Cycle-skipping Issues Research Articles

Characteristic reflection wavefield-based wave equation traveltime inversion

Introducing a high-precision deep-depth velocity model is essential for accurately imaging seismic data in complex areas such as deepwater and deep depth in oil and gas exploration. Conventional methods, such as reflection full-waveform inversion (RFWI), face challenges, notably the cycle-skipping issue, especially when the initial velocity model is far away from the true model. To address this, we develop a novel wave equation traveltime inversion (WETI) using the characteristic reflection wavefield (CRW) to update the low-wavenumber components of the velocity model. The CRWs are primary reflections in seismic data that contribute significantly to velocity model updates. We implement a multistep approach, including migration, characteristic reflector structure (CRS) picking, and demigration to extract the CRW. This extraction, constrained by CRS, ensures accurate matching between the observed and the simulated CRWs, even with an inaccurate initial velocity model, thus enabling an accurate traveltime shift estimation during inversion. Unlike conventional methods that use all reflection data simultaneously, CRW-WETI strategically selects fewer CRWs, enhancing inversion convexity and avoiding local minima from the cycle-skipping problem. This selective inversion begins with a limited number of CRWs to ensure stability and progressively incorporates more as the WETI process advances, eventually transitioning into conventional RFWI. Moreover, with the CRS constraint, CRW can be selected based on the target region. The target-oriented velocity inversion can also be implemented easily to update a local anomalous velocity model. Numerical examples of synthetic and field data demonstrate the effectiveness and validity of our method.

The importance of data preconditioning strategies for envelope full-waveform inversion methods: Demonstration on marine seismic data

Envelope full-waveform inversion (EI) of seismic data has been proposed to overcome the cycle-skipping issue and recover long-wavelength velocity components for over a decade. However, we are aware of few published successful applications of EI on real data examples (except for some correlation- or phase-based EI methods). We implement EI methods (EI with [Formula: see text], EI with p = 1, and improved EI) on 2D marine seismic data and find that the amplitude mismatching between the modeled and observed data is a critical factor that prevents the successful application of EI methods to real data. To match the amplitude better, we include a data weighting preconditioner in the objective function of the EI methods. The preconditioner term acts as a weighting factor to compensate for the amplitude mismatch between the observed and modeled data. We develop a method for calculating the preconditioner term using the amplitude of the head waves as a reference. Furthermore, we derive the adjoint source and gradient of the EI using the data preconditioning method. We illustrate the successful application of EI methods with the data preconditioning method using the 2D marine seismic data example. In comparison to the EI methods without the data preconditioning method, those using data preconditioning yield much more geophysically reasonable velocity models and Kirchhoff image sections and common-image gathers.

Joint wave-equation inversion of Rayleigh and Love dispersion curves

Wave-equation dispersion (WD) inversion techniques for surface waves have proven to be a robust way to invert the S-wave velocity model. Unlike 1D dispersion curve inversion, WD method obviates the need for a layered model assumption and reduces the susceptibility to cycle-skipping issues in surface wave full-waveform inversion. Previous WD inversion experiments conducted on Rayleigh and Love waves have highlighted that inverting Love waves yields better stability due to their independence from the P-wave velocity model. Nevertheless, Rayleigh waves possess the advantage of greater penetration depth compared with Love waves with similar wavelengths. Therefore, combining the two types of surface waves is a feasible way to improve the accuracy of S velocity tomograms. In light of this, we develop a novel approach: a joint WD inversion encompassing Rayleigh and Love waves. This innovative technique adjusts the weighting of individual WD gradients using the sensitivity factor of an equivalent layered model, offering a significant advancement in subsurface characterization. Synthetic model tests demonstrate that the joint WD inversion method can generate a more accurate S-wave velocity model, particularly in the presence of complex low-velocity layers or high-velocity layers, when compared with individual wave WD inversion techniques. Simultaneously, the results of field tests validate the effectiveness of the proposed joint WD inversion strategy in producing a more dependable S-wave velocity distribution that aligns closely with the actual geologic structure.

