A Rayleigh Wave Globally Optimal Full Waveform Inversion Framework Based on GPU Parallel Computing

Conventional gradient-based full waveform inversion (FWI) is a local optimization, which is highly dependent on the initial model and prone to trapping in local minima. Globally optimal FWI that can overcome this limitation is particularly attractive, but is currently limited by the huge amount of calculation. In this paper, we propose a globally optimal FWI framework based on GPU parallel computing, which greatly improves the efficiency, and is expected to make globally optimal FWI more widely used. In this framework, we simplify and recombine the model parameters, and optimize the model iteratively. Each iteration contains hundreds of individuals, each individual is independent of the other, and each individual contains forward modeling and cost function calculation. The framework is suitable for a variety of globally optimal algorithms, and we test the framework with particle swarm optimization algorithm for example. Both the synthetic and field examples achieve good results, indicating the effectiveness of the framework. .

Time Domain Full Waveform Inversion Based on Gradient Preconditioning with an Angle-Dependent Weighting Factor

There are lots of low wavenumber noises in the gradients of time domain full waveform inversion(FWI),which can seriously reduce the accuracy and convergence speed of FWI.Thus,we introduce an angle-dependent weighting factor to precondition the gradients so as to suppress the low wavenumber noises when the multi-scale FWI is implemented in the high frequency.Model experiments show that the FWI based on the gradient preconditioning with an angle-dependent weighting factor has faster convergence speed and higher inversion accuracy than the conventional FWI.The tests on real marine seismic data show that this method can adapt to the FWI of field data,and provide high-precision velocity models for the actual data processing.

Three-dimensional frequency-domain full waveform inversion based on the nearly-analytic discrete method

The nearly analytic discrete(NAD)method is a kind of finite difference method with advantages of high accuracy and stability.Previous studies have investigated the NAD method for simulating wave propagation in the time-domain.This study applies the NAD method to solving three-dimensional(3D)acoustic wave equations in the frequency-domain.This forward modeling approach is then used as the“engine”for implementing 3D frequency-domain full waveform inversion(FWI).In the numerical modeling experiments,synthetic examples are first given to show the superiority of the NAD method in forward modeling compared with traditional finite difference methods.Synthetic 3D frequency-domain FWI experiments are then carried out to examine the effectiveness of the proposed methods.The inversion results show that the NAD method is more suitable than traditional methods,in terms of computational cost and stability,for 3D frequency-domain FWI,and represents an effective approach for inversion of subsurface model structures.

Time-Domain Full Waveform Inversion Using the Gradient Preconditioning Based on Transmitted Wave Energy

The gradient preconditioning approach based on seismic wave energy can effectively avoid the huge memory consumption of the gradient preconditioning algorithms based on the Hessian matrix. However, the accuracy of this approach is prone to be influ- enced by the energy of reflected waves. To tackle this problem, the paper proposes a new gradient preconditioning method based on the energy of transmitted waves. The approach scales the gradient through a precondition factor, which is calculated by the ‘ap- proximate transmission wavefield’ simulation based on the nonreflecting acoustic wave equation. The method requires no computing nor storing of the Hessian matrix and its inverse matrix. Furthermore, the proposed method can effectively eliminate the effects of geometric spreading and disproportionality in the gradient illumination. The results of model experiments show that the time-domain full waveform inversion (FWI) using the gradient preconditioning based on transmitted wave energy can achieve higher inversion accuracy for deep high-velocity bodies and their underlying strata in comparison with the one using the gradient preconditioning based on seismic wave energy. The field marine seismic data test shows that our proposed method is also highly applicable to the FWI of field marine seismic data.

Acoustic Based Crosshole Full Waveform Slowness Inversion in the Time Domain

We develop a new full waveform inversion (FWI) method for slowness with the crosshole data based on the acoustic wave equation in the time domain. The method combines the total variation (TV) regularization with the constrained optimization together which can inverse the slowness effectively. One advantage of slowness inversion is that there is no further approximation in the gradient derivation. Moreover, a new algorithm named the skip method for solving the constrained optimization problem is proposed. The TV regularization has good ability to inverse slowness at its discontinuities while the constrained optimization can keep the inversion converging in the right direction. Numerical computations both for noise free data and noisy data show the robustness and effectiveness of our method and good inversion results are yielded.

