Localized adaptive waveform inversion: theory and numerical verification

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.

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.

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.

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) 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.

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.

The European continent undergoes a series of complicated tectonic processes since the closure of the Tethyan Ocean, including oceanic plate subduction, continental plate subduction, continental collision and orogeny. High resolution 3-D mantle structure under the European plate is key to investigate its geodynamic evolution history. Full waveform seismic inversion for mantle structure has become feasible with the advent of accurate 3D wave propagation codes and the use of adjoint sources to compute the gradient of misfit functions between data and synthetics. The adjoint source, in this approach, depends on how a misfit is defined between data and synthetics. The time or phase shift between data and synthetic has been used in most full waveform inversions for mantle structure. Waves that sample the upper mantle, however, are almost always multi-pathed due to discontinuities, the low velocity zone as well as large amplitude lateral variations, leading to complex waveforms that cannot be fully captured by time/phase shifts. Here we use the normalized correlation coefficient between data and synthetics as a misfit function to simultaneously capture both the phase and relative amplitude information of the waveform, and perform full waveform inversion on a large data set of three component seismic data from Europe. The global 3-D tomography model S362WMANI combined with crustal model EPcrust comprises our starting model. The adopted numerical solver for the wave equation is SPECFEM3D_GLOBAL, a high-accuracy numerical simulation package based on the spectral-element method. We adopt multi-stage inversion method to iteratively enhance the frequency range of the inversion. Our preliminary results show significant improved resolution of the upper mantle structure and provide key constraints on the deep subduction processes of micro-plates in the Europe region.

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.

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

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.

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.

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.

Seismic acquisitions in urban areas represent a challenge due to the noisy environment during the recording of the data. This often prevents obtaining high-resolution images of the subsurface in the area of interest. One way to achieve better results, despite the challenging circumstances, is full-waveform inversion (FWI). This method iteratively updates a starting model until the simulated data and the observed field data matches to a satisfactory level. This eventually yields the desired subsurface model. However, FWI comes with some major limitations. Because each iteration requires a calculation of the wave propagation several times for each shot, it is highly computationally expensive. Furthermore, due to storage limitations, especially if high performance computing (HPC) clusters are not available, the model size is restricted as well. This work aims to find a solution for these limitations in an elastic 2D case by implementing the random-shot workflow (RSW). In a conventional FWI workflow which is called full-shot workflow (FSW) in this work, each iteration uses all shots of the recorded data before updating the whole model at once which leads to the high computational and storage costs. RSW randomly picks only one shot, applies FWI within the area of the shot and its corresponding receivers and then updates the model only within this subset. This should, in my expectation, split the workload into several smaller subsets without sacrificing the effectiveness of FWI. After the implementation, I compare RSW to FSW in a synthetic benchmark where I verify the performance. Afterwards, the workflows are compared to each other by using a fraction of a field data set that was acquired in Salt Lake City. These data represents a large urban seismic land data set with a low signal-to-noise ratio. They also have suboptimal conditions such as the end-on spread geometry due to the use of a land streamer and occasionally bad coupling of the geophones. This turns out to be a big challenge for the FWI algorithm. The starting model used for the field data inversion is based on the multichannel analysis of surface waves which deals with the inversion of surface-wave dispersion curves. It is a cost-efficient way to obtain an initial model for FWI.With a subset of this model and the data, I investigate the influence of traces near the sources which suffer heavily from the suboptimal acquisition conditions, and if more iterations can improve the results. Removing such traces improve the major part of the results, while more iterations are not beneficial with this particular data set. At the end I show a final comparison of both workflows by applying them to the complete Salt Lake City field data set. An application of FWI on such a large-scale land streamer acquisition with such a large number of shots has, to the best of my knowledge, not been done yet, most likely because of the large computational costs, even in 2D. The comparisons and investigations show that while RSW has a major computational advantage in storage usage and per-CPU performance, it is too sensitive to bad traces and shots which significantly degrade the final model. Therefore, it is unsuitable for field data with a low signal-to-noise ratio or that have not been preprocessed well. Furthermore, RSW cannot utilise the whole power of HPC clusters while FSW is able to do so by optimal shot parallelisation. Thus, FSW could potentially outperform RSW by raw computing power if such excessive power is available.

Viscoelastic full-waveform inversion is applied to walk-away vertical seismic profile data acquired at a producing heavy-oil field in Western Canada for the determination of subsurface velocity models (P-wave velocity [Formula: see text] and S-wave velocity [Formula: see text]) and attenuation models (P-wave quality factor [Formula: see text] and S-wave quality factor [Formula: see text]). To mitigate strong velocity-attenuation trade-offs, a two-stage approach is adopted. In Stage I, [Formula: see text] and [Formula: see text] models are first inverted using a standard waveform-difference (WD) misfit function. Following this, in Stage II, different amplitude-based misfit functions are used to estimate the [Formula: see text] and [Formula: see text] models. Compared to the traditional WD misfit function, the amplitude-based misfit functions exhibit stronger sensitivity to attenuation anomalies and appear to be able to invert [Formula: see text] and [Formula: see text] models more reliably in the presence of velocity errors. Overall, the root-mean-square amplitude-ratio and spectral amplitude-ratio misfit functions outperform other misfit function choices. In the final outputs of our inversion, significant drops in the [Formula: see text] to [Formula: see text] ratio (~1.6) and Poisson’s ratio (~0.23) are apparent within the Clearwater Formation (depth ~0.45–0.50 km) of the Mannville Group in the Western Canada Sedimentary Basin. Strong [Formula: see text] (~20) and [Formula: see text] (~15) anomalies are also evident in this zone. These observations provide information to help identify the target attenuative reservoir saturated with heavy-oil resources.

Traditional full waveform inversion objective functions typically rely on discrepancies between waveforms, making full waveform inversion highly dependent on the initial model and the low-frequency components in the seismic data. The instantaneous phase of the seismic signal reflects the kinematic information of seismic waves and has a more linear relationship with the subsurface velocity structure, aiding in the retrieval of low-wavenumber components of velocity models. Conversely, waveform information is instrumental in achieving high-resolution inversion results while maintaining algorithmic stability. In addition, automatic differentiation provides an efficient and accurate mechanism for computing gradients by systematically applying the chain rule across the computational graph, yielding reliable derivative information for optimization. To balance the contributions of waveform and phase information in velocity inversion, within the framework of automatic differentiation, we integrate instantaneous phase and waveform information to propose an automatic differentiation-based weighted instantaneous phase inversion method. Numerical tests using the Marmousi model and the Overthrust model demonstrate that the proposed method achieves more accurate velocity inversion results.