A progressive data assimilation strategy in full-waveform adjoint tomography

Elastic full-waveform inversion (FWI) is powerful in resolving fine-scale structures in complex models, but suffering from strong nonlinearity and slow convergence rate, especially in the presence of cycle-skipping issues when the initial model is not well constrained. We present a progressive data assimilation (PDA) strategy to mitigate the cycle-skipping issue and boost the convergence rate. Specifically, we split each seismogram into several time windows with a constant length after the first arrival. At each iteration, the PDA strategy automatically screens the windows that are not cycle skipped for inversion according to the criteria of similarity measured between the observed and predicted waveform within a given window. Through the iterations, the model improves and more data will automatically be assimilated into the inversion, forming a positive feedback of model update, and speed up the convergence rate. To provide additional constraints and also stabilise the inversion at the beginning, we include low-frequency surface waves and examine their contribution to the inversion. We validate the effectiveness of the PDA strategy in FWI through numerical experiments using the 1D linear gradient velocity model and the Foothill model. Compared with the conventional FWI strategy, the proposed PDA strategy in an elastic FWI can effectively mitigate the cycle-skipping problem and significantly accelerate the convergence rate. We further apply this method to the field data, and it improves the shallow part of the model with only a fraction of the entire dataset, demonstrating the great potential for practical usage.

A Graph-Space Optimal Transport Approach Based on Kaniadakis κ-Gaussian Distribution for Inverse Problems Related to Wave Propagation.

Data-centric inverse problems are a process of inferring physical attributes from indirect measurements. Full-waveform inversion (FWI) is a non-linear inverse problem that attempts to obtain a quantitative physical model by comparing the wave equation solution with observed data, optimizing an objective function. However, the FWI is strenuously dependent on a robust objective function, especially for dealing with cycle-skipping issues and non-Gaussian noises in the dataset. In this work, we present an objective function based on the Kaniadakis κ-Gaussian distribution and the optimal transport (OT) theory to mitigate non-Gaussian noise effects and phase ambiguity concerns that cause cycle skipping. We construct the κ-objective function using the probabilistic maximum likelihood procedure and include it within a well-posed version of the original OT formulation, known as the Kantorovich-Rubinstein metric. We represent the data in the graph space to satisfy the probability axioms required by the Kantorovich-Rubinstein framework. We call our proposal the κ-Graph-Space Optimal Transport FWI (κ-GSOT-FWI). The results suggest that the κ-GSOT-FWI is an effective procedure to circumvent the effects of non-Gaussian noise and cycle-skipping problems. They also show that the Kaniadakis κ-statistics significantly improve the FWI objective function convergence, resulting in higher-resolution models than classical techniques, especially when κ=0.6.

Localized adaptive waveform inversion: regularizations for Gabor deconvolution and 3-D field data application

SUMMARYRecently, we have developed a localized adaptive waveform inversion (LAWI) method to tackle the cycle-skipping issue in velocity reconstruction through seismic waveform inversion. The LAWI method employs a local matching filter, computed using Gabor deconvolution, to measure the instantaneous time-shift between observed and calculated data. Unlike the adaptive waveform inversion (AWI) approach, the LAWI method can take the non-stationarity between observed and calculated data into account. In this work, we investigate two types of regularization based on prior information about the expected filter, which could be a minimum-norm filter or a delta-shape filter, with regard to their effects on the robustness and resolution of inversion. We demonstrate on synthetic data the advantages and disadvantages of these two types of prior information, where the delta-type LAWI may handle multiple observed phases not initially predicted by the starting velocity model. Therefore, we apply the delta-type LAWI to a high-quality 3-D field data set in the North Sea, eliminating the need for data-windowing tuning, which can be tedious and time-consuming for 3-D data. Under different workflows with varying reliable initial models and frequency bands of the pressure data considered, we show that the LAWI approach is robust, effective and efficient for reconstructing the P-wave velocity, while other approaches such as AWI and graph-space optimal-transport method may require meticulous data-tuning strategies to converge to the correct model. Well logs and data fits, primarily from early arrivals, give us confidence that this LAWI approach could be applied to various acquisitions and subsurface targets, thanks to its phase-driven principle.

