Fundamental Issues in Full Waveform Inversion

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.

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.

PreviousNext No AccessSEG Technical Program Expanded Abstracts 2004Efficient calculation of steepest descent direction for source‐independent waveform inversion using normalized wavefield by convolutionAuthors: Soonhong CheongChangsoo ShinSukjoon PyunDong‐Joo MinSangyon SuhSoonhong CheongCollege of Civil, Urban & Geosystem Eng., Seoul National Univ.Search for more papers by this author, Changsoo ShinCollege of Civil, Urban & Geosystem Eng., Seoul National Univ.Search for more papers by this author, Sukjoon PyunCollege of Civil, Urban & Geosystem Eng., Seoul National Univ.Search for more papers by this author, Dong‐Joo MinCollege of Civil, Urban & Geosystem Eng., Seoul National Univ.Search for more papers by this author, and Sangyon SuhKorea Institute of Geoscience & Mineral ResourcesSearch for more papers by this authorhttps://doi.org/10.1190/1.1851168 SectionsAboutPDF/ePub ToolsAdd to favoritesDownload CitationsTrack CitationsPermissions ShareFacebookTwitterLinked InRedditEmail Abstract In conventional waveform inversion, geophysicists usually invert a velocity model as well as a source signature simultaneously. For a source‐independent waveform inversion, we define a new objective function by multiplying both observed data and forward modeled data by the respective reference wavefields on the cross. For computation of the steepest descent of the new objective function, we exploit a matrix formalism originated from the symmetry of Green's function of wave equation. In this case, we calculate the steepest descent without explicity computing the Jacobian matrix. Numerical structure of our algorithm resembles that of prestack reverse‐time migration and waveform inversion.Permalink: https://doi.org/10.1190/1.1851168FiguresReferencesRelatedDetailsCited BySource-independent seismic envelope inversion based on the direct envelope Fréchet derivativePan Zhang, Ru-Shan Wu, and Liguo Han1 October 2018 | GEOPHYSICS, Vol. 83, No. 6Robust source-independent elastic full-waveform inversion in the time domainQingchen Zhang, Hui Zhou, Qingqing Li, Hanming Chen, and Jie Wang3 March 2016 | GEOPHYSICS, Vol. 81, No. 2Laplace-domain full-waveform inversion of seismic data lacking low-frequency informationWansoo Ha and Changsoo Shin8 August 2012 | GEOPHYSICS, Vol. 77, No. 5 SEG Technical Program Expanded Abstracts 2004ISSN (print):1052-3812 ISSN (online):1949-4645Copyright: 2004 Pages: 2586 publication data© 2004 Copyright © 2004 Society of Exploration GeophysicistsPublisher:Society of Exploration Geophysicists HistoryPublished: 03 Jan 2005 CITATION INFORMATION Soonhong Cheong, Changsoo Shin, Sukjoon Pyun, Dong‐Joo Min, and Sangyon Suh, (2004), "Efficient calculation of steepest descent direction for source‐independent waveform inversion using normalized wavefield by convolution," SEG Technical Program Expanded Abstracts : 1842-1845. https://doi.org/10.1190/1.1851168 Plain-Language Summary PDF DownloadLoading ...

plex shallow subsurface. This workflow incorporates dispersion curve inversion, traveltime tomography and full waveform inversion into a single framework to optimize the advantages of each. An accurate near-surface velocity is derived using the dispersion curve inversion technique that overcomes limitations of traditional traveltime tomography. This velocity is then incorporated into layer-stripping full waveform or RMO inversion schemes by “freezing” the ultra-shallow part of the model in order to retrieve the deeper velocity structure. We illustrate the workflow on a test data set from Inner Mongolia, China, whose long-offsets and low-frequency content provide the necessary input for both the dispersion curve and full waveform inversion techniques. Using this data set, we demonstrate the efficacy of the integrated methodology in determining both the macro-velocity model and near-surface velocities in structurally complex geological setting.

