Application Of Full Waveform Inversion Research Articles

Full waveform inversion of cross-hole radio frequency electromagnetic data

Summary We consider application of full waveform inversion (FWI) to radio-frequency electromagnetic (EM) data. Radio-frequency imaging (RIM) is a cross-borehole technique to image electromagnetic subsurface properties from measurements of transmitted radio-frequency waves. It is used in coal seam imaging, ore exploration and various engineering and civil engineering applications. RIM operates at frequencies from 50 kHz to several tens of MHz. It differs from other geophysical EM methods, because the frequency band includes the transition between the wave propagation and diffusion regimes. RIM data are acquired in two-dimensional cross-hole sections in a reciprocal manner. Traditionally, radio-frequency data are inverted by straight ray tomography because it is inexpensive and easy to implement. It is argued that due to attenuation, the sensitivity of the transmitted electric field is the strongest within the first Fresnel zone of the ray connecting the transmitter and receiver. While straight ray tomography is a simple to implement and fast method, the non-linearity in the relationship between model parameters and data is often strong enough to warrant non-linear inversion techniques. FWI is an iterative high resolution technique, in which the physical properties are updated to minimize the misfit between the measured and modelled wavefields. Full waveform techniques have been used and extensively studied for the inversion of seismic data, and more recently, they have been applied to the inversion of GPR data. Non-linear inversion methods for RIM data are less advanced. Their use has been hindered by the high cost of full wave modelling and the high conductivity contrasts of many RIM targets, and, to some extent, by the limitations of the measuring instruments. We present the first application of this methodology to simultaneous conductivity and permittivity inversion of RIM data. We implement the inversion in the frequency domain in two dimensions using L-BFGS optimization. We analyze the sensitivity of the data to the model parameters and the parameter trade-off and validate the proposed methodology on a synthetic example with moderate conductivity variations and localized highly conductive targets. We then apply the FWI methodology to a field data set from Sudbury, Canada. For the field data set, we determine the most appropriate preprocessing steps that take into account specific peculiarities of RIM: the insufficient prior information about the subsurface and the limitations of the measuring equipment. We show that FWI is applicable under the conditions of RIM and is robust to imperfect prior knowledge: we obtain satisfactory model recoveries starting from homogeneous initial models in all of our examples. Just as other methods, FWI underestimates large conductivity contrasts due to the loss of sensitivity of the transmitted electric field to the conductivity variations as the conductivity increases above a certain level. The permittivity inside high conductors can not be recovered, however, recovering permittivity variations in the resistive zones helps obtain better focused conductivity images with fewer artifacts. Overall, FWI produces cleaner, less noisy and higher resolution reconstructions than the methods currently used in practice.

Depth imaging of multicomponent seismic data through the application of 2D full‐waveform inversion to P‐ and SH‐wave data: SEAM II Barrett model study

AbstractWe examine the value of the nine‐component seismic survey by generating the Kirchhoff depth migration images of compressional wave (P‐wave), converted wave (PS‐wave) and horizontally polarized shear wave (SH shear wave) data simulated from the SEAM II Barrett unconventional model. We first utilize full waveform inversion to obtain a P‐wave velocity model from P‐wave data and an S‐wave velocity model from SH‐wave data. Both P‐wave and SH‐wave data are generated with the maximum frequency of 20 Hz while assuming that the subsurface is isotropic. To implement full waveform inversion, we use a two‐dimensional time‐domain finite‐difference method and the L2 norm to measure the data misfit. We use both refractions and reflections in P‐ and SH‐wave data to reconstruct the P‐ and S‐wave velocity models from the surface to the reservoir. The inverted P‐ and S‐wave velocities contain the main features of the model (e.g., major faults and channels) but have some difficulties in estimating high‐frequency velocity variation within the first 300‐m depth of the model due to the frequency constraint. We then use the inverted P‐ and S‐wave velocities to generate Kirchhoff depth migration gathers and images from the P‐, PS‐ and SH‐wave data. The flat P‐ and SH‐wave common‐image offset gathers suggest that SH‐ and P‐wave full waveform inversion can generate adequate S‐ and P‐wave velocities for migration. Flat PS‐wave gathers and the clear PS‐wave migration image are also obtained using the inverted P‐ and S‐wave velocities simultaneously. This result indicates that obtaining S‐wave velocities from SH‐wave data can aid PS‐wave data processing and imaging. Moreover, the SH‐wave images and S‐wave images of the radial component provide better delineation of fault planes and small‐scale geobodies within the reservoir since the wavelength of the S‐wave is smaller compared to P‐wave when similar frequency ranges are recorded. Therefore, our study shows that S‐wave velocities can be successfully constructed by the two‐dimensional full waveform inversion application of the SH‐wave data. The subsequent imaging of multicomponent seismic data improves the delineation of certain unconventional reservoirs compared to the traditional P‐wave imaging.