Full Waveform Inversion Using a High-Dimensional Local-Coherence Misfit Function

Conventional full-waveform seismic inversion (FWI) tries to estimate a subsurface model that can accurately predict surface records by minimizing an L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -norm misfit between observed and synthetic data. If the initial model is far away from the true model, the cycle-skipping issue might occur and the L2-norm based FWI produces a spurious model. To mitigate this problem, we present a novel FWI scheme using a high-dimensional local coherence misfit function. A 2D/3D window is first used to extract local seismic waveform from the common-shot gathers. Then, we apply a normalized cross-correlation to measure the coherence of local synthetic and observed records, which is used as the misfit to iteratively update subsurface velocity model. The new misfit function enhances the contribution of phase fitting while reducing the amplitude contribution, which helps to increase the tolerance of FWI to an inaccurate initial velocity model. In addition, the computation of local waveform coherence along the temporal and spatial axes can adaptively balance the adjoint source amplitudes for strong near-offset reflections and weak far-offset refractions, which improves the low-wavenumber updates. Numerical experiments for synthetic and field data demonstrate that the proposed FWI scheme has a better tolerance to inaccurate starting models and is less sensitive to cycle-skipping issue compared with conventional FWI method.

Elastic Reflection Waveform Inversion With a Nonlinear Born Scattering Operator for Multiparameter Reconstruction

Elastic full waveform inversion (FWI) is one of the most popular tools in recovering the subsurface P- and S-velocity models with high resolution. However, it requires either a good starting model or low-frequency data to avoid the cycle-skipping issue. Elastic reflection waveform inversion (ERWI) uses the migration/demigration process to compute the low-wavenumber gradient. One drawback of conventional ERWI is that it mainly relies on the first-order Born scattering and only utilizes primary reflection wave paths, providing limited wavenumber illumination in the update. In this letter, a nonlinear Born scattering operator is incorporated with ERWI for multiparameter reconstruction to mitigate the drawback of the conventional linear Born approximation. Finally, we use numerical examples to show that the nonlinear Born scattering operator allows us to take advantage of the multiples and enlarge the wavenumber illumination in ERWI results.

ML-misfit: A neural network formulation of the misfit function for full-waveform inversion

A robust misfit function is essential for mitigating cycle-skipping in full-waveform inversion (FWI), leading to stable updates of the velocity model in this highly nonlinear optimization process. State-of-the-art misfit functions, including matching filter or optimal transport misfits, are all hand-crafted and developed from first principles. With the growth of artificial intelligence in geoscience, we propose learning a robust misfit function for FWI, entitled ML-misfit, based on machine learning. Inspired by the recently introduced optimal transport of the matching filter objective function, we design a specific neural network architecture for the misfit function in a form that allows for global comparison of the predicted and measured data. The proposed neural network architecture also guarantees that the resulting misfit is a pseudo-metric for efficient training. In the framework of meta-learning, we train the network by running FWI to invert for randomly generated velocity models and update the parameters of the neural network by minimizing the meta-loss, which is defined as the accumulated difference between the true and inverted velocity models. The learning and improvement of such an ML-misfit are automatic, and the resulting ML-misfit is data-adaptive. We first illustrate the basic principles behind the ML-misfit for learning a convex misfit function using a travel-time shifted signal example. Furthermore, we train the neural network on 2D horizontally layered models and apply the trained neural network to the Marmousi model; the resulting ML-misfit provides robust updating of the model and mitigates the cycle-skipping issue successfully.

Curvelet-Based Joint Waveform and Envelope Inversion of Early-Arrival Imaging Shallow Geological Structure

Abstract Near-surface imaging structures often plays a significant role in the field of environmental and engineering geophysics. Early-arrival waveform inversion (EWI) is state-of-the-art method to imaging near-surface structures due to its high resolution. However, the method faces with cycle-skipping issue which might lead to an unexpected local minimum. Envelope inversion (EI) could deal with this issue which contributes to the ultralow-frequency information extracted from the envelope but has a low resolution. We have developed a curvelet-based joint waveform and envelope inversion (CJWEI) method for inverting imaging near-surface velocity structures. By inverting two types of data, we are able to recover the low- and high-wavenumber structures and mitigate the cycle-skipping problem. Curvelet transform was used to decompose seismic data into different scales and provide a multiscale inversion strategy to further reduce non-uniqueness of waveform inversion efficiently. With synthetic test and real data application, we demonstrate that our method can constrain the anomalies and hidden layers in the shallow structure more efficiently as well as is robust in terms of noise. The proposed multiscale joint inversion offers a computational efficiency and high precision to imaging fine-scale shallow underground structures.

