Misfit Function Research Articles

Methodology of elastic full‐waveform inversion of multicomponent ocean‐bottom data for anisotropic media

AbstractFull‐waveform inversion of multicomponent data can provide an improved estimation of medium parameters using both compressional‐ and shear‐wave information. However, most earlier studies that involved a full‐waveform inversion of ocean‐bottom data are based on acoustic anisotropic or elastic isotropic approximations. Here, we consider realistic elastic anisotropic media and develop an efficient full‐waveform inversion framework for estimating model parameters. We simulate seismic wavefields using a previously developed coupled acoustic/elastic wave propagator that implements a mimetic finite‐difference method with fully staggered grids to accurately handle the fluid/solid boundary conditions. The algorithm employs a multiscale approach starting from low frequencies and incorporating higher frequency bands in the later inversion stages. We analyse the influence of different types of input data on the accuracy of the inverted anisotropy parameters for hard and soft water bottoms. The employed misfit function incorporates information from both hydrophones and ocean‐bottom geophones. Numerical examples indicate that injecting multiple data components simultaneously increases the complexity of the objective function and often degrades the quality of the estimated medium parameters. Thus, we propose a sequential strategy using a single data component at a time. Pressure (hydrophone) data alone can provide satisfactory results if long offsets (i.e., with the offset/depth ratio ≥ 3) are available. Adding the horizontal particle‐displacement or ‐velocity components increases the accuracy of the estimated shear‐wave vertical velocity () and P‐wave normal‐moveout () velocity, especially for strongly heterogeneous sub‐water‐bottom models.

Subsoil density field reconstruction through 3-D FWI: a systematic comparison between vertical- and horizontal-force seismic sources

SUMMARY Bulk-density (ρ) of soil is an important indicator of soil compaction and type. A knowledge of the spatial variability of in situ soil density is important in geotechnical engineering, hydrology and agriculture. Surface geophysical methods have so far shown limited success in providing an accurate and high-resolution image of 3-D soil-density distribution. In this pursuit, 3-D seismic full-waveform inversion (FWI) is promising, provided the robustness and accuracy of density inversion via this approach can be established in the near-surface scale. However, simultaneous reconstruction of ρ and seismic wave velocities through multiparameter FWI remains a challenging task. Near-surface seismic data are commonly dominated by dispersive surface waves whose velocities are controlled by the value and distribution of shear-wave velocity (VS). One major difficulty in estimating reliably ρ from near-surface seismic data is due to the relatively low sensitivity of the seismic wavefield to ρ compared to seismic velocities. Additionally, the accuracy of the estimated ρ decreases due to error in VS—an issue known as parameter coupling. Parameter coupling makes it difficult to estimate accurately ρ within the framework of conventional gradient-based FWI. More sophisticated optimization approaches (e.g. truncated Newton) can reduce the effect of parameter coupling, but these approaches are commonly not affordable in near-surface applications due to heavy computational burden. In this research, we have investigated how choosing correctly the force direction of the seismic source can contribute to a higher accuracy of ρ estimates through 3-D FWI. Using scattered wavefields, the Hessian, and inversion tests, an in-depth and systematic investigation of data sets corresponding to different force directions has been carried out. A comparison of the scattered wavefields due to a point-localized ρ perturbation for different force directions shows the robustness of the horizontal-force data set to noise compared to the vertical-force data set. Furthermore, for a point-scatterer model, an analysis of the gradients of the misfit function using the Hessian shows that utilizing a horizontal-force source enables one to reconstruct the high-resolution gradient with relatively small parameter coupling. Finally, inversion tests for two different subsoil models demonstrate that 3-D FWI on a horizontal-force-source seismic data set is capable of providing a more accurate 3-D ρ distribution in soil compared to a vertical-force-source data set. Our results show that the use of a horizontal-force source might allow avoiding computationally demanding, costly optimization approaches in 3-D FWI.

