DL-AWI: Adaptive Full Waveform Inversion Using a Deep Twin Neural Network

Achieving high-resolution intracranial imaging in a safe and portable manner is critical for the diagnosis of intracranial diseases, preoperative planning of craniotomies and intraoperative management during craniotomy procedures. Adaptive waveform inversion (AWI), a variant of full waveform inversion (FWI), has shown potential in intracranial ultrasound imaging. However, the robustness of AWI is affected by the parameterization of the Gaussian penalty matrix and the challenges posed by transcranial scenarios. Conventional AWI struggles to produce accurate images in these cases, limiting its application in critical medical settings. To address these issues, we propose a stabilized adaptive waveform inversion (SAWI) method, which introduces a user-defined zero-lag position for the Wiener filter. Numerical experiments demonstrate that SAWI can achieve accurate imaging under Gaussian penalty matrix parameter settings where AWI fails, perform successful transcranial imaging in configurations where AWI cannot, and maintain the same imaging accuracy as AWI. The advantage of this method is that it achieves these advancements without modifying the AWI framework or increasing computational costs, which helps to promote the application of AWI in medical fields, particularly in transcranial scenarios.

Adaptive waveform inversion (AWI) provides a means of performing full-waveform inversion (FWI) that appears to be immune to the effects of cycle skipping. However, the form of the AWI algorithm suggests that it could have increased sensitivity to errors in the assumed source wavelet, to noise in the field data, and to inadequacies in the physics used to simulate wave propagation. We examine each of these for a synthetic model. We show that AWI is in fact less sensitive than FWI to errors in the source wavelet, and is no more sensitive to errors in the data and in the modelling than is FWI. It appears likely that the immunity that AWI displays to cycle skipping also contributes to its reduced sensitivity to errors in the assumed source wavelet.

SUMMARYCorrectly interpreting phase events thanks to data processing techniques based on correlation or deconvolution has been the focus of numerous studies in the field of high-resolution seismic imaging using full-waveform inversion. To mitigate the non-convexity of the misfit function and the risk to converge towards non-informative local minima, correlation and deconvolution techniques make it possible to focus on phase information instead of amplitude information and to design more convex misfit function, alleviating the dependency of the full-waveform inversion process on the accuracy of initial models. Such techniques however rely on the assumption that phase events can be compared one by one, or that all the phase events are shifted in time in a similar way. This assumption is not satisfied in practice, which limits the effectiveness of these correlation/deconvolution-based methods. To overcome this issue, we propose to account for the non-stationary relation between observed and predicted data through a local in-time deconvolution technique, based on time–frequency analysis of the signal using a Gabor transform. This makes it possible to estimate instantaneous time-shift between locally coherent phase events. This strategy generalizes the conventional normalized deconvolution technique, which has been popularized under the name of adaptive waveform inversion. To support the introduction of our novel method, we compare it with four misfit functions based respectively on classical cross-correlation, penalized cross-correlation, penalized deconvolution, and adaptive waveform inversion. We analyse the behaviour of these methods on specific scenarios, and then propose a comparison on 2-D synthetic benchmarks. We show how our ‘localized’ adaptive waveform inversion applies in these realistic tests and overcomes some of the limitations of the aforementioned techniques.

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.

SUMMARYRecently, we have developed a localized adaptive waveform inversion (LAWI) method to tackle the cycle-skipping issue in velocity reconstruction through seismic waveform inversion. The LAWI method employs a local matching filter, computed using Gabor deconvolution, to measure the instantaneous time-shift between observed and calculated data. Unlike the adaptive waveform inversion (AWI) approach, the LAWI method can take the non-stationarity between observed and calculated data into account. In this work, we investigate two types of regularization based on prior information about the expected filter, which could be a minimum-norm filter or a delta-shape filter, with regard to their effects on the robustness and resolution of inversion. We demonstrate on synthetic data the advantages and disadvantages of these two types of prior information, where the delta-type LAWI may handle multiple observed phases not initially predicted by the starting velocity model. Therefore, we apply the delta-type LAWI to a high-quality 3-D field data set in the North Sea, eliminating the need for data-windowing tuning, which can be tedious and time-consuming for 3-D data. Under different workflows with varying reliable initial models and frequency bands of the pressure data considered, we show that the LAWI approach is robust, effective and efficient for reconstructing the P-wave velocity, while other approaches such as AWI and graph-space optimal-transport method may require meticulous data-tuning strategies to converge to the correct model. Well logs and data fits, primarily from early arrivals, give us confidence that this LAWI approach could be applied to various acquisitions and subsurface targets, thanks to its phase-driven principle.

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.

Summary Extended waveform inversion provides an effective way to mitigate cycle skipping that usually occurs in conventional full waveform inversion (FWI), resulting in an inaccurate local minimum model. A matching filter between the predicted and observed data can provide an additional degree of freedom to avoid the cycle skipping. We extend the search space to treat the matching filter as an independent variable that we use to bring the compared data within a half cycle to obtain accurate direction of velocity updates. In this case, the objective function with a reasonable penalty parameter has a larger region of convexity compared to conventional FWI. The normalization of the data can bring us an equivalent normalization of the filter, and a more effective convergence. A Marmousi example demonstrates these features.

