Multiscale Inversion Research Articles

Multiscale Direct Envelope Inversion: Algorithm and Methodology for Application to the Salt Structure Inversion

AbstractFor the full‐waveform inversion, obtaining a globally optimal solution is difficult when the seismic data do not contain sufficient low‐frequency information and the inversion object is a salt structure with a large‐scale salt dome. The large salt dome presents difficulty for the inversion of the salt structure; however, it also confers a convenience that can be utilized: A large salt dome makes the events in the seismic record sparse and isolated. We use the envelope to obtain the arrival time of the events in the seismic data and apply multiscale strategies to extract the low‐frequency information of the seismic data that relates to the long‐wavelength components of the subsurface. In calculating the gradient, the envelope Fréchet derivative is used to take advantage of the multiscale envelope for the characteristics of the salt structure. Therefore, the multiscale direct envelope inversion using the window‐averaged envelope and the direct envelope Fréchet derivative is proposed. To further improve the computational efficiency and inversion quality of the multiscale direct envelope inversion, several auxiliary strategies are proposed: A joint misfit function is designed to make the inversion method more adaptable, and an offset weighting factor is introduced to implement the multioffset inversion method to improve the inversion quality of subsalt. We analyze the antinoise performance of the multiscale direct envelope inversion. The SEG/EAGE salt model is used to test the application effect of the multiscale direct envelope inversion. The antinoise ability and insensitivity to low‐frequency data of the method are verified in the numerical experiments.

Generalized Multiscale Inversion for Heterogeneous Problems

In this work, we propose a generalized multiscale inversion algorithm for heterogeneous problems that aims at solving an inverse problem on a computational coarse grid. Previous inversion techniques for multiscale problems seek a coarse-grid medium properties, e.g., permeability and conductivity, and by doing so, they assume that there exists a homogenized representation of the underlying fine-scale permeability field on a coarse grid. Generally such assumptions do not hold for highly heterogeneous fields, e.g., fracture media or channelized fields, where the width of channels are very small compared to the scale of coarse grids. In these cases, grid refinement can lead to many degrees of freedom, and thus numerically unattractive to apply. The proposed algorithm is based on the Generalized Multiscale Finite Element Method (GMsFEM), which uses local spectral problems to identify non-localized features, i.e., channels (high-conductivity inclusions that connect the boundaries of the coarse-grid block). The inclusion of these features in the coarse space enables one to achieve a good accuracy. The approach is valid under the assumption that the solution can be well represented in a reduced-dimensional space spanned by multiscale basis functions. In practice, these basis functions are non-observable as we do not identify the fine-scale features of the permeability field. Our inversion algorithm finds the discretization parameters of the resulting system on the coarse grid. By doing so, we identify the appropriate coarse-grid parameters representing the permeability field instead of fine-grid permeability field. We illustrate the potential of the approach by numerical results for fractured media.

Seismic Waveform Tomography With Simplified Restarting Scheme

For shot-encoded seismic waveform tomography, the restarted limited-memory BFGS (L-BFGS) algorithm is an effective technique to suppress the crosstalk effect among encoded seismic shots. It restarts the L-BFGS calculation at each iteration segment, consisted of a group of iterations, and recodes the individual shots randomly not only at the beginning but also at the inside of the iteration segment. Here, we simplified this scheme using an invariant shot-encoding within each iteration segment and recoding individual shots only at the beginning of the segment. This simplification did compromise the image quality at the early stage of inversion, as the crosstalk effect appeared on the inversion result of the low-frequency data. However, it eventually achieved both the computation efficiency and the good quality for a multiscale inversion procedure, which inverts the seismic data from low-frequency components to high-frequency components in sequence.

