Cycle-skipping Problem Research Articles

Full waveform inversion with smoothing of dilated convolutions

Abstract 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.

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.

A collocated inversion of sources and early arrival waveforms for credible tomograms: Synthetic and field data examples

AbstractWaveform inversion is theoretically a powerful tool to reconstruct subsurface structures, but a usually encountered problem is that accurate sources are very rare, causing the computation to be unstable or divergent. This challenging practical problem, although sometimes ignored and even imperceptible, can easily create discrepancies in calculated shot gathers, which will then lead to wrong residuals that will be smeared back to the gradients, hence jeopardizing the inverted tomograms. For any real dataset, every shot gather corresponds to its unique source even if some gathers can be transformed alike after data processing. To resolve this problem, we propose a collocated inversion of sources and early arrival waveforms with the two submodules executing successively. Not only can this method reconstruct a decent source wavelet that approaches the ground truth, but also it can produce credible background tomograms with optimized sources. Part of the cycle skipping problems can also be mitigated because it avoids the trial and error experiments on various sources. Numerical tests on a synthetic and a land dataset validate the effectiveness of this method. Restrictions on initial sources or starting velocity models will be relaxed, and this method can be extended to any other applications for engineering or exploration purposes.

Full waveform inversion using frequency shift envelope-based global correlation norm for ultrasound computed tomography

Many studies have been carried out on ultrasound computed tomography (USCT) for its ability to offer quantitative measurements of tissue sound speed. Full waveform inversion (FWI) is a technique for reconstructing high-resolution sound speed images by iteratively minimizing the difference between the observed ultrasound data and the synthetic data based on the waveform equation. However, FWI suffers from cycle-skipping, which usually causes FWI convergence at a local minimum. Cycle-skipping occurs when the phase difference between the observed data and the synthetic data exceeds half a cycle. The simplest way to avoid cycle-skipping is to use low-frequency information for reconstruction. Nevertheless, in imaging systems, the response bandwidth of the probe is limited, and reliable low-frequency information often exceeds the response band. Therefore, it is a challenge to perform FWI imaging and avoid cycle-skipping problems without low-frequency information. In this paper, we propose a frequency shift envelope-based global correlation norm (FSEGCN), where an artificial source wavelet with a lower frequency is adopted to calculate synthetic data. FSEGCN compared with FWI, envelope inversion (EI), global correlation norm (GCN), envelope-based global correlation norm (EGCN) through concentric circle phantom without low-frequency information. The experimental results demonstrated the capability of the proposed method to recover the sound speed close to the exact model in the absence of low-frequency information, whereas FWI, EI, GCN, and EGCN cannot. Experiments on phantoms of the human head and calf show that artificial source wavelets can reduce image artifacts and enhance reconstruction robustness, when original low-frequency information is absent.

Novel Phase-Sensitive Full-Waveform Tomography for Seismic Imaging

Abstract Full-waveform tomography (FWT) is increasingly recognized as a pivotal technique for delineating high-resolution subsurface properties. Despite its significant potential, practical applications of FWT encounter persistent challenges, particularly in dealing with local minima and cycle-skipping problems. These difficulties often arise and are intensified by the least-squares (L2) norm’s intrinsic insensitivity to phase mismatches. To address these challenges, we have redefined the traditional L2 norm misfit function by incorporating a time shift within the synthetic waveform. This shift is determined by the temporal discrepancies between the observed and synthetic waveforms, identified through a cross-correlation technique. This approach, termed phase-sensitive FWT, integrates phase differences into the new misfit function, thus significantly mitigating the cycle-skipping problem. Numerical experiments demonstrate that PSFWT reduces dependence on the initial model and achieves more accurate inversion results compared with the traditional L2 norm method, highlighting its potential for enhancing the precision and reliability of seismic imaging.

Unbalanced optimal transport for full waveform inversion in visco-acoustic media

Abstract As a high-precision parameter inversion method, visco-acoustic full waveform inversion (QFWI) is widely used in the inversion of parameters such as velocity and quality factor Q in visco-acoustic media. Conventional QFWI, using the L2 norm as the objective function, is susceptible to face the cycle-skipping problem, especially with inaccurate initial models. Lately, adopting the optimal transportation (OT) distance as the objective function in QFWI (OT-QFWI) has become one of the most promising solutions. In OT-QFWI, converting oscillatory seismic data into a probability distribution that satisfies equal-mass and non-negativity conditions is essential. However, seismic data in visco-acoustic media face challenges in meeting the equal-mass assumption, primarily due to the attenuation effect (amplitude attenuation and phase distortion) associated with the quality factor Q. Unbalanced optimal transportation (UOT) has shown potential in solving equal-mass assumption. It offers the advantage of relaxing equal-mass requirements through entropy regularization. Owing to this advantage, UOT can mitigate the attenuation effect caused by inaccurate quality factor Q during the inversion. Simultaneously, the Sinkhorn algorithm can quickly solve the UOT distance through CUDA programming. Accordingly, we propose a UOT-based QFWI (UOT-QFWI) method to improve the accuracy of two-parameter inversion. The proposed method mitigates the impact of inaccurate quality factor Q by introducing the UOT distance to calculate the objective function, thereby helping to obtain more accurate inverted parameters. Experimental tests on the 1D Ricker wavelet and 2D synthetic model are used to validate the effectiveness and robustness of our proposed method.

