Tomography-mode full-waveform inversion with structural shaping regularization

In recent years the elastic full waveform inversion (FWI) was successfully applied to synthetic and field data to compute high resolution velocity models. While seismic velocities are derived from recorded phase information, density models can be estimated from the amplitudes. However, due to the complexity of the inverse problem a long wavelength initial model is required for a good reconstruction of the density. The inclusion of gravity data into the FWI concept can solve this problem. In this study a two-step hierarchic joint inversion of seismic waveforms and gravity data is tested using the Marmousi-II model. In step 1 FWI is performed for all elastic parameters. Gaussian filtered velocity models of the true model and a constant halfspace density model (CDH) are used as initial models. While the velocities can be reconstructed well, the density shows large deviations from the true model. In step 2 joint inversion is applied to optimize only the density model, while the velocity inversion results of the first step and the CDH are used as initial models. The results of this combined approach show a significant improvement of the density model compared to the results of a pure FWI.

<p>Full waveform inversion (FWI) is one of the most attractive geophysical inversion methods that reconstruct models with higher quality by exploiting the information of full wave-field. Despite its high resolution and successful practical applications, there still exist several obstacles to the successful application of FWI for passive earthquake sources, such as the high non-linearity for model convergence and demand for accurate source information, such as the moment tensor, the source time function, etc. To alleviate the requirement for a priori source information in waveform inversion, we propose a new method called Waveform Energy Focusing Tomography (WEFT), which backpropagates the observed wavefield from the receivers, not the data residuals like in conventional FWI, and tries to maximize the back-propagated wavefield energy around the source location over a short period around the origin time. Therefore, there is no need to provide the focal mechanism and source time function in advance. To better reconstruct the passive sources, the least-squares moment tensor migration approach is used, and the Hessian matrix is approximated using either analytic expression or raytracing. Since waveform fitting is superseded by simpler energy maximization, the nonlinearity of WEFT is weaker than that of FWI, and even less-accurate initial velocity model can be used. These advantages of WEFT make it more practical  for challenging earthquake data, especially for local small magnitude earthquakes where both velocity model and earthquake source information are unknown.</p>

Full-waveform inversion (FWI) is a highly nonlinear and nonconvex problem. To mitigate the dependence of FWI on the quality of starting model and on the low frequencies in the data, we apply the gradient sampling algorithm (GSA) introduced for nonsmooth, nonconvex optimization problems to FWI. The search space is hugely expanded to have more freedom to accommodate large velocity errors in the starting model. The original implementation of GSA requires explicit calculation of the gradient at each sampled vector, which is prohibitively expensive. Based on the observation that a slight perturbation in the velocity model causes a small spatial shift of the wavefield, we have approximated the sampled gradients by crosscorrelating the space-shifted source- and receiver-side wavefields. Theoretical derivation suggests that the two wavefields should be shifted in the same direction to obtain reasonable low-wavenumber updates. The final descent search direction is obtained by summing all the shifted gradients. For practical implementation, we only take one random space shift at each time step during the gradient calculation. This simplification provides an efficient realization in which the computational costs and memory requirements are the same as conventional FWI. Multiple numerical examples demonstrate that the proposed method alleviates the cycle-skipping problem of conventional FWI when starting from very crude initial velocity models without low-frequency 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.

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.

Wavefield reconstruction inversion (WRI) mitigates cycle skipping by using an inaccurate initial velocity. This attractive technique is usually implemented with shot records. However, if large numbers of shot records are used, WRI can become computationally burdensome due to the many over-determined linear systems that need to be solved. To alleviate this computational issue, we propose an efficient WRI scheme involving plane-wave encoding (WRI-PW) in the frequency domain. Plane-wave encoding can dramatically reduce the number of relevant datasets by transforming shot records into common ray-parameter gathers with time shifting. Therefore, plane-wave encoding is widely used in many aspects of seismic data processing (e.g. waveform inversion, reverse time migration, etc.). Initially, we performed a simple numerical experiment using a velocity model with a box-shaped anomaly. WRI-PW also could generate scattering wavefields in a homogeneous model. Next, computational efficiency was checked with a modified Marmousi-2 model. The results show that the usage of a sufficient plane-wave angle can achieve satisfactory inversion results. It indicates that WRI-PW requires small datasets compared to WRI. Thus, the computational costs for solving the augmented system can be reduced. Further experiments were conducted to evaluate the robustness of WRI-PW to random noise and to compare WRI-PW and conventional full waveform inversion (FWI) with a modified SEG/EAGE salt velocity model. We verify that WRI-PW is more robust to random noise than WRI, it exhibited less dependency on the accuracy of the initial velocity model than conventional FWI and it is computationally efficient.

