Full waveform inversion of blended surface-related multiple orders

Traditional full-waveform inversion (FWI) seeks to find the best model by minimizing an objective function defined as the difference between the model-predicted and observed data in amplitude and phase. In principle, FWI should fit all wave types including direct waves, diving waves, primaries, and multiples. However, when an initial model is far from the true model, FWI will encounter difficulties in matching multiples. Physically, multiples may contain more subsurface information compared to primary and diving waves. Multiples cover a wide range of reflection angles during wave propagation and offer the advantage of imaging the shadow zones that cannot be reached or are poorly illuminated by primary reflections. We have developed a new method of waveform inversion using multiples. We first separate the multiples into different orders. The objective function we seek to minimize consists of the data difference between the modeled data using a lower order multiple as the source and the higher order multiple as data. This method is called controlled-order multiple waveform inversion (CMWI). Our numerical examples determined that the CMWI is a promising method to improve velocity updates.

Simultaneous-source techniques have been proposed to reduce the acquisition cost and alleviate the computational overburden of data processing, and they have been applied to full waveform inversion (FWI) in recent years. These techniques are mainly based on the l2-norm objective function (least-squares criterion). However, if we consider that real field data contain noise such as outliers, it would be preferable to use the robust objective function in simultaneous-source FWI. In this study, we propose combining the simultaneous-source FWI with the l1-norm objective function (least-absolute criterion), which is known to be robust to data containing noise, specifically outliers. For the l1-norm-based simultaneous-source (SS-1) FWI, we first verify the crosstalk reduction and the robustness to data containing outliers. The expectations of the gradient direction directly give an evidence for the crosstalk reduction, and a signal-to-noise ratio analysis supports the convergence of our algorithm. The analysis shows that the crosstalk noise of the SS-1 FWI can be suppressed by random phase encoding. In addition, the spectrum of the weighted residuals indicates that the main property of the l1-norm objective function is preserved in the SS-1 FWI. Numerical examples show that the SS-1 FWI produces reliable results as the l2-norm simultaneous-source FWI does, similar to the case of the individual-source FWI. These results support the idea that combining the simultaneous-source FWI with the l1-norm objective function improves the computational efficiency and preserves the intrinsic characteristics of the l1-norm objective function.

Full-waveform inversion (FWI) is firmly established within our industry as a powerful velocity model building tool. FWI carries significant theoretical advantages over conventional velocity model building methods such as refraction and reflection tomography. Specifically, by solving a nonlinear inverse problem through the wave equation, FWI is able to recover a broadband velocity model containing both high and low spatial wavenumbers, thus extending the approximation of residual moveout correction inherent in traditional velocity model building approaches. Moreover, FWI is capable of inverting information from the entire wavefield (i.e., early arrivals, reflections, refractions, and multiple energy) rather than from a subset as in conventional approaches (i.e., first break and primary reflections), thereby availing itself of more information to better constrain its model estimate. However, these theoretical benefits cannot be realized easily in practice because various complexities of real seismic data often conspire to violate algorithmic assumptions, leading to unsatisfactory results. Dynamic matching FWI (DMFWI) is a newly developed algorithm that solves an inversion problem that maximizes the cross correlation of two dynamically matched data sets — one recorded and the other synthetic. Dynamic matching of the two data sets de-emphasizes the amplitude impact, which allows the algorithm to focus on minimizing their kinematic differences rather than amplitude in the data-fitting process. The multichannel correlation makes the algorithm robust for data with low signal-to-noise ratio. Applications of DMFWI across different types of acquisition and geologic settings demonstrate that this novel FWI approach can resolve complex velocity errors and provide high-quality migrated images that exhibit a high degree of geologic plausibility. Additionally, reflectivity images can be obtained in a straightforward manner as natural byproducts through computation of the directional derivative of the inverted FWI velocity models.

Full waveform inversion (FWI) is a method for building subsurface models with high resolution by optimizing a data-fitting problem. However, it is still a challenge to apply the FWI method to complicated subsurface models, because FWI is a highly ill-fitting inversion problem. When the simulated data differ from the observed data by more than half a cycle, FWI tends to suffer from the cycle-skipping problems and get trapped in a local minimum. To help the inversion to converge to the accurate subsurface model, we typically build a reasonable initial model or use low-frequency data to start our inversion. Here, we developed a novel technique called smoothing of dilated convolutions inversion (SDCI) to generate low-frequency components from seismic data to recover low-wavenumber components of the subsurface. In the theory part, we first present the objective function of the SDCI method and then derive the gradient of the objective function relative to the velocity using the adjoint-state approach. We then apply the method to a set of simulated data from the Marmousi model. The SDCI method can reasonably estimate the subsurface model even if the simulated seismic record does not contain information below 5 Hz. The numerical examples prove the effectiveness and feasibility of the proposed method. From the SDCI results, the FWI method recovers the velocity with higher accuracy and resolution.

