Frequency-domain Full Waveform Inversion Research Articles

Edge-preserving smoothing for simultaneous-source full-waveform inversion model updates in high-contrast velocity models

Full-waveform inversion (FWI) can provide accurate estimates of subsurface model parameters. In spite of its success, the application of FWI in areas with high-velocity contrast remains a challenging problem. Quadratic regularization methods are often adopted to stabilize inverse problems. Unfortunately, edges and sharp discontinuities are not adequately preserved by quadratic regularization techniques. Throughout the iterative FWI method, an edge-preserving filter, however, can gently incorporate sharpness into velocity models. For every point in the velocity model, edge-preserving smoothing assigns the average value of the most uniform window neighboring the point. Edge-preserving smoothing generates piecewise-homogeneous images with enhanced contrast at boundaries. We adopt a simultaneous-source frequency-domain FWI, based on quasi-Newton optimization, in conjunction with an edge-preserving smoothing filter to retrieve high-contrast velocity models. The edge-preserving smoothing filter gradually removes the artifacts created by simultaneous-source encoding. We also have developed a simple model update to prevent disrupting the convergence of the optimization algorithm. Finally, we perform tests to examine our algorithm.

Frequency-domain full-waveform inversion with non-linear descent directions

Frequency-domain full-waveform inversion with non-linear descent directions

Gas hydrate quantification using full-waveform inversion of sparse ocean-bottom seismic data: A case study from Green Canyon Block 955, Gulf of Mexico

We present a case study of gas hydrate quantification using dense short-offset multichannel seismic (MCS) and sparse long-offset ocean-bottom-seismometer (OBS) data in lease block Green Canyon 955 (GC955), Gulf of Mexico (GOM), where the presence of gas hydrate was interpreted using logging while drilling (LWD) data acquired by the GOM Gas Hydrate Joint Industry Project Leg II expedition. We use frequency-domain full-waveform inversion (FWI) of seven OBS gathers to invert for a P-wave velocity model of an approximately 7 km long MCS profile connecting two LWD sites, GC955-H and GC955-Q. We build the starting model for FWI using traveltime inversion (TI) of the MCS and OBS data. In addition, we use the TI model for depth migrating the MCS stack. At the LWD sites, we constrain the hydrate saturation ([Formula: see text]) using sonic and resistivity logs. Unfortunately, as is typical of seismic quantification problems, the FWI model resolution is not sufficient to extrapolate the LWD-based [Formula: see text]. Therefore, we apply Backus averaging to the sonic log, at 60 m wavelength, bringing it within approximately 8% of the FWI model and make the assumption that averaging the sonic log is same as redistributing the gas hydrate within the Backus wavelength. In this manner, instead of [Formula: see text], the FWI model is able to estimate the total gas hydrate volume. In the end, we use the FWI model and the migrated stack to constrain the locations and bulk volumes of free gas and gas hydrate. Our results demonstrate that with careful processing, reasonable estimates on locations and bulk volumes of submarine gas hydrate accumulations can be achieved even with sparse seismic data that are not adequate for amplitude-based assessments.

A developed nearly analytic discrete method for forward modeling in the frequency domain

High-efficiency forward modeling methods play a fundamental role in full waveform inversion (FWI). In this paper, the developed nearly analytic discrete (DNAD) method is proposed to accelerate frequency-domain forward modeling processes. We first derive the discretization of frequency-domain wave equations via numerical schemes based on the nearly analytic discrete (NAD) method to obtain a linear system. The coefficients of numerical stencils are optimized to make the linear system easier to solve and to minimize computing time. Wavefield simulation and numerical dispersion analysis are performed to compare the numerical behavior of DNAD method with that of the conventional NAD method. The results demonstrate the superiority of our proposed method. Finally, the DNAD method is implemented in frequency-domain FWI, and high-resolution inverse results are obtained.

Multi-parameter full waveform inversion using Poisson’s ratio for elastic media

In multi-parameter full waveform inversion (FWI), the success of recovering each parameter is dependent on characteristics of the partial derivative wavefields (or virtual sources), which differ according to parameterisation. Elastic FWIs based on the two conventional parameterisations (one uses Lamé constants and density; the other employs P- and S-wave velocities and density) have low resolution of gradients for P-wave velocities (or λ). Limitations occur because the virtual sources for P-wave velocity or λ (one of the Lamé constants) are related only to P–P diffracted waves, and generate isotropic explosions, which reduce the spatial resolution of the FWI for these parameters. To increase the spatial resolution, we propose a new parameterisation using P-wave velocity, Poisson’s ratio, and density for frequency-domain multi-parameter FWI for isotropic elastic media. By introducing Poisson’s ratio instead of S-wave velocity, the virtual source for the P-wave velocity generates P–S and S–S diffracted waves as well as P–P diffracted waves in the partial derivative wavefields for the P-wave velocity. Numerical examples of the cross–triangle–square (CTS) model indicate that the new parameterisation provides highly resolved descent directions for the P-wave velocity. Numerical examples of noise-free and noisy data synthesised for the elastic Marmousi-II model support the fact that the new parameterisation is more robust for noise than the two conventional parameterisations.