In seismic exploration, establishing a reliable velocity model is crucial for seismic migration imaging and reservoir characterization. Full-waveform inversion (FWI) is used to recover subsurface physical parameters by iteratively minimizing the misfit between the observed and synthetic data, theoretically offering the highest modeling accuracy. FWI is a combination of migration imaging and tomography. However, for reflection data, the update of the tomographic component is notably weaker than the migration component. When the gradient is dominated by the migration component, FWI can easily fall into local minima, which leads to bad inversion results. Hence, it is necessary to extract the tomographic component from the original FWI gradient to build a robust background velocity model when the initial model is inaccurate. In this paper, we develop to use the bidimensional empirical mode decomposition (BEMD) method to separate the original FWI gradient into the tomography and migration component. The BEMD method first adaptively decomposes the gradient into different-scale subcomponents. Then, the small-scale bidimensional intrinsic mode functions are combined into the migration component, and the large-scale residual component is considered to be the tomographic component. At the beginning of FWI, the background velocity is updated using the tomographic component, and with iterations, small-scale components are gradually added to update the velocity model. Numerical experiments show that the BEMD method can effectively separate the two components. The BEMD-based FWI can reconstruct the background velocity model, improving the FWI accuracy.

Full waveform inversion (FWI) makes full use of the seismic waveforms to find high-resolution velocity and density models by minimising residuals between the calculated and recorded data. Q attenuation widely exists in the subsurface media, leading to weak amplitude and misplacement of reflectors. However, the commonly used Q-compensated FWI (QFWI) based on the second-order wave equation has difficulties in simultaneously inverting velocity and density fields. A QFWI method based on new first-order viscoacoustic quasi-differential equations is proposed to simultaneously produce velocity and density fields. Based on the adjoint state inversion theory, Q-compensated forward-propagated operators, adjoint operators, and gradient equations are derived using the newly derived first-order viscoacoustic quasi-differential wave equations. The time-domain multi-scale decomposition method is introduced to update the velocity and density models from a low to a high wavenumber. Numerical examples on an actual work area model and a modified attenuating Marmousi model show that the proposed QFWI method produces higher-accuracy velocity and density models with iterations by correcting the Q attenuation than the conventional acoustic FWI. Even when the Q model is extremely inaccurate, the proposed QFWI obtains acceptable inversion results. Compared to the conventional QFWI, our QFWI better inverts velocity field in the case of an inaccurate density model. Finally, we verify the adaptability of our QFWI to field data.

Summary Full waveform inversion of full-azimuth broadband ocean bottom nodes is emerging as the go-to strategy for high-resolution seismic imaging in marine settings. The areal extent of node acquisitions and the high signal-to-noise ratio at low frequencies fosters the illumination and undershooting of sought structures by deeply penetrating diving waves and post-critical reflections even in complex environments. In this work, we present full waveform inversion results of a newly provisioned full-azimuth broadband ocean bottom node data from the Gorgon field located off the northwest coast of Australia. We show that even at the first stage of the inversion, using conventional full waveform inversion, the P-wave vertical velocity model is significantly updated and reveals expected structural features in accordance with the geological setting of the Gorgon basin. The enhanced data fit and the excellent match of velocity profiles with well logs validates this result which will serve as baseline for ongoing and future multi-component and multi-parameter FWI of this dataset from offshore North West Australia.

<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>

SUMMARY Full waveform inversion (FWI) is a high-precision subsurface imaging technique that inverts subsurface parameter models by minimizing the discrepancy between observed and synthetic seismic data. However, complicated wave propagation mechanisms, non-convexity of the loss function and limited seismic acquisition system necessitate the incorporation of sufficient prior and physical constraints to alleviate the ill-posedness and cycle-skipping. Although the implicit FWI (IFWI) can encode implicit spectral bias (i.e. inverting model parameters from low frequencies to high frequencies) to reduce the dependence on an accurate initial model, its limited high-frequency inversion capability results in thousands of iterations for the final results. In this paper, we indicate that the frequency hyperparameters of the sine activation function in IFWI modulate the spectral bias, making a trade-off between inversion accuracy and stability, that is, lower frequencies yield robust FWI but lower accuracy, while higher frequencies achieve higher accuracy on the premise of an accurate initial model. To improve both the stability and accuracy of IFWI, we propose a novel implicit FWI method with an adaptive Fourier reparametrization strategy (termed FR-IFWI), which explicitly encodes multifrequency information by reparametrizing the network weights using a learnable coefficient matrix and fixed Fourier frequency bases. The role of learnable matrices in neural networks can evolve from determining frequencies in IFWI to actively selecting frequencies from fixed frequency bases through FR-IFWI, which alleviates the dependence on activation function frequency and obtains more robust and accurate inversion results. Extensive numerical experiments on the modified Marmousi, 2-D SEG/EAGE Salt and Overthrust models confirm that FR-IFWI successfully achieves superior inversion efficiency and accuracy compared with conventional FWI and IFWI methods.