Accelerating full-waveform inversion using source stacking: synthetic experiments at the global scale in a realistic 3-D earth model

SUMMARY The spectral element method is currently the method of choice for computing accurate synthetic seismic wavefields in realistic 3-D earth models at the global scale. However, it requires significantly more computational time, compared to normal mode-based approximate methods. Source stacking, whereby multiple earthquake sources are aligned on their origin time and simultaneously triggered, can reduce the computational costs by several orders of magnitude. We present the results of synthetic tests performed on a realistic radially anisotropic 3-D model, slightly modified from model SEMUCB-WM1 with three component synthetic waveform ‘data’ for a duration of 10 000 s, and filtered at periods longer than 60 s, for a set of 273 events and 515 stations. We consider two definitions of the misfit function, one based on the stacked records at individual stations and another based on station-pair cross-correlations of the stacked records. The inverse step is performed using a Gauss–Newton approach where the gradient and Hessian are computed using normal mode perturbation theory. We investigate the retrieval of radially anisotropic long wavelength structure in the upper mantle in the depth range 100–800 km, after fixing the crust and uppermost mantle structure constrained by fundamental mode Love and Rayleigh wave dispersion data. The results show good performance using both definitions of the misfit function, even in the presence of realistic noise, with degraded amplitudes of lateral variations in the anisotropic parameter ξ. Interestingly, we show that we can retrieve the long wavelength structure in the upper mantle, when considering one or the other of three portions of the cross-correlation time series, corresponding to where we expect the energy from surface wave overtone, fundamental mode or a mixture of the two to be dominant, respectively. We also considered the issue of missing data, by randomly removing a successively larger proportion of the available synthetic data. We replace the missing data by synthetics computed in the current 3-D model using normal mode perturbation theory. The inversion results degrade with the proportion of missing data, especially for ξ, and we find that a data availability of 45 per cent or more leads to acceptable results. We also present a strategy for grouping events and stations to minimize the number of missing data in each group. This leads to an increased number of computations but can be significantly more efficient than conventional single-event-at-a-time inversion. We apply the grouping strategy to a real picking scenario, and show promising resolution capability despite the use of fewer waveforms and uneven ray path distribution. Source stacking approach can be used to rapidly obtain a starting 3-D model for more conventional full-waveform inversion at higher resolution, and to investigate assumptions made in the inversion, such as trade-offs between isotropic, anisotropic or anelastic structure, different model parametrizations or how crustal structure is accounted for.

Constructing a 3‐D Radially Anisotropic Crustal Velocity Model for Oklahoma Using Full Waveform Inversion

AbstractOver the past decade, the seismicity rate in the state of Oklahoma has increased significantly, which has been linked to industrial operations, such as saltwater injection and hydraulic fracturing. Taking advantage of induced earthquakes and recently deployed seismometers, we construct a 3‐D radially anisotropic seismic velocity model for the crust of Oklahoma by using full waveform inversion. To mitigate the well‐known cycle‐skipping problem, we use misfit functions based on phase and waveform differences in several frequency bands. Relative velocity perturbations in the inverted model allow us to delineate major geological provinces in Oklahoma, such as the Anadarko Basin and the Cherokee Platform/Shelf. In addition, radial anisotropy in the inverted model reflects deformation within the crust of Oklahoma, which might correlate with sedimentary layering, microcracks/fractures, as well as dominant orientation of anisotropic minerals. The crystalline basement beneath Oklahoma can be inferred from the new velocity model, which enables us to better classify induced seismicity in current earthquake catalogs. Furthermore, synthetic experiments suggest that the new velocity model enables us to better constrain earthquake locations in Oklahoma, especially for determining their depths, which are important for investigating induced seismicity.

Full waveform inversion with combined misfit functions and application in land seismic data

Full waveform inversion reconstructs subsurface structures by matching the synthetic waveform to the observed waveform. Inaccuracy of the source wavelets can, thus, easily lead to an inaccurate model. Simultaneously updating source wavelets and model parameters is a conventionally used strategy. However, when the initial model is very far from the true model, cycle skipping exists, and estimating a reliable source wavelet is very difficult. We propose a combinatory inversion workflow based on seismic events. We apply a Gaussian time window around the first break and gradually increase its width to include more seismic events. The influence of inaccurate source wavelets is alleviated by applying a Gaussian time window around the first break to evaluate the normalized cross-correlation-based objective function. There are inevitable small model artifacts caused by inter-event interactions when calculating cross-correlations. As a result, we switch to the optimal transport function to clean the model and update the source wavelets simultaneously. The combinatory strategy has been applied to models with different types of geological structures. Starting from a crude initial model, we recovered a high-resolution and high-fidelity model and the source wavelets in two synthetic experiments. Finally, we apply our inversion strategy to a real-land seismic dataset in Southeast China and obtain a higher-resolution velocity model. By comparing an inversion velocity profile with well log information and the recorded data with the simulated data, we conclude that our inversion results for the field data are accurate and this new strategy is effective.