Quantification of thickness loss in a liquid-loaded plate using ultrasonic guided wave tomography

Ultrasonic guided wave tomography (GWT) provides an attractive solution to map thickness changes from remote locations. It is based on the velocity-to-thickness mapping employing the dispersive characteristics of selected guided modes. This study extends the application of GWT on a liquid-loaded plate. It is a more challenging case than the application on a free plate, due to energy of the guided waves leaking into the liquid. In order to ensure the accuracy of thickness reconstruction, advanced forward models are developed to consider attenuation effects using complex velocities. The reconstruction of the thickness map is based on the frequency-domain full waveform inversion (FWI) method, and its accuracy is discussed using different frequencies and defect dimensions. Validation experiments are carried out on a water-loaded plate with an irregularly shaped defect using S0 guided waves, showing excellent performance of the reconstruction algorithm.

Comparison of weighting techniques for acoustic full waveform inversion

To reconstruct long-wavelength structures in full waveform inversion (FWI), the wavefield-damping and weighting techniques have been used to synthesize and emphasize low-frequency data components in frequency-domain FWI. However, these methods have some weak points. The application of wavefield-damping method on filtered data fails to synthesize reliable low-frequency data; the optimization formula obtained introducing the weighting technique is not theoretically complete, because it is not directly derived from the objective function. In this study, we address these weak points and present how to overcome them. We demonstrate that the source estimation in FWI using damped wavefields fails when the data used in the FWI process does not satisfy the causality condition. This phenomenon occurs when a non-causal filter is applied to data. We overcome this limitation by designing a causal filter. Also we modify the conventional weighting technique so that its optimization formula is directly derived from the objective function, retaining its original characteristic of emphasizing the low-frequency data components. Numerical results show that the newly designed causal filter enables to recover long-wavelength structures using low-frequency data components synthesized by damping wavefields in frequency-domain FWI, and the proposed weighting technique enhances the inversion results.

Frequency-domain full waveform inversion with an angle-domain wavenumber filter

To mitigate the local minima problem in full waveform inversion, the iteration process can be carried out in a multi-scale way, with the inversion starting from large-scale perturbations and gradually changing to small-scale structures. However, this technique is often limited due to the lack of low-frequency information in the observed data. In a scattering process, the scale length of the velocity perturbation is not only related to the frequency, but also associated with the scattering angles. We introduce a scattering angle filter in frequency-domain full waveform inversion. Choosing proper scattering angles in the iteration process can control the wavenumber components entering into the velocity updating, and accomplish multi-scale waveform inversion. Using the slant-stacking method, we decompose the source- and receiver-side waves into local plane waves, from which the scattering angles between the incident and scattering directions are calculated and used as the constraint to determine the wavenumber components to be retrieved. Small scattering angles are related to large-scale model perturbations. Therefore, by filtering out large scattering angles, we can invert the large-scale background perturbations first. Then, by gradually increasing the band pass of scattering angles, we can retrieve fine-scale structures. Numerical examples demonstrate that when the initial model has large velocity errors and the low-frequency information is unavailable in the data, the multi-scale strategy based on the angle-domain wavenumber filter can largely improve the convergence in the initial iteration stage and make the searching towards the global minimum.

Depth Scaling Strategy Using a Flexible Damping Factor for Frequency-Domain Elastic Full Waveform Inversion

We introduce a depth scaling strategy to improve the accuracy of frequency-domain elastic full waveform inversion (FWI) using the new pseudo-Hessian matrix for seismic data without low frequency components. The depth scaling strategy is based on the fact that the damping factor in the Levenberg-Marquardt method controls the energy concentration in the gradient. In other words, a large damping factor makes the Levenberg Marquardt method similar to the steepest-descent method, by which shallow structures are mainly recovered. With a small damping factor, the Levenberg-Marquardt method becomes similar to the Gauss-Newton methods by which we can resolve deep structures as well as shallow structures. In our depth scaling strategy, a large damping factor is used in the early stage and then decreases automatically with the trend of error as the iteration goes on. With the depth scaling strategy, we can gradually move the parameter-searching region from shallow to deep parts. This flexible damping factor plays a role in retarding the model parameter update for shallow parts and mainly inverting deeper parts in the later stage of inversion. By doing so, we can improve deep parts in inversion results. The depth scaling strategy is applied to synthetic data without lowfrequency components for a modified version of the SEG/EAGE overthrust model. Numerical examples show that the flexible damping factor yields better results than the constant damping factor when reliable low frequency components are missing.

