Sukjoon Pyun

Refraction traveltime tomography using damped monochromatic wavefield

For complicated earth models, wave-equation–based refraction-traveltime tomography is more accurate than ray-based tomography but requires more computational effort. Most of the computational effort in traveltime tomography comes from computing traveltimes and their Fr ´ echet derivatives, which for ray-based methods can be computed directly. However, in most wave-equation traveltime-tomography algorithms, the steepest descent direction of the objective function is computed by the backprojection algorithm, without computing a Fr ´ echet derivative directly. We propose a new wave-based refraction-traveltime– tomography procedure that computes Fr ´ echet derivatives directly and efficiently. Our method involves solving a damped-wave equation using a frequency-domain, finite-element modeling algorithm at a single frequency and invoking the reciprocity theorem. A damping factor, which is commonly used to suppress wraparound effects in frequency-domain modeling, plays the role of suppressing multievent wavefields. By limiting the wavefield to a single first arrival, we are able to extract the first-arrival traveltime from the phase term without applying a time window. Computing the partial derivative of the damped wave-equation solution using the reciprocity theorem enables us to compute the Fr ´ echet derivative of amplitude, as well as that of traveltime, with respect to subsurface parameters. Using the Marmousi-2 model, we demonstrate numerically that refraction traveltime tomography with large-offset data can be used to provide the smooth initial velocity model necessary for prestack depth migration.

Frequency-domain waveform inversion using an l1-norm objective function

In general, seismic waveform inversion adopts an objective function based on the l2-norm. However, waveform inversion using the l2-norm produces distorted results because the l2-norm is sensitive to statistically invalid data such as outliers. As an alternative, there have been several studies applying l1-norm-based objective functions to waveform inversion. Although waveform inversion based on the l1-norm is known to produce robust inversion results against specific outliers in the time domain, its effectiveness and characteristics are yet to be studied in the frequency domain. The present study proposes an algorithm for l1-norm-based waveform inversion in the frequency domain. The proposed algorithm employs a structure identical to those used in conventional frequency-domain waveform inversion algorithms that exploit the back-propagation technique, but displays robustness against outliers, which has been confirmed based on inversion of the synthetic Marmousi model. The characteristics and advantages of the l1-norm were analysed by comparing it with the l2-norm. In addition, inversion was performed on data containing outliers to examine the robustness against outliers. The effectiveness of removing outliers was verified by using the l1-norm to calculate the residual wavefield and its spectrum for the data containing outliers.

Efficient electric resistivity inversion using adjoint state of mixed finite-element method for Poisson's equation

We propose an electric resistivity inversion method that is similar to the reverse time migration technique applied to seismic data. For calculating model responses and inversion, we use the mixed finite-element method with the standard P1-P0 pair for triangular decompositions, which makes it possible to compute both the electric potential and the electric field vector economically. In order to apply the adjoint state of the Poisson equation in the resistivity inverse problem, we introduce an apparent electric field defined as the dot product between the computed electric field vector and a weighting factor and then defining a virtual source to compute the partial derivative of the electric field vector. We exploit the adjoint state (the symmetry of Greens function) of matrix equations derived from solving the Poisson equation by the mixed finite-element method, for the calculation of the steepest descent direction of our objective function. By computing the steepest descent direction by a dot product of backpropagated residual and virtual source, we can avoid the cumbersome and expensive process of computing the Jacobian matrix directly. We calibrate our algorithm on a synthetic of a buried conductive block and obtain an image that is compatible with the limits of the resistivity method.

3D Elastic Full Waveform Inversion In the Laplace Domain

We implement the Laplace-domain waveform inversion algorithm for 3D elastic media. In the Laplace domain, it is straightforward to obtain numerical solutions of the wave equation using an iterative solver, since the Laplacedomain wave equation behaves similarly to the Poisson equation. In this paper, the 3D elastic wave equation was formulated in a weak form by the finite element method and solved using the preconditioned conjugate gradient method. We adopted the logarithmic objective function in order to enhance the sensitivity of highly damped wavefields for a successful inversion, and then used the back-propagation algorithm to perform the inversion by the iterative local descent method. Through the synthetic examples, we confirmed that our algorithm was more efficient than the direct solver and resulted in a reasonable velocity model.

Efficient calculation of steepest descent direction for source-independent waveform inversion using normalized wavefield by convolution

Summary In conventional waveform inversion, geophysicists usually invert a velocity model as well as a source signature simultaneously. For a source-independent waveform inversion, we define a new objective function by multiplying both observed data and forward modeled data by the respective reference wavefields on the cross. For computation of the steepest descent of the new objective function, we exploit a matrix formalism originated from the symmetry of Green’s function of wave equation. In this case, we calculate the steepest descent without explicitly computing the Jacobian matrix. Numerical structure of our algorithm resembles that of prestack reverse-time migration and waveform inversion.

Application of Refraction Tomography Algorithms to Real Data

We applied two wave-based refraction traveltime tomography algorithms to a real, field data set obtained in Congo, Africa. The two wave-based refraction tomography methods extract traveltimes by taking the logarithm of damped wavefield. The damped wavefield can be assumed to have a single, first arrival when we choose an optimal damping factor in complex angular frequency. The main differences between two methods exist in computing the steepest-descent directions. One method computes Frechet derivative of traveltime directly; The other method introduces a backpropagation algorithm and virtual source concept, which is commonly employed in waveform inversion and reverse time migration. The two methods have already proven to give similar, good results for synthetic data. After applying the two wavebased tomography method to the real, field data, we can know that both of them describe high-velocity salt dome structure very clearly and give reliable solutions for real data. Having used the inverted velocity model in prestack Kirchhoff migration successfully, we can also confirm that the inverted velocity model obtained by the refraction tomography method can be effectively applied in prestack Kirchhoff migration.