Toward an automatic full-wave inversion: Synthetic study cases

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.

Summary Full-waveform inversion (FWI) fails to converge when starting with a poor initial model, especially for complex structures such as salt models. Building a good initial salt model requires manual interpretation from seismic images, which is time-consuming and depends on human decisions. Also, for FWI to have a stable convergence, the seismic data should have low frequencies and long offsets to penetrate the deeper structure and build an accurate salt body. Here, we automatically reconstruct the salt body with the aid of deep learning in a 2D vintage field data from the Gulf of Mexico area that lacks frequencies below 6 Hz and have a maximum offset of 4.8 Km. We perform FWI starting from a linearly increasing model in a multiscale approach. We train three 1D U-net networks that are applied subsequently after each FWI frequency scale: the first two U-nets are responsible for salt flooding and improving the top of the salt, and the third one is to unflood the salt to the base. The inversion result reconstructs a large salt body structure, although the data lacks low frequencies and long offsets.

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.

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.

The Arabian Gulf near-surface geology is complex, with extremely shallow waters and a hard water bottom generating high amplitude short period multiples and thinly bedded high and low velocity layers creating high apparent anisotropy in the bandwidth of seismic surveys. Obtaining an accurate description of the velocity variations in the near-surface and at intermediate depths is a necessity for reliable imaging and positioning of the reservoir layers located underneath. We propose a two-step full-waveform inversion (FWI) of ocean-bottom node (OBN) seismic data from offshore Abu Dhabi. We update a velocity model, using both diving and reflected waves, to reach the required depth of penetration. FWI has become an industry standard for velocity model building. However, due to the oscillatory nature of seismic data, FWI is known to be subject to cycle-skipping, where the inversion process falls into a local minimum. This risk is mitigated by using an accurate enough initial model and the use of low frequencies. In the shallow waters of offshore Abu Dhabi, near-offset data suffer from strong mud-roll and guided-wave energy that are not properly modeled with acoustic FWI. We exclude these offsets from the input data and use diving waves, starting at 3.5Hz, to update the near surface. The diving waves penetration is limited to approximately one kilometer in this area and corresponds to the base of a shallow high velocity layer. To reconcile the kinematics of reflected waves, travelling mostly vertically and used for imaging, and diving waves, travelling mostly horizontally, and used in the velocity update, we need an accurate estimation of the anisotropy. This is obtained using Backus averaging from available well logs. For deeper updates, the data are processed to remove the mud-roll and guided wave energy. This allows for the inclusion of reflections and near offsets. The FWI update is performed to 10Hz and penetrates about 3km into the sub-surface. We applied this FWI workflow to a recent node survey acquired offshore Abu Dhabi. The velocity model obtained follows the main geological structures and accurately describes the velocity variations in the shallow sub-surface. The estimation of anisotropy is important to ensure good convergence of the FWI and for imaging and vertical positioning of the migrated events. The reverse-time migration (RTM) image obtained with the updated model shows improved focusing and simplified depth structures compared to the RTM image obtained with the smooth initial model. To the best of our knowledge, this is the first successful implementation of FWI, here combining diving and reflected waves, on a dataset from offshore Abu Dhabi. It is a step towards resolving buried anomalies such as karst features, that cause imaging distortions at deeper reservoir levels.

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.

Summary The cycle-skipping issue is the main challenge in the field application of Full Waveform Inversion (FWI) when the initial velocity model is inaccurate and the data is lack of low frequency. In this abstract, we propose a Joint-Domain FWI (JDFWI) framework to overcome the cycle-skipping issue by solving a least-squares objective function minimising the travel time misfit in both data and model domains. In the model domain, we adapt the delay time common image gather (DTCIG) to measure the travel time inaccuracy. In the data domain, we extract the travel time misfit between the field and simulated data without adapting their amplitude information. Application of the proposed JDFWI on the 2D example with complex salt bodies demonstrates that it can overcome the cycle skipping issue due to both poor initial model and lack of low frequency. The results of the Gulf of Mexico ocean-bottom node (OBN) application show improved salt boundary estimation without introducing a halo and the recovery of detailed velocity feature that matches with the geology structure on the reflectivity image.

Full Waveform Inversion (FWI) is a seismic imaging technique that utilizes the entire content of seismic trace. This method aims to minimizes differences between real and synthetic seismograms. FWI consists of three main steps: forward modelling to simulate synthetic seismograms, adjoint simulation to calculate the gradient misfit, and optimization to obtain an appropriate model perturbation. FWI is an extremely non-linear inversion technique and computationally intensive since it involves many model parameters. In the context of geophysics, finding parameter models that describe the structure of the earth’s subsurface is the main goal. Geophysical measurements in the field only obtain data that is influenced by the parameter model, in this case seismic recording data, therefore mathematical techniques are studied and applied to determine the correct parameter model from the data obtained, also known as the inversion process. In this study, main research area is on optimization step which include mathematical approach to find perturbation model effectively. The Newton method can be used in the optimization step of FWI. However, its implementation is not promising due to high computational cost in calculating Hessian matrix. To reduce computation cost, an identity matrix is chosen to replace the Hessian matrix, as it usually done in the steepest descent method. Since FWI has a high level of non-linearity, steepest descent method requires many iterations to minimize the objective function. Therefore, this study uses a conjugate gradient method to make the data misfit minimized more quickly. Apart from that, the Line search technique is used to find an appropriate step-length at each iteration on both methods. The FWI workflow is applied to the synthetic acoustic velocity model. We conduct FWI using steepest descent and conjugate gradient methods. The results show that the conjugate gradient method requires fewer iterations than the steepest descent method to minimize the objective function. FWI model result using conjugate gradient method is closer with the true model compared to steepest descent method, showed on misfit result and model image. By comparing these methods we can see that conjugate gradient method can be chosen for application on FWI optimization step for better result on seismic imaging. On the industry scale, better results provide less uncertainty on the model parameters that useful for energy exploration or engineering.

