Yunseok Choi

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.

Frequency‐domain waveform inversion using the unwrapped phase

Phase wrapping in the frequency-domain (or cycle skipping in the time-domain) is the major cause of the local minima problem in the waveform inversion. The unwrapped phase has the potential to provide us with a robust and reliable waveform inversion, with reduced local minima. We propose a waveform inversion algorithm using the unwrapped phase objective function in the frequencydomain. The unwrapped phase, or what we call the instantaneous traveltime, is given by the imaginary part of dividing the derivative of the wavefield with respect to the angular frequency by the wavefield itself. As a result, the objective function is given a traveltime-like function, which allows us to smooth it and reduce its nonlinearity. The gradient of the objective function is computed using the back-propagation algorithm based on the adjoint-state technique. We apply both our waveform inversion algorithm using the unwrapped phase and the conventional waveform inversion and show that our inversion algorithm gives better convergence to the true model than the conventional waveform inversion.

Automatic picking of the first arrival event using the unwrapped‐phase of the Fourier transformed wavefield

First-arrival picking has long suffered from cycle skipping, especially when the first arrival is contaminated with noise or have experienced complex near surface phenomena. We propose a new algorithm for automatic picking of first arrivals using an approach based on unwrapping the phase. We unwrap the phase by taking the derivative of the Fourier-transformed wavefield with respect to the angular frequency and isolate its amplitude component. To do so, we first apply a damping function to the seismic trace, calculate the derivative of the wavefield with respect to the angular frequency, divide the derivative of wavefield by the wavefield itself, and finally take its imaginary part. We compare our derivative approach to the logarithmic one and show that the derivative approach does not suffer from the phase wrapping or cycle-skipping effects. Numerical examples show that our automatic picking algorithm gives convergent and reliable results for the noise-free synthetic data and noisy field data.

Frquency‐domain elastic waveform inversion with irregular surface topography

In order to describe the irregular topography that is commonly encountered in land-based seismic exploration, we present a frequency-domain elastic wave modeling algorithm for this phenomenon, incorporated into an elastic waveform inversion algorithm. We use a finite-element method, both for the modeling and for the inversion algorithms, in which the main body is approximated by rectangular elements and irregular topography is described by triangular elements. In common with conventional finite-element modeling algorithms, our finiteelement irregular topography modeling algorithm also naturally satisfies the free-surface boundary condition due to the Neumann boundary condition, which is incorporated when we construct the finite-element formulae. For the inversion algorithm, we use the steepest-descent method, and scale the gradient direction using the diagonal of the pseudo-Hessian matrix rather than the approximate or full Hessian matrix. We apply the inversion technique to the AA-line of the SEG/EAGE salt dome model, modified to account for irregular surface topography. Through numerical examples, we demonstrate that our elastic waveform inversion can reproduce subsurface structures and elastic parameters fairly well, even for a model with irregular topography.