Full waveform inversion (FWI) is an established precise velocity estimation tool for seismic exploration. Machine learning-based FWI could plausibly circumvent the long-standing cycle-skipping problem of traditional model-driven methods. The physics-guided self-supervised FWI is appealing in that it avoids having to make tedious efforts in terms of label generation for supervised methods. One way is to employ an inversion network to convert the seismic shot gathers into a velocity model. The objective function is to minimize the difference between the recorded seismic data and the synthetic data by solving the wave equation using the inverted velocity model. To further improve the efficiency, we propose a two-stage training strategy for the self-supervised learning FWI. The first stage is to pretrain the inversion network using a simultaneous source for a large-scale velocity model with high efficiency. The second stage is switched to modeling the separate shot gathers for an accurate measurement of the seismic data to invert the velocity model details. The inversion network is a partial convolution attention modified UNet (PCAMUNet), which combines local feature extraction with global information integration to achieve high-resolution velocity model estimation from seismic shot gathers. The time-domain 2D acoustic wave equation serves as the physical constraint in this self-supervised framework. Different loss functions are used for the two stages, that is, the waveform loss with time weighting for the first stage (simultaneous source) and the hybrid waveform with time weighting and logarithmic envelope loss for the second stage (separate source). Comparative experiments demonstrate that the proposed approach improves both inversion accuracy and efficiency on the Marmousi2 model, Overthrust model, and BP model tests. Moreover, the method exhibits excellent noise resistance and stability when low-frequency data component is missing.

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.

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

The extension of the velocity-model domain to subsurface offsets solves the local-minima problem of data-fitting waveform inversion. By regularizing the extended-model data-fitting inversion with the addition of an image-focusing term to the objective function, we achieve robust global convergence of the waveform inversion problem. The method shares with full waveform inversion the advantage of simultaneously solving for all the wavelengths of the model, but it also has the global convergence characteristics of wave-equation migration velocity analysis. The numerical implementation of the proposed inversion method requires the solution of an extended waveequation where velocity is a convolutional, instead of scalar, operator. The resulting method is therefore computationally intensive, and more computationally efficient approximations would be beneficial. Numerical tests performed on synthetic data modeled assuming a modified Marmousi model demonstrate the global convergence as well the high-resolution potential of the method.

Long-wavelength velocity model building is a nonlinear process. It has traditionally been achieved without appealing to wave-equation-based approaches for combined refracted and reflected waves. We developed a cascaded wave-equation tomography method in the data domain, taking advantage of the information contained in the reflected and refracted waves. The objective function was the traveltime residual that maximized the crosscorrelation function between real and synthetic data. To alleviate the nonlinearity of the inversion problem, refracted waves were initially used to provide vertical constraints on the velocity model, and reflected waves were then included to provide lateral constraints. The use of reflected waves required scale separation. We separated the long- and short-wavelength subsurface structures into velocity and density models, respectively. The velocity model update was restricted to long wavelengths during the wave-equation tomography, whereas the density model was used to absorb all the short-wavelength impedance contrasts. To improve the computation efficiency, the density model was converted into the zero-offset traveltime domain, where it was invariant to changes of the long-wavelength velocity model. After the wave-equation tomography has derived an optimized long-wavelength velocity model, full-waveform inversion was used to invert all the data to retrieve the short-wavelength velocity structures. We developed our method in two synthetic tests and then applied it to a marine field data set. We evaluated the results of the use of refracted and reflected waves, which was critical for accurately building the long-wavelength velocity model. We showed that our wave-equation tomography strategy was robust for the real data application.

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

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.

Depth imaging in the Gulf of Mexico becomes increasingly challenging when we explore under a complex overburden such as salt. Traditional marine streamer surveys offer limited illumination of deep reservoirs under thick salt bodies. In addition, errors in modelled salt geometry cause significant kinematic mismatches between field and synthetic data, leading to the problem of cycle-skipping in classic least squares Full Waveform Inversion (FWI). In this paper we combine long offset Ocean Bottom Node (OBN) surveys with a waveform inversion technique that uses a time shift objective function to address these challenges.

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.

Full-waveform inversion (FWI) is capable of handling various wave types, e.g., primary reflections, direct waves, diving waves, and multiple reflections. However, FWI often encounters high nonlinearity when the starting velocity model significantly deviates from the true velocity, and the existence of multiples can further worsen this situation. As a result, the objective function may be trapped in a local minimum, causing the inversion process to fail to converge to the global minimum. Constructing an objective function by decomposing multiples into different controlled orders and performing the FWI on multiples with consecutive orders can expedite convergence. However, conducting the FWI using multiples in an order-by-order way may result in high computational cost. In this study, we address this challenge by creating virtual sources and supergathers. These supergathers phase-encode many orders of multiples and integrate them into one for FWI, requiring only one FWI for all orders of multiples. By employing such a random phase-encoding scheme, crosstalk artifacts arising from interferences between mismatched multiple orders in the source and receiver wavefields can be largely mitigated. The proposed method is validated through numerical experiments with a synthetic dataset, as well as using marine field data from Madagascar. The results demonstrate that FWI employing phase-encoded multiples significantly improves convergence while maintaining computational efficiency.

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.