Geobody estimation by Bhattacharyya method utilizing nonlinear inverse modeling of magnetic data in Baba-Ali iron deposit, NW Iran

One of the essential geophysical concerns is the estimation of the physical and geometrical parameters of the reserve (geobody), which is done by exploiting the nonlinear inverse modeling of magnetic data. The present study includes preparing and modeling magnetic data to suggest the Baba Ali Iron ore deposit's drilling locations in NW Iran. The area is covered with 1000 points of geomagnetic reading with an almost 5 × 10 m2 regularly spaces grid trending WE. The areal and depth extent of the iron ore geobody was unknown. The Bhattacharyya method by MATLAB software coding was employed to underestimate the misfit function and re-construct potential field data, providing the most suitable fitting with measured magnetic data. In this order, the residual calculated anomaly exhibited an excellent two-dimensional conformation with forward modeling. Furthermore, 3D modeling correctly reconstructs the anomalies' productive resources' properties. After preparing full magnetic maps, the magnetic lenses distinguished in four anomalies of surface depths, 20, 50, and deeper than 50 m for this zone. This magnetite lens for the first zone was estimated based on analytical signal filters applied on the entire magnetic map so that the lens's depth is trivial and almost zero. Due to specific gravity calculated as 4.77 t/m3, initial storage capacity is suggested to be about 95,400 tons of magnetite, pyrite, and hematite minerals at most in an area about 6 Km2. Finally, to complete the preliminary explorations of the specified area, exploratory drilling is suggested for three points by inverse modeling. Regarding this study as the first try in magnetic reconnaissance step of Iron mineral exploration in the study area, there is no geological constraints available based on drilling evidences. However, the model is well satisfies the surface anomalies considering residual magnetic property.

Ambient‐Noise Wave‐Equation Tomography of the Alps and Ligurian‐Provence Basin

AbstractTaking benefit of the AlpArray temporary network and permanent networks in W‐Europe, we construct a 3‐D onshore‐offshore velocity model of the crust and upper mantle using ambient‐noise wave‐equation tomography. We use a frequency‐dependent phase traveltime misfit function in an iterative procedure to refine a recent 3‐D Vs model computed from a Bayesian two‐step ambient noise tomography. Observed waveforms consist of vertical‐component noise correlations from 600 broadband stations in the Alps and surroundings, including ocean‐bottom seismometers in the Ligurian sea. We perform 3‐D inversion in the 5–85 s period range. In the long‐period band (20–85 s), an elastic approximation is considered, while in the 5–20 s band, we account for the effect of water layer in the Ligurian sea by applying a fluid‐solid coupling for acoustic‐elastic waveform simulations. The resulting Vs model enhances the shape and contrast of velocity structures, accounting for 3‐D and finite‐frequency effects. It emphasizes the deep sediments of the Ligurian‐Provence basin and focuses on the low‐velocity anomalies of the crust in the W‐Alps. We obtain a high‐resolution Moho depth map covering the Alps and Ligurian sea. In the W‐Alps, this map confirms the deepening of the European crust following the subduction beneath Adria and the existence of major structures such as the Moho jump beneath the external crystalline massifs and shallow depths associated with the Ivrea Body. It provides further constraints on the deep structure beneath the Ligurian‐Provence basin, regarding the lateral and along‐strike crustal‐thickness variations from the oceanic domain to the conjugate margins.

Estimating Young’s moduli based on ultrasound and full-waveform inversion

Ultrasound is commonly utilized to determine the dynamic Young’s moduli for estimating material deterioration or quantifying material parameters. This is either done with a resonance frequency analysis or ultrasound for rock or concrete specimens in geophysics or civil engineering. For ultrasound, these are typically cylindrical specimens measured in a through-transmission arrangement. However, this method presents some challenges, such as the need to accurately determine the onset of each wave mode and to calibrate system latency. To overcome these challenges, this study introduces a novel method that utilizes wavefield simulation and full-waveform inversion to estimate p- and s-wave, also called compression and shear wave, velocities. Where the traditional method requires two measurements, this innovative approach requires only one measurement with a single p-wave transducer. The s-wave velocity is estimated considering mode conversions within the specimen. High-fidelity ultrasound simulations are necessary for this approach. For that reason, the spatially distributed excitation of the ultrasound transducer is characterized by a laser Doppler vibrometer measurement. This led to a good estimate of the directivity pattern of the ultrasound transducer. The inverse problem for determining p- and s-wave velocity was solved successfully using a suitable misfit or cost function, the so-called graph-optimal-transport misfit. The specimens for the proof of concept study include six different metals. The precision of the estimated velocities using full-waveform inversion was compared with manual picking. As the simulated and measured waveforms matched well, this study can be seen as a starting point for material parameter determination for more complex geometries and heterogeneous materials using full-waveform inversion.