Wasserstein distance-based full waveform inversion method for density reconstruction

Reliable density information is an essential factor in lithologic interpretation and reservoir evaluation. However, density reconstruction is hampered by a lack of long-wavelength information and the crosstalk between different parameters. The powerful nonlinearity in density inversion constrains the multi-parameter full waveform inversion application. This paper introduces the Wasserstein distance into the full waveform inversion for velocity and density based on the optimal transport theory. We construct an objective function with better convexity to avoid the cycle-skipping caused by the poor initial model. Furthermore, due to the sensitivity of the quadratic Wasserstein distance to low-frequency, we add the low-wavenumber component absent in the conventional density inversion. We acquire a long-wavelength density background using optimal transport inversion and then apply it as an initial model for least-squares inversion. Numerical examples show that our optimal transport inversion effectively builds a reasonable density background model for FWI. Density inversion can be improved by incorporating optimal transport and least-squares theory, leading to a reliable quantitative description for petroleum exploration and development.

Estimation of rock physics properties via full waveform inversion of vertical seismic profile data recorded by accelerometer and fibre-optic sensors

SUMMARY Combining elastic full waveform inversion (FWI) with rock physics holds promise for expanding the application of FWI beyond seismic imaging to predicting and monitoring reservoir properties. Distributed acoustic sensing (DAS), a rapidly developing seismic acquisition technology, is being explored for its potential in supporting FWI applications. In this study, we implement a sequential inversion scheme that integrates elastic FWI and Bayesian rock physics inversion, using a vertical seismic profile (VSP) data set acquired with accelerometer and collocated DAS fibre at the Carbon Management Canada’s Newell County Facility. Our aim is to establish a baseline model of porosity and lithology parameters to support later monitoring of CO2 storage. Key strategies include an effective source approach for addressing near-surface complications, a modelling strategy to simulate DAS data comparable to field data, and a Gaussian mixture approach to capture the bimodality of rock properties. We conduct FWI tests on accelerometer, DAS, and combined accelerometer-DAS data. While our inversion results accurately reproduce either data set, the elastic models inverted from accelerometer data outperform the other two in matching well logs and identifying the target reservoir. We attribute this outcome to the limited complementarity of DAS data with accelerometer data in our experiment, along with the limitations imposed by single-component measurements on DAS. The porosity and lithology models predicted from accelerometer-derived elastic models are reasonably accurate at the well location and exhibit geologically meaningful spatial distribution.

A decade of technology advancement in seismic processing: A case study from reprocessing legacy sparse OBC data in Kashagan Field

Ceaseless advancements in seismic data processing technology and workflow enable the extraction of increasing amounts of subsurface information essential for derisking and optimizing natural resource development. The Kashagan oil field, one of the world's largest carbonate reservoirs, is faced with significant development optimization challenges due to a combination of complex geology and suboptimal seismic data coverage. The latest processing technologies were applied to the 2001–2002 vintage sparse ocean-bottom cable data and produced step-change improvements over the 2010–2011 legacy processing and imaging results. Such significant image uplift is representative of technology advancements over the span of a decade fully leveraging the newly preprocessed input data, including (1) an integrated model building workflow, (2) adaptive application of full-waveform inversion, (3) geologically constrained reflection tomography, and (4) least-squares imaging. The new seismic results improved the structural and stratigraphic imaging of multiple layers of shallow carbonates, mitigated fault shadow and other complex overburden effects, increased the resolution of subsalt carbonate reservoir features such as karst and internal fractures, improved well marker depth misfit, and ultimately influenced the placement of upcoming wells and reservoir development plans. Further improvements in seismic imaging would be feasible with a modern acquisition of more densely sampled seismic data, which would allow the full potential of the latest seismic processing technologies and workflows to be unlocked.