Model inversion of BCVL based on DWD of MISR image

This paper puts forward a novel algorithm for the inversion of biophysical components of vegetation leaves (BCVL) by means of Remote Sensing (RS) image based on multi-scale two-dimension Discrete Wavelet Decomposition (DWD). With the aid of the MISR (multi-angle imaging spectro-radiometer) image, the multi-scale BCVL inversion is implemented with the algorithm on each scale of wavelet decomposition based on DWD of MISR image. The retrieved results of the rough-scale wavelet decomposition are regarded as a priori knowledge of the fine-scale inversion, and then the retrieved results of the rough-scale are also considered as the initial values of the fine-scale inversion. In this way, wavelet decomposition of MISR image is gradually refined, and the retrieved result is likewise gradually refined until the final retrieved result of BCVL is achieved in the end. The algorithm provides a novel solution scheme for the model inversion of BCVL with the aid of DWD of MISR image. Furthermore, this article also discusses the arithmetic principle of the model inversion of BCVL by employing the radiative transfer model of vegetation leaves (RTMV). On this basis, this manuscript builds the inversion model of BCVL based on RTMV. The proposed algorithm is applied specially to the model inversion of BCVL by using MISR image in this experiment, and satisfactory results are finally achieved. Taking the model inversion of LAI (leaf area index) as an example, the retrieved result of a higher precision is achieved in the experiment. Where, correlation coefficient between measured LAI and retrieved LAI is R=0.84, and root-mean-square error is RMSE=0.26. Experiment demonstrates that the proposed algorithm is very effectual and reliable in the model inversion of BCVL. The proposed algorithm opens up a novel algorithmic pathway for the model inversion of BCVL by means of DWD of MISR image.

Insight Into Frontal Seismogenic Zone in the Mentawai Locked Region From Seismic Full Waveform Inversion of Ultralong Offset Streamer Data

AbstractSumatra subduction zone is one of the most seismically active zones on Earth. After having produced three Mw > 8.4 earthquakes and several Mw > 7.5 earthquakes, including the Mw = 7.8 2010 tsunami earthquake, the northern Mentawai segment is still locked and is capable of generating a great earthquake, possibly a disastrous tsunami. We analyzed ultralong offset seismic reflection data from this locked zone to characterize the nature of the accretionary prism and the plate interface using a combination of traveltime tomography, full waveform inversion, and prestack depth migration. In order to enhance the refractions, we downward extrapolate the streamer data to the seafloor, allowing the refraction arrivals to be observed from near‐zero offset up to far offset, and use a traveltime tomography to determine the background velocity in the upper sediments. Starting from these velocities, we perform a multiscale elastic full waveform inversion to determine the detailed P wave velocity structure of the subsurface. Based on this velocity, we perform a prestack depth migration to obtain seismic image in the depth domain and compute the porosity of the sediments to determine fluid content along faults. Our results show a low‐velocity subduction channel with high porosity at the plate interface that connects the likely active frontal thrusts at the toe of accretionary wedge, suggesting that the frontal section of the prism is seismogenic. We have also observed a low‐velocity layer in the middle of wedge separating old sediments below from new sediments above, defining the roots of bivergent thrust faults up to the seafloor, which can be interpreted as a psudo‐décollement.

Crustal Magmatism and Deformation Fabrics in Northeast Japan Revealed by Ambient Noise Tomography

AbstractMapping both isotropic and anisotropic velocity structures of the crust provides insight into the dominant mechanisms that produce the deformation. We performed ambient noise tomography for shear velocity Vs in the Tohoku region, Japan, where plate kinematics remains simple as the Pacific plate is subducting under the Okhotsk plate. Cross‐correlation functions were retrieved from Hi‐net recordings and phase velocities of 3–16 s were measured, followed by a one‐step, wavelet‐based multiscale inversion for both Vs and its azimuthal anisotropy. The isotropic model shows that high Vs anomalies underlie the Cretaceous aged Kitakami and Abukuma Mountains from shallow crust to lower crust. Low Vs is found to track the active volcanoes with the strongest anomalies concentrated at roughly 10 km depth, which we interpret as the shallow magma reservoirs of the Quaternary magmatic system. The resolved azimuthal anisotropy exhibits a switch of anisotropy in the upper crust from island‐parallel to a random or suppressed state in 5–15 km depth interval and back to island‐parallel. The island‐parallel fast direction may map the structural fabrics of the crustal deformation, and the spatial correlation between velocity and anisotropy patterns suggests that continuous volcanic activities and magma ponding disrupt the otherwise consistent fabrics in the upper crust. Below 20 km depth, the anisotropy turns to convergence‐parallel, which may result from shearing either imposed by the return flow in the mantle wedge or frozen‐in from the last stage of extension of the continental margin that opened the Sea of Japan.