Self-mixing interferometry (SMI) is superior to other laser interferometry methods due to its simplicity and compactness. However, SMI signals are often complex and difficult to process due to interference in the form of variations in the effective reflectivity of the target, noisy signals, complex signal shapes, and other dependencies. Deep neural networks have been a very popular area of research in computer artificial intelligence in recent years, allowing more implicit features to be uncovered than traditional shallow machine learning. It has been shown that convolutional and back-propagation neural networks can be used for SMI signal processing. There are also studies that have used machine learning genetic algorithms for absolute distance measurement. Based on the above, this study used convolutional neural networks to form a deep neural network for absolute distance measurement based on SMI technology. We first trained the deep convolutional neural network at different feedback strengths. The results of the Convolutional Neural Network (CNN) model showed a coefficient of determination of 0.9987. which is consistent with the required model. The trained network was then used to estimate absolute distances with and without the addition of noise. The comparison proves that the proposed method is noise-proof and has high adaptability for measurements under different conditions.

Ultrasound computed tomography (USCT) receives increasing attention because of its capability to reconstruct quantitative information about the material property distribution as images with superior resolution. However, one roadblock for the wide adoption of relevant techniques is the high demand for computational resources and the long processing time for solving a large inverse problem in imaging. To alleviate the associated challenges, a two-stage inversion scheme is proposed: 1) the ultrasound scanning signals are first processed using a full waveform inversion (FWI) technique with a single iteration to rapidly create a model (image) with embedded wave speed distribution; 2) the corresponding image will be further improved by feeding into a pre-trained deep neural network. The deep learning models presented in this paper are built upon two architectures to instantaneously solve the associated inverse problems and to produce a high-resolution image in real-time. The first is based on 1D convolutional neural network (1D-CNN) layers with an autoencoder structure. The second implements additional layers and skip connections inspired by a U-Net architecture. The resultant superior reconstructions from both CNNs demonstrate that the proposed framework produces a high-resolution image from a rapidly-generated, low-resolution image in real-time, with dramatically improved results.

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.

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.

We have proposed a misfit function based on adaptive matching filtering (AMF) to tackle challenges associated with cycle skipping and local minima in full-waveform inversion (FWI). This AMF is designed to measure time-varying phase differences between observations and predictions. Compared with classical least-squares waveform differences, our misfit function behaves as a smooth, quadratic function with a broad basin of attraction. These characters are important because local gradient-based optimization approaches used in most FWI schemes cannot guarantee convergence toward true models if misfit functions include local minima or if the starting model is far away from the global minimum. The 1D and 2D synthetic experiments illustrate the advantages of the proposed misfit function compared with the classical least-squares waveform misfit. Furthermore, we have derived adjoint sources associated with the proposed misfit function and applied them in several 2D time-domain acoustic FWI experiments. Numerical results found that the proposed misfit function can provide good starting models for FWI, particularly when low-frequency signals are absent in recorded data.

Early arrival waveform inversion (EWI) is an efficient waveform inversion method to resolve the near-surface velocity structures with high resolution. However, EWI suffers from cycle-skipping issues, which is similar to the full waveform inversion (FWI), resulting in artifacts produced in the reconstructed near-surface velocity model. These artifacts denote that the inversion process drops to a local minimum and are challenging to remove because it typically couples with the velocity layers or objects. We suggest that the artifacts produced by EWI typically correspond to velocities with large variations, and such velocities could cause local variations in common offset gather (COG) traveltimes after applying static corrections. We develop an annealing static-constrained early arrival waveform inversion (ASEWI) by minimizing the COG first-arrival variances after applying both long- and short-wavelength statics with an annealed process. The static-constrained term can guide the objective function to converge well and avoids dropping into local minima in the early stage of the iterations. As the inversion progresses, the weight of the static-constrained term is gradually weakened for the better fittings of early arrivals. ASEWI does not introduce additional undetermined coefficients due to an annealed scaling factor that is only associated with the number of iterations. Thus, the velocity artifacts produced by the local minimum in EWI can be suppressed by removing the corresponding variations in COG traveltimes. The effectiveness of the ASEWI method is demonstrated by the applications of both synthetic and field data examples. We also introduce the conventional first-arrival traveltime tomography (FTT) and EWI for comparisons. Both synthetic and field data results show that SEWI can improve the imaging resolution, suppress the artifacts, and produce near-surface velocity and static solutions with higher accuracy compared with two other methods.

Summary Considering the complex nonlinearity between seismic data and perturbations in the model, a waveform optimization problem based on the least square sample-to-sample comparison has proven to be inadequate in dealing with such nonlinearity. A slew of more global comparison based optimizations have proven their values in circumventing such nonlinearity. Among these methods in particular, adaptive waveform inversion (AWI), which is based on the deconvolution operator, offers the right balance between phase matching and amplitude normalization. The key element in the AWI success is the normalization factor, which is inherently offered by the instantaneous traveltime measure used to unwrap the phase. Thus, we recast the AWI problem using a misfit function based on the original generalized instantaneous travel time, which highlights some of the features of AWI and traces their roots. We demonstrate that the misfit function of AWI is actually the least square averaged instantaneous travel time over multiple frequencies. We use the Marmousi model and Chevron 2014 FWI benchmark dataset to verify the effectiveness of the proposed method.