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.

Forge ahead in acquisition and model building via full-waveform inversion: Gulf of Mexico case study

Full-waveform inversion (FWI) has proven itself an essential tool for velocity model building using seismic data. In recent years, the geophysical community has made substantial progress in developing FWI to overcome some long-standing limitations, such as handling cycle skipping, better using reflection energy, including more physics in the inversion algorithms, and increasing inverted frequencies to achieve higher resolution. FWI does not only target the larger and midscale model updates, more and more FWI examples were presented with increased resolution which allows for extraction of FWI derived reflectivity. Interrogating the FWI kernel with given geology and acquisition geometry can provide critical information on how the acquisition design should be optimized to provide FWI a better opportunity to update the velocity model at target depth, especially in the deep part of the model. When the acquisition geometry is different in terms of offsets, azimuthal coverage, and the minimum frequency in the recording, it can be analyzed to design workflows to enable FWI to optimally update the model parameters. Obviously, recent ocean-bottom node acquisitions that record long offsets, full azimuth, and low frequencies have made FWI shine in the seismic industry. Modifying and reshaping a complex salt geometry is one of the ultimate goals of FWI. In addition to the salt boundary being an issue, there is a potential cycle-skipping problem associated with uncertainties of large salt bodies missing or misplaced even though the frequencies used to start FWI are becoming lower due to the advancement in seismic acquisition and FWI algorithms. Furthermore, if the FWI-predicted data are simulated with an acoustic engine, it could pose amplitude and phase discrepancies at high-velocity contrast interfaces. Elastic FWI alone has been proposed as a means of overcoming these challenges associated with salt and regions with high-velocity contrast. We determine the FWI progress of the past few decades with the latest examples from different acquisition geometries.

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.

Efficient 1.5D full waveform inversion in the Laplace-Fourier domain

Accurate near-surface characterization and velocity model building are important for a number of geotechnical and geophysical applications. 3D full waveform inversion (FWI) can be used to generate a detailed velocity model, but must be provided with a good initial velocity model. A method for producing such an initial velocity model is explored in this paper. Assuming that the near-subsurface can be locally approximated by an effective flat-layered medium, a specialized form of acoustic FWI is proposed. The inversion is based on regularized least-squares in the Laplace-Fourier domain as a measure to mitigate the cycle-skipping problem. Forward modeling is carried out by solving a number of independent finite-difference problems in parallel, for a set of horizontal wavenumbers. The set is determined adaptively, using a pair of algorithms discussed in detail. A numerical implementation of the inverse Hankel transform, used to synthesize data in the frequency-offset domain, is also described. The forward modeling scheme is validated against analytic solutions of the Helmholtz equation, showing good agreement between the two. FWI is tested by inverting a synthetic dataset consisting of flat layers with embedded velocity anomalies. Important features are recovered after inversion at a small number of complex frequencies, providing a detailed initial model for successive 3D FWI.

Robust full-waveform inversion based on automatic differentiation and differentiable dynamic time warping

AbstractFull waveform inversion is a methodology that determines high-resolution parameters. The widely used L2-norm misfit function has local minima if the low wavenumber components are not accurate. Suffering from a cycle skipping problem, the solution of waveform inversion will be trapped in the local minima. Dynamic time warping aims to find an optimal alignment between two signals, which is a more robust measure to avoid cycle-skipping challenges. However, the discontinuity makes the conventional dynamic time warping distance unsuitable for waveform inversion. We introduce a soft dynamic time warping distance as the misfit function, which is differentiable in that the inverted solution can converge to the accurate global minimum. We compare the convexity of the L2-norm and soft dynamic time warping distance and show that the soft dynamic time warping distance has a wider convexity range with different time shift and amplitudes. It can alleviate the half-wavelength limitation of the conventional L2-norm. We calculate the gradient using the automatic differentiation technique and the minibatch strategy and then analyse the alignment paths of different smooth parameters. A significant smooth parameter γ makes the Soft-DTW distance tending to the L2-norm, which generates new local minima. We recommend a small smooth parameter to ensure the convexity of the Soft-DTW distance. Numerical examples show that the soft dynamic time warping can effectively reconstruct the deep velocity parameters of the BG Compass and Marmousi models with noise robustness and lower dependence on the initial model.