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 l 2 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 l 2 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 l 2 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.

A Bayesian approach to estimate uncertainty for full-waveform inversion using a priori information from depth migration

Full-waveform inversion (FWI) enables us to obtain high-resolution subsurface images; however, estimating model uncertainties associated with this technique is still a challenging problem. We have used a Bayesian inference framework to estimate model uncertainties associated with FWI. The uncertainties were assessed based on an a posteriori covariance operator, evaluated at the maximum a posteriori model. For the prior distribution, we have used a spatially nonstationary covariance operator based on a plane-wave construction with local dips measured from migrated images. Preconditioned frequency-domain FWI was used to estimate the maximum a posteriori model. Efficient manipulation of the posterior covariance was based on a low-rank approximation of the data misfit Hessian preconditioned by the prior covariance operator. The strong decay of the singular values indicated that data were mostly informative about a low-dimensional subspace of model parameters. To reduce computational cost of the randomized singular value decomposition, we have used a Hessian approximation based on point-spread functions. The 2D numerical examples with synthetic data confirmed that the method can effectively estimate uncertainties for FWI. Visual comparisons of random samples drawn from the prior and posterior distributions have allowed us to evaluate model uncertainties. Furthermore, we found out how statistical quantities, such as means and pointwise standard deviation fields, can be efficiently extracted from the prior and posterior distributions. These fields helped us to objectively assess subsurface images provided by FWI.

Fast 3D frequency-domain full-waveform inversion with a parallel block low-rank multifrontal direct solver: Application to OBC data from the North Sea

Wide-azimuth long-offset ocean bottom cable (OBC)/ocean bottom node surveys provide a suitable framework to perform computationally efficient frequency-domain full-waveform inversion (FWI) with a few discrete frequencies. Frequency-domain seismic modeling is performed efficiently with moderate computational resources for a large number of sources with a sparse multifrontal direct solver (Gauss-elimination techniques for sparse matrices). Approximate solutions of the time-harmonic wave equation are computed using a block low-rank (BLR) approximation, leading to a significant reduction in the operation count and in the volume of communication during the lower upper (LU) factorization as well as offering great potential for reduction in the memory demand. Moreover, the sparsity of the seismic source vectors is exploited to speed up the forward elimination step during the computation of the monochromatic wavefields. The relevance and the computational efficiency of the frequency-domain FWI performed in the viscoacoustic vertical transverse isotropic (VTI) approximation was tested with a real 3D OBC case study from the North Sea. The FWI subsurface models indicate a dramatic resolution improvement relative to the initial model built by reflection traveltime tomography. The amplitude errors introduced in the modeled wavefields by the BLR approximation for different low-rank thresholds have a negligible footprint in the FWI results. With respect to a standard multifrontal sparse direct factorization, and without compromise of the accuracy of the imaging, the BLR approximation can bring a reduction of the LU factor size by a factor of up to three. This reduction is not yet exploited to reduce the effective memory usage (ongoing work). The flop reduction can be larger than a factor of 10 and can bring a factor of time reduction of around three. Moreover, this reduction factor tends to increase with frequency, namely with the matrix size. Frequency-domain viscoacoustic VTI FWI can be viewed as an efficient tool to build an initial model for elastic FWI of 4C OBC data.

A synthetic study to assess the applicability of full-waveform inversion to infer snow stratigraphy from upward-looking ground-penetrating radar data

Snow stratigraphy and liquid water content are key contributing factors to avalanche formation. Upward-looking ground-penetrating radar (upGPR) systems allow nondestructive monitoring of the snowpack, but deriving density and liquid water content profiles is not yet possible based on the direct analysis of the reflection response. We have investigated the feasibility of deducing these quantities using full-waveform inversion (FWI) techniques applied to upGPR data. For that purpose, we have developed a frequency-domain FWI algorithm in which we additionally took advantage of time-domain features such as the arrival times of reflected waves. Our results indicated that FWI applied to upGPR data is generally feasible. More specifically, we could show that in the case of a dry snowpack, it is possible to derive snow densities and layer thicknesses if sufficient a priori information is available. In case of a wet snowpack, in which it also needs to be inverted for the liquid water content, the algorithm might fail, even if sufficient a priori information is available, particularly in the presence of realistic noise. Finally, we have investigated the capability of FWI to resolve thin layers that play a key role in snow stability evaluation. Our simulations indicate that layers with thicknesses well below the GPR wavelengths can be identified, but in the presence of significant liquid water, the thin-layer properties may be prone to inaccuracies. These results are encouraging and motivate applications to field data, but significant issues remain to be resolved, such as the determination of the generally unknown upGPR source function and identifying the optimal number of layers in the inversion models. Furthermore, a relatively high level of prior knowledge is required to let the algorithm converge. However, we feel these are not insurmountable and the new technology has significant potential to improve field data analysis.

