For Full Waveform Inversion Research Articles

Full waveform inversion based on deep learning and the phase-preserving symplectic partitioned Runge-Kutta method

To obtain more accurate full waveform inversion results, we present a forward modeling method with minimal phase error, low numerical dispersion, and high computational efficiency. To solve the 2D acoustic wave equation, we utilize a finite-difference (FD) scheme with optimized coefficients for spatial discretization, combined with a phase-preserving symplectic partitioned Runge-Kutta method for temporal discretization. This results in the development of the optimized symplectic partitioned Runge-Kutta (OSPRK) forward modeling method. We further apply the OSPRK method in conjunction with a recurrent neural network (RNN) for full waveform inversion (FWI). Our study explores the impact of various loss functions, Nadam optimizer parameters, and the incorporation of physical information operators on inversion performance. Numerical experiments demonstrate that the OSPRK method significantly reduces numerical dispersion compared to traditional FD methods. The Log-Cosh loss function offers superior stability across different learning rates, while the Nesterov-accelerated Adaptive Moment Estimation (Nadam) optimizer with optimized parameters greatly enhances convergence speed and inversion accuracy. Furthermore, the inclusion of physical information operators markedly improves inversion outcomes.

Full-waveform inversion by model extension: Practical applications

Producing reliable acoustic subsurface velocity models still remains the main bottleneck of the oil and gas industry’s traditional imaging sequence. In complex geologic settings, the output of conventional ray-based or wave-equation-based tomographic methods may not be sufficiently accurate for full-waveform inversion (FWI) to converge to a geologically satisfactory earth model. We create a new method referred to as full-waveform inversion by model extension (FWIME) in which a wave-equation migration velocity analysis (WEMVA) technique is efficiently paired with a modified version of FWI. We find that our method is more powerful than applying WEMVA and FWI sequentially, and that it can converge to accurate solutions without the use of a good initial guess or low-frequency energy. We determine FWIME’s potential on five realistic and challenging numerical examples that simulate complex geologic scenarios often encountered in hydrocarbon exploration. We guide the reader step by step throughout the optimization process. We find that our method can simultaneously invert all wave types with the same simple mechanism and without the need for a user-intensive hyperparameter tuning process. In an open-source online repository, we provide our C++/compute unified device architecture (CUDA) numerical implementation accelerated with graphics processing units, encapsulated in a Python interface. All the numerical examples developed are accessible through Python notebooks and fully reproducible.

Spectral recomposition for optimizing starting points in Full-Waveform Inversion

By optimizing the wavelet fitting of the arrivals in a reflection event, it is possible to perform significant enhancements for Full-Waveform Inversion (FWI) predictions and its processing time. For this reason, approaches capable of reducing the number of iterations for FWI — without decreasing the quality of the prediction — are of interest. Once the initial guess for FWI is better estimated, it is possible to predict the velocity model within a determined accuracy with fewer iterations. We thus propose an approach which can perform this estimation, based on spectral recomposition of seismic data. We design an inversion scheme to reconstruct the seismic spectrum of wavelets of a reflection event, which subsequently allows estimating the position in time of each wavelet in a seismogram. After finding the position in time of each wavelet, we can guide the calculated wavelet to fit the corresponding observed signal, starting from a closer initial point. Our approach leads to quite accurate predictions of velocity models with fewer iterations.

Acquisition and near-surface impacts on VSP mini-batch FWI and RTM imaging in desert environment

The SEG Advanced Modeling (SEAM) Arid benchmark model was designed to simulate an extremely heterogeneous low-velocity near surface (NS), which is typical of desert environments and typically not well characterized or imaged. Imaging of land seismic data is highly sensitive to errors in the NS velocity model. Vertical seismic profiling (VSP) partly alleviates the impact of the NS as the receivers are located at depth in the borehole. Deep learning (DL) offers a flexible optimization framework for full-waveform inversion (FWI), often outperforming typically used optimization methods. We investigate the quality of images that can be obtained from SEAM Arid VSP data by acoustic mini-batch reverse time migration (RTM) and full-waveform imaging. First, we focus on the effects of seismic vibrator and receiver array positioning and imperfect knowledge of the NS model when inverting 2D acoustic data. FWI imaging expectedly and consistently outperforms RTM in our tests. We find that the acquisition density is critical for RTM imaging and less so for FWI, while NS model accuracy is critical for FWI and has less effect on RTM imaging. Distributed acoustic sensing along the full length of the well provides noticeable improvement over a limited aperture array of geophones in imaging deep targets in both RTM and FWI imaging scenarios. Finally, we compare DL-based FWI imaging with inverse scattering RTM using the upgoing wavefield from the original SEAM data. Use of significantly more realistic 3D elastic physics for the simulated data generation and simple 2D acoustic inversion engine makes our inverse problem more realistic. We observe that FWI imaging in this case produces an image with fewer artifacts.