<p>The seismic full waveform inversion (FWI), as one of important ways to obtain the seismic wave velocity, has made rapid development in the last decade. In response to problems of cycle-skipping artifacts, dependence on the initial model, and low-frequency information in FWI, researchers have made many improvements, such as multi-scale envelope inversion and low-frequency extension. Recently, deep learning has been also adopted seismic data processing and interpretation, because of its strong nonlinear mapping ability. However, these works depend on labels used for training heavily, especially for the velocity model in the inversion, which prevents them from real application. Referring to these studies, this work combines low-frequency extension commonly as well as multiscale inversion with deep learning, and proposes a multi-scale FWI gradient optimization method based on CNN. CNN we designed is trained to predict the inversion gradient corresponding to the low-frequency band data in FWI, so that multi-scale gradient optimization can be directly used in multi-scale inversion, expanding the low-frequency information in the actual data and reducing the calculation in FWI. With a specially designed dataset, CNN is trained to optimize the gradients computed from the high-frequency band data by predicting the gradients corresponding to the low-frequency band data and the gradients corresponding to the mid-frequency band data, respectively. The predicted gradients are used in different stages of the multi-scale inversion. The low-frequency gradients are used to invert the initial structural construction so as not to rely on a good initial model, and the high-frequency gradients are used to improve the accuracy of the inversion results. In this way, low-frequency expansion and multiscale inversion can be achieved simultaneously. Our method achieves good results on the initial model for a given uniform wave velocity, effectively alleviating the reliance on the initial model in FWI. This study provides a new idea of combining deep learning and full waveform inversion, which will be effectively used in seismic data processing.</p>

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.

Summary While full waveform inversion (FWI) can provide high resolution velocity models, the method is often let down by the cycle skipping problem. Several solutions based on the estimation of the time shifts have been proposed. Among them, dynamic warping FWI (D-FWI) combines the dynamic warping for the computation of the time shifts between the observed and the predicted data, and a technique which partially warps the observed data to connect the predicted data in a cycle skipping-free way. However, the D-FWI still risks falling into local minima, in particular when the initial model is inaccurate and low-frequency data are missing. Here, we propose a more robust process for estimating the time shifts in which dynamic warping is repeated with offset dependent constraints. The more reliably detected time shifts, can be used in D-FWI with a waveform-based or a traveltime-based objective function to better manage the cycle-skipping problem. We demonstrate the method with a synthetic data from the Marmousi model as well as one challenging OBC dataset from Indonesia.

In seismic exploration, 3D acoustic full waveform inversion (FWI) is nowadays commonly used to estimate large-to-intermediate scale velocity features with low-frequency and wide-aperture data. Velocity maps estimated with this technique can have higher resolution than the ones obtained by travel-time inversion. However, since full waveform inversion fits both the phases and the amplitudes of seismic data, the acoustic assumption may not be adequate to reproduce the data's amplitude versus offset variations in presence of hard velocity contrasts. This could limit the applications of acoustic FWI and put some constrains on the choice of the initial model reducing model's resolution. To better understand the behaviour of acoustic FWI in presence of hard layers and study the potential elastic effects, we built a synthetic model that mimics the geology of North Oman. Then we inverted the data generated with this model starting from an almost exact initial model and also from a smooth one. We show that acoustic FWI diverges when starting from an initial model with too high resolution. The inversion over-fits the elastic data leading to erroneous velocity updates. This synthetic experiment illustrates the limitations of acoustic FWI when used to obtain a high resolution velocity image in complex geology.

The development of an accurate velocity model is the significant target in the Full Waveform Inversion (FWI) process where the data fitting process is carried out based on an ill-posed technique. In the FWI technique optimization process plays a crucial role through which objective function minimizes, which is related to the misfit function between observed and modelled data. However, the influence of external factors such as data fitting errors (local minima) and the presence of noise in data are involved in the success of this processing technique. The artefacts that arise during gradient computation also affect this processing technique. This study presents a strategy to mitigate the influence of these local minima and other artefacts based on the velocity variation coefficient where an angle-dependent gradient conditioning approach has been proposed. It is an auto-controlled process in which the primary mechanism updates the velocity model from a large angle scale to a smaller angle scale when iteration begins. At each iteration, it preserves the previous result whereby it does not scatter or overlap with the previous one. It covers all the angles smoothly which helps in minimizing the data misfit and providing a high-resolution velocity model. The proposed conditioning approach is demonstrated by implementing the Marmousi model, and the result proves that the method provides a much-improved velocity inversion result which is attained with reasonable iterations. This study represents of a suitable procedure for the FWI processing technique where less sensitive artefacts are identified with negligible time consumption. Furthermore, it also helps to reduce the cycle skips and improve convergence in any complex scenario.

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

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.