Full waveform inversion comparison of conventional and broadband marine seismic streamer data, NW Shelf Australia

In this paper, we illustrate the application of frequency- domain FWI to two 2D data sets acquired simultaneously offshore North West Australia (NWA); one data set was recorded with a conventional streamer, and the other with the use of variable depth streamer configuration. First we discuss the two data sets, and our frequency-domain FWI algorithm. We then present each FWI result, and discuss the superiority of the broadband FWI results obtained with the variable depth streamer data. SUMMARY The lack of low-frequency information in conventional marine seismic streamer data inhibits the success of frequency-domain full waveform inversion (FWI). Low frequencies are typically absent in marine seismic data due to the low-cut spectral responses of airgun sources and hydrophone receivers, and the fact that the air-water interface produces source and receiver ghost reflections which create notch frequencies in the data amplitude spectrum. Advances in broadband streamer acquisition, such as the variable depth towed streamer, allow us to extend the low and high bounds of the useful frequency bandwidth in the seismic data spectrum. We illustrate the application of frequency-domain FWI to two 2D seismic data sets acquired simultaneously offshore North West Australia. Both data sets were acquired together, one with a conventional streamer, and the other with a variable depth streamer configuration. Our examples demonstrate that the FWI results are clearly superior when using the broadband variable depth streamer data, compared to using the conventional streamer data.

Velocity model estimation by full waveform inversion of time-lapse 4D passive seismic array data

SUMMARY In passive-source monitoring, an accurate velocity model is important to precisely estimate microseismic source locations, and to understand changes in reservoir properties. In this study, we employ frequency-domain full waveform inversion in order to obtain a high- resolution velocity model by exploiting full wavefields. We demonstrate the feasibility of the method for a surface geophone array by inverting for time-lapse 4D velocity changes in a realistic subsurface model. Our method successfully estimates the small velocity changes of a few percent within layers of 10s of meters, even for a single passive seismic source event. The analysis of wavepaths and gradients suggests that keys for the successful inversion are the use of full wavefields (both first and scattered arrivals), the vicinity of velocity changes to the source, and the wide-aperture surface array.

Two-dimensional frequency-domain acoustic full-waveform inversion with rugged topography

We studied finite-element-method-based two-dimensional frequency-domain acoustic FWI under rugged topography conditions. The exponential attenuation boundary condition suitable for rugged topography is proposed to solve the cutoff boundary problem as well as to consider the requirement of using the same subdivision grid in joint multifrequency inversion. The proposed method introduces the attenuation factor, and by adjusting it, acoustic waves are sufficiently attenuated in the attenuation layer to minimize the cutoff boundary effect. Based on the law of exponential attenuation, expressions for computing the attenuation factor and the thickness of attenuation layers are derived for different frequencies. In multifrequency-domain FWI, the conjugate gradient method is used to solve equations in the Gauss–Newton algorithm and thus minimize the computation cost in calculating the Hessian matrix. In addition, the effect of initial model selection and frequency combination on FWI is analyzed. Examples using numerical simulations and FWI calculations are used to verify the efficiency of the proposed method.

An improved scattering-integral approach for frequency-domain full waveform inversion

An improved scattering-integral approach for frequency-domain full waveform inversion

Efficient 3-D frequency-domain mono-parameter full-waveform inversion of ocean-bottom cable data: application to Valhall in the visco-acoustic vertical transverse isotropic approximation