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.

Illumination guided sparse geometry optimization for target-oriented full-waveform inversion: An ocean bottom node synthetic study

For full-waveform inversion (FWI) with sparse survey layout, it is crucial to optimize the source and receiver geometry toward full-wave illumination. We propose a sparse geometry optimization method and workflow for target-oriented FWI using finite-frequency illumination analysis. The frequency-dependent impact of a source-receiver pair to the wavefield sensitivity kernel and diagonal terms of approximate Hessian matrix are derived under visco-acoustic model assumptions. Based on the reciprocity principle, we use virtual sources in target zone to efficiently calculate the Green's function and illumination from acquisition surface. Optimal locations of sparse receivers are determined by comparing band-limited illumination at reference stations and their surrounding region through an illumination attribute scan. FWI model residual errors are used to evaluate the performance of optimized geometry layout for a given target zone. The performances of different geometries are compared to determine a suitable frequency band for FWI geometry optimization. Synthetic deep water model and ocean bottom node acquisition geometry are used to verify the feasibility of the workflow. Numerical tests in target-oriented FWI show that optimized geometry with constraints of band-limited illumination below10 Hz has better performance than that above 10 Hz. An extension of our geometry optimization method to sparse source configuration is straightforward.

A Hybrid Optimization Framework for Seismic Full Waveform Inversion

AbstractA hybrid optimization framework is proposed for full waveform inversion (FWI) problems by incorporating derivative information into the model update rule of a global optimization method called Very Fast Simulated Annealing (VFSA). The proposed optimization framework tackles the local minima issue of non‐linear inverse problems. Additionally, it can converge to the close neighborhood of the global solution from different starting points with an improved convergence speed compared to traditional global optimization methods. Applied to large‐scale FWI problems, the proposed framework greatly relaxes the issue of the dependence of FWI on starting models. Given a proper tuning and sufficient number of iterations, hybrid optimization based FWI can render a good background model, which is a good approximation to the ground truth, even with uninformative prior constraints and poor starting models. The output of hybrid optimization based FWI can be used as the starting model for a subsequent local optimization based FWI run to improve the spatial resolution of the result. The proposed hybrid optimization framework is very general, and can be applied to other linear and non‐linear problems that need optimization loops.

An efficient symplectic stereo-modeling method for seismic inversion by using deep learning technique

AbstractThis paper proposes an efficient symplectic stereo-modeling (SSTEM) method for full waveform inversion (FWI) by using a deep learning technique. To solve the 2D acoustic equation, the SSTEM method uses a third-order optimal symplectic partitioned Runge–Kutta approach as a time-stepping method. An eighth-order stereo-modeling operator is used for spatial discretization. The SSTEM method is then expressed with a recurrent neural network (RNN). This is realized mainly because the time advancing format of the SSTEM method is similar to that of RNN, and they both use the information from the previous time step to obtain information from the current time step. With SSTEM as the forward modeling method, FWI is implemented using Tensorflow. The well-known adaptive moment estimation (Adam) optimizer and Nesterov adaptive moment estimation (Nadam) optimizer with mini-batch are used. The applicability of the developed code is also verified on GPUs. The numerical results show that the SSTEM method is more efficient and produces less numerical dispersion than the conventional finite-difference (FD) method when the same sampling rate in a wavelength is used. We compare several loss functions. The mean square (MSE) error and absolute (ABS) error loss functions are first tested. Another loss function that adds a physical differential operator to the original loss function is then considered. The FWI results show that this loss function has some improvements. Finally, we implement FWI on the complex Marmousi and SEG/EAGE models, and the inversion results demonstrate that the proposed method is suitable for seismic imaging in complex media.

Developing Realistic FDTD GPR Antenna Surrogates by Means of Particle Swarm Optimization

The antenna is the most important part of a ground-penetrating radar (GPR) system and defines the employed electromagnetic pulse and how it is transferred to the ground. It is crucial to account for these coupling effects in numerical simulations and to implement realistic antenna models, e.g., for full-waveform inversion (FWI). We present a method of developing and adapting 3-D finite-difference time-domain (FDTD) models of GPR antennas, complete with electric components, dielectric material properties, and feed pulse details. We exemplify this with a commercially available, shielded 400 MHz GPR antenna, a model of which was set up by fitting synthetic data to an experimental signal of the antenna reflected at a metal plate in the air. For this FWI, we used a particle swarm optimization (PSO) algorithm because the fit parameters show complex individual effects on the GPR waveform. The resulting antenna model is then validated against data measured in air, water, and with a metal plate in the near field of the antenna. Overall, the synthetic data reproduce the validation data very accurately. Signals of objects placed in the near field of the antenna and the change of the shape and frequency content of the radiated wavelet with varying subsurface properties are emulated correctly.

