Seismic waveform modelling and inversion with velocity-deviatoric stress-isotropic pressure formulation

2021 ◽  
Author(s):  
Yi Zhang ◽  
Luca de Siena ◽  
Alexey Stovas
Keyword(s):  
Waveform Inversion ◽  
Real Data ◽  
Deviatoric Stress ◽  
Model Parameters ◽  
Perfectly Matched Layers ◽  
Current Form ◽  
Anisotropic Models ◽  
Seismic Waveform ◽  
Adjoint State ◽  
Isotropic Pressure

<p>In waveform inversion, most of the existing adjoint-state methods are based on the second-order elastic wave equations subject to displacement. The implementation of the acoustic-elastic coupling problem and free-surface in this formulation is not explicit, especially for arbitrary boundaries. The formulation of velocity-deviatoric stress-isotropic pressure can tackle the above issue. We firstly review the difference between velocity stress equations and velocity-deviatoric stress-isotropic pressure equations. Then the adjoint state of the velocity-stress equations are derived, decomposing stresses into their deviatoric and isotropic parts. To simulate the unbounded wavefield, perfectly matched layers (PML) are embedded into the system of equations. It is modified for cheap computation, which avoids PML-related memory variables by applying complex coordinate stretch to three Cartesian axes in parallel.</p><p>A 3D velocity-deviatoric stress-isotropic stress formulation is implemented with the staggered finite-difference method for several synthetic models (including anisotropic models). And inversions are then performed to reconstruct the model parameters, which is followed by a sensitivity analysis.</p><p>This method has the potential to be used with real data, both for active and passive seismics. However, in its current form, since it does not treat fluid/anisotropic solid interfaces correctly, it is limited to fluid or isotropic solid problems.</p>

A new parameter set for anisotropic multiparameter full-waveform inversion and application to a North Sea data set

Geophysics ◽  
2016 ◽  
Vol 81 (4) ◽  
pp. U25-U38 ◽  
Author(s):  
Nuno V. da Silva ◽  
Andrew Ratcliffe ◽  
Vetle Vinje ◽  
Graham Conroy
Keyword(s):  
North Sea ◽  
Waveform Inversion ◽  
Real Data ◽  
Model Parameters ◽  
Full Waveform Inversion ◽  
Radiation Patterns ◽  
Data Set ◽  
Full Waveform ◽  
Seismic Image ◽  
Central North Sea

Parameterization lies at the center of anisotropic full-waveform inversion (FWI) with multiparameter updates. This is because FWI aims to update the long and short wavelengths of the perturbations. Thus, it is important that the parameterization accommodates this. Recently, there has been an intensive effort to determine the optimal parameterization, centering the fundamental discussion mainly on the analysis of radiation patterns for each one of these parameterizations, and aiming to determine which is best suited for multiparameter inversion. We have developed a new parameterization in the scope of FWI, based on the concept of kinematically equivalent media, as originally proposed in other areas of seismic data analysis. Our analysis is also based on radiation patterns, as well as the relation between the perturbation of this set of parameters and perturbation in traveltime. The radiation pattern reveals that this parameterization combines some of the characteristics of parameterizations with one velocity and two Thomsen’s parameters and parameterizations using two velocities and one Thomsen’s parameter. The study of perturbation of traveltime with perturbation of model parameters shows that the new parameterization is less ambiguous when relating these quantities in comparison with other more commonly used parameterizations. We have concluded that our new parameterization is well-suited for inverting diving waves, which are of paramount importance to carry out practical FWI successfully. We have demonstrated that the new parameterization produces good inversion results with synthetic and real data examples. In the latter case of the real data example from the Central North Sea, the inverted models show good agreement with the geologic structures, leading to an improvement of the seismic image and flatness of the common image gathers.