A review of misfit functions for adjoint full waveform inversion in seismology

SUMMARY In seismological full waveform inversion, the choice of misfit functions plays a critical role in quantifying the discrepancy between observed and synthetic data, affecting convergence rate and also final results. We revisit and compare six commonly used misfit functions, including cross-correlation time-shift (CC), least-square waveform difference (L2), multitaper time-shift (MT), exponentiated phase shift (EP), time–frequency phase shift (TF) and zero-lag cross-correlation coefficient (CCC), with respect to their definitions, adjoint sources and misfit kernels for velocity perturbations. Synthetic tests are performed for several canonical models. First, we simulated wave propagation in a model with a single rectangular anomaly with sharp boundaries and a smoothed variant of that model. We analysed the resulting misfit kernels first for the P-wave phase, which is highly distorted in the sharp model due to strong heterogeneities, and mostly experiences traveltime perturbations in the smooth model. Second, we considered a model where a laterally limited region is subject to layered anomalies (low velocity in the middle crust and high velocity in the lower crust) and determine misfit kernels for S and surface waves in this model. Based on these two simplified seismological scenarios, we further perform iterative test inversions using different misfit functions. Combining the features of misfit kernels and synthetic inversion results, we find that CCC, L2 and EP are the most effective at identifying the sharpness of velocity anomalies from the direct body waves and their scattered phases. Consequently, inversion based on those misfit measures yielded the best recovery in the inversion test. For surface and S waves from crustal sources, TF appears to be the most effective in constraining the heterogeneous structure in the crust but needs more iterations for convergence than other misfit functions.

HMCLab: a framework for solving diverse geophysical inverse problems using the Hamiltonian Monte Carlo method

SUMMARY The use of the probabilistic approach to solve inverse problems is becoming more popular in the geophysical community, thanks to its ability to address nonlinear forward problems and to provide uncertainty quantification. However, such strategy is often tailored to specific applications and therefore there is a need for common platforms to solve different geophysical inverse problems and showing potential and pitfalls of the methodology. In this work, we demonstrate a common framework within which it is possible to solve such inverse problems ranging from, for example, earthquake source location to potential field data inversion and seismic tomography. This allows us to fully address nonlinear problems and to derive useful information about the subsurface, including uncertainty estimation. This approach can, in fact, provide probabilities related to certain properties or structures of the subsurface, such as histograms of the value of some physical property, the expected volume of buried geological bodies or the probability of having boundaries defining different layers. Thanks to its ability to address high-dimensional problems, the Hamiltonian Monte Carlo (HMC) algorithm has emerged as the state-of-the-art tool for solving geophysical inverse problems within the probabilistic framework. HMC requires the computation of gradients, which can be obtained by adjoint methods. This unique combination of HMC and adjoint methods is what makes the solution of tomographic problems ultimately feasible. These results can be obtained with ‘HMCLab’, a numerical laboratory for solving a range of different geophysical inverse problems using sampling methods, focusing in particular on the HMC algorithm. HMCLab consists of a set of samplers (HMC and others) and a set of geophysical forward problems. For each problem its misfit function and gradient computation are provided and, in addition, a set of prior models can be combined to inject additional information into the inverse problem. This allows users to experiment with probabilistic inverse problems and also address real-world studies. We show how to solve a selected set of problems within this framework using variants of the HMC algorithm and analyse the results. HMCLab is provided as an open source package written both in Python and Julia, welcoming contributions from the community.