Computationally efficient 3-D frequency-domain full waveform inversion (FWI) is applied to ocean-bottom cable data from the Valhall oil field in the visco-acoustic vertical transverse isotropic (VTI) approximation. Frequency-domain seismic modelling is performed with a parallel sparse direct solver on a limited number of computer nodes. A multiscale imaging is performed by successive inversions of single frequencies in the 3.5–10 Hz frequency band. The vertical wave speed is updated during FWI while density, quality factor QP and anisotropic Thomsen's parameters δ and ϵ are kept fixed to their initial values. The final FWI model shows the resolution improvement that was achieved compared to the initial model that was built by reflection traveltime tomography. This FWI model shows a glacial channel system at 175 m depth, the footprint of drifting icebergs on the palaeo-seafloor at 500 m depth, a detailed view of a gas cloud at 1 km depth and the base cretaceous reflector at 3.5 km depth. The relevance of the FWI model is assessed by frequency-domain and time-domain seismic modelling and source wavelet estimation. The agreement between the modelled and recorded data in the frequency domain is excellent up to 10 Hz although amplitudes of modelled wavefields propagating across the gas cloud are overestimated. This might highlight the footprint of attenuation, whose absorption effects are underestimated by the homogeneous background QP model (QP = 200). The match between recorded and modelled time-domain seismograms suggests that the inversion was not significantly hampered by cycle skipping. However, late arrivals in the synthetic seismograms, computed without attenuation and with a source wavelet estimated from short-offset early arrivals, arrive 40 ms earlier than the recorded seismograms. This might result from dispersion effects related to attenuation. The repeatability of the source wavelets inferred from data that are weighted by a linear gain with offset is dramatically improved when they are estimated in the FWI model rather than in the smooth initial model. The two source wavelets, estimated in the FWI model from data with and without offset gain, show a 40 ms time-shift, which is consistent with the previous analysis of the time-domain seismograms. The computational efficiency of our frequency-domain approach is assessed against a recent time-domain FWI case study performed in a similar geological environment. This analysis highlights the efficiency of the frequency-domain approach to process a large number of sources and receivers with limited computational resources, thanks to the efficiency of the substitution step performed by the direct solver. This efficiency can be further improved by using a block-low rank version of the multifrontal solver and by exploiting the sparsity of the source vectors during the substitution step. Future work will aim to update attenuation and density at the same time of the vertical wave speed.

3D hybrid-domain full waveform inversion on GPU

The traditional frequency-domain full waveform inversion (FWI) method has limited application for 3D case due to its significant computational challenges and huge memory cost. The time-frequency hybrid-domain FWI is thus utilized to combine the computational efficiency of time-domain FWI and multi-scale inversion of frequency-domain FWI. We present a simplified hybrid-domain FWI method. Compared with the previous hybrid-domain FWI, our method can take one less Discrete Fourier Transform (DFT) and one less inverse Discrete Fourier Transform (IDFT) for single iteration and achieved the equivalent inversion effect. Meanwhile, the hybrid-domain FWI is very suitable for fine-grained parallel computation, and the application of graphic processing unit (GPU) has significantly enhanced the computation efficiency. Modeling, boundary absorption and DFT are the most time-consuming modules. The occupancy of the three kernels achieved a good level as Compute Unified Device Architecture (CUDA) Visual Profile shows and the bandwidth usage has been much improved by introducing the tiling method for 3D finite difference problem. Finally, our hybrid-domain FWI is applied to a 3D model on our personal computer equipped with GTX 680 graphic card, and complexity analysis of our algorithm is presented. The results further confirm the feasibility of this technique.

Fast full waveform inversion with source encoding and second-order optimization methods

Full waveform inversion (FWI) of 3-D data sets has recently been possible thanks to the development of high performance computing. However, FWI remains a computationally intensive task when high frequencies are injected in the inversion or more complex wave physics (viscoelastic) is accounted for. The highest computational cost results from the numerical solution of the wave equation for each seismic source. To reduce the computational burden, one well-known technique is to employ a random linear combination of the sources, rather that using each source independently. This technique, known as source encoding, has shown to successfully reduce the computational cost when applied to real data. Up to now, the inversion is normally carried out using gradient descent algorithms. With the idea of achieving a fast and robust frequency-domain FWI, we assess the performance of the random source encoding method when it is interfaced with second-order optimization methods (quasi-Newton l-BFGS, truncated Newton). Because of the additional seismic modelings required to compute the Newton descent direction, it is not clear beforehand if truncated Newton methods can indeed further reduce the computational cost compared to gradient algorithms. We design precise stopping criteria of iterations to fairly assess the computational cost and the speed-up provided by the source encoding method for each optimization method. We perform experiment on synthetic and real data sets. In both cases, we confirm that combining source encoding with second-order optimization methods reduces the computational cost compared to the case where source encoding is interfaced with gradient descent algorithms. For the synthetic data set, inspired from the geology of Gulf of Mexico, we show that the quasi-Newton l-BFGS algorithm requires the lowest computational cost. For the real data set application on the Valhall data, we show that the truncated Newton methods provide the most robust direction of descent.