Full Waveform Inversion for NDT using ultrasonic linear arrays

Ultrasonic (UT) imaging is a widespread technique for nondestructive testing (NDT). The state-of-the-art UT image reconstruction algorithms are based on delay-and-sum (DAS) operations, which assume constant acoustic velocity across the tested objects and also neglects nonlinear effects such as diffractions and multiple reflections. These assumptions limit the reconstruction capabilities of DAS-based algorithms, especially for complex objects composed by several materials. In seismology, Full Waveform Inversion (FWI) methods have been used to obtain subsurface properties based on scattered and transmitted sound waves - which is a very similar problem to ultrasonic imaging in NDT - using the full wave information, showing promising results. In this paper, we present the application of FWI in NDT using simulated data representing acquisitions with a common linear array transducer. We review the theoretical formulation of FWI and discuss some difficulties that arise when it is applied to NDT. Image reconstruction is performed using both a state-of-the-art DAS algorithm, namely the Total Focusing Method, and FWI. The implementation targets a GPU platform, using the CUDA API. This leverages the highly parallelizable nature of the simulations and the FWI algorithm. The obtained results show that FWI can show more internal structures than TFM, with less information about the specimen. To foster further development, all source codes are provided.

Elastic full-waveform inversion on Caesar-Tonga — Case study

Full-waveform inversion (FWI) has been widely used on 3D data sets to build detailed velocity models over the past 10 years. Most of these projects used pressure data and an acoustic approximation, with the assumption that the field data were dominated with P-waves. This approach of FWI can determine parameters related to the acoustic wave equation. It focuses on updating velocities by minimizing the misfit between observed and model data. Acoustic FWI, using the pressure component of collected data, has shown tremendous potential in simple geologic settings. Successful FWI projects, using wide-azimuth streamer, ocean bottom, and land survey geometries, have convinced the oil industry to pursue the next step by involving more physical properties. However, questions remain on how far we can properly describe field data with the acoustic approximation and at what point we need to switch to a much more expensive elastic wave equation implementation. In a complicated geologic region such as the Gulf of Mexico (GoM), the seismic wavefield can be complex and elastic FWI is needed to achieve a better velocity model, even when using mostly pressure data alone. We demonstrate the application of elastic FWI on sparse-node ocean-bottom-node data from the GoM and show comparisons to the acoustic solution. The comparisons demonstrate the benefits of the elastic FWI implementation when applied to image steeply dipping Miocene sands beneath a complex salt canopy, despite the increased computational expense. Furthermore, we demonstrate that when elastic FWI is applied to sufficiently high frequencies, the FWI-derived reflectivity product and velocity model are reliable interpretation products.

Atlantis — 20 years of seismic innovation finally removes the shroud of mystery

The Atlantis Field has gone through more than two decades of continuous seismic imaging efforts, during which time many innovative technologies were incubated, the most recent one being the successful application of full-waveform inversion (FWI) in salt environments. This technique led to a significant improvement in the subsalt image. However, imaging challenges remain for the Atlantis reservoirs, primarily due to the complex overburden salt geometries and the highly compartmentalized reservoir. Even with an improved velocity model from FWI, the conventional reverse time migration (RTM) images still suffer from illumination issues and contain strong migration swings that hinder the subsalt imaging and subsequent interpretations. Furthermore, early versions of FWI employed an acoustic assumption, leading to visible salt halos at the salt boundaries in the velocity model, which adversely impacted the reservoir imaging. In the last 12 months, elastic time-lag FWI (TLFWI) and FWI-derived reflectivity (FDR) imaging using long-offset ocean-bottom node data have minimized these imaging issues at Atlantis, providing another step change in subsalt understanding. Although the 3D RTM images using the elastic FWI velocity model are similar overall to their acoustic counterparts, the 4D time-lapse RTM images at Atlantis show noticeable improvements. Furthermore, FDR images derived from elastic FWI velocities show obvious benefits over the acoustic ones. With a more accurate modeling engine that allows for better match between synthetic and real data, FDR imaging shows improved illumination, higher signal-to-noise ratio, and better reservoir details over acoustic FDR imaging. This recent advancement in using elastic TLFWI has had immediate positive effects in facilitating the Atlantis Field's current and future development.