Reflection-based traveltime and waveform inversion with second-order optimization

Reflection-based inversion that aims to reconstruct the low-to-intermediate wavenumbers of the subsurface model, can be a complementary to refraction-data-driven full-waveform inversion (FWI), especially for the deep target area where diving waves cannot be acquired at the surface. Nevertheless, as a typical nonlinear inverse problem, reflection waveform inversion may easily suffer from the cycle-skipping issue and have a slow convergence rate, if gradient-based first-order optimization methods are used. To improve the accuracy and convergence rate, we introduce the Hessian operator into reflection traveltime inversion (RTI) and reflection waveform inversion (RWI) in the framework of second-order optimization. A practical two-stage workflow is proposed to build the velocity model, in which Gauss-Newton RTI is first applied to mitigate the cycle-skipping problem and then Gauss-Newton RWI is employed to enhance the model resolution. To make the Gauss-Newton iterations more efficiently and robustly for large-scale applications, we introduce proper preconditioning for the Hessian matrix and design appropriate strategies to reduce the computational costs. The example of a real dataset from East China Sea demonstrates that the cascaded Hessian-based RTI and RWI have good potential to improve velocity model building and seismic imaging, especially for the deep targets.

A graph-space optimal transport objective function based on q-statistics to mitigate cycle-skipping issues in FWI

SUMMARY Full-waveform inversion (FWI) is a powerful seismic imaging methodology to estimate geophysical parameters that honour the recorded waveforms (observed data), and it is conventionally formulated as a least-squares optimization problem. Despite many successful applications, least-squares FWI suffers from cycle skipping issues. Optimal transport (OT) based FWI has been demonstrated to be a useful strategy for mitigating cycle skipping. In this work, we introduce a new Wasserstein metric based on q-statistics in the context of the OT distance. In this sense, instead of the data themselves, we consider the graph of the seismic data, which are positive and normalized quantities similar to probability functions. By assuming that the difference between the graphs of the modelled and observed data obeys the q-statistics, we introduce a robust q-generalized graph-space OT objective function in the FWI context namely q-GSOT-FWI, in which the standard GSOT-FWI based on l2-norm is a particular case. To demonstrate how the q-GSOT-FWI deals with cycle skipping, we present two numerical examples involving 2-D acoustic wave-equation modelling. First, we investigate the convexity of q-GSOT objective function regarding different time-shifts, and, secondly, we present a Brazilian pre-salt synthetic case study, from a crude initial model which generates significant cycle-skipping seismic data. The results reveal that the q-GSOT-FWI is a powerful strategy to circumvent cycle skipping issues in FWI, in which our objective function proposal presents a smoother topography with a wider attraction valley to the optimal minimum. They also show that q-statistics leads to a significant improvement of FWI objective function convergence, generating higher resolution acoustic models than classical approaches. In addition, our proposal reduces the computational cost of calculating the transport plan as the q-value increases.

Geometrical-Feature-Preserving Adjoint Tomography of Near-Surface Structure with Seismic Early Arrival

Early arrival waveform inversion (EWI) is an essential approach to obtaining velocity structures in near-surface. Due to suffering from a cycle‐skipping issue, it is difficult to reach the global minima for conventional EWI with the misfit function of least-squares norm (L2‐norm). Following the optimal transportation theory, we developed an EWI solution with a new objective function based on quadratic‐Wasserstein‐metric (W2-norm) to maintain the geometric characteristics of the distribution and improve the stability and convexity of the inverse problem. First, we gave the continuous form of the adjoint source and the Frechet gradient of the Wasserstein metric for seismic early arrival, which leads to an easy and efficient way to implement in the adjoint-state method. Then, we conducted two synthetic experiments on the target model containing some velocity anomalies and hidden layers to test its effectiveness in mapping accurate and high-resolution near-surface velocity structure. The results show that the W2-normed EWI can mitigate cycle-skipping issues compared with the L2-normed EWI. In addition, it can deal with hidden layers and is robust in terms of noise. The application to a real dataset indicates that this new solution can recover more details in the shallow structure, especially in the aspect of dealing with hidden layers.