Wavelet packet envelope multi-scale full waveform inversion

Full waveform inversion( FWI) is an effective tool for constructing high resolution velocity models,but it is affected by a local minima problem. Without long offsets and low frequency data,it is difficult to apply the conventional multi-scale FWI to actual seismic data. In this study,the large offset and low frequency information are provided by the method of wavelet packet envelope for the conventional FWI. The gradient can be computed efficiently with the adjoint state method without any additional computational cost. Marmousi synthetic data is used to illustrate that,compared with Hilbert envelope-based FWI,wavelet packet envelope FWI can provide an adequately accurate model for the conventional FWI approach even when the initial model is far from the true model and the low-frequency data are missing.

Full waveform inversion based on improved MLQN method

Full waveform inversion( FWI) is a challenging data-fitting procedure between model wave field value and theoretical wave field value. The essence of FWI is an optimization problem,and therefore,it is important to study optimization method. The study is based on conventional Memoryless quasi-Newton( MLQN)method. Because the Conjugate Gradient method has ultra linear convergence,the authors propose a method by using Fletcher-Reeves( FR) conjugate gradient information to improve the search direction of the conventional MLQN method. The improved MLQN method not only includes the gradient information and model information,but also contains conjugate gradient information. And it does not increase the amount of calculation during every iterative process. Numerical experiment shows that compared with conventional MLQN method,the improved MLQN method can guarantee the computational efficiency and improve the inversion precision.

Full waveform inversion based on initial model built from envelope inversion

Full waveform inversion is a fitting process based on full seismic wave field simulation data using the full waveform information in seismic records and theoretically it is the ultimate goal of seismic inversion. However,there are many problems to be solved in practical application. Firstly,it is the strong nonlinear problem between the seismic wave field and inversion parameters; secondly,the lack of low-frequency information in seismic records. In this study,the envelope is used as objective function inversion to provide the inversion result for the multi-scale full waveform inversion as the initial model,solving the lack of low-frequency information in seismic records. Taking the envelope of seismic records as the objective function in combination of multi-scale full waveform inversion became a new inversion strategy,which naturally achieved the compensation of shortage of low-frequency information and inversion from low frequency to high frequency,reducing the non-linearity in the inversion process. The comparison of the result of full waveform inversion of the initial model built through envelope inversion with the result of the conventional multi-scale full waveform inversion indicates the effectiveness of envelope inversion for the recovery of low-frequency information in seismic records.

Study of frequency domain full waveform inversion based on Huber norm and L-BFGS algorithm

Full waveform inversion( FWI) is a high resolution inversion method,which can reveal detailed information of the structure and lithology under complex geological background. It is limited by many kinds of noises when the method applied to the real seismic data. Based on Huber function criterion,the objective function combinates the anti-noise of L1 norm and the stability of L2 norm in theory,the authors derive the gradient formula of the Huber function by using L-BFGS algorithm for FWI. The new method is proved by synthetic seismic data with the Gaussian noise and the impulse noise. Numerical test results show that L-BFGS algorithm is applied to the frequency domain FWI with the convergence speed and high calculation accuracy,and can effectively reduce computer memory usage; and the Huber function is more robust and stable than L2 norm even with the noises.

Study of full waveform inversion based on L-BFGS algorithm

Full waveform inversion is mainly used to obtain high resolution velocity models of subsurface. The size of full waveform inversion will lead to a gigantic computation cost. Under the available computer resource and the limitation of full waveform inversion,the authors propose L-BFGS algorithm as the optimization method to solve this problem. In order to demonstrate the flexibility of the method,three different numerical experiments have been done to analyze the properties of full waveform inversion based on L-BFGS.

Elastic Full Waveform Inversion Based on the Trust Region Strategy

In this paper, we investigate the elastic wave full-waveform inversion (FWI) based on the trust region method. The FWI is an optimization problem of minimizing the misfit between the observed data and simulated data. Usually, the line search method is used to update the model parameters iteratively. The line search method generates a search direction first and then finds a suitable step length along the direction. In the trust region method, it defines a trial step length within a certain neighborhood of the current iterate point and then solves a trust region subproblem. The theoretical methods for the trust region FWI with the Newton type method are described. The algorithms for the truncated Newton method with the line search strategy and for the Gauss-Newton method with the trust region strategy are presented. Numerical computations of FWI for the Marmousi model by the L-BFGS method, the Gauss-Newton method and the truncated Newton method are completed. The comparisons between the line search strategy and the trust region strategy are given and show that the trust region method is more efficient than the line search method and both the Gauss-Newton and truncated Newton methods are more accurate than the L-BFGS method.