Incorporating geotechnical and geophysical investigations for underground obstruction detection: A case study

Determining the location and boundary of underground obstructions and/or anomalies is a common problem and often a great challenge for tunneling and underground construction. In this study, geotechnical investigations (penetration tests and borehole drilling/sampling) and geophysical investigations (surface wave method and cross-hole seismic method) were conducted with the aim of identifying the location and boundary of rock obstructions in Changi East, Singapore. The surface wave method is frequently used in the sites with lateral homogeneity in previous studies, but its application in the sites with rock obstructions is rare. The experimental results of this study indicate that the surface wave method is also able to determine the upper surface of rock obstructions, but difficult to identify the lateral and bottom boundaries of rock obstructions. To improve the precision of detection, the full waveform inversion (FWI) method was used to process the data from the cross-hole seismic survey. The results indicate that the inversion precision of P-wave is higher than that of S-wave. The horizontal and vertical ranges of rock obstructions in the P-wave inversion results are 14–26 m and 7.5–11.0 m respectively, roughly consistent with the results of penetration tests (about 15–25 m) and borehole log (8.85–10.80 m). This result proves that the sequential application of first-arrival time analysis and FWI can effectively delineate the boundary of rock obstructions. Finally, the results of various detection methods were analyzed and compared in this study. Considering the advantages of various methods, we propose a cost-effective and high-precision workflow containing both geotechnical and geophysical investigations.

Seismic full-waveform inversion based on superresolution for high-precision prediction of reservoir parameters

The frequency band limitation of seismic data limits the resolution of full-waveform inversion (FWI) results. In addition, the high computational cost seriously affects the practical application of FWI. To alleviate these concerns, an FWI method based on superresolution (SR-FWI) is developed to improve the prediction efficiency and accuracy of the reservoir parameters. A channel attention mechanism is introduced for the multifrequency characteristics of the model images. A constrained residual channel attention network (CRCAN) is built for superresolution (SR) by adding structural constraints to the loss function of a deep learning network. A total of 65,000 sets of geologic models and natural images constitute the network training data, 90% of which are used for training with the rest used for testing. The iterative calculation for FWI is time-consuming; hence, SR is applied to the iterative process to reduce the number of iterations and accelerate the model update. Low-resolution images along with the synthetic and field data are used for the evaluation of the CRCAN and SR-FWI algorithms, respectively. The test results find that CRCAN can effectively improve the image resolution, whereas SR-FWI is beneficial due to its high efficiency and precision, especially in predicting the stratum edge and small-scale anomalies. Therefore, SR-FWI is a powerful means of reservoir static and dynamic detection and can provide high-resolution information for projects, such as resource development and CO2 storage.

Enhancing salt model resolution and subsalt imaging with elastic FWI

Full-waveform inversion (FWI) has become the centerpiece of velocity model building (VMB) in seismic data processing in recent years. It has proven capable of significantly improving the velocity model and, thus, the migration image for different acquisition types and geologic settings, including complex environments such as salt. With the advent of FWI imaging, the scope of FWI applications has extended further from VMB into the imaging landscape. However, current FWI applications in the industry prevalently employ the acoustic approximation. One common problem of acoustic FWI (A-FWI) is the apparent salt halos at the salt-sediment interface in the resulting FWI velocity and FWI image, the presence of which hinders direct interpretation and imaging focusing around salt bodies. With synthetic and field data examples, we demonstrate that this salt halo is caused mainly by the large mismatch between the elastic recorded data and the acoustic modeled data, particularly at middle to long offsets. To overcome limitations imposed by acoustic assumptions, we developed an elastic FWI (E-FWI) algorithm that combines an elastic modeling engine with the time-lag cost function, which we call elastic time-lag FWI (E-TLFWI). With a more accurate modeling engine, E-TLFWI significantly reduces the salt halo observed in its acoustic counterpart. However, we also observe that the images migrated using the A-FWI and E-FWI velocity models remain similar overall, with some slight improvements around and beneath salt boundaries, particularly near steep salt flanks, as a result of the reduced salt halo. By contrast, FWI images derived from E-TLFWI show considerable benefits over those from acoustic time-lag FWI, such as improved event focusing, better structural continuity, and higher signal-to-noise ratio. The sharpened salt boundaries and enhanced quality of the FWI images reveal the significant value of E-FWI and provide the justification for its greatly increased cost.

