Building good starting models for full-waveform inversion using adaptive matching filtering misfit

Summary Full-waveform inversion can be severely compromised by problems of cycle skipping; this occurs when predicted and observed data differ by more than half a cycle, and it leads the inversion to recover a local rather than the global minimum model. Overcoming cycle skipping normally requires both a good starting model and low-frequency content in the field data. Here we present a scheme that uses a non-linear extrapolation to add missing low- frequencies into the field data. We demonstrate the scheme using a 3D OBC field dataset, and show that it can invert to recover the global minimum model even when the original un-extrapolated field dataset is significantly cycle skipped. Introduction Full-waveform inversion (FWI) is an optimization scheme that seeks to find the global minimum of the misfit between field seismic data and synthetic data generated by a model. It proceeds towards the global minimum by a series of local linearized iterations from a starting model. The true relationship between model and data however is non-linear, and several of these non-linearities can lead the inversion scheme to become trapped within local minima rather than reaching the desired global solution. Amongst these non-linearities, the most important is that produced by the oscillatory nature of the seismic data themselves. A cycle-skipped starting model, that is one that produces data that differ by more than half a cycle with respect to the field data, will normally lead the inversion in entirely the wrong direction towards a local minimum. This problem is well known; its practical solution typically requires a highly accurate starting model combined with low-frequency field data. In these circumstances, the predicted starting data are not cycle skipped at the lowest frequencies, and the global solution can be found by starting FWI using the lowest frequencies in the field data. In this paper, we present a FWI scheme that relaxes the requirement for low-frequency field data and a good starting model, and that is able to reach the global minimum solution even when the starting data are cycle skipped. We do this by using the observed field data to synthesize low-frequency data that were not present in the original field data. We then invert these invented low- frequencies, which has the effect of dragging the starting model towards the global minimum. We then switch to invert the true field data which are now no longer cycle skipped, and so reach the global minimum.

A full-waveform inversion (FWI) is a highly nonlinear inversion methodology. FWI tends to converge to a local minimum rather than a global one. We refer to this phenomenon as “cycle skipping” in FWI. A cost-effective solution for resolving this issue is to design a more convex misfit function for the optimization problem. A global comparison based on using a matching filter (MF) admits more robust misfit functions. In this case, we would compute an MF first by deconvolving the predicted data from the measured ones. When the velocity model is accurate, the predicted data resemble the measured ones, and the resulting MF would be an approximated Dirac delta function. If the velocity produces data that are different from the observed ones, a misfit function can be formulated by penalizing the energy away from the zero-lag time (the center). Here, we develop a general mechanism for an evolution of the MF to our objective in FWI. Specifically from the statistics point of view, rather than using a penalty, we propose a novel misfit by minimization of the mean and information entropy of the MF distribution. We show that the resulting misfit function can mitigate the “cycle skipping” as well as reduce the mean and variance of the resulting MF distribution. We use a modified Marmousi example to demonstrate the features of the proposed misfit. We also evaluate the robustness of the proposed method using inaccurate (rotation in phase) source wavelets and measured data with different levels of Gaussian random noise.

Crustal‐scale imaging by the full‐waveform inversion (FWI) of long‐offset seismic data is inherently difficult because the large number of wavelengths propagating through the crust makes the inversion prone to cycle skipping. Therefore, efficient crustal‐scale FWI requires an accurate starting model and a stable workflow minimizing the nonlinearity of the inversion. Here we attempt to reprocess a challenging 2‐D ocean‐bottom seismometer (OBS) data set from the eastern Nankai Trough. The starting model is built by first‐arrival traveltime tomography (FAT), which is FWI assisted for tracking cycle skipping. We iteratively refine the picked traveltimes and then reiterate the FAT until the traveltime residuals remain below the cycle‐skipping limit. Subsequently, we apply Laplace‐Fourier FWI, in which progressive relaxation of time damping is nested within frequency continuation to hierarchically inject more data into the inversion. These two multiscale levels are complemented by a layer‐stripping approach implemented through offset continuation. The reliability of the FWI velocity model is assessed by means of source wavelet estimation, synthetic seismogram modeling, ray tracing modeling, dynamic warping, and checkerboard tests. Although the viscoacoustic approximation is used for wave modeling, the synthetic seismograms reproduce most of the complexity of the data with a high traveltime accuracy. The revised FWI scheme produces a high‐resolution velocity model of the entire crust that can be jointly interpreted with migrated images derived from multichannel seismic data. This study opens a new perspective on the design of OBS crustal‐scale experiments amenable to FWI; however, a further assessment of the optimal OBS spacing is required for reliable FWI.

plex shallow subsurface. This workflow incorporates dispersion curve inversion, traveltime tomography and full waveform inversion into a single framework to optimize the advantages of each. An accurate near-surface velocity is derived using the dispersion curve inversion technique that overcomes limitations of traditional traveltime tomography. This velocity is then incorporated into layer-stripping full waveform or RMO inversion schemes by “freezing” the ultra-shallow part of the model in order to retrieve the deeper velocity structure. We illustrate the workflow on a test data set from Inner Mongolia, China, whose long-offsets and low-frequency content provide the necessary input for both the dispersion curve and full waveform inversion techniques. Using this data set, we demonstrate the efficacy of the integrated methodology in determining both the macro-velocity model and near-surface velocities in structurally complex geological setting.