SUMMARY The accurate and fast estimation of velocity models is crucial in seismic imaging. Conventional methods, such as tomography, stereotomography, migration velocity analysis and full-waveform inversion (FWI), obtain appropriate velocity models; however, they require intense and specialized human supervision and consume much time and computational resources. In recent years, some works investigated deep learning (DL) algorithms to obtain the velocity model directly from shots or migrated angle panels, obtaining encouraging predictions of synthetic models. This paper proposes a new flow to recover structurally complex velocity models with DL. Inspired by the conventional geophysical velocity model building methods, instead of predicting the entire model in one step, we predict the velocity model iteratively. We implement the iterative nature of the process when, at each iteration, we train the DL algorithm to determine the velocity model with a certain level of precision/resolution for the next iteration; we name this process as ‘Deep-Tomography’. Starting from an initial model, that is an ultrasmooth version of the true model, Deep-Tomography is able to predict an appropriate final model, even in complete unseen during the training data, like the Marmousi model. When used as the initial model for FWI, the models estimated by Deep-Tomography can also improve substantially the final results obtained with FWI.

Summary Ultra-long offset stationary-receiver acquisitions implemented with multi-component Ocean Bottom Nodes (OBN) are emerging as a suitable technology for deep offshore subsalt imaging by full waveform inversion (FWI). Ultra-long offsets allow to undershoot the deepest targeted structures with diving waves. In this framework, first-arrival traveltime + slope tomography (FASTT) provides a simple and efficient tool to build kinematically-accurate initial velocity model for full waveform inversion (FWI). In particular, the incorporation of slopes in the first-arrival tomography in addition to traveltimes removes artifacts in poorly illuminated areas and better reconstruct structural dips. Either the FASTT or the FWI velocity models can be used as reliable background models for prestack depth migration of co-incident towed-streamer data. However, the high resolution of the FWI velocity model facilitates the integrated interpretation of the reflectivity mapped by migration and the intermediate-scale velocity variations reconstructed by waveform inversion. We illustrate the workflow combining first-arrival tomography + FWI of ultra-long offset OBN data and migration of co-incident towed-streamer data with a challenging deep crustal case study in the eastern Nankai subduction zone.

Full-waveform inversion (FWI) includes migration and tomography modes. The tomographic component of the gradient from reflection data is usually much weaker than the migration component. To use the tomography mode to fix background velocity errors, it is necessary to extract the tomographic component from the gradient. Otherwise, the inversion will be dominated by the migration mode. We have developed a method based on nonstationary smoothing to extract the tomographic component from the raw gradient. By analyzing the characteristics of the scattering angle filtering, the wavenumber of the tomographic component at a given frequency is seen to be smaller than that of the migration component. Therefore, low-wavenumber-pass filtering can be applied to extract the tomographic component. The low-wavenumber-pass smoothing filters are designed with Gaussian filters that are determined by the frequency of inversion, the model velocity, and the minimum scattering angle. Thus, this filtering is nonstationary smoothing in the space domain. Because this filtering is carried out frequency by frequency, it works naturally and efficiently for FWI based on frequency-domain modeling. Furthermore, because the maximum opening angle of the reflections in a typical acquisition geometry is much smaller than the minimum scattering angle for the tomographic component, which is generally set at 160°, there is a relatively large gap between the wavenumbers of the tomographic and migration components. In other words, the nonstationary smoothing can be applied once to a group of frequencies for time-domain FWI without leaking the migration component into the tomographic component. Analyses and numerical tests indicate that two frequency groups are generally sufficient to extract the tomographic component for the typical frequency range of time-domain FWI. The numerical tests also demonstrate that the nonstationary smoothing method is effective and efficient at extracting the tomographic component for reflection waveform inversion.

Velocity model building and depth imaging are known to be complicated endeavors in numerous onshore areas in the Middle East. Despite recent advancements toward denser, wide-azimuth, and low-frequency seismic acquisitions, the application of full-waveform inversion (FWI) remains challenging. The main difficulties arise from complex near-surface geology that creates wave mode conversions (P to S waves but also strong multiples) and very energetic and complex ground roll overlaying reflected waves and dominating low frequencies, with the diving waves being less energetic and noisier. This paper presents recent progress in land FWI toward higher-resolution velocity models, with successful applications to case studies from the Sultanate of Oman. The first example shows an application of land FWI imaging. It shows that we are able to invert jointly diving and reflected waves in acoustic FWI. With proper input data conditioning and initial model preparation, we successfully ran acoustic FWI up to 45 Hz, resulting in a high-resolution velocity model and derived FWI image that significantly improved subsurface imaging and geologic understanding. The inverted high-frequency velocity could also be more reliably used for subsequent reservoir characterization steps. Given the important role of a good-quality near-surface model for stabilizing high-frequency FWI updates, the second example highlights the creation of a detailed shallow velocity model. We propose using elastic FWI of surface waves for characterization of the near surface in complex geologic settings. Here, we demonstrate the effectiveness of this workflow to enhance imaging of the shallow subsurface. Compared to conventional methods such as surface-wave inversion, elastic FWI of surface waves has the advantage of 3D full-wavefield modeling with no requirement for picking. Our workflow also benefits from the ultra-low frequencies recovered by interferometry to obtain a more stable deeper update.