Application of full-waveform inversion to land data: Case studies in onshore Mexico

Velocity model building and imaging for land surveys are often challenging due to near-surface complexity contaminating the reflection signal. Incorporating full-waveform inversion (FWI) in the velocity model building workflow for land surveys offers benefits not achieved with traditional model building tools, but it also brings difficulties. We have developed an effective model building workflow for land seismic data that incorporates dynamic matching FWI (DMFWI). DMFWI employs an objective function that uses multidimensional local windowed crosscorrelations between the dynamically matched version of observed and synthetic data. Dynamic matching de-emphasizes the impact of amplitudes, allowing the algorithm to focus on using kinematic information for velocity updates. The proposed workflow produces a geologically plausible and consistent model for data acquired with limited offsets. Refraction and reflection tomography may also be included in the workflow. The workflow is applied to onshore surveys in Mexico. Despite challenges of the near-surface geology and limitations of the acquisition parameters in the study areas, the proposed model building workflow successfully derives a high-resolution velocity model that significantly improves the migrated depth image.

Near-Surface 2D Imaging via FWI of DAS Data: An Examination on the Impacts of FWI Starting Model

Full waveform inversion (FWI) and distributed acoustic sensing (DAS) are powerful tools with potential to improve how seismic site characterization is performed. FWI is able to provide true 2D or 3D images of the subsurface by inverting stress wave recordings collected over a wide variety of scales. DAS can be used to efficiently collect high-resolution stress wave recordings from long and complex fiber optic arrays and is well-suited for large-scale site characterization projects. Due to the relative novelty of combining FWI and DAS, there is presently little published literature regarding the application of FWI to DAS data for near-surface (depths < 30 m) site characterization. We perform 2D FWI on DAS data collected at a well-characterized site using four different, site-specific 1D and 2D starting models. We discuss the unique benefits and challenges associated with inverting DAS data compared to traditional geophone data. We examine the impacts of using the various starting models on the final 2D subsurface images. We demonstrate that while the inversions performed using all four starting models are able to fit the major features of the DAS waveforms with similar misfit values, the final subsurface images can be quite different from one another at depths greater than about 10 m. As such, the best representation(s) of the subsurface are evaluated based on: (1) their agreement with borehole lithology logs that were not used in the development of the starting models, and (2) consistency at shallow depths between the final inverted images derived from multiple starting models. Our results demonstrate that FWI applied to DAS data has significant potential as a tool for near-surface site characterization while also emphasizing the significant impact that starting model selection can have on FWI results.

Time-Domain Elastic Full Waveform Inversion With Frequency Normalization

Time-domain elastic full waveform inversion (FWI) uses seismic data to recover the high-resolution subsurface medium properties for structural imaging and lithologic identification. The approximate pulse generated by the explosion source in seismic exploration evolves into a limited bandwidth seismic wavelet through the propagation of the seismic wave in the underground medium, exhibiting strong energy near the dominant frequency and weak energy far away. Therefore, time-domain FWI using the band-limited seismic wavelet can only match the energy of the dominant frequencies present in a dataset to a large extent, resulting in insufficient low-wavenumber updates of model parameters because of the weak energy of low frequencies. To remove the effect of finite-frequency-band wavelet spectra from the time-domain FWI, we propose a time-domain elastic FWI without wavelet spectral limitation. In our method, a newly refined seismic wavelet with a normalized amplitude and accurate phase is used to propagate seismic waves in the time domain. Data residual measurement is performed based on the summation of single-frequency residuals between the normalized synthetic and observed frequencies. The adjoint-state method is used to approximate the gradients of the model parameters, and the decoupled wavefields obtained by the phase-sensitive detection method were involved in the gradient calculation. Overall, the proposed method helps FWI avoid falling into local minima by enhancing the low wavenumber reconstruction of the velocity models. The elastic FWI numerical tests and field data FWI application demonstrate that our approach can reliably recover high-precision inversion results.