Newton optimization and the Hessian matrix in scale-separation waveform inversion problems with reflected and transmitted waves

Summary Full waveform inversion (FWI) is often utilized for determining the medium parameters of the Earth by minimizing the misfit between observed and modelled waves. The conventional formulation of FWI is well suited to utilize transmitted waves to determine long-to-intermediate wavelength properties of the medium. Hovewer, the penetration depth of transmitted waves may be limited for a given acquisition. As a remedy, reflection waveform inversion (RWI) exploits the kinematic information in reflected waves to retrieve these long-wavelength properties, particularly below the maximum illumination depth of transmitted waves. Previous works have demonstrated the benefits of utilizing both transmitted and reflected waves, thereby considering a reconciliation of conventional FWI and RWI. One such formulation is known as joint full waveform inversion (JFWI), where reflected and transmitted waves are separately compared in the misfit functional. A disadvantage of gradient-based optimization schemes in RWI and JFWI is the presence of strong crosstalk due to mapping of reflected data residuals along the entire reflection wavepaths. Because RWI and JFWI require frequent least-squares migration, we are incentivized to determine well-focused search directions that allow large movements between the migration steps. Thus, we propose an improvement to RWI and JFWI by utilizing Newton methods to directly account for crosstalk between parameters. The crosstalk effects are described by the second-order sensitivities of the misfit functional, i.e. the Hessian matrix. We adopt the adjoint-state method to compute the first- and second order sensitivities of the misfit functional for any of these waveform inversion procedures in a unified framework and for arbitrary misfit functionals. Furthermore, we conceptually study the Hessian matrix in least-squares based full-, reflection- and joint full waveform inversion from a data-domain point of view. Synthetic examples and a three-dimensional field data example demonstrate that the truncated Gauss-Newton method is effective in attenuating artefacts due to crosstalk between spatially separated variables, thus substantially improving the spatial focusing of the search directions.

Comparison of force and moment tensor estimations of subevents during the 2022 Hunga–Tonga submarine volcanic eruption

SUMMARYThe 2022 January 15 Hunga–Tonga submarine volcanic eruption was unprecedented in the modern era for its size, in terms of its plume height and atmospheric waves, including sound. Global seismic stations recorded maximal radiated energy during 30 min of the plume-forming phase of the eruption (04:00 to 04:30 UTC), with the largest subevents occurring over a 5-min interval starting at 04:15:17 UTC. Here, we consider two simple point-source models—force and moment tensor—and separately consider the single main subevent and the sequence of four subevents. Estimation of source models for the first subevent in the sequence is achieved with a complete search of model parameter space to find the global minimum of a waveform misfit function (body or surface waves). We performed 25 runs to explore the impact of depth, source model (force or moment tensor), wave type (body or surface), and component (vertical, radial, and transverse) on the waveform fits and estimated best-fitting source. Visualization of the misfit function reveals complex trade-offs among model parameters, highlighting the importance of characterizing uncertainties and parameter trade-offs. The four-subevent source model has up to 28 model parameters and requires an efficient search algorithm to find the best-fitting source. For this, we use the covariance matrix adaptation evolution strategy implemented on a high-performance computing cluster. The 10 four-subevent runs for each source model return sequences of subvertical downward forces and explosive-like moment tensors for each subevent. Our results show that these two simple source models provide comparable fits to regional and global seismic waveforms and that the source types for each subevent—either force direction or moment tensor source type on the eigenvalue lune—are similar enough to each other to consider that the subevents originate from the same process. Our estimation of the source mechanisms, sizes, and relative timing should benefit a physical interpretation of the eruption sequence.

High-precision and high-efficiency first-arrival slope tomography via eikonal solvers and the adjoint-state method

Abstract First-arrival slope tomography (FAST) introduces first-arrival slopes, corresponding to the horizontal components of the slowness vectors at the receiver and source positions to supplement first-arrival traveltime for better guiding ray propagation in the media until the best match is achieved with the observed data. FAST can recover the velocity model with higher resolution and precision than first-arrival traveltime tomography (FATT) but is computationally intensive. In this context, we propose an improved approach, referred to as high-precision and high-efficiency first-arrival slope tomography (HFAST). HFAST redefines one of the slopes using the reciprocity principle and simultaneously employs the first-arrival traveltime and slopes to ensure high-quality model building. On the other hand, HFAST extracts calculated data and derives the gradient of the misfit function from the solutions of relatively limited forward and inverse problems, resulting in a low computational cost. The cost of HFAST is proportional to the minimum between the receivers and sources, whereas the cost of FAST is scaled to the sum of the receivers and sources. Numerical experiments involving the checkerboard and SEAM II Foothill models demonstrate that HFAST can achieve a higher inversion precision than FATT, especially in the recovery of small-scale anomalies and the presence of velocity reversal. Moreover, HFAST is more computationally efficient than FAST and suitable for managing large data sets. Therefore, HFAST can be regarded as a valuable supplement to current first-arrival-based model building methods and has the potential to be applied in static corrections, prestack depth migration and waveform inversion in the future.