Joint Traditional and Reflection Envelope Inversion

The envelope inversion (EI) is an effective method to recover low-wavenumber components, which helps to produce a good initial model for full-waveform inversion (FWI). However, when the initial model is not capable of generating reflections, it brings enormous challenges for EI to invert deep low-wavenumber components, especially for short offset data. In contrast, reflection waveform inversion (RWI) uses demigration data to fit the observed reflection, which focuses on the transmission information with short offset seismic data. However, the cycle skipping and high nonlinearity of the RWI misfit still exist when the low-frequency information is absent. In this letter, we develop a joint traditional and reflection envelope inversion (JREI) that utilizes both reflection and transmission waves with envelope low-frequencies to recover low-wavenumber components in the shallow and deep regions simultaneously. We then use the FWI with high-frequency seismic data to obtain the high-wavenumber components. Applications to the modified Marmousi and Overthrust models demonstrate that the JREI can invert a better starting velocity model for the FWI to achieve a high-resolution inversion result.

The value of very low frequencies and far offsets for seismic data in the Permian Basin: Case study on a new dense survey from the Central Basin Platform

The depth of penetration and multidimensional characteristics of seismic waves make them an essential tool for subsurface exploration. However, their band-limited nature can make it difficult to integrate them with other types of ground measurements. Consequently, far offsets and very low-frequency components are key factors in maximizing the information jointly inverted from all recorded data. This explains why extending seismic bandwidth and available offsets has become a major industry focus. Although this requirement generally increases the complexity of acquisition and has an impact on its cost, improvements have been clearly and widely demonstrated on marine data. Onshore seismic data have generally followed the same trend but face different challenges, making it more difficult to maximize the benefits, especially for full-waveform inversion (FWI). This paper describes a new dense survey acquired in 2020 in the Permian Basin and aims to objectively assess the quality and benefits brought by a richer low end of the spectrum and far offsets. For this purpose, we considered several aspects, from acquisition design and field data to FWI imaging and quantitative interpretation.

SPiRaL: a multiresolution global tomography model of seismic wave speeds and radial anisotropy variations in the crust and mantle

SUMMARYSPiRaL is a joint global-scale model of wave speeds (P and S) and anisotropy (vertical transverse isotropy, VTI) variations in the crust and mantle. The model is comprised of >2.1 million nodes with five parameters at each node that capture velocity variations for P- and S-waves travelling at arbitrary directions in transversely isotropic media with a vertical symmetry axis (VTI). The crust (including ice, water, sediments and crystalline layers) is directly incorporated into the model. The default node spacing is approximately 2° in the lower mantle and 1° in the crust and upper mantle. The grid is refined with ∼0.25° minimum node spacing in highly sampled regions of the crust and upper mantle throughout North America and Eurasia. The data considered in the construction of SPiRaL includes millions of body wave traveltimes (crustal, regional and teleseismic phases with multiples) and surface wave (Rayleigh and Love) dispersion. A multiresolution inversion approach is employed to capture long-wavelength heterogeneities commonly depicted in global-scale tomography images as well as more localized details that are typically resolved in more focused regional-scale studies. Our previous work has demonstrated that such global-scale models with regional-scale detail can accurately predict both teleseismic and regional body wave traveltimes, which is necessary for more accurate location of small seismic events that may have limited signal at teleseismic distances. SPiRaL was constructed to predict traveltimes for event location and long-period waveform dispersion for seismic source inversion applications in regions without sufficiently tuned models. SPiRaL may also serve as a starting model for full-waveform inversion (FWI) with the goal of fitting waves with periods 10–50 s over multiple broad regions (thousands of kilometres) and potentially the globe. To gain insight to this possibility, we simulated waveforms for a small set of events using SPiRaL and independent waveform-based models for comparison. For the events tested, the performance of the traveltime-based SPiRaL model is shown to be generally on par with regional 3-D waveform-based models in three regions (western United States, Middle East, Korean Peninsula) suggesting SPiRaL may serve as a starting model for FWI over broad regions.