Waveform Energy Focusing Tomography With Passive Seismic Sources

By taking advantage of the information carried by the entire seismic wavefield, Full Waveform Inversion (FWI) is able to yield higher resolution subsurface velocity models than seismic traveltime tomography. However, FWI heavily relies on the knowledge of source information and good initial models, and could be easily trapped into local minima caused by cycle skipping issue because of its high nonlinearity. To mitigate these issues in FWI, we propose a novel method called Waveform Energy Focusing Tomography (WEFT) for passive seismic sources. Unlike conventional FWI, WEFT back-propagates the seismic records directly instead of the data residuals, and updates the velocity models by maximizing the stacking energy for all the moment tensor components from back-propagated wavefields around the sources. Therefore, except for source locations and origin times, WEFT does not require other source attributes in advance for the inversion. Since WEFT does not aim at fitting synthetic and observed waveforms, it has lower nonlinearity and is less prone to the cycle skipping issue compared to FWI. For the proof of concept, we have validated WEFT using several 2D synthetic tests to show it is less affected by inaccurate source locations and data noise. These advantages render WEFT more applicable for tomography using passive seismic sources when the source information is generally not accurately known. Although the inverted model from WEFT is inevitably influenced by the source distribution as well as its radiation patterns, and its resolution is likely lower than that of FWI, it can act as an intermediate step between traveltime tomography and FWI by providing a more reliable and accurate velocity model for the latter.

Optimal transport full-waveform inversion: from theory to industrial applications with examples from the Sultanate of Oman

Summary The optimal transport problem was formulated more than 200 years ago to calculate the optimal way of transporting piles of sand. Due to interesting properties of its solutions with respect to shifts between the compared distributions, optimal transport has recently been adapted to full-waveform inversion to mitigate the cycle-skipping issue. Various formalisms have been proposed. Here we present an overview of these approaches, emphasizing more specifically the approach based on the bi-dimensional Kantorovich-Rubinstein norm, which has led to numerous successful full-waveform inversion applications. We illustrate these successes with two onshore case studies from the Sultanate of Oman.

A reflection-based efficient wavefield inversion

Full-waveform inversion (FWI) is popularly used to obtain a high-resolution subsurface velocity model. However, it requires either a good initial velocity model or low-frequency data to mitigate the cycle-skipping issue. Reflection-waveform inversion (RWI) uses a migration/demigration process to retrieve a background model that can be used as a good initial velocity in FWI. The drawback of conventional RWI is that it requires the use of least-squares migration, which is often computationally expensive, and it is still prone to cycle skipping at far offsets. To improve the computational efficiency and overcome cycle skipping in the original RWI, we have incorporated it into a recently introduced method called efficient wavefield inversion (EWI) by inverting for the Born-scattered wavefield instead of the wavefield itself. In this case, we use perturbation-related secondary sources in the modified source function. Unlike conventional RWI, the perturbations are calculated naturally as part of the calculation of the scattered wavefield in an efficient way. Because the sources in the reflection-based EWI (REWI) are located in the subsurface, we are able to update the background model along the reflection wavepath. In the background velocity inversion, we calculate the background perturbation by a deconvolution process at each frequency. After obtaining the REWI inverted velocity model, a sequential FWI or EWI is needed to obtain a high-resolution model. We determine the validity of our approach using synthetic data generated from a section of the Sigsbee2A model. To further demonstrate the effectiveness of our approach, we test it on an ocean-bottom cable data set from the North Sea. We find that our methodology leads to improved velocity models as evidenced by flatter angle gathers.