Dynamic Convolution-based Misfit Function for Time Domain Full Waveform Inversion

Under the same propagation operator, the precision of the seismic wavelet determines whether synthetic data can match field data accurately. By constructing the convolution wavefield objective function, the source difference between simulated and observed data is ignored, thus avoiding the source wavelet estimation. Theoretically, this process has no restriction on the accuracy of the wavelet, and a multi-scale inversion strategy can be implemented by using a source wavelet with different dominant frequencies. When the propagation operator used does not conform with reality, even if both the wavelet and the parameter mode are accurate, it is impossible to simulate the full waveforms of the observed data. Meanwhile, the difference in the Green function brings new discrepancy to the convolution wavefield objective function and affects the final inversion results. The method presented in this paper is based on the convolution wavefield objective function. On the basis of an original single reference trace, we discuss the performance of the convolution wavefield-type objective function under multiple reference traces selected from different offsets. After introducing the changed wavefield information, the objective function has the ability to adapt to different types of data. The analysis shows that nonlinearity is significantly increased after introducing the different wavefield information and also increases with inversion frequency. Even for anisotropic data, it is still possible to give a relatively accurate structure at the low-frequency stage, which shows that the wavefield information from different offsets can help to weaken the artifacts introduced by operator mismatch, thus providing more possibilities for the application of acoustic wave equation inversion.

A robust multi-scale approach to quantitative susceptibility mapping

Quantitative Susceptibility Mapping (QSM), best known as a surrogate for tissue iron content, is becoming a highly relevant MRI contrast for monitoring cellular and vascular status in aging, addiction, traumatic brain injury and, in general, a wide range of neurological disorders. In this study we present a new Bayesian QSM algorithm, named Multi-Scale Dipole Inversion (MSDI), which builds on the nonlinear Morphology-Enabled Dipole Inversion (nMEDI) framework, incorporating three additional features: (i) improved implementation of Laplace's equation to reduce the influence of background fields through variable harmonic filtering and subsequent deconvolution, (ii) improved error control through dynamic phase-reliability compensation across spatial scales, and (iii) scalewise use of the morphological prior. More generally, this new pre-conditioned QSM formalism aims to reduce the impact of dipole-incompatible fields and measurement errors such as flow effects, poor signal-to-noise ratio or other data inconsistencies that can lead to streaking and shadowing artefacts. In terms of performance, MSDI is the first algorithm to rank in the top-10 for all metrics evaluated in the 2016 QSM Reconstruction Challenge. It also demonstrated lower variance than nMEDI and more stable behaviour in scan-rescan reproducibility experiments for different MRI acquisitions at 3 and 7 Tesla. In the present work, we also explored new forms of susceptibility MRI contrast making explicit use of the differential information across spatial scales. Specifically, we show MSDI-derived examples of: (i) enhanced anatomical detail with susceptibility inversions from short-range dipole fields (hereby referred to as High-Pass Susceptibility Mapping or HPSM), (ii) high specificity to venous-blood susceptibilities for highly regularised HPSM (making a case for MSDI-based Venography or VenoMSDI), (iii) improved tissue specificity (and possibly statistical conditioning) for Macroscopic-Vessel Suppressed Susceptibility Mapping (MVSSM), and (iv) high spatial specificity and definition for HPSM-based Susceptibility-Weighted Imaging (HPSM-SWI) and related intensity projections.