Deep-Learning-Based Low-Frequency Reconstruction in Full-Waveform Inversion

Low frequencies are vital for full-waveform inversion (FWI) to retrieve long-scale features and reliable subsurface properties from seismic data. Unfortunately, low frequencies are missing because of limitations in seismic acquisition steps. Furthermore, there is no explicit expression for transforming high frequencies into low frequencies. Therefore, low-frequency reconstruction (LFR) is imperative. Recently developed deep-learning (DL)-based LFR methods are based on either 1D or 2D convolutional neural networks (CNNs), which cannot take full advantage of the information contained in 3D prestack seismic data. Therefore, we present a DL-based LFR approach in which high frequencies are transformed into low frequencies by training an approximately symmetric encoding-decoding-type bridge-shaped 3D CNN. Our motivation is that the 3D CNN can naturally exploit more information that can be effectively used to improve the LFR result. We designed a Hanning-based window for suppressing the Gibbs effect associated with the hard splitting of the low- and high-frequency data. We report the significance of the convolutional kernel size on the training stage convergence rate and the performance of CNN’s generalization ability. CNN with reasonably large kernel sizes has a large receptive field and is beneficial to long-wavelength LFR. Experiments indicate that our approach can accurately reconstruct low frequencies from bandlimited high frequencies. The results of 3D CNN are distinctly superior to those of 2D CNN in terms of precision and highly relevant low-frequency energy. FWI on synthetic data indicates that the DL-predicted low frequencies nearly resemble those of actual low frequencies, and the DL-predicted low frequencies are accurate enough to mitigate the FWI’s cycle-skipping problems. Codes and data of this work are shared via a public repository.

Low-Frequency Prediction Based on Multiscale and Cross-Scale Deep Networks in Full-Waveform Inversion

The well-known cycle-skipping problem in full-waveform inversion (FWI) can make the iterative solution fall into local minima and produce an undesired inverted result when reliable low-frequency components in seismic data and a good initial model are not available. The recovery of low-frequency data can effectively solve the cycle-skipping problem. However, hardware limitations have made it difficult to obtain reliable low-frequency components in seismic data. Thus, we adopt a multi-scale and cross-scale convolutional neural network (MCCNN) to build the nonlinear mapping between high-frequency and low-frequency data from synthetic training data sets. The major benefit of MCCNN is that it can fully use the multi-scale and cross-scale information in the high-frequency data to predict the low-frequency data. Several numerical experiments show the effectiveness and benefits of the low-frequency recovery of MCCNN. On the one hand, introducing the in-stage multi-scale and across-stage cross-scale information can accelerate the convergence rate in the training process and improve the low-frequency prediction accuracy. On the other hand, MCCNN has good generalization abilities in predicting the low-frequency data from the Marmousi and Overthrust models, different model sizes, and wavelet types and frequencies. The acoustic FWI results show that the predicted low-frequency data can effectively prevent the inversion from falling into a local minimum and help FWI obtain an accurate velocity model even if we start from a poor initial model.

Imaging near-surface S-wave velocity and attenuation models by full-waveform inversion with distributed acoustic sensing-recorded surface waves

Distributed acoustic sensing (DAS) technology is, increasingly, the seismic acquisition mode of choice for its high spatial sampling rate, low cost, and nonintrusive deployability. It is being widely evaluated as an enabler of seismic monitoring for [Formula: see text] sequestration in building subsurface time-lapse images and in characterizing near-surface environments. To advance this evaluation, field seismic surveys with optical fibers have been conducted at the Containment and Monitoring Institute’s Field Research Station (CaMI.FRS) in Newell County, Alberta, Canada. In comparison to the standard geophones, optical fibers deployed in surface trenches at CaMI.FRS have recorded high-quality surface waves, rich in low frequencies and exhibiting limited spatial aliasing. These benefits have motivated us to apply the full-waveform inversion (FWI) approach to image the S-wave velocity ([Formula: see text]) and attenuation (quality factor [Formula: see text]) models at shallow site using the surface waves recorded by optical fibers. Compared to the conventional surface-wave dispersion approach, FWI can intrinsically incorporate fundamental and high-order modes and produce [Formula: see text] model with high spatial resolution that resolves horizontal variations. The low-frequency components below 10 Hz measured in the DAS recordings are helpful to overcome the cycle-skipping problem of FWI. Following the adjoint-state method, [Formula: see text] sensitivity kernel can be calculated efficiently with memory strain variables. The [Formula: see text] model is iteratively estimated with a new misfit function measuring root-mean-square amplitude differences, which helps to reduce the trade-off artifacts. The synthetic data obtained from the inverted models are consistent with the observed data in amplitude and phase. The inversion results provide valuable information to characterize the near-surface environments at CaMI.FRS and are expected to support seismic imaging in deeper [Formula: see text] injection zones.