Evaluation of Multi-Scale Full Waveform Inversion with Marine Vertical Cable Data

Seismic illumination plays an important role in subsurface imaging. A better image can be expected either through optimizing acquisition geometry or introducing more advanced seismic mi- gration and/or tomographic inversion methods involving illumination compensation. Vertical cable survey is a potential replacement of traditional marine seismic survey for its flexibility and data quality. Conventional vertical cable data processing requires separation of primaries and multiples before migration. We proposed to use multi-scale full waveform inversion （FWI） to improve illumination coverage of vertical cable survey. A deep water velocity model is built to test the capability of multi-scale FWI in detecting low velocity anomalies below seabed. Synthetic results show that multi-scale FWI is an effective model building tool in deep-water exploration. Geometry optimization through target ori- ented illumination analysis and multi-scale FWI may help to mitigate the risks of vertical cable survey. The combination of multi-scale FWI, low-frequency data and multi-vertical-cable acquisition system may provide both high resolution and high fidelity subsurface models.

Layer-Stripping Full Waveform Inversion with Damped Seismic Reflection Data

Full waveform inversion（FWI） directly minimizes errors between synthetic and observed data.For the surface acquisition geometry,reflections generated from deep reflectors are sensitive to overburden structure,so it is reasonable to update the macro velocity model in a top-to-bottom manner.For models dominated by horizontally layered structures,combination of offset/time weighting and constant update depth control（CUDC） is sufficient for layer-stripping FWI.CUDC requires ray tracing to determine reflection traveltimes at a constant depth.As model complexity increases,the multi-path effects will have to be considered.We developed a new layer-stripping FWI method utilizing damped seismic reflection data,which does not need CUDC and ray tracing.Numerical examples show that effective update depth（EUD） can be controlled by damping constants even in complex regions and the inversion result is more accurate than conventional methods.

Full Waveform Inversion Method for Horizontally Inhomogeneous Stratified Media

Full waveform inversion method is an approach to grasp the physical property parameters of un- derground media in geotechnical nondestructive detection and testing field. Using finite-diference time domain(FDTD) method for elastic wave equations, the full-wave field in horizontally inhomogeneous stratified media for elastic wave logging was calculated. A numerical 2D model with three layers was computed for elastic wave propagation in horizontally inhomogeneous media. The full waveform inversion method was verified to be feasible for evaluating elastic parameters in lateral inhomogeneous stratified media and showed well accuracy and conver- gence. It was shown that the time cost of inversion had certain dependence on the choice of starting initial model. Furthermore, this method was used in the detection of nonuniform grouting in the construction of immersed tube tunnel. The distribution of nonuniform grouting was clearly evaluated by the S-wave velocity profile of grouted mortar base below the tunnel floor.

Quantitative ultrasound brain imaging with multiscale deconvolutional waveform inversion

High-resolution images of human brain are critical for monitoring the neurological conditions in a portable and safe manner.Sound speed mapping of brain tissues provides unique information for such a purpose.In addition,it is particularly important for building digital human acoustic models,which form a reference for future ultrasound research.Conventional ultrasound modalities can hardly image the human brain at high spatial resolution inside the skull due to the strong impedance contrast between hard tissue and soft tissue.We carry out numerical experiments to demonstrate that the time-domain waveform inversion technique,originating from the geophysics community,is promising to deliver quantitative images of human brains within the skull at a sub-millimeter level by using ultra-sound signals.The successful implementation of such an approach to brain imaging requires the following items:signals of sub-megahertz frequencies transmitting across the inside of skull,an accurate numerical wave equation solver simulating the wave propagation,and well-designed inversion schemes to reconstruct the physical parameters of targeted model based on the optimization theory.Here we propose an innovative modality of multiscale deconvolutional waveform inversion that improves ultrasound imaging resolution,by evaluating the similarity between synthetic data and observed data through using limited length Wiener filter.We implement the proposed approach to iteratively update the parametric models of the human brain.The quantitative imaging method paves the way for building the accurate acoustic brain model to diagnose associated diseases,in a potentially more portable,more dynamic and safer way than magnetic resonance imaging and x-ray computed tomography.

Full waveform inversion based on the ensemble Kalman filter method using uniform sampling without replacement

Full waveform inversion(FWI) has been increasingly more and more important in seismology to better understand the interior structure of the Earth. FWI, by taking advantage of both the traveltime and amplitude in the data, provides high-resolution model parameters of the earth which can produce images with high resolution. However, this inversion method conventionally suffers from non-uniqueness due to many local minima of the objective function and large computing costs. In this study, we propose a new FWI method in a semi-random framework by integrating the ensemble Kalman filter and uniform sampling without replacement. Numerical results demonstrate that the new method can achieve highresolution results and a wider convergence domain. Accordingly, the new method overcomes the disadvantage of conventional FWIs that depend strongly on the initial model.