A practical guide of the 2D acoustic full waveform inversion on synthetic land seismic data

Full Waveform Inversion (FWI) has been successfully applied in seismic hydrocarbon exploration as it results in the highest resolution inverted velocity and density models. However, the application of FWI on land datasets is a much more challenging process than its application on marine datasets. This paper suggests a step-by-step guide for the 2D acoustic FWI on a synthetic dataset, in particular, in which we used the SEAM Phase II-Foothills dataset that exhibits many of the characteristics of real land data. The methodology includes tools for the processing of seismic data based on the features of acquisition geometry, source estimation, first arrival tomography, plus the software strategies to meet the requirements of the FWI (memory and computation requirements). The document also includes a discussion of the results of velocity models obtained for a low-resolution Dip 2D line of the SEAM data, using a workstation with low-to-intermediate hardware requirements.

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.

Trust-region versus line search globalization strategies for inexact Newton method and application in full waveform inversion

In this study, we consider the inexact Newton method using the trust-region and line search globalization strategies when solving the large-scale full waveform inversion problem. An elaborate stopping criterion or forcing term is introduced in order to avoid oversolvings of Newton equation. We develop an efficient method to compute (pseudo) Hessian vector products. In addition, a diagonal preconditioner based on the pseudo-Hessian approach is employed to further accelerate the solution of Newton equation. We discuss the specific parameter choices for these present methods. Numerical experiments based on the Marmousi2, BP 2004, and Sigsbee models are conducted to show the numerical performance of proposed inexact Newton methods. Numerical experiments demonstrate that the trust-region can perform better than the line search globalization strategy in the presence of nonlinearity and nonconvexity of the full waveform problem, especially for high contrast velocity models.

Innovative application of full-waveform inversion applied to extended wide-azimuth marine streamer seismic data in a complex salt environment

We demonstrate the application of full-waveform inversion (FWI) guided velocity model building to an extended wide-azimuth towed streamer (EWATS) seismic data set in the Gulf of Mexico. Field data are collected over a historically challenging imaging area, colloquially called the “grunge zone” due to the formation of a compressional allosuture emplaced between two colliding salt sheets. These data have a poor subsalt image below the suture with conventional narrow-azimuth data. Additional geologic complexities are observed including high-velocity carbonate carapace near the top of salt and multiple intrasalt sedimentary inclusions. As such, improved seismic imaging is required to plan and execute wells targeting subsalt strata. Significant improvements to the velocity model and subsalt image have been evident with wide-azimuth towed streamer and later EWATS data using conventional top-down velocity model building approaches. Subsequently, high-impact improvements have been made using EWATS data with an FWI velocity model building workflow; this study represents an early successful application of FWI used to update salt body geometries from streamer seismic data, wherein many prior applications were limited to improving sedimentary velocities. Later petrophysical data have verified the new FWI-derived model, which has significantly increased confidence in the structural and stratigraphic interpretation of subsalt reservoir systems below the grunge zone.

Wavefield Reconstruction Inversion via Physics-Informed Neural Networks

Wavefield reconstruction inversion (WRI) formulates a PDE-constrained optimization problem to reduce cycle skipping in full-waveform inversion (FWI). WRI is often implemented by solving for the frequency-domain representation of the wavefield using the finite-difference method. The approach requires matrix inversions and affords limited flexibility to accommodate irregular model geometries. On the other hand, the physics-informed neural network (PINN) uses the underlying physical laws as loss functions to train the neural network (NN) to provide flexible continuous functional approximations of the solutions without matrix inversions. By including a data-constrained term in the loss function, the trained NN can reconstruct a wavefield that simultaneously fits the recorded data and satisfies the Helmholtz equation for a given initial velocity model. Using the predicted wavefields, we rely on a small-size NN to predict the velocity using the reconstructed wavefield. In this velocity prediction NN, spatial coordinates are used as input data to the network, and the scattered Helmholtz equation is used to define the loss function. After we train this network, we are able to predict the velocity in the domain of interest. We develop this PINN-based WRI method and demonstrate its potential using a part of the Sigsbee2A model and a modified Marmousi model. The results show that the PINN-based WRI is able to invert for a reasonable velocity with very limited iterations and frequencies, which can be used in a subsequent FWI application.

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.