Shallow geological structures building by joint waveform inversion based on curvelet decomposition

Shallow geological imaging structures often play an essential role in the field of environmental and engineering geophysics. Early arrival waveform inversion (EWI) is high-resolution method to imaging near-surface structures. We proposed a joint waveform and envelope inversion method based on curvelet transform (CJWEI) to build shallow geological structures. By taking advantage of their complementary sensitivities to near-surface structure, we can constrain both low- and high-wavenumber components of structures, as well as mitigating the cycle-skipping problem. Curvelet transform was used to decompose seismic data into different scales and provide a multi-scale inversion strategy to further reduce non-uniqueness of the inversion. With synthetic test, we demonstrate that our method can constrain the anomalies and hidden layers in the shallow structure more efficiently as well as robust in terms of noise.

The connection of velocity and impedance sensitivity kernels with scattering-angle filtering and its application in full waveform inversion

Multi-scale strategies such as starting from the low-frequency and early-arrival part of recorded data are commonly used in full waveform inversion (FWI) to maneuver complex nonlinearity. An alternative way is to apply appropriate filtering and conditioning to the misfit gradient in the model domain. In acoustic constant-density media, we prove that velocity and impedance sensitivity kernels are equivalent to applying a high-pass and a low-pass scattering-angle filter to a conventional single-parameter velocity (CSV) kernel. The high-pass scattering-angle filter allows the velocity kernel to include low-wavenumber updates (tomography component). In contrast, the low-pass scattering-angle filter helps the impedance kernel to yield high-wavenumber updates (migration component). The velocity model can be updated using a hybrid gradient of two components combined with appropriate weights. This FWI scheme is able to overcome the potential nonlinearity and partially mitigate the cycle-skipping problem. Numerical examples for the SEG/EAGE overthrust model and the Marmousi model demonstrate that the hybrid gradient facilitates FWI to converge faster to the true model even in cases when conventional CSV-based FWI fails.

Shape Dynamic Time Warping for Seismic Waveform Inversion

ABSTRACT We illustrate shape dynamic time warping (ShapeDTW) for 1D structure waveform inversion with examples from Tanzania, East Africa, and Northeastern Oklahoma. Seismograms of the Mw 5.9 Lake Rukwa, Tanzania, earthquake (18 August 1994) recorded by regional broadband seismographic stations of the Tanzania Broadband Seismic Experiment are used to demonstrate the method. We also test the inversion through solving for 1D crustal velocity structure of the Cherokee Platform in Oklahoma using seismograms of the 2011 Mw 5.7 Prague, Oklahoma, earthquake recorded by the EarthScope Transportable Array. Dynamic time warping (DTW) bypasses the time expense required for manual identification of seismic phases and shows potential for linking discrete seismic phases between two similar waveforms. ShapeDTW improves inversion performance by characterizing the character of the time series near each amplitude point. Applying Sakoe-Chiba band limits to ShapeDTW and an additional amplitude constraint improves the overfitting problem that occurs with the classic DTW method. We apply ShapeDTW to estimate optimal time shifts between observed and synthetic waveforms with time shifts reduced through inversion. The crustal velocity model of Tanzania found using ShapeDTW is close to that found using “phase time inversion” of numerous regional seismic phases observed in the three-component waveforms in a previous study but avoids the necessity of isolating and interpreting different seismic phases in the waveforms. Reducing the ShapeDTW functional misfits can overcome the cycle-skipping problem that often occurs with direct comparison of amplitudes in waveform inversion. This method may be useful in generating starting models for other 1D and 3D inversion methods.

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.

High-frequency localized elastic full-waveform inversion for time-lapse seismic surveys

Seismic full-waveform inversion (FWI) is a powerful method used to estimate the elastic properties of the subsurface. To mitigate the nonlinearity and cycle-skipping problems, in a hierarchical manner, one first inverts the low-frequency content to determine long- and medium-wavelength structures and then increases the frequency content to obtain detailed information. However, the inversion of higher frequencies can be computationally very expensive, especially when the target of interest, such as oil/gas reservoirs and axial melt lens, is at a great depth, far away from source and receiver arrays. To address this problem, we have developed a localized FWI algorithm in which iterative modeling is performed locally, allowing us to extend inversions for higher frequencies with little computation effort. Our method is particularly useful for time-lapse seismic, where the changes in elastic parameters are local due to fluid extraction and injection in the subsurface. In our method, the sources and receivers are extrapolated to a region close to the target area, allowing forward modeling and inversion to be performed locally after low-frequency full-model inversion for the background model, which by nature only represents long- to medium-wavelength features. Numerical tests show that the inversion of low-frequency data for the overburden is sufficient to provide an accurate high-frequency estimation of elastic parameters of the target region.