Laplace-domain Waveform Inversion Research Articles

Handling Negative Values for the Logarithmic Objective Function in Acoustic Laplace-Domain Full-Waveform Inversion Using Real Variables

The Laplace-domain waveform inversion is a full-waveform inversion method that recovers large-scale subsurface models. The inversion updates subsurface model parameters to minimize the differences between the modeled and the observed wavefields in the Laplace domain. The inversion results can be used as an accurate initial model for subsequent high-resolution waveform inversions. Pure Laplace-domain wavefields can be obtained by transforming the time-domain signals using the Laplace transform of real variables. The real Laplace transform is mathematically identical to the Fourier transform using the imaginary angular frequency; however, the Laplace transform using only real variables is computationally more efficient than that using complex variables. The Laplace-transformed wavefields are real-valued signals, and thus, it is natural to use real values in the Laplace-domain waveform inversions. However, the real logarithm function in the logarithmic objective function cannot handle negative values. Inversions using complex logarithms can solve this problem, but they demand more memory and computations than those required for inversions using real variables only. We suggest a simple method to overcome the negative-value problem for the real logarithm in the objective function. By taking the absolute values of the negative signals in the logarithmic objective function, we can obtain inversion results from inversions using real variables only that are equivalent to those from inversions using complex variables. We demonstrate the proposed method using the Society of Exploration Geophysicists (SEG)/European Associations of Geoscientists & Engineers (EAGE) salt model and a field data set. The inversions using real variables only took less than 22% of the time of the inversion using complex variables in the numerical examples.

A new method for numerical dispersion analysis of Laplace-domain 2-D elastic wave equation

Elastic wave modelling in the Laplace domain is the foundation of Laplace-domain elastic wave full waveform inversion. In order to use Laplace-domain numerical modelling schemes efficiently, appropriate grid intervals must be chosen correctly. The determination of grid intervals is based on numerical dispersion analysis of Laplace-domain elastic wave equation. A new method of numerical dispersion analysis is developed for Laplace-domain 2-D elastic wave equation. The novelty of the method lies in two aspects: (1) Based on the concept of the pseudo-wavelength for both P- and S-waves, I introduce a general quantity which is a combination of the Laplace constant, velocity and grid interval. Therefore, the dispersion relations no longer directly depend on the concrete Laplace constant, velocity and grid interval. Just like the frequency-domain dispersion analysis, a general conclusion with regard to the general quantity can be made; (2) The ratio of numerical eigenvalue to analytical eigenvalue can be interpreted as the normalised attenuation propagation velocity. The number of grid points per S-wave pseudo-wavelength is determined by the accuracy of normalised P- and S-wave attenuation propagation velocities. Based on a commonly used finite-element scheme, the numbers of grid points per S-wave pseudo-wavelength are determined for different Poisson ratios and for different ratios of directional grid intervals. These results are important in applying the finite-element scheme to Laplace-domain elastic wave modelling and full waveform inversion. Comparisons with the analytical solution validate the criterion on the numbers of grid points per S-wave pseudo-wavelength.

Interrelation between Laplace constants and the gradient distortion effect in Laplace-domain waveform inversion

Laplace-domain waveform inversion (WI) is generally used to generate smooth initial velocity models for frequency- or time-domain full-waveform inversion. However, in the inversion results of Laplace-domain WI, anomalies such as salt domes are sometimes shifted. We evaluate the “gradient-distortion effect” that causes undesirable changes in parameter updates and found that this is caused by the relationship between the partial derivatives of Laplace wavefields with respect to two different parameters. By analyzing the gradient of the Laplace-domain misfit function, we found that the gradient distortion effect increases as the Laplace constants used in the Laplace-domain WI decrease. The velocity model inverted in the Laplace domain is generally blurred from shallower parameters to deeper parameters because the partial derivatives of the Laplace wavefields with respect to shallower parameters tend to be larger than those of deeper parameters. We found two solutions for suppressing the gradient distortion effect. The first one is the sequentially ordered Laplace constant approach with multiple Laplace constants. We discover that a dense, broad set of Laplace constants should be sequentially used in this approach. The second solution is the Gauss-Newton method, in which the Hessian matrix is considered. Numerical tests performed using a four-layer model and the BP benchmark model show the gradient distortion effect appeared in the inversion results and the effectiveness of the sequentially ordered Laplace constant approach. In addition, tests using an inverted BP benchmark model determine that the inversion results can be improved by applying a broad, dense set of Laplace constants to synthetic data. Finally, we verify the effectiveness of the Gauss-Newton method at suppressing the gradient distortion effect using the BP benchmark model.