Progressive transfer learning for low-frequency data prediction in full-waveform inversion

To effectively overcome the cycle-skipping issue in full-waveform inversion (FWI), we have developed a deep neural network (DNN) approach to predict the absent low-frequency (LF) components by exploiting the hidden physical relation connecting the LF and high-frequency (HF) data. To efficiently solve this challenging nonlinear regression problem, two novel strategies are proposed to design the DNN architecture and to optimize the learning process: (1) the dual data feed structure and (2) progressive transfer learning. With the dual data feed structure, not only the HF data, but also the corresponding beat tone data, are fed into the DNN to relieve the burden of feature extraction. The second strategy, progressive transfer learning, enables us to train the DNN using a single evolving training data set. Within the framework of progressive transfer learning, the training data set continuously evolves in an iterative manner by gradually retrieving the subsurface information through the physics-based inversion module, progressively enhancing the prediction accuracy of the DNN and propelling the inversion process out of the local minima. The synthetic numerical experiments suggest that, without any a priori geologic information, the LF data predicted by the progressive transfer learning are sufficiently accurate for an FWI engine to produce reliable subsurface velocity models free of cycle-skipping artifacts.

Visco-acoustic full waveform inversion using decoupled fractional Laplacian constant-Q wave equation and optimal transport-based misfit function

Full waveform inversion (FWI) has the potential to recover high-resolution physical parameters of the Earth from seismic data. Considering the attenuation effects of the subsurface, we develop a new visco-acoustic FWI method in the time domain with a known Q model. The seismic wavefield is modelled by solving a decoupled fractional Laplacian visco-acoustic wave equation, which can better describe amplitude attenuation and phase distortion during wave propagation. To weaken the initial model dependence of visco-acoustic FWI, we introduce the optimal transport-based misfit functional which can measure the data residual globally. Numerical examples on the 2D Marmousi model demonstrate the superiority of the proposed method compared with the L2 norm-based method which measures the misfit of observed and synthetic seismic data locally, generating cycle-skipping issue. The inversion results of seismic data without low-frequency components further demonstrate the validity and effectiveness of the proposed method, which is insensitive towards low-frequency components, and may achieve reasonable inversion results for field data application.

Strong Scattering Elastic Full Waveform Inversion With the Envelope Fréchet Derivative

Full waveform inversion (FWI) is, as an optimization problem, strongly nonconvex and is influenced intensely by the cycle skipping issue, especially for multiparameter inversions like the elastic case. When there are strong scattering heterogeneities in the target media, there will be more challenges for the inversion problem. The direct envelope inversion strategy uses the envelope Fréchet derivative to tackle the cycle skipping problem for strong scattering inversion and has been effectively used for the acoustic case. We extend the direct envelope inversion method to the elastic situation in this letter. We derive the elastic envelope Fréchet derivative and show how the strong scattering multiparameter elastic inversion is accomplished under the direct envelope inversion framework. Numerical tests with the SEG/EAGE salt velocity model proved the effectiveness of this method for the strong scattering elastic medium.

Source-independent efficient wavefield inversion

SUMMARYFull-waveform inversion (FWI) is an effective tool to retrieve a high-resolution subsurface velocity model. The source wavelet accuracy plays an important role in reaching that goal. So we often need to estimate the source function before or within the inversion process. Source estimation requires additional computational cost, and an inaccurate source estimation can hamper the convergence of FWI. We develop a source-independent waveform inversion utilizing a recently introduced wavefield reconstruction based method, which we refer to as efficient wavefield inversion (EWI). In EWI, we essentially reconstruct the wavefield by fitting it to the observed data as well as a wave equation based on iterative Born scattering. However, a wrong source wavelet will induce errors in the reconstructed wavefield, which may lead to a divergence of this optimization problem. We use a convolution-based source-independent misfit function to replace the conventional data fitting term in EWI to formulate a source-independent EWI (SIEWI) objective function. By convolving the observed data with a reference trace from the predicted data and convolving the predicted data with a reference trace from the observed data, the influence of the source wavelet on the optimization is mitigated. In SIEWI, this new formulation is able to mitigate the cycle-skipping issue and the source wavelet uncertainty simultaneously. We demonstrate those features on the Overthrust model and a modified Marmousi model. Application on a 2-D real data set also shows the effectiveness of the proposed method.