Multiscale full-waveform inversion based on shot subsampling

Conventional full-waveform inversion is computationally intensive because it considers all shots in each iteration. To tackle this, we establish the number of shots needed and propose multiscale inversion in the frequency domain while using only the shots that are positively correlated with frequency. When using low-frequency data, the method considers only a small number of shots and raw data. More shots are used with increasing frequency. The random-in-group subsampling method is used to rotate the shots between iterations and avoid the loss of shot information. By reducing the number of shots in the inversion, we decrease the computational cost. There is no crosstalk between shots, no noise addition, and no observational limits. Numerical modeling suggests that the proposed method reduces the computing time, is more robust to noise, and produces better velocity models when using data with noise.

Gravity compression forward modeling and multiscale inversion based on wavelet transform

The main problems in three-dimensional gravity inversion are the non-uniqueness of the solutions and the high computational cost of large data sets. To minimize the high computational cost, we propose a new sorting method to reduce fluctuations and the high frequency of the sensitivity matrix prior to applying the wavelet transform. Consequently, the sparsity and compression ratio of the sensitivity matrix are improved as well as the accuracy of the forward modeling. Furthermore, memory storage requirements are reduced and the forward modeling is accelerated compared with uncompressed forward modeling. The forward modeling results suggest that the compression ratio of the sensitivity matrix can be more than 300. Furthermore, multiscale inversion based on the wavelet transform is applied to gravity inversion. By decomposing the gravity inversion into subproblems of different scales, the non-uniqueness and stability of the gravity inversion are improved as multiscale data are considered. Finally, we applied conventional focusing inversion and multiscale inversion on simulated and measured data to demonstrate the effectiveness of the proposed gravity inversion method.

The Collaborative Seismic Earth Model: Generation 1.

We present a general concept for evolutionary, collaborative, multiscale inversion of geophysical data, specifically applied to the construction of a first‐generation Collaborative Seismic Earth Model. This is intended to address the limited resources of individual researchers and the often limited use of previously accumulated knowledge. Model evolution rests on a Bayesian updating scheme, simplified into a deterministic method that honors today's computational restrictions. The scheme is able to harness distributed human and computing power. It furthermore handles conflicting updates, as well as variable parameterizations of different model refinements or different inversion techniques. The first‐generation Collaborative Seismic Earth Model comprises 12 refinements from full seismic waveform inversion, ranging from regional crustal‐ to continental‐scale models. A global full‐waveform inversion ensures that regional refinements translate into whole‐Earth structure.

Multi-scale signed envelope inversion

Envelope inversion based on modulation signal mode was proposed to reconstruct large-scale structures of underground media. In order to solve the shortcomings of conventional envelope inversion, multi-scale envelope inversion was proposed using new envelope Fréchet derivative and multi-scale inversion strategy to invert strong contrast models. In multi-scale envelope inversion, amplitude demodulation was used to extract the low frequency information from envelope data. However, only to use amplitude demodulation method will cause the loss of wavefield polarity information, thus increasing the possibility of inversion to obtain multiple solutions. In this paper we proposed a new demodulation method which can contain both the amplitude and polarity information of the envelope data. Then we introduced this demodulation method into multi-scale envelope inversion, and proposed a new misfit functional: multi-scale signed envelope inversion. In the numerical tests, we applied the new inversion method to the salt layer model and SEG/EAGE 2-D Salt model using low-cut source (frequency components below 4 Hz were truncated). The results of numerical test demonstrated the effectiveness of this method.

Acceleration for 2D time-domain elastic full waveform inversion using a single GPU card

Full waveform inversion (FWI) is a challenging procedure due to the high computational cost related to the modeling, especially for the elastic case. The graphics processing unit (GPU) has become a popular device for the high-performance computing (HPC). To reduce the long computation time, we design and implement the GPU-based 2D elastic FWI (EFWI) in time domain using a single GPU card. We parallelize the forward modeling and gradient calculations using the CUDA programming language. To overcome the limitation of relatively small global memory on GPU, the boundary saving strategy is exploited to reconstruct the forward wavefield. Moreover, the L-BFGS optimization method used in the inversion increases the convergence of the misfit function. A multiscale inversion strategy is performed in the workflow to obtain the accurate inversion results. In our tests, the GPU-based implementations using a single GPU device achieve >15 times speedup in forward modeling, and about 12 times speedup in gradient calculation, compared with the eight-core CPU implementations optimized by OpenMP. The test results from the GPU implementations are verified to have enough accuracy by comparing the results obtained from the CPU implementations.