Laplace-domain waveform modeling and inversion for the 3D acoustic–elastic coupled media

Laplace-domain waveform inversion reconstructs long-wavelength subsurface models by using the zero-frequency component of damped seismic signals. Despite the computational advantages of Laplace-domain waveform inversion over conventional frequency-domain waveform inversion, an acoustic assumption and an iterative matrix solver have been used to invert 3D marine datasets to mitigate the intensive computing cost. In this study, we develop a Laplace-domain waveform modeling and inversion algorithm for 3D acoustic–elastic coupled media by using a parallel sparse direct solver library (MUltifrontal Massively Parallel Solver, MUMPS). We precisely simulate a real marine environment by coupling the 3D acoustic and elastic wave equations with the proper boundary condition at the fluid-solid interface. In addition, we can extract the elastic properties of the Earth below the sea bottom from the recorded acoustic pressure datasets. As a matrix solver, the parallel sparse direct solver is used to factorize the non-symmetric impedance matrix in a distributed memory architecture and rapidly solve the wave field for a number of shots by using the lower and upper matrix factors. Using both synthetic datasets and real datasets obtained by a 3D wide azimuth survey, the long-wavelength component of the P-wave and S-wave velocity models is reconstructed and the proposed modeling and inversion algorithm are verified. A cluster of 80CPU cores is used for this study.

2D Laplace-Domain Waveform Inversion of Field Data Using a Power Objective Function

The wavefield in the Laplace domain has a very small amplitude except only near the source point. In order to deal with this characteristic, the logarithmic objective function has been used in many Laplace domain inversion studies. The Laplace-domain waveform inversion using the logarithmic objective function has fewer local minima than the time- or frequency domain inversion. Recently, the power objective function was suggested as an alternative to the logarithmic objective function in the Laplace domain. Since amplitudes of wavefields are very small generally, a power <1 amplifies the wavefields especially at large offset. Therefore, the power objective function can enhance the Laplace-domain inversion results. In previous studies about synthetic datasets, it is confirmed that the inversion using a power objective function shows a similar result when compared with the inversion using a logarithmic objective function. In this paper, we apply an inversion algorithm using a power objective function to field datasets. We perform the waveform inversion using the power objective function and compare the result obtained by the logarithmic objective function. The Gulf of Mexico dataset is used for the comparison. When we use a power objective function in the inversion algorithm, it is important to choose the appropriate exponent. By testing the various exponents, we can select the range of the exponent from 5 × 10−3 to 5 × 10−8 in the Gulf of Mexico dataset. The results obtained from the power objective function with appropriate exponent are very similar to the results of the logarithmic objective function. Even though we do not get better results than the conventional method, we can confirm the possibility of applying the power objective function for field data. In addition, the power objective function shows good results in spite of little difference in the amplitude of the wavefield. Based on these results, we can expect that the power objective function will produce good results from the data with a small amplitude difference. Also, it can partially be utilized at the sections where the amplitude difference is very small.

3D acoustic modelling and waveform inversion in the Laplace domain for an irregular sea floor using the Gaussian quadrature integration method

Abstract This paper explores the concept of 3D acoustic modelling and inversion in the Laplace domain. We used the Gaussian quadrature integration method to assemble the element mass matrix of a coarse regular rectangular element containing both water and subsurface parts, approximating the properties of these two parts without refining the element into smaller components. The purpose of this investigation was to model an irregular sea floor using 3D Laplace-domain modelling and inversion without increasing the computational cost incurred when using a small grid interval to precisely delineate the sea floor. The algorithm was verified by comparing the analytical solution to the modelled wavefield obtained using the conventional Laplace-domain modelling method and the modelled wavefield obtained using the Gaussian quadrature integration in a two-layer model (with 50-m, 100-m, and 200-m grid spacing and 5, 10, 20, and 40 Gaussian quadrature integration points) where an oblique water–subsurface interface is given. The results show that, under certain conditions, the Gaussian quadrature integration method can generate relatively accurate and consistent wavefields compared to the conventional modelling method. Moreover, we performed 3D Laplace-domain inversion with this Gaussian quadrature modelling technique to obtain synthetic data with a flat and irregular interface using a 100-m and 200-m grid. The results obtained under the conditions herein show that 3D Laplace-domain waveform inversion with Gaussian quadrature integration can generate a more accurate and cost-efficient velocity model compared to that obtained using conventional Laplace-domain waveform inversion.