Full waveform inversion (FWI) suffers from the cycle skipping problem, because the observed data usually lack low-frequency components or due to errors in the wavelet estimation. In addition, the strong low-frequency non-zero-mean noise can have a large impact on FWI results. Thus, we propose a local waveform traveltime correction scheme to solve the situations when the observed data lack low-frequency components or when the estimation for the wavelet is incorrect. We use a sliding time window, which is used to decrease the traveltime differences between the calculated and observed data to increase the cross-correlation between them. Besides, we propose a zero-mean normalized cross-correlation misfit function to reduce the interference of the low-frequency non-zero-mean noise. Therefore, we propose new approaches to improve FWI results whether the observed data lack low-frequency components or the observed data are contaminated by the non-zero-mean low-frequency noise. Numerical examples on Marmousi model show the feasibility of a FWI based on the zero-mean normalized cross-correlation misfit function and a FWI based on the local traveltime correction method.

The full waveform inversion (FWI) of strongly dispersive Love wave data is a challenging task. Amplitude, phase and dispersion information not only depends on the density and shear modulus distribution in the subsurface, but also significantly on intrinsic damping. This is especially a problem in near surface data applications with complex underground structures and low Qs values. Therefore, the FWI of a dispersive Love wavefield demands an accurate initial visco-elastic model and careful data pre-processing. Another key ingredient of a successful time-domain FWI is the sequential inversion of frequency filtered data in order to mitigate the non-linearity of the inverse problem. Common FWI strategies are based solely on either low- or bandpass filtered data. In this study we introduce a workflow consisting of a combined low- and bandpass filter strategy to achieve an appropriate data fit of the low-frequency Love wave and high-frequency refracted SH-wavefield. The applicability of this FWI strategy and the importance of a visco-elastic medium description is demonstrated for SH field data from the transect over a medieval 2D canal structure in southern Germany. The resolved canal shape and small scale structures in the inversion results are verified by an archaeological excavation.

ABSTRACTTo obtain a physical understanding of gradient-based descent methods in full-waveform inversion (FWI), we find a connection between the FWI gradient and the image provided by reverse time migration (RTM). The gradient uses the residual data as a virtual source, and RTM uses the observed data directly as the boundary condition, so the FWI gradient is similar to a time integration of the RTM image using the residual data, which physically converts the phase of the reflectivity to the phase of the velocity. Therefore, gradient-based FWI can be connected to the classical reflectivity-to-velocity/impedance inversion (RVI). We have developed a new FWI scheme that provides a self-contained and physically intuitive derivation, which naturally establishes a connection among the amplitude-preserved RTM, the Zoeppritz equations (amplitude variation with angle inversion), and RVI, and combines them into a single framework to produce a preconditioned inversion formula. In this scheme, the relative velocity upd...

To obtain a physical understanding of gradient-based descent methods in full-waveform inversion (FWI), we find a connection between the FWI gradient and the image provided by reverse time migration (RTM). The gradient uses the residual data as a virtual source, and RTM uses the observed data directly as the boundary condition, so the FWI gradient is similar to a time integration of the RTM image using the residual data, which physically converts the phase of the reflectivity to the phase of the velocity. Therefore, gradient-based FWI can be connected to the classical reflectivity-to-velocity/impedance inversion (RVI). We have developed a new FWI scheme that provides a self-contained and physically intuitive derivation, which naturally establishes a connection among the amplitude-preserved RTM, the Zoeppritz equations (amplitude variation with angle inversion), and RVI, and combines them into a single framework to produce a preconditioned inversion formula. In this scheme, the relative velocity update is a phase-modified and deconvolved RTM image obtained with the residual data. Consistent with the deconvolution, the multiscale approach applies a gradually widening low-pass frequency filter to the deconvolved wavelet at early iterations, and then it uses the unfiltered deconvolved wavelet for the final iterations. Our numerical testing determined that the new method makes a significant improvement to the quality of the inversion result.

Traditional iteration-based full-waveform inversion (FWI) methods encounter serious challenges if the initial velocity model is far from the true model or if the observed data are lacking low-frequency content. As such, the optimization algorithm may be trapped in local minima and fail to go to a global optimal model. In addition, the traditional FWI method requires long-offset data to update the deep structure of a velocity model with diving waves. To overcome the disadvantages of traditional FWI under these circumstances, we have developed a reflection intensity waveform inversion method. This method aims to minimize the seismic intensity differences between the modeled reflection data and field data. Our method is less dependent on the starting model, and long-offset data are no longer required. The wave intensity, proportional to the square of the original data amplitude, can have a low-frequency band and a higher frequency band, even for waveforms without initial low-frequency content. Our multiscale intensity inversion starts from the low-frequency information in the intensity data, and it can largely avoid the cycle-skipping problem. Synthetic and field data examples demonstrate that our method is able to overcome cycle skipping in handling data with no low-frequency information.