Reflection full waveform inversion

Because of the combination of optimization algorithms and full wave equations, full-waveform inversion(FWI) has become the frontier of the study of seismic exploration and is gradually becoming one of the essential tools for obtaining the Earth interior information. However, the application of conventional FWI to pure reflection data in the absence of a highly accurate starting velocity model is difficult. Compared to other types of seismic waves, reflections carry the information of the deep part of the subsurface. Reflection FWI, therefore, is able to improve the accuracy of imaging the Earth interior further. Here, we demonstrate a means of achieving this successfully by interleaving least-squares RTM with a version of reflection FWI in which the tomographic gradient that is required to update the background macro-model is separated from the reflectivity gradient using the Born approximation during forward modeling. This provides a good update to the macro-model. This approach is then followed by conventional FWI to obtain a final high-fidelity high-resolution result from a poor starting model using only reflection data.Further analysis reveals the high-resolution result is achieved due to a deconvolution imaging condition implicitly used by FWI.

Q full-waveform inversion based on the viscoacoustic equation

Presently, most full-waveform inversion methods are developed for elastic media and ignore the effect of attenuation. The calculation of the quality factor Q is based on velocity parameter inversion under the assumption of a given Q-model that is obtained by tomographic inversion. However, the resolution of the latter is low and cannot reflect the amplitude attenuation and phase distortion during wave propagation in viscoelastic media. Thus, a Q waveform inversion method is proposed. First, we use standard linear body theory to describe attenuation and then we derive the simplified viscoacoustic equation that characterizes amplitude attenuation and phase distortion. In comparison with conventional equations, the simplifi ed equation involves no memory variables and therefore requires less memory during computation. Moreover, the implementations of the attenuation compensation are easier. The adjoint equation and the corresponding gradient equation with respect to either L2-norm or the zero-lag cross-correlation objective function are then derived and the regularization strategy for overcoming the instability during numerical solution of the adjoint equation is proposed. The Q waveform inversion is developed using the limited-memory Broyden–Fletcher– Goldfarb–Shanno (L-BFGS) iteration method for known velocity. To alleviate the dependence of the waveform inversion on the initial model and overcome cycle skipping to some extent, we adopt multiscale analysis. Furthermore, anti-noise property and double-parameter inversion are assessed based on the results of numerical modeling.

Full-waveform Velocity Inversion Based on the Acoustic Wave Equation

Full-waveform velocity inversion based on the acoustic wave equation in the time domain is investigated in this paper. The inversion is the iterative minimization of the misfit between observed data and synthetic data obtained by a numerical solution of the wave equation. Two inversion algorithms in combination with the CG method and the BFGS method are described respectively. Numerical computations for two models including the benchmark Marmousi model with complex structure are implemented. The inversion results show that the BFGS-based algorithm behaves better in inversion than the CG-based algorithm does. Moreover, the good inversion result for Marmousi model with the BFGS-based algorithm suggests the quasi-Newton methods can provide an important tool for large-scale velocity inversion. More computations demonstrate the correctness and effectives of our inversion algorithms and code.

全波形发射电流激发下地面瞬变电磁法磁场特征

在过去四十年,大部分关于瞬变电磁法的三维建模和反演研究都聚焦于磁场时间导数,而对于磁场数据研究并不多见.如今,随着超导量子干涉仪(SQUID)技术日益成熟,基于磁场数据的瞬变电磁法探测案例逐渐增多.基于矢量有限单元法,采用频谱法和时步法分别实现了适用于瞬变电磁法磁场数据的三维正演.以低阻覆盖层下的良导体和均匀半空间中板状良导体模型为例,研究了全波形发射电流激发的瞬变电磁法磁场及其时间导数特征,并探讨了二者对深部良导体的探测能力.正演计算结果表明:由于受发射电流全波形效应影响,晚延时磁场随目标体电导率增大而先增强、后减弱,然而即使不考虑全波形效应,磁场时间导数也呈现上述规律;在高阻和低阻围岩情形中,磁场比其时间导数对深部良导体反映更加灵敏,并且晚延时磁场比其时间导数更不易受地磁场背景噪声干扰,说明磁场比其时间导数更适合于深部良导体探测.需要注意的是,发射电流全波形效应降低了瞬变电磁法对深部良导体的识别能力,建议在瞬变电磁法对块状或脉状金属硫化物矿床探测时,尽量采用低基频发射电流波形.