Laplace-domain full-waveform inversion of seismic data lacking low-frequency information

The lack of the low-frequency information in field data prohibits the time- or frequency-domain waveform inversions from recovering large-scale background velocity models. On the other hand, Laplace-domain waveform inversion is less sensitive to the lack of the low frequencies than conventional inversions. In theory, frequency filtering of the seismic signal in the time domain is equivalent to a constant multiplication of the wavefield in the Laplace domain. Because the constant can be retrieved using the source estimation process, the frequency content of the seismic data does not affect the gradient direction of the Laplace-domain waveform inversion. We obtained inversion results of the frequency-filtered field data acquired in the Gulf of Mexico and two synthetic data sets obtained using a first-derivative Gaussian source wavelet and a single-frequency causal sine function. They demonstrated that Laplace-domain inversion yielded consistent results regardless of the frequency content within the seismic data.

Laplace-domain waveform inversion versus refraction-traveltime tomography

SUMMARY Geophysicists and applied mathematicians have proposed a rich suite of long-wavelength velocity estimation algorithms to construct starting velocity models for subsequent pre-stack depth migration and inversion. Refraction-traveltime tomography derives subsurface velocity models from picked first-arrival traveltimes. In contrast, Laplace-domain waveform inversion recovers long-wavelength velocity structure using the weighted amplitudes of first and later arrivals. There are several implementations of first-arrival traveltime inversion, with most based on ray tracing, and some based on the damped monochromatic wave equation, which accurately represent simple and finite-frequency first arrivals. Computationally, Laplace-domain wavefield inversion is quite similar to refraction-traveltime tomography using damped monochromatic wavefield, but the objective functions used in inversion are radically different. As in classical ray trace-based traveltime inversion, the objective of refraction-traveltime tomography using damped monochromatic wavefield is to match the phase (traveltime) of the first arrival of each measured seismic trace. In contrast, the objective of Laplace-domain wavefield inversion is to match the weighted amplitudes of both first and later arrivals to the weighted amplitudes of the measured seismic trace. Principles of refraction-traveltime tomography were used to generate velocity models of the earth one century ago. Laplace-domain waveform inversion is a more recently introduced algorithm and has been less rigorously studied by the seismic research community, with many workers believing it be equivalent to finite-frequency first-arrival traveltime tomography. We show that Laplace-domain waveform inversion is both theoretically and empirically different from finite-frequency first-arrival traveltime tomography. Specifically, we examine the Jacobian (sensitivity) kernels used in the two inversion schemes to quantify the different sensitivities (and hence the inversion results) of the two methods. Analysing both surface responses and sensitivity results, we show that the Laplace-domain waveform inversion's sensitivity to later arrivals provides significantly improved resolution of deeper velocity structure than the first-arrival traveltime tomography. We demonstrate this capability using numerical inversion examples using a synthetic five-layer model and the synthetic BP benchmark model. Because of the similar algorithmic structure, Laplace-domain waveform inversion fits neatly as a starting velocity model pre-processing component of a larger (multi) frequency-domain wave equation inversion solution package.

2-D acoustic Laplace-domain waveform inversion of marine field data

SUMMARY The Laplace-domain full waveform inversion method can build a macroscale subsurface velocity model that can be used as an accurate initial model for a conventional full waveform inversion. The acoustic Laplace-domain inversion produced is promising for marine field data examples. Although applying an acoustic inversion method to the field data generally requires several pre-processing steps, pre-processing for the Laplace-domain inversion has not been explained in detail. We provide a detailed explanation of how to apply the Laplace-domain waveform inversion to field data through numerical tests with Gulf of Mexico data sets. The pre-processing includes bandpass filtering, muting of the noise before the first arrival, and extraction of the water depth. We choose the range and the interval between the Laplace damping constants empirically by applying a threshold value to the damped time traces and the Laplace-domain wavefields. The observed data are transformed to the Laplace domain using the selected damping; this method yielded a long-wavelength inversion result. The damping constant and the maximum offset affect the penetration depth of the inversion result. The maximum recording time is important for a stable Laplace-transformation and affects the inversion result; however, the latter effect is not significant.

Source estimation and direct wave reconstruction in Laplace-domain waveform inversion for deep-sea seismic data

SUMMARY We propose a strategy to overcome the high sensitivity to early-time noise of the Laplace-domain waveform inversion. In deep-sea seismic data, this problem is particularly crucial to obtaining accurate velocity structures. To this end, rather than simply filtering or muting early-time data, we propose replacing the original, noise-contaminated direct waves with analytically computed noise-free waves. To reconstruct the noise-free direct waves, we compute Green's functions for half-space media, estimate the source wavelet from the original direct waves using the full Newton method in the frequency domain and then convolve the Green's functions with the estimated source wavelet. The data obtained by merging the reconstructed direct waves with the original late-time data set can then be used for Laplace- and Laplace-Fourier-domain waveform inversions. To verify the source estimation and direct wave reconstruction strategy, we applied it to field data acquired in a deep-sea environment and obtained a realistic 2-D velocity model. The source estimation and direct wave reconstruction methods can also be applied to 3-D Laplace-domain waveform inversion.