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.

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.

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.

Source estimation is essential for all wave-equation-based seismic inversions, including full-waveform inversion (FWI) and the recently proposed wavefield-reconstruction inversion (WRI). When the source estimation is inaccurate, errors will propagate into the predicted data and introduce additional data misfit. As a consequence, inversion results that minimize this data misfit may become erroneous. To mitigate the errors introduced by the incorrect and preestimated sources, an embedded procedure that updates sources along with medium parameters is necessary for the inversion. So far, such a procedure is still missing in the context of WRI, a method that is, in many situations, less prone to local minima related to so-called cycle skipping, compared with FWI through exact data fitting. Although WRI indeed helps to mitigate issues related to cycle skipping by extending the search space with wavefields as auxiliary variables, it relies on having access to the correct source functions. To remove the requirement of having the accurate source functions, we have developed a source-estimation technique specifically designed for WRI. To achieve this task, we consider the source functions as unknown variables and arrive at an objective function that depends on the medium parameters, wavefields, and source functions. During each iteration, we apply the so-called variable projection method to simultaneously project out the source functions and wavefields. After the projection, we obtain a reduced objective function that only depends on the medium parameters and invert for the unknown medium parameters by minimizing this reduced objective. Numerical experiments illustrate that this approach can produce accurate estimates of the unknown medium parameters without any prior information of the source functions.

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.

When reliable a priori information is not available, it is difficult to correctly predict near-surface S-wave velocity models from Rayleigh waves through existing techniques, especially in the case of complex geology. To tackle this issue, we have developed a new method: two-grid genetic-algorithm Rayleigh-wave full-waveform inversion (FWI). Adopting a two-grid parameterization of the model, the genetic algorithm inverts for unknown velocities and densities at the nodes of a coarse grid, whereas the forward modeling is performed on a fine grid to avoid numerical dispersion. A bilinear interpolation brings the coarse-grid results into the fine-grid models. The coarse inversion grid allows for a significant reduction in the computing time required by the genetic algorithm to converge. With a coarser grid, there are fewer unknowns and less required computing time, at the expense of the model resolution. To further increase efficiency, our inversion code can perform the optimization using an offset-marching strategy and/or a frequency-marching strategy that can make use of different kinds of objective functions and allows for parallel computing. We illustrate the effect of our inversion method using three synthetic examples with rather complex near-surface models. Although no a priori information was used in all three tests, the long-wavelength structures of the reference models were fairly predicted, and satisfactory matches between “observed” and predicted data were achieved. The fair predictions of the reference models suggest that the final models estimated by our genetic-algorithm FWI, which we call macromodels, would be suitable inputs to gradient-based Rayleigh-wave FWI for further refinement. We also explored other issues related to the practical use of the method in different work and explored applications of the method to field data.

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.

ABSTRACTNumerical implementation of the gradient of the cost function in a gradient‐based full‐ waveform inversion (FWI) is essentially a migration operator used in wave equation migration. In FWI, minimizing different data residual norms results in different weighting strategies of data residuals at receiver locations prior to back‐propagation into the medium. In this paper, we propose different scaling methods to the receiver wavefield and compare their performances. Using time‐domain reverse‐time migration (RTM), we show that compared to conventional algorithms, this type of scaling is able to significantly suppress non‐Gaussian noise, i.e., outliers. Our tests also show that scaling by its absolute norm produces better results than other approaches.

Numerical implementation of the gradient of the cost function in a gradient-based full waveform inversion (FWI) is a migration operator used in wave equation migration. In FWI, minimizing different data residual norms results in different weighting strategies of data residuals at receiver locations prior to back propagation into the medium. In this paper, we propose different scaling methods to the receiver wavefield and compare their performances. Using time domain reverse time migration (RTM), we show that this type of scaling is able to significantly suppress outliers compared to conventional algorithms. Our tests also show that scaling by its absolute norm produces better results than other approaches.

Summary Reverse-Time Migration (RTM) is routinely used for depth velocity model building and imaging in deepwater exploration around the world. In this study, we employed analytical redatuming to improve the efficiency of true-amplitude RTM for deepwater Earth model. This method analytically redatums both the source and receiver wavefields from surface to water bottom and injects the redatumed wavefield into the computational domain while preserving true-amplitude in the migrated common-shot image. By using analytic redatuming in deepwater area, the total runtime, memory and disk space requirements are significantly reduced. In addition, the image quality is improved by overcoming the singularity of point-source injection in the finite difference method, accurate positioning of source and receivers, and the reduction of numerical dispersion noise. The underlying theories developed in this paper can be extended to some other important applications such as Full-Waveform Inversion (FWI), true-amplitude layer-stripping RTM/FWI.

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.