 5 Hz are removed from the data, the velocity structure is not as good as those for the noise-free and noisy data. For field data, we compare the time-domain synthetic seismograms generated for the velocity model inverted by our algorithm with real seismograms and find that the results show that our inversion algorithm reveals short-period features of the subsurface. Although we use wrapped phases in our examples, we still obtain reasonable results. We expect that if we were to use correctly unwrapped phases in the inversion algorithm, we would obtain better results.

Efficient calculation of a partial‐derivative wavefield using reciprocity for seismic imaging and inversion

Linearized inversion of surface seismic data for a model of the earth’s subsurface requires estimating the sensitivity of the seismic response to perturbations in the earth’s subsurface. This sensitivity, or Jacobian, matrix is usually quite expensive to estimate for all but the simplest model parameterizations. We exploit the numerical structure of the finite-element method, modern sparse matrix technology, and source–receiver reciprocity to develop an algorithm that explicitly calculates the Jacobian matrix at only the cost of a forward model solution. Furthermore, we show that we can achieve improved subsurface images using only one inversion iteration through proper scaling of the image by a diagonal approximation of the Hessian matrix, as predicted by the classical Gauss-Newton method. Our method is applicable to the full suite of wave scattering problems amenable to finiteelement forward modeling. We demonstrate our method through some simple 2-D synthetic examples.

A comparison between the behavior of objective functions for waveform inversion in the frequency and Laplace domains

In the frequency domain, gradient-based local-optimization methods of waveform inversions have been unsuccessful at inverting subsurface parameters without an accurate starting model. Such methods could not correct automatically for poor starting models because multiple local minima madeitdifficulttoapproachthetrueglobalminimum.Inthis study,wecomparedthebehaviorofobjectivefunctionsinthe frequency and Laplace domains. Wavefields in the Laplace domain correspond to the zero-frequency component of a damped wavefield; thus, the Laplace-domain waveform inversion can image smooth velocity models. Objective functions in the Laplace-domain inversion have a smoother surface and fewer local minima than in the frequency-domain inversion.Weappliedthewaveforminversiontoa2Dsliceof the acoustic SEG/EAGE salt model in the Laplace domain and recovered smooth velocity models from inaccurate initial velocity conditions.We also successfully imaged velocities of the salt, SEG overthrust, and Institut Francais du PetroleMarmousimodelswiththefrequency-domaininversion method by using the inverted velocity model of the Laplacedomaininversionastheinitialmodel.

Waveform inversion using a back-propagation algorithm and a Huber function norm

Waveforminversionfacesdifficultieswhenappliedtoreal seismic data, including the existence of many kinds of noise. The 1 -norm is more robust to noise with outliers than the least-squares method. Nevertheless, the least-squares method is preferred as an objective function in many algorithms because the gradient of the 1 -norm has a singularity when the residual becomes zero. We propose a complex-valued Huber function for frequency-domain waveform inversion that combines the 2 -norm for small residuals with the 1 -norm for large residuals. We also derive a discretized formula for the gradient of the Huber function. Through numericaltestsonsimplesyntheticmodelsandMarmousidata, we find the Huber function is more robust to outliers and coherent noise.We apply our waveform-inversion algorithm to field data taken from the continental shelf under the East Sea in Korea. In this setting, we obtain a velocity model whose syntheticshotprofilesaresimilartotherealseismicdata.

A frequency‐space 2-D scalar wave extrapolator using extended 25-point finite‐difference operator

Finite‐difference frequency‐domain modeling for the generation of synthetic seismograms and crosshole tomography has been an active field of research since the 1980s. The generation of synthetic seismograms with the time‐domain finite‐difference technique has achieved considerable success for waveform crosshole tomography and for wider applications in seismic reverse‐time migration. This became possible with the rapid development of high performance computers. However, the space‐frequency (x,ω) finite‐difference modeling technique is still beyond the capability of the modern supercomputer in terms of both cost and computer memory. Therefore, finite‐difference time‐domain modeling is much more popular among exploration geophysicists. A limitation of the space‐frequency domain is that the recently developed nine‐point scheme still requires that G, the number of grid points per wavelength, be 5. This value is greater than for most other numerical modeling techniques (for example, the pseudospectral scheme). ...