2D Elastic Waveform Inversion in the Laplace Domain

Abstract There are many obstacles to applying waveform inversion to seismic data. However, the most critical factor is the absence of the low-frequency components that are needed for constructing long-wavelength structure. This problem stems from the highly nonlinear property of waveform inversion, which causes the algorithm to be trapped in a local minimum. The waveform inversion in the Laplace domain, rather than the usual frequency domain, is capable of producing velocity models with long-wavelength information. A study on this method was recently published, which was limited to the problem of acoustic media. In this paper, we extend Laplace-domain waveform inversion to elastic media. Unlike acoustic inversion, elastic inversion requires sophisticated manipulation of the gradient direction. We suggest a method to modify pseudo-Hessian matrices by using a heuristic weighting function. We test our inversion algorithm on synthetic seismic data generated using the Society of Exploration Geophysicists/European Association of Geoscientists & Engineers (SEG/EAGE) salt-dome model and the Commission on Controlled-Source Seismology (CCSS) model. Inversion results using these data sets also produce the long-wavelength velocity model and demonstrate that Laplace-domain waveform inversion is robust to the initial velocity model. Furthermore, we provide an example showing that our inverted result is a suitable initial model for the frequency-domain waveform inversion.

3D acoustic waveform inversion in the Laplace domain using an iterative solver

In this paper we propose a 3D acoustic full waveform inversion algorithm in the Laplace domain. The partial differential equation for the 3D acoustic wave equation in the Laplace domain is reformulated as a linear system of algebraic equations using the finite element method and the resulting linear system is solved by a preconditioned conjugate gradient method. The numerical solutions obtained by our modelling algorithm are verified through a comparison with the corresponding analytical solutions and the appropriate dispersion analysis. In the Laplace-domain waveform inversion, the logarithm of the Laplace transformed wavefields mainly contains long-wavelength information about the underlying velocity model. As a result, the algorithm smoothes a small-scale structure but roughly identifies large-scale features within a certain depth determined by the range of offsets and Laplace damping constants employed. Our algorithm thus provides a useful complementary process to time- or frequency-domain waveform inversion, which cannot recover a large-scale structure when low-frequency signals are weak or absent. The algorithm is demonstrated on a synthetic example: the SEG/EAGE 3D salt-dome model. The numerical test is limited to a Laplace-domain synthetic data set for the inversion. In order to verify the usefulness of the inverted velocity model, we perform the 3D reverse time migration. The migration results show that our inversion results can be used as an initial model for the subsequent high-resolution waveform inversion. Further studies are needed to perform the inversion using time-domain synthetic data with noise or real data, thereby investigating robustness to noise.

Two-dimensional Laplace-domain waveform inversion using adaptive meshes: an experience of the 2004 BP velocity-analysis benchmark data set

SUMMARY To delineate the Earth's structure precisely by time- or frequency-domain waveform inversions, an appropriate initial model that contains long-wavelength structures of the true model is essential. A Laplace-domain waveform inversion, which can provide a smooth velocity model that is equivalent to a long-wavelength velocity model from scratch, was recently proposed. The Laplace-domain waveform inversion can be used for obtaining a good initial model for full-waveform inversions. To improve the applicability of the 2-D Laplace-domain waveform inversion to field data, we adopted the adaptive finite element not only to deal with sources and receivers at shallow depths but also to reduce the computational burden using larger elements below the sources and receivers. For the boundary condition on the bounded domain for the Laplace-domain modelling and inversion, the unique property of the Laplace-domain wavefield, which was damped rapidly from a source, was taken into account. The computational domain was extended further from a source and then the Dirichlet boundary condition was applied to the outer edges of the extended model. With these two adaptations, our Laplace-domain waveform inversion could then be tested on a very large 2004 BP velocity-analysis benchmark data set. Numerical tests of the BP model, which contains three salt domes of high-velocity contrast, were conducted on a self-made synthetic data set by Laplace-domain modelling and a time-domain original data set. For the two data sets, a Laplace-domain waveform inversion using the adaptive mesh successfully provided a smooth velocity model that contained long-wavelength structures of the true velocity model. The smooth velocity model obtained by the Laplace-domain waveform inversion was used as a good initial model for the frequency-domain waveform inversion of the original BP data set. The resulting frequency-domain inversion recovered almost every feature of the salt domes except for some subsalt structures.

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 made it difficult to approach the true global minimum. In this study, we compared the behavior of objective functions in the 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. We applied the waveform inversion to a 2D slice of 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 Petrole Marmousi models with the frequency-domain inversion method by using the inverted velocity model of the Laplace-domain inversion as the initial model.