Least‐squares reverse time migration using the most energetic source wavefields based on excitation amplitude imaging condition

AbstractLeast‐squares reverse time migration, a linearized inversion problem, can provide high‐quality migration image by minimizing the misfit function, which is defined by predicted and observed data. According to the theory of data‐domain least‐squares reverse time migration, a forward source wavefield that is simulated with a fixed background velocity does not change during iterations. However, storing the forward source wavefield directly into computer memory involves substantial memory consumption. Although a source wavefield reconstruction technique can be applied during least‐squares reverse time migration iterations, this approach can increase the computational cost because of the need for additional wavefield simulations. To alleviate this computational issue in storing the forward source wavefield, we propose an efficient least‐squares reverse time migration scheme based on an excitation amplitude method. Unlike conventional excitation amplitude imaging conditions, the proposed least‐squares reverse time migration scheme enables one to reconstruct the forward source wavefield by convolving a source wavelet with the excitation amplitude of Green's function at the excitation time. With this excitation amplitude method, the forward source wavefield can be efficiently stored in the computer memory because the sizes of the excitation amplitude and excitation time maps are equal to the size of one snapshot. To validate the feasibility of our least‐squares reverse time migration scheme, we perform dot‐product tests and compare forward source wavefields, demigrated data and gradient vectors obtained by conventional least‐squares reverse time migration and our proposed least‐squares reverse time migration. Using numerical tests on synthetic data, we confirm that our least‐squares reverse time migration can produce high‐quality migration results with a significant improvement in computational efficiency with respect to performance time and memory consumption.

The Mean Moment of Inertia for Irregularly Shaped Phobos and Its Application to the Constraint for the Two-Layer Interior Structure for the Martian Moon

The interior structure of Phobos has been the subject of debate in recent years, with the moment of inertia being a determining factor. To study this structure, we modeled Phobos with a two-layer structure and calculated its mean density and moment of inertia using updated gravity coefficients of degree-2 and forced libration amplitudes. By minimizing the misfit between modeled and derived moment of inertia, and observed and modeled mean density, we determined the frequency distribution for estimated parameters, including the core radius rc, core density ρc, and density ρm of the outer layer. Our results indicate that the optimized core radius is around 8.2 km for our models, along with a core density compromise of approximately 2500 kg·m−3, and an outer layer density of around 1400 kg·m−3. These values have remarkable sensitivity to the misfit function, implying a higher density likely inside Phobos compared to the outer layer. Given that the large core density was associated with ice content, it suggested that the fractional ice content in the outer layer is approximately 11% with a rock density of 2200 kg·m−3, while the content in the core is lower at 2.4% with a rock density of 3000 kg·m−3. The methodology introduced in this study can be further used to study the interior structure of irregularly shaped asteroids.

The back-and-forth method for the quadratic Wasserstein distance-based full-waveform inversion

The conventional least-squares misfit function compares synthetic data to observed data in a point-by-point style. The Wasserstein distance function, also called the optimal transport function, matches patterns. The kinematic information of seismograms is therefore efficiently extracted. This property makes it more convex than the conventional least-squares function. Computing the 1D Wasserstein function is fast. Processing the 2D or 3D seismic data volume trace by trace, however, loses the interreceiver coherency. The main difficulty of extending to the high-dimensional Wasserstein function is the heavy computation cost. This computational challenge can be alleviated by a back-and-forth method. After explaining the computation strategy and incorporating it into full-waveform inversion, we illustrate the superior performances of the high-dimensional Wasserstein function with a Camembert model and the Marmousi model. The superiority is also demonstrated with the Chevron 2014 blind test.