3D reverse-time migration using the acoustic wave equation: An experience with the SEG/EAGE data set

Kirchhoff is the most commonly used 3D prestack migration algorithm because of its speed and other economic advantages, but it uses a high-frequency ray approximation to the wave equation and, therefore, has difficulties in imaging complex geologic structures where multipathing occurs (e.g., beneath rugose horizons such as faulted salt domes where traveltime calculations become difficult).

Improved frequency‐domain elastic wave modeling using weighted‐averaging difference operators

We develop a new finite-difference scheme that reduces the number of grid points per wavelength required in frequency-domain elastic modeling. Our approach computes weighted averages of the spatial second-order derivative and the mass acceleration terms using a 25-point computational stencil. By determining the weighting coefficients to minimize numerical dispersion and numerical anisotropy, we reduce the number of grid points to 3.3 per shear wavelength, with a resulting error in velocities smaller than 1%. Our choice of grid points reduces the computer memory needed to store the complex impedance matrix to 4% of that for a conventional second-order scheme and to 54% of that for a combined second-order scheme. The 25-point weighted averaging scheme of this paper makes it possible to accurately simulate realistic models. Numerical examples show that this technique can achieve the same accurate solutions with fewer grid points than those from previous frequency-domain second-order schemes. Our technique can be extended directly to 3-D elastic modeling; the computational efficiency will be even greater than that realized for 2-D models.

Frequency-Domain Elastic Full Waveform Inversion Using the New Pseudo-Hessian Matrix: Experience of Elastic Marmousi-2 Synthetic Data

Abstract A proper scaling method allows us to find better solutions in waveform inversion, and it can also provide better images in true-amplitude migration methods based on a least-squares method. For scaling the gradient of a misfit function, we define a new pseudo-Hessian matrix by combining the conventional pseudo-Hessian matrix with amplitude fields. Because the conventional pseudo-Hessian matrix is assumed to neglect the zero-lag autocorrelation terms of impulse responses in the approximate Hessian matrix of the Gauss–Newton method, it has certain limitations in scaling the gradient of a misfit function relative to the approximate Hessian matrix. To overcome these limitations, we introduce amplitude fields to the conventional pseudo-Hessian matrix, and the new pseudo-Hessian matrix is applied to the frequency-domain elastic full waveform inversion. This waveform inversion algorithm follows the conventional procedures of waveform inversion using the backpropagation algorithm. A conjugate-gradient method is employed to derive an optimized search direction, and a backpropagation algorithm is used to calculate the gradient of the misfit function. The source wavelet is also estimated simultaneously with elastic parameters. The new pseudo-Hessian matrix can be calculated without the extra computational costs required by the conventional pseudo-Hessian matrix, because the amplitude fields can be readily extracted from forward modeling. Synthetic experiments show that the new pseudo-Hessian matrix provides better results than the conventional pseudo-Hessian matrix, and thus, we believe that the new pseudo-Hessian matrix is an alternative to the approximate Hessian matrix of the Gauss–Newton method in waveform inversion.

Traveltime and amplitude calculations using the damped wave solution

Because of its computational efficiency, prestack Kirchhoff depth migration remains the method of choice for all but the most complicated geological depth structures. Further improvement in computational speed and amplitude estimation will allow us to use such technology more routinely and generate better images. To this end, we developed a new, accurate, and economical algorithm to calculate first-arrival traveltimes and amplitudes for an arbitrarily complex earth model. Our method is based on numerical solutions of the wave equation obtained by using well-established finite-difference or finite-element modeling algorithms in the Laplace domain, where a damping term is naturally incorporated in the wave equation. We show that solving the strongly damped wave equation is equivalent to solving the eikonal and transport equations simultaneously at a fixed reference frequency, which properly accounts for caustics and other problems encountered in ray theory. Using our algorithm, we can easily calculate first-arrival traveltimes for given models. We present numerical examples for 2-D acoustic models having irregular topography and complex geological structure using a finite-element modeling code.