Reflection Full Waveform Inversion with Decoupled Elastic-wave Equations in Inhomogeneous Medium

Reflection full-waveform inversion (RFWI) can reduce the nonlinearity of inversion providing an accurate initial velocity model for full-waveform inversion (FWI) through the tomographic components (low-wavenumber). However, elastic-wave reflection full-waveform inversion (ERFWI) is more vulnerable to the problem of local minimum due to the complicated multi-component wavefield. Our algorithm first divides kernels of ERFWI into four subkernels based on the theory of decoupled elastic-wave equations. Then we try to construct the tomographic components of ERFWI with only single-component wavefields, similarly to acoustic inversions. However, the S-wave velocity is still vulnerable to the coupling effects because of P-wave components contained in the S-wavefield in an inhomogeneous medium. Therefore we develop a method for decoupling elastic- wave equations in an inhomogeneous medium, which is applied to the decomposition of kernels in ERFWI. The new decoupled system can improve the accuracy of S-wavefield and hence further reduces the high-wavenumber crosstalk in the subkernel of S-wave velocity after kernels are decomposed. The numerical examples of Sigsbee2A model demonstrate that our ERFWI method with decoupled elastic-wave equations can efficiently recover the low-wavenumber components of the model and improve the precision of the S-wave velocity.

Application of Envelope in Salt Structure Velocity Building: From Objective Function Construction to the Full-Band Seismic Data Reconstruction

For full-waveform inversion (FWI), building the salt structure velocity model is a challenging problem. In the absence of a high-fidelity initial model, low-frequency seismic data are indispensable for accurate reconstruction of the long-wavelength components of salt domes, which are often missing in field data. The envelope isolates the amplitude information from the seismic data and has a stronger linear relationship with the velocity than the seismic data. It is used to enhance the convexity of the objective function in an envelope inversion (EI). However, the properties of the envelope are not effectively utilized in EI due to the lack of their proper understanding. In order to solve the problems that EI has in building the salt structure velocity, such as the cycle-skipping problem, the gradient calculation problem caused by the waveform Frechet derivative, and the multisolution problem caused by the missing polarity information of the envelope, we propose a multiscale direct signed EI (MSDSEI) based on the direct envelope Frechet derivative and multiscale signed envelope. The development process from EI to MSDSEI also inspired us to understand the physical mechanism involved in the low-frequency components of the envelope: the signed envelope can be regarded as a low-pass filter of the reflection sequences in the subsurface. Thus, we use the smoothness and phase-independent properties of the envelope to propose an envelope-based full-band seismic data reconstruction method for multiscale FWI. Finally, the performance of various EI methods is verified on the Sigsbee2A model.

CADD: A seamless solution to the Domain Decomposition problem of subdomain boundaries and cross-points

The solution of wave problems using Domain Decomposition (DD) requires that the subdomain boundaries should be virtually non-existent, so that waves are not affected by the boundaries. This is a primary problem in DD, and it intensifies in the case of cross-points at which three or more subdomains meet. This topic has received a lot of attention in recent years, with special treatment of cross-points. This paper explains and demonstrates that this problem does not exist in Component-Averaged Domain Decomposition (CADD). CADD is implemented here with the authors’ CARP-CG algorithm, but it is shown that other implementations are also possible. The reason for the non-existence of this problem in CARP-CG is that in some superspace of the problem space, CARP-CG is mathematically equivalent to the CG acceleration of the Kaczmarz algorithm with cyclic relaxation parameters, applied to a single linear system. Due to its advantages, CARP-CG was adopted by some geophysics researchers as the solver of the Helmholtz and the elastic wave equations for full waveform inversion (FWI).

A numerical study of multi-parameter full waveform inversion with iterative regularization using multi-frequency vibroseis data

We study the inverse boundary value problem for time-harmonic elastic waves, for the recovery of P- and S-wave speeds from vibroseis data or the Neumann-to-Dirichlet map. Our study is based on our recent result pertaining to the uniqueness and a conditional Lipschitz stability estimate for parametrizations on unstructured tetrahedral meshes of this inverse boundary value problem. With the conditional Lipschitz stability estimate, we design a procedure for full waveform inversion (FWI) with iterative regularization. The iterative regularization is implemented by projecting gradients, after scaling, onto subspaces associated with the mentioned parametrizations yielding Lipschitz stability. The procedure is illustrated in computational experiments using the continuous Galerkin finite element method of recovering the rough shapes and wave speeds of geological bodies from simple starting models, near and far from the boundary, that is, the free surface.