Multiscale phase inversion of seismic data

We present a scheme for multiscale phase inversion (MPI) of seismic data that is less sensitive than full-waveform inversion (FWI) to the unmodeled physics of wave propagation and to a poor starting model. To avoid cycle skipping, the multiscale strategy temporally integrates the traces several times, i.e., high-order integration, to produce low-boost seismograms that are used as input data for the initial iterations of MPI. As the iterations proceed, lower frequencies in the data are boosted by using integrated traces of lower order as the input data. The input data are also filtered into different narrow frequency bands for the MPI implementation. Numerical results with synthetic acoustic data indicate that, for the Marmousi model, MPI is more robust than conventional multiscale FWI when the initial model is moderately far from the true model. Results from synthetic viscoacoustic and elastic data indicate that MPI is less sensitive than FWI to some of the unmodeled physics. Inversion of marine data indicates that MPI is more robust and produces modestly more accurate results than FWI for this data set.

Effect of surface-related Rayleigh and multiple waves on velocity reconstruction with time-domain elastic FWI

It is critically important to assess the effectiveness of elastic full waveform inversion (FWI) algorithms when FWI is applied to real land seismic data including strong surface and multiple waves related to the air-earth boundary. In this paper, we review the realization of the free surface boundary condition in staggered-grid finite-difference (FD) discretization of elastic wave equation, and analyze the impact of the free surface on FWI results. To reduce inputs/outputs (I/O) operations in gradient calculation, we adopt the boundary value reconstruction method to rebuild the source wavefields during the backward propagation of the residual data. A time-domain multiscale inversion strategy is conducted by using a convolutional objective function, and a multi-GPU parallel programming technique is used to accelerate our elastic FWI further. Forward simulation and elastic FWI examples without and with considering the free surface are shown and analyzed, respectively. Numerical results indicate that no free surface incorporated elastic FWI fails to recover a good inversion result from the Rayleigh wave contaminated observed data. By contrast, when the free surface is incorporated into FWI, the inversion results become better. We also discuss the dependency of the Rayleigh waveform incorporated FWI on the accuracy of initial models, especially the accuracy of the shallow part of the initial models.

Bayesian seismic multi-scale inversion in complex Laplace mixed domains

Seismic inversion performed in the time or frequency domain cannot always recover the long-wavelength background of subsurface parameters due to the lack of low-frequency seismic records. Since the low-frequency response becomes much richer in the Laplace mixed domains, one novel Bayesian impedance inversion approach in the complex Laplace mixed domains is established in this study to solve the model dependency problem. The derivation of a Laplace mixed-domain formula of the Robinson convolution is the first step in our work. With this formula, the Laplace seismic spectrum, the wavelet spectrum and time-domain reflectivity are joined together. Next, to improve inversion stability, the object inversion function accompanied by the initial constraint of the linear increment model is launched under a Bayesian framework. The likelihood function and prior probability distribution can be combined together by Bayesian formula to calculate the posterior probability distribution of subsurface parameters. By achieving the optimal solution corresponding to maximum posterior probability distribution, the low-frequency background of subsurface parameters can be obtained successfully. Then, with the regularization constraint of estimated low frequency in the Laplace mixed domains, multi-scale Bayesian inversion in the pure frequency domain is exploited to obtain the absolute model parameters. The effectiveness, anti-noise capability and lateral continuity of Laplace mixed-domain inversion are illustrated by synthetic tests. Furthermore, one field case in the east of China is discussed carefully with different input frequency components and different inversion algorithms. This provides adequate proof to illustrate the reliability improvement in low-frequency estimation and resolution enhancement of subsurface parameters, in comparison with conventional Bayesian inversion in the frequency domain.

