Hyunggu Jun

3D Laplace-domain full waveform inversion using a single GPU card

Abstract The Laplace-domain full waveform inversion is an efficient long-wavelength velocity estimation method for seismic datasets lacking low-frequency components. However, to invert a 3D velocity model, a large cluster of CPU cores have commonly been required to overcome the extremely long computing time caused by a large impedance matrix and a number of source positions. In this study, a workstation with a single GPU card (NVIDIA GTX 580) is successfully used for the 3D Laplace-domain full waveform inversion rather than a large cluster of CPU cores. To exploit a GPU for our inversion algorithm, the routine for the iterative matrix solver is ported to the CUDA programming language for forward and backward modeling parts with minimized modification of the remaining parts, which were originally written in Fortran 90. Using a uniformly structured grid set, nonzero values in the sparse impedance matrix can be arranged according to certain rules, which efficiently parallelize the preconditioned conjugate gradient method for a number of threads contained in the GPU card. We perform a numerical experiment to verify the accuracy of a floating point operation performed by a GPU to calculate the Laplace-domain wavefield. We also measure the efficiencies of the original CPU and modified GPU programs using a cluster of CPU cores and a workstation with a GPU card, respectively. Through the analysis, the parallelized inversion code for a GPU achieves the speedup of 14.7 – 24.6 x compared to a CPU-based serial code depending on the degrees of freedom of the impedance matrix. Finally, the practicality of the proposed algorithm is examined by inverting a 3D long-wavelength velocity model using wide azimuth real datasets in 3.7 days.

Smooth 2‐D ocean sound speed from Laplace and Laplace‐Fourier domain inversion of seismic oceanography data

In seismic oceanography, processed images highlight small temperature changes, but inversion is needed to obtain absolute temperatures. Local search-based full waveform inversion has a lower computational cost than global search but requires accurate starting models. Unfortunately, most marine seismic data have little associated hydrographic data and the band-limited nature of seismic data makes extracting the long wavelength sound speed trend directly from seismic data inherently challenging. Laplace and Laplace-Fourier domain inversion (LDI) can use rudimentary starting models without prior information about the medium. Data are transformed to the Laplace domain, and a smooth sound speed model is extracted by examining the zero and low frequency components of the damped wavefield. We applied LDI to five synthetic data sets based on oceanographic features and recovered smoothed versions of our synthetic models, showing the viability of LDI for creating starting models suitable for more detailed inversions.

Regularized Laplace–Fourier-Domain Full Waveform Inversion Using a Weighted l 2 Objective Function

Full waveform inversion (FWI) can be applied to obtain an accurate velocity model that contains important geophysical and geological information. FWI suffers from the local minimum problem when the starting model is not sufficiently close to the true model. Therefore, an accurate macroscale velocity model is essential for successful FWI, and Laplace–Fourier-domain FWI is appropriate for obtaining such a velocity model. However, conventional Laplace–Fourier-domain FWI remains an ill-posed and ill-conditioned problem, meaning that small errors in the data can result in large differences in the inverted model. This approach also suffers from certain limitations related to the logarithmic objective function. To overcome the limitations of conventional Laplace–Fourier-domain FWI, we introduce a weighted l2 objective function, instead of the logarithmic objective function, as the data-domain objective function, and we also introduce two different model-domain regularizations: first-order Tikhonov regularization and prior model regularization. The weighting matrix for the data-domain objective function is constructed to suitably enhance the far-offset information. Tikhonov regularization smoothes the gradient, and prior model regularization allows reliable prior information to be taken into account. Two hyperparameters are obtained through trial and error and used to control the trade-off and achieve an appropriate balance between the data-domain and model-domain gradients. The application of the proposed regularizations facilitates finding a unique solution via FWI, and the weighted l2 objective function ensures a more reasonable residual, thereby improving the stability of the gradient calculation. Numerical tests performed using the Marmousi synthetic dataset show that the use of the weighted l2 objective function and the model-domain regularizations significantly improves the Laplace–Fourier-domain FWI. Because the Laplace–Fourier-domain FWI is improved, the frequency-domain FWI, in which the Laplace–Fourier-domain FWI result is used as the starting model, yields inversion result much closer to the true velocity.

Frequency-domain waveform modelling and inversion for coupled media using a symmetric impedance matrix

ABSTRACT To simulate the seismic signals that are obtained in a marine environment, a coupled system of both acoustic and elastic wave equations is solved. The acoustic wave equation for the fluid region simulates the pressure field while minimizing the number of degrees of freedom of the impedance matrix, and the elastic wave equation for the solid region simulates several elastic events, such as shear waves and surface waves. Moreover, by combining this coupled approach with the waveform inversion technique, the elastic properties of the earth can be inverted using the pressure data obtained from the acoustic region. However, in contrast to the pure acoustic and elastic cases, the complex impedance matrix for the coupled media does not have a symmetric form because of the boundary (continuity) condition at the interface between the acoustic and elastic elements. In this study, we propose a manipulation scheme that makes the complex impedance matrix for acoustic–elastic coupled media to take a symmetric form. Using the proposed symmetric matrix, forward and backward wavefields are identical to those generated by the conventional approach; thus, we do not lose any accuracy in the waveform inversion results. However, to solve the modified symmetric matrix, LDLT factorization is used instead of LU factorization for a matrix of the same size; this method can mitigate issues related to severe memory insufficiency and long computation times, particularly for large‐scale problems.

Application of full waveform inversion algorithms to seismic data lacking low-frequency information from a simple starting model

Full waveform inversion (FWI) is a method that is used to reconstruct velocity models of the subsurface. However, this approach suffers from the local minimum problem during optimisation procedures. The local minimum problem is caused by several issues (e.g. lack of low-frequency information and an inaccurate starting model), which can create obstacles to the practical application of FWI with real field data. We applied a 4-phase FWI in a sequential manner to obtain the correct velocity model when a dataset lacks low-frequency information and the starting velocity model is inaccurate. The first phase is Laplace-domain FWI, which inverts the large-scale velocity model. The second phase is Laplace-Fourier-domain FWI, which generates a large- to mid-scale velocity model. The third phase is a frequency-domain FWI that uses a logarithmic wavefield; the inverted velocity becomes more accurate during this step. The fourth phase is a conventional frequency-domain FWI, which generates an improved velocity model with correct values. The detailed methods of applying each FWI phase are explained, and the proposed method is validated via numerical tests with a SEG/EAGE salt synthetic dataset and Gulf of Mexico field dataset. The numerical tests show that the 4-phase FWI inverts the velocity correctly despite the lack of low-frequency information and an inaccurate starting velocity model both in synthetic data and field data. The low-frequency information and correct starting velocity are important for full waveform inversion (FWI). However, obtaining low-frequency information and accurate starting velocity from the field seismic exploration is difficult. This paper suggests a 4-phase FWI to invert the correct velocity model when a dataset lacks low-frequency information and accurate starting velocity.