Projection methods and applications for seismic nonlinear inverse problems with multiple constraints

Nonlinear inverse problems are often hampered by local minima because of missing low frequencies and far offsets in the data, lack of access to good starting models, noise, and modeling errors. A well-known approach to counter these deficiencies is to include prior information on the unknown model, which regularizes the inverse problem. Although conventional regularization methods have resulted in enormous progress in ill-posed (geophysical) inverse problems, challenges remain when the prior information consists of multiple pieces. To handle this situation, we have developed an optimization framework that allows us to add multiple pieces of prior information in the form of constraints. The proposed framework is more suitable for full-waveform inversion (FWI) because it offers assurances that multiple constraints are imposed uniquely at each iteration, irrespective of the order in which they are invoked. To project onto the intersection of multiple sets uniquely, we use Dykstra’s algorithm that does not rely on trade-off parameters. In that sense, our approach differs substantially from approaches, such as Tikhonov/penalty regularization and gradient filtering. None of these offer assurances, which makes them less suitable to FWI, where unrealistic intermediate results effectively derail the inversion. By working with intersections of sets, we avoid trade-off parameters and keep objective calculations separate from projections that are often much faster to compute than objectives/gradients in 3D. These features allow for easy integration into existing code bases. Working with constraints also allows for heuristics, where we built up the complexity of the model by a gradual relaxation of the constraints. This strategy helps to avoid convergence to local minima that represent unrealistic models. Using multiple constraints, we obtain better FWI results compared with a quadratic penalty method, whereas all definitions of the constraints are in terms of physical units and follow from the prior knowledge directly.

An effective absorbing layer for the boundary condition in acoustic seismic wave simulation

Efficient numerical simulation of seismic wavefields generally involves truncating the Earth model in order to keep computing time and memory requirements down. Absorbing boundary conditions, therefore, are applied to remove the boundary reflections caused by this truncation, thereby allowing for accurate modeling of wavefields. In this paper, we derive an effective absorbing boundary condition for both acoustic and elastic wave simulation, through the simplification of the damping term of the split perfectly matched layer (SPML) boundary condition. This new boundary condition is accurate, cost-effective, and easily implemented, especially for high-performance computing. Stability analysis shows that this boundary condition is effectively as stable as normal (non-absorbing) wave equations for explicit time-stepping finite differences. We found that for full-waveform inversion (FWI), the strengths of the effective absorbing layer—a reduction of the computational and memory cost coupled with a simplistic implementation—significantly outweighs the limitation of incomplete absorption of outgoing waves relative to the SPML. More importantly, we demonstrate that this limitation can easily be overcome through the use of two strategies in FWI, namely variable cell size and model extension thereby fully compensating for the imperfectness of the proposed absorbing boundary condition.

Multi-scale full waveform inversion for areas with irregular surface topography in an auxiliary coordinate system

For land exploration areas with irregular surface topography, there are many challenges and problems for full waveform inversion (FWI); for example, which type of wave equation should be used to calculate high-accuracy seismic wavefields, how to deal with diffraction of irregular surface topography, what initial velocity model should be utilised to improve the inversion accuracy and how to enhance the computational efficiency of iterative FWI. Aiming at these difficulties, we first simulate the seismic waves with the first-order acoustic wave equation in an auxiliary coordinate system, which easily describes irregular surface topography. Then, we apply this wavefield simulation frame to FWI to improve inversion quality of near-surface regions with strong elevation and velocity variation. Furthermore, to enhance the robustness and computational efficiency, a time-domain multi-scale decomposition method based on the Wiener filter and an optimised encoding strategy are introduced to the proposed inversion frame, and are critical to promoting the practical application of our method. Typical numerical tests prove that the proposed method can obtain more accurate inversion results than the traditional time-domain FWI.

Elastic full-waveform inversion for VTI media: A synthetic parameterization study

One of the main challenges for full-waveform inversion (FWI) is taking into account both anisotropy and elasticity. Here, we perform elastic FWI for a synthetic 2D VTI (transversely isotropic with a vertical symmetry axis) model based on the geologic section at Valhall field in the North Sea. Multicomponent surface data are generated by a finite-difference code. We apply FWI in the time domain using a multiscale approach with three frequency bands. An approximate inverse Hessian matrix, computed using the L-BFGS-B algorithm, is employed to scale the gradients of the objective function and improve the convergence. In the absence of significant diving-wave energy in the deeper part of the section, the model is updated primarily with reflection data. An oblique displacement source, which excites sufficiently intensive shear waves in the conventional offset range, helps provide more accurate updates in the Shear-wave vertical velocity, especially in the shallow layers. We test three model parameterizations, which exhibit different radiation patterns and, therefore, create different parameter trade-offs. Whereas most examples are for a constant-density model, we also generate a density field using Gardner’s relationship and invert for the density along with the velocity parameters. The parameterizations that combine velocities and anisotropy coefficients generally yield superior results to the one that includes only velocities, provided that a reasonably accurate initial model is available.