FULL-WAVEFORM INVERSION WITH SOURCE AND RECEIVER COUPLING EFFECTS CORRECTION

Geophysics ◽  
2021 ◽  
pp. 1-37
Author(s):  
Xinhai Hu ◽  
Wei Guoqi ◽  
Jianyong Song ◽  
Zhifang Yang ◽  
Minghui Lu ◽  
...  
Keyword(s):  
Waveform Inversion ◽  
Nonlinear Least Squares ◽  
Real Data ◽  
Model Parameters ◽  
Full Waveform Inversion ◽  
Linear Inversion ◽  
Near Surface ◽  
Model Parameter ◽  
Full Waveform ◽  
Coupling Factors

Coupling factors of sources and receivers vary dramatically due to the strong heterogeneity of near surface, which are as important as the model parameters for the inversion success. We propose a full waveform inversion (FWI) scheme that corrects for variable coupling factors while updating the model parameter. A linear inversion is embedded into the scheme to estimate the source and receiver factors and compute the amplitude weights according to the acquisition geometry. After the weights are introduced in the objective function, the inversion falls into the category of separable nonlinear least-squares problems. Hence, we could use the variable projection technique widely used in source estimation problem to invert the model parameter without the knowledge of source and receiver factors. The efficacy of the inversion scheme is demonstrated with two synthetic examples and one real data test.

Simultaneous determination of time delays and stacking weights in seismic array beamforming

Geophysics ◽  
1995 ◽  
Vol 60 (2) ◽  
pp. 491-502 ◽  
Author(s):  
Weijian Mao ◽  
David Gubbins
Keyword(s):  
Conjugate Gradient ◽  
Time Delays ◽  
Random Noise ◽  
Waveform Inversion ◽  
Real Data ◽  
Seismic Array ◽  
Model Parameters ◽  
Static Corrections ◽  
Array Beamforming ◽  
Value Decomposition

An algorithm for the estimation of time delays and weights in an arbitrary‐single or three‐component seismic array is developed by the use of a linearized waveform inversion technique. This algorithm differs from conventional crosscorrelation methods in its ability to simultaneously obtain time delays and weights by minimizing residuals of all possible waveform fittings, and by its robustness in the presence of high random noise levels and local geological scattering. There are N stations in an array, and for each station, a beam is formed by a weighted linear combination of the remaining (N − 1) seismic traces. The time delays and weights are model parameters to be found by minimizing the sum of N objective functions. Two optimization algorithms for solving the least‐squares problem, singular‐value decomposition and conjugate gradient, are compared, and the conjugate gradient method is found to be satisfactory and faster for large arrays. The algorithm was tested using synthetic array data with high noise, real data from shots in a borehole to a linear array on land, and Ms 6.7 earthquake data recorded with a broadband three‐component array. The success with synthetic and real data shows the algorithm to be useful for seismic data stacking, residual static corrections, and phase picking when the data quality is poor.

scholarly journals First-arrival traveltime tomography for anisotropic media using the adjoint-state method

Geophysics ◽  
2016 ◽  
Vol 81 (4) ◽  
pp. R147-R155 ◽  
Author(s):  
Umair bin Waheed ◽  
Garret Flagg ◽  
Can Evren Yarman
Keyword(s):  
Waveform Inversion ◽  
Surface Model ◽  
Model Parameters ◽  
Mathematical Framework ◽  
Near Surface ◽  
Traveltime Tomography ◽  
Adjoint State ◽  
First Arrival ◽  
State Method ◽  
Adjoint State Method

Traveltime tomography using transmission data has been widely used for static corrections and for obtaining near-surface models for seismic depth imaging. More recently, it is also being used to build initial models for full-waveform inversion. The classic traveltime tomography approach based on ray tracing has difficulties in handling large data sets arising from current seismic acquisition surveys. Some of these difficulties can be addressed using the adjoint-state method, due to its low memory requirement and numerical efficiency. By coupling the gradient computation to nonlinear optimization, it avoids the need for explicit computation of the Fréchet derivative matrix. Furthermore, its cost is equivalent to twice the solution of the forward-modeling problem, irrespective of the size of the input data. The presence of anisotropy in the subsurface has been well established during the past few decades. The improved seismic images obtained by incorporating anisotropy into the seismic processing workflow justify the effort. However, previous literature on the adjoint-state method has only addressed the isotropic approximation of the subsurface. We have extended the adjoint-state technique for first-arrival traveltime tomography to vertical transversely isotropic (VTI) media. Because [Formula: see text] is weakly resolvable from surface seismic alone, we have developed the mathematical framework and procedure to invert for [Formula: see text] and [Formula: see text]. Our numerical tests on the VTI SEAM model demonstrate the ability of the algorithm to invert for near-surface model parameters and reveal the accuracy achievable by the algorithm.