Full-waveform inversion (FWI) in seismic scenarios continues to be a complex procedure for subsurface imaging that might require extensive human interaction in terms of model setup, constraints, and data preconditioning. The underlying reason is the strong nonlinearity of the problem that forces the addition of a priori knowledge (or bias) in order to obtain geologically sound results. In particular, when the use of a long-offset receiver is not possible or may not favor the reconstruction of the fine structure of the model, one needs to rely on reflection data. As a consequence, the inversion process is more prone to becoming stuck in local minima. Nevertheless, misfit functionals can be devised that can either cope with missing long-wavenumber features of initial models (e.g., cross-correlation-based misfit) or invert reflection-dominated data whenever the models are sufficiently good (e.g., normalized offset-limited least-squares misfit). By combining both, high-frequency data content with poor initial models can be successfully inverted. If one can figure out simple parameterizations for such functionals, the amount of uncertainty and manual work related to tuning FWI would be substantially reduced. Thus, FWI might become a semiautomatized imaging tool.

SeisElastic2D: An open-source package for multiparameter full-waveform inversion in isotropic-, anisotropic- and visco-elastic media

Full waveform inversion (FWI) is the combination of the tomography mode and the migration mode. It is essential to emphasize tomographic component during the early stage of inversion when the initial model is far away from the true model. To prevent the inverted model from falling into local minima, we develop a method to update tomographic lowwavenumber components under the frame of FWI with structural shaping regularization. We firstly separate the conventional FWI gradient into tomography and migration modes based on an implicit wavefield decomposition method. Then the tomography mode is used to emphasize low-wavenumber component updates, while the migration mode is used to estimate local-dip information. Then we introduce structural shaping regularization using seislet transform to mitigate the artifacts caused by the leakage of migration component in tomographic updates. Numerical results show that tomography-mode FWI with structural shaping regularization (RSFWI) is stable and effective to recover a background velocity model. The final inverted velocity model is more accurate than the conventional FWI result.

Full waveform inversion (FWI) iteratively improves the accuracy of the model by minimizing the discrepancies between the predicted and the observed data. However, FWI commonly suffers from cycle skipping when the initial model is poor, leading to an erroneous result. To mitigate this problem, we propose deep-learning-backed adaptive waveform inversion (DL-AWI), which introduces a deep twin neural network to precondition the waveforms and compare the ratio of two signals with a zero-lag spike, thereby enhancing the stability of the inversion process. DL-AWI can project the synthetic and observed signals into an extended latent space via several convolutional neural networks (CNNs) with shared weights, which can accelerate the data matching. Compared with classic FWI methods, the proposed DL-AWI provides a wider space for model updates, significantly decreasing the risk of being trapped in local minima. We use synthetic and field examples to validate its efficiency in subsurface model inversion, and the results show that DL-AWI is robust even when a poor initial model is provided.

Seismic full waveform inversion (FWI) is able to build high-resolution velocity model based on the full information carried by seismic wave. However, FWI requires an accurate enough initial model to ensure convergence. In this paper, we propose a new nonlinear FWI method to mitigate the initial model dependence problem. Specifically, we firstly propose a nonlinear operator within the hybrid model- and data-driven framework based on the frequency controllable envelope operator (FCEO) and a deep learning architecture U-Net. FCEO is used to obtain the envelope of a band-limited data and U-Net realizes the mapping from this envelope to that corresponding to a lower frequency band. The U-Net is trained in a self-supervised manner that avoids the reliance on labeled data and benefits the generalization ability. Based on the nonlinear operator, a nonlinear FWI method is proposed by defining a new misfit function. In addition, the calculation of gradient is derived using the adjoint-state method. Using numerical examples, we investigate the performance of the proposed nonlinear operator and the new nonlinear FWI method. The results clearly demonstrate that the proposed nonlinear operator is effective in obtaining low-frequency envelope data, and the new nonlinear FWI method has advantages over common method in mitigating cycle-skipping and building an initial model for conventional FWI.

Summary Full Waveform Inversion (FWI) can create on an inaccurate model as a result of cycle skipping, if the initial model is not close enough to the true one, or there is insufficient low frequencies in the data. Furthermore, FWI model updates can be affected by a reflectivity imprint prior to the resolution of long-wavelength features. Imaging with the resulting incorrect model will create structural uncertainty, and will hamper an evaluation of potential prospects. Cycle skipping can be mitigated by using a robust norm for measuring the data misfit (W2-norm), instead of a traditional L2-norm. Used with a velocity gradient that removes the imprint of the reflectivity, we demonstrate an application to data resolving a high-velocity layer that was not present in the inital model. Corroborated by well data, the resulting earth model accurately reflects the subsurface, which, in turn, reduces uncertainty in the final structural image.