scholarly journals Time Domain Waveform Inversion for theQModel Based on the First-Order Viscoacoustic Wave Equations

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Guowei Zhang ◽  
Jinghuai Gao
Keyword(s):  
Objective Function ◽  
Wave Equations ◽  
Seismic Waves ◽  
Waveform Inversion ◽  
Model Parameters ◽  
Model Inversion ◽  
First Order ◽  
Standard Linear Solid ◽  
Standard Linear ◽  
Absorptive Property

Propagating seismic waves are dispersed and attenuated in the subsurface due to the conversion of elastic energy into heat. The absorptive property of a medium can be described by the quality factorQ. In this study, the first-order pressure-velocity viscoacoustic wave equations based on the standard linear solid model are used to incorporate the effect ofQ. For theQmodel inversion, an iterative procedure is then proposed by minimizing an objective function that measures the misfit energy between the observed data and the modeled data. The adjoint method is applied to derive the gradients of the objective function with respect to the model parameters, that is, bulk modulus, density, andQ-related parameterτ. Numerical tests on the crosswell recording geometry indicate the feasibility of the proposed approach for theQanomaly estimation.

Experimental identification of the material standard linear solid model parameters by means of dynamical measurements

2021 ◽  
pp. 107754632110371
Author(s):  
Stefano Amadori ◽  
Giuseppe Catania
Keyword(s):  
Frequency Domain ◽  
Solid Material ◽  
Model Parameters ◽  
Solid Model ◽  
Material Parameters ◽  
Experimental Identification ◽  
Standard Linear Solid Model ◽  
Standard Linear Solid ◽  
Standard Linear ◽  
Material Standard

A procedure for the experimental identification of the material standard linear solid model parameters by means of dynamic mechanical analysis test instrument measurements is presented. Since the standard linear solid material stress–strain functional D( ω) relationship in the frequency domain formally depends on the standard linear solid material parameters, a procedure able to identify these parameters from test measurement estimates is proposed in this work. Nevertheless, a critical, nonlinear and non-parametric approach is to be followed since the number of the material standard linear solid block components is generally unknown, and the material D( ω) shows a highly nonlinear dependency on the unknown standard linear solid material parameters. For these reasons, measurement and test model noise is expected to strongly influence the accuracy of the identification results. A multi-step procedure is presented, consisting first in the non-parametric identification of a frequency dependent, two degrees of freedom model instrument frame by means of a polynomial rational function, where polynomial order and parameters, such as polynomial coefficients and pole-residue couples, are optimally identified by means of an algebraic numerical technique and of an iterative stabilization procedure. Another procedure able to identify the material D( ω) polynomial rational functional relationship in the frequency domain is also proposed, taking into account the dynamic contribution of the instrument frame, of the inertial contribution of the distributed mass of the beam and of the lumped mass of the instrument force measuring system. An effective procedure, able to identify the standard linear solid material model parameters in the time domain from the identified material physical poles, is finally proposed. Some application examples, concerning the identification of the standard linear solid model of a known material and of an unknown composite material, are shown and discussed as well.

scholarly journals Full-waveform inversion, Part 3: Optimization

2018 ◽  
Vol 37 (2) ◽  
pp. 142-145 ◽  
Author(s):  
Philipp Witte ◽  
Mathias Louboutin ◽  
Keegan Lensink ◽  
Michael Lange ◽  
Navjot Kukreja ◽  
...  
Keyword(s):  
Objective Function ◽  
Wave Equations ◽  
Waveform Inversion ◽  
Full Waveform Inversion ◽  
Acoustic Wave Equation ◽  
Automated Generation ◽  
Domain Specific ◽  
Full Waveform ◽  
Adjoint State ◽  
Adjoint State Method

This tutorial is the third part of a full-waveform inversion (FWI) tutorial series with a step-by-step walkthrough of setting up forward and adjoint wave equations and building a basic FWI inversion framework. For discretizing and solving wave equations, we use Devito ( http://www.opesci.org/devito-public ), a Python-based domain-specific language for automated generation of finite-difference code ( Lange et al., 2016 ). The first two parts of this tutorial ( Louboutin et al., 2017 , 2018 ) demonstrated how to solve the acoustic wave equation for modeling seismic shot records and how to compute the gradient of the FWI objective function using the adjoint-state method. With these two key ingredients, we will now build an inversion framework that can be used to minimize the FWI least-squares objective function.

The generalized standard-linear-solid model and the corresponding viscoacoustic wave equations revisited

2019 ◽  
Author(s):  
Qi Hao ◽  
Stewart Greenhalgh
Keyword(s):  
Frequency Domain ◽  
Relaxation Function ◽  
Wave Equations ◽  
Complex Modulus ◽  
Spectral Element ◽  
Solid Model ◽  
Standard Linear Solid Model ◽  
Standard Linear Solid ◽  
Standard Linear ◽  
Forward And Inverse Modeling

Summary The generalized standard-linear-solid model, also called the Zener model, is widely used in viscoacoustic/viscoelastic wavefield forward and inverse modeling, because the wave equations in this model can be written in differential equation form, which can be solved efficiently by time-domain numerical methods such as finite difference method, spectral element method, etc. For this model, however, two different expressions for the relaxation function (or complex modulus) appear in the literature somewhat confusingly. In addition to this confusion, the time- and