Simultaneous random plus outlier attenuation and deblending based on L1‐norm misfit function

AbstractDeblending technology aims at separating simultaneous source seismic data between adjacent shots by allowing multiple sources to be shot simultaneously. Conventional deblending methods based on sparse inversion assume that the primary source is coherent, and the secondary source is randomized. The L2‐norm minimization constraint can effectively minimize the Gaussian random noise while deblending in the transform domain. Nonetheless, the L2‐norm misfit function is highly sensitive to outliers, negatively influencing the deblending performance. An effective optimization strategy is developed with deblending in pre‐stack seismic data to eliminate outliers and enhance deblending accuracy. For this reason, we introduce the deblending algorithm in the morphological component analysis framework, modify the L2‐norm misfit function to outlier‐robust L1‐norm and provide the corresponding derivation in detail via the alternating direction method of multipliers. Applications to synthetic and field data sets prove the improved robustness and efficiency of our deblending method.

Full‐Waveform Tomography of the African Plate Using Dynamic Mini‐Batches

AbstractWe present a seismic model of the Africa Plate, constructed with the technique of full‐waveform inversion. The purpose of our model is to serve as a foundation for quantitative geodynamic and geochemical interpretation, earthquake‐induced ground motion predictions, and earthquake source inversion. Starting from the first‐generation Collaborative Seismic Earth Model, we invert seismograms filtered to a minimum period of 35 s and compute gradients of the misfit function with respect to the model parameters using the adjoint state method. We use dynamically changing mini‐batches of the complete data set to compute approximate gradients at each iteration. This approach has three significant advantages: (a) it reduces computational costs for model updates and the inversion, (b) it enables the use of larger datasets without increasing iteration costs, and (c) it makes it trivial to assimilate new data since we can extend the data set without changing the misfit function. We invert data from 397 unique earthquakes and 184,356 unique source‐receiver pairs. We clearly image tectonic features such as the Afar triple junction and low‐velocity zones below the Hoggar, Aïr, and Tibesti Mountains, pronounced more than in earlier works. Finally, we introduce a new strategy to assess model uncertainty. We deliberately perturb the final model, perform additional mini‐batch iterations, and compare the result with the original final model. This test uses actual seismic data instead of artificially generated synthetic data and requires no assumptions about the linearity of the inverse problem.

Full waveform inversion guided wave tomography with a recurrent neural network

Corrosion quantitative detection of plate or plate-like structure materials is crucial in industrial Non-Destructive Testing (NDT) for determining their remaining life. For doing that, a novel ultrasonic guided wave tomography method, incorporating recurrent neural network (RNN) into full waveform inversion (FWI) called as RNN-FWI, is proposed in this paper. When the wave equation of an acoustic model is solved by a forward model with the cyclic calculation units of an RNN, it is shown that the inversion of the forward model can be obtained iteratively by minimizing a waveform misfit function of quadratic Wasserstein distance between the modeled and measured data. It is also demonstrated that the gradient of the objective function can be obtained by automatic differentiation while the parameters of the waveform velocity model are updated by the adaptive momentum estimation algorithm (Adam). The U-Net deep image prior (DIP) is used as the velocity model regularization in each iteration. The final thickness maps of the plate or plate-like structure materials shown can be archived by the dispersion characteristics of guided waves. Both the numerical simulation and experimental results show that the proposed RNN-FWI tomography method performs better than the conventional time-domain FWI in terms of convergence rate, initial model requirement, and robustness.

Elastic vertically transversely isotropic full-waveform inversion: From synthetic to field data

Representing the earth as an elastic medium leads to a more accurate approximation of the seismic response than can be obtained by assuming an acoustic medium. However, this representation requires a larger number of unknown parameters to be introduced and a more complex seismogram to be considered. Consequently, it is important to carefully deal with two challenging issues, the impact of strong surface waves and the crosstalk among parameters, for land seismic multiparameter full-waveform inversion (FWI). Under the elastic vertically transversely isotropic assumption, a robust FWI approach is developed by combining the optimal transport (OT) misfit function, the Gaussian time window, and the suitable parameterization [Formula: see text]. Without specific preprocessing, surface waves usually dominate the inversion process, leaving the body waves not fully exploited. Correspondingly, the subsurface model is only updated at a shallow depth. The OT misfit function can balance the contributions of surface and body waves to model updates in an automatic manner. In addition, the Gaussian time window further improves the weights of the body waves during inversion. We efficiently exploit the entire seismogram with these two approaches. In this configuration, the performances of several parameterizations chosen from radiation pattern analysis are evaluated. Our inversion strategy is applied to the Middle East model and a field data application. In both cases, it outperforms other inversion strategies. The entire land seismic data are efficiently exploited and multiparameters are simultaneously reconstructed.