Full waveform inversion with sparse structure constrained regularization

Abstract Full waveform inversion is a large-scale nonlinear and ill-posed problem. We consider applying the regularization technique for full waveform inversion with structure constraints. The structure information was extracted with difference operators with respect to model parameters. And then we establish an l p {l_{p}} - l q {l_{q}} -norm constrained minimization model for different choices of parameters p and q. To solve this large-scale optimization problem, a fast gradient method with projection onto convex set and a multiscale inversion strategy are addressed. The regularization parameter is estimated adaptively with respect to the frequency range of the data. Numerical experiments on a layered model and a benchmark SEG/EAGE overthrust model are performed to testify the validity of this proposed regularization scheme.

On sequential multiscale inversion and data assimilation

Multiscale approaches are very popular for example for solving partial differential equations and in many applied fields dealing with phenomena which take place on different levels of detail. The broad idea of a multiscale approach is to decompose your problem into different scales or levels and to use these decompositions either for constructing appropriate approximations or to solve smaller problems on each of these levels, leading to increased stability or increased efficiency. The idea of sequential multiscale is to first solve the problem in a large-scale subspace and then successively move to finer scale spaces. Our goal is to analyzethe sequential multiscale approach applied to an inversion or state estimation problem. We work in a generic setup given by a Hilbert space environment. We work out the analysis both for an unregularized and a regularized sequential multiscale inversion. In general the sequential multiscale approach is not equivalent to a full solution, but we show that under appropriate assumptions we obtain convergence of an iterative sequential multiscale version of the method. For the regularized case we develop a strategy to appropriately adapt the regularization when an iterative approach is taken. We demonstrate the validity of the iterative sequential multiscale approach by testing the method on an integral equation as it appears for atmospheric temperature retrieval from infrared satellite radiances.

A wavelet multiscale method for the inverse problem of a nonlinear convection–diffusion equation

This paper is concerned with the problem of identifying the diffusion parameters in a nonlinear convection–diffusion equation, which arises as the saturation equation in the fractional flow formulation of the two-phase porous media flow equations. The forward problem is discretized using finite-difference methods and the inverse problem is formulated as a minimization problem with regularization terms. In order to overcome disturbance of local minimum, a wavelet multiscale method is applied to solve this parameter identification inverse problem. This method works by decomposing the inverse problem into multiple scales with wavelet transform so that the original inverse problem is reformulated to a set of sub-inverse problems relying on scale variables, and successively solving these sub-inverse problems according to the size of scale from the smallest to the largest. The stable and fast regularized Gauss–Newton method is applied to each scale. Numerical simulations show that the proposed algorithm is widely convergent, computationally efficient, and has the anti-noise and de-noising abilities. • Permeability identification of a nonlinear convection–diffusion equation is studied. • The finite difference scheme for nonlinear convection–diffusion equation is given. • A wavelet multiscale method is proposed for parameter identification. • This method is one of multiscale inversion method. • Numerical simulations testify the effectiveness of the wavelet multiscale method.

Acoustic Waveform Inversion for Ocean Turbulence

AbstractThe seismic oceanography method is based on extracting and stacking the low-frequency acoustic energy scattered by the ocean heterogeneity. However, a good understanding on how this acoustic wavefield is affected by physical processes in the ocean is still lacking. In this work an acoustic waveform modeling and inversion method is developed and applied to both synthetic and real data. In the synthetic example, the temperature field is simulated as a homogeneous Gaussian isotropic random field with the Kolmogorov–Obukhov spectrum superimposed on a background stratified ocean structure. The presented full waveform inversion method is based on the ray-Born approximation. The synthetic seismograms computed using the ray-Born scattering method closely match the seismograms produced with a more computationally expensive finite-difference method. The efficient solution to the inverse problem is provided by the multiscale nonlinear inversion approach that is specifically stable with respect to noise. Full waveform inversion tests are performed using both the stationary and time-dependent sound speed models. These tests show that the method provides a reliable reconstruction of both the spatial sound speed variation and the theoretical spectrum due to fully developed turbulence. Finally, the inversion approach is applied to real seismic reflection data to determine the heterogeneous sound speed structure at the west Barents Sea continental margin in the northeast Atlantic. The obtained model illustrates in more detail the processes of diapycnal mixing near the continental slope.