On the computation of the Fréchet derivatives for seismic waveform inversion in 3D general anisotropic, heterogeneous media

Geophysics ◽  
2009 ◽  
Vol 74 (5) ◽  
pp. WB153-WB163 ◽  
Author(s):  
Bing Zhou ◽  
Stewart Greenhalgh
Keyword(s):  
Anisotropic Medium ◽  
Heterogeneous Media ◽  
Jacobian Matrix ◽  
Displacement Vector ◽  
Waveform Inversion ◽  
Model Parameters ◽  
Seismic Waveform ◽  
Model Parameterization ◽  
Frechet Derivatives ◽  
Fréchet Derivatives

We present a perturbation method and a matrix method for formulating the explicit Fréchet derivatives for seismic body-wave waveform inversion in 3D general anisotropic, heterogeneous media. Theoretically, the two methods yield the same explicit formula valid for any class of anisotropy and are completely equivalent if the model parameterization in the inversion is the same as that used in the discretization scheme (unstructured or structured mesh) for forward modeling. Explicit formulas allow various model parameterization schemes that try to match the resolution capability of the data and possibly reduce the dimensions of the Jacobian matrix. Based on the general expressions, relevant formulas for isotropic and 2.5D and 3D tilted transversely isotropic (TTI) media are derived. Two computational schemes, constant-point and constant-block parameterization, offer effective and efficient means of forming the Jacobian matrix from the explicit Fréchet derivatives. The sensitivity patterns of the displacement vector to the independent model parameters in a weakly anisotropic medium clearly convey the imaging capability possible with seismic waveform inversion in such an anisotropic medium.

Elastic full-waveform inversion for VTI media: Methodology and sensitivity analysis

Geophysics ◽  
2016 ◽  
Vol 81 (2) ◽  
pp. C53-C68 ◽  
Author(s):  
Nishant Kamath ◽  
Ilya Tsvankin
Keyword(s):  
Objective Function ◽  
Waveform Inversion ◽  
Transversely Isotropic ◽  
Model Parameters ◽  
Valuable Insight ◽  
Full Waveform Inversion ◽  
Full Waveform ◽  
Adjoint State ◽  
P And Sv Waves ◽  
Vti Media

Most existing implementations of full-waveform inversion (FWI) are limited to acoustic approximations. In this paper, we present an algorithm for time-domain elastic FWI in laterally heterogeneous VTI (transversely isotropic with a vertical symmetry axis) media. The adjoint-state method is employed to derive the gradients of the objective function with respect to the stiffness coefficients and then to a chosen set of VTI parameters. To test the algorithm, we introduce Gaussian anomalies in the Thomsen parameters of a homogeneous VTI medium and perform 2D FWI of multicomponent transmission data for two different model parameterizations. To analyze the sensitivity of the objective function to the model parameters, the Fréchet kernel of FWI is obtained by linearizing the elastic wave equation using the Born approximation and employing the asymptotic Green’s function. The amplitude of the kernel (“radiation pattern”) yields the angle-dependent energy scattered by a perturbation in a certain model parameter. Then we convert the general expressions into simple approximations for the radiation patterns of P- and SV-waves in VTI media. These analytic developments provide valuable insight into the potential of multicomponent elastic FWI and help explain the numerical results for models with Gaussian anomalies in the VTI parameters.

Multiparameter estimation with acoustic vertical transverse isotropic full-waveform inversion of surface seismic data

2016 ◽  
Vol 4 (4) ◽  
pp. SU1-SU16 ◽  
Author(s):  
Xin Cheng ◽  
Kun Jiao ◽  
Dong Sun ◽  
Denes Vigh
Keyword(s):  
Vertical Velocity ◽  
Seismic Data ◽  
Joint Inversion ◽  
Waveform Inversion ◽  
Real Data ◽  
Vertical Axis ◽  
Model Parameters ◽  
Full Waveform Inversion ◽  
Full Waveform ◽  
Wavefield Simulation

Obtaining accurate depth-migrated images demands an anisotropic representation of the earth. As a prominent tool for building high-resolution earth models, full-waveform inversion (FWI) therefore must not only account for anisotropy during wavefield simulation but also reconstruct the anisotropy fields. We have developed an inversion strategy to perform acoustic multiparameter FWI of surface seismic data in transversely isotropic media with a vertical axis of symmetry (VTI). During the early era of FWI practice, most studies only invert for the most dominant parameter, that is, the vertical velocity, and the rest of the model parameters are either ignored or kept constant. Recently, more and more emphases focus on inverting for more parameters, such as for the vertical velocity and the anisotropy fields; these are referred to as multiparameter inversion. Due to the dominant influence of the vertical velocity on the kinematics of surface seismic data, we have developed a hierarchical approach to invert for the vertical velocity first, but we kept the anisotropy fields unchanged and only switched to joint inversion of the vertical velocity and the anisotropy fields when the inversion for the vertical velocity approaches convergence. In addition, we have illustrated the necessity of incorporating the diving and reflection energy during inversion to mitigate the nonuniqueness of the solutions caused by the coupling between the vertical velocity and the anisotropy fields. We also demonstrate the success of our method for VTI FWI using synthetic and real data examples based on marine surface seismic acquisition. Our results show that incorporation of multiparameter anisotropy inversion produced better focused migration images.

Analysis of optimal transport and related misfit functions in full-waveform inversion

Geophysics ◽  
2018 ◽  
Vol 83 (1) ◽  
pp. A7-A12 ◽  
Author(s):  
Yunan Yang ◽  
Björn Engquist
Keyword(s):  
Optimal Transport ◽  
Waveform Inversion ◽  
Wasserstein Distance ◽  
Model Parameters ◽  
Full Waveform Inversion ◽  
Wasserstein Metric ◽  
Full Waveform ◽  
Adjoint State ◽  
State Method ◽  
Adjoint State Method

Full-waveform inversion has evolved into a powerful computational tool in seismic imaging. New misfit functions for matching simulated and measured data have recently been introduced to avoid the traditional lack of convergence due to cycle skipping. We have introduced the Wasserstein distance from optimal transport for computing the misfit, and several groups are currently further developing this technique. We evaluate three essential observations of this new metric with implication for future development. One is the discovery that trace-by-trace comparison with the quadratic Wasserstein metric works remarkably well together with the adjoint-state method. Another is the close connection between optimal transport-based misfits and integrated techniques with normalization as, for example, the normalized integration method. Finally, we study the convexity with respect to selected model parameters for different normalizations and remark on the effect of normalization on the convergence of the adjoint-state method.

scholarly journals Element-by-element parallel spectral-element methods for 3-D acoustic-wave-equation-based teleseismic wave modeling

2017 ◽  
Author(s):  
Shaolin Liu ◽  
Dinghui Yang ◽  
Xingpeng Dong ◽  
Qiancheng Liu ◽  
Yongchang Zheng
Keyword(s):  
Boundary Condition ◽  
Waveform Inversion ◽  
Spectral Element ◽  
Computational Effort ◽  
Model Parameters ◽  
Perfectly Matched Layers ◽  
Time Step ◽  
Absorbing Boundary ◽  
Adjoint Tomography ◽  
Increasing Demand

Abstract. The increasing demand for the high-resolution imaging of deep lithosphere structures requires the utilization of a teleseismic dataset for waveform inversion. The construction of an efficient algorithm for the teleseismic wavefield modeling is valuable for the calculation of misfit kernels or Fréchet derivatives when the teleseismic waveform is used for adjoint tomography. Here, we introduce an element-by-element parallel spectral-element method (EBE-SEM) for the efficient modeling of teleseismic wavefield propagation in a localized geology model. Under the assumption of the plane wave, the frequency-wavenumber (FK) technique is implemented to compute the boundary wavefield used for constructing the boundary condition of the teleseismic wave incidence. To reduce the memory required for the storage of the boundary wavefield for the incidence boundary condition, an economical strategy is introduced to store the boundary wavefield on the model boundary. The perfectly matched layers absorbing boundary condition (PML ABC) is formulated by the EBE-SEM to absorb the scattered wavefield from the model interior. The misfit kernel (derivatives of the waveform misfit with respect to model parameters) can be easily constructed without extra computational effort for the calculation of the element stiffness matrix per time step during the calculation of the adjoint wavefield. Three synthetic examples demonstrate the validity of EBE-SEM for use in teleseismic wavefield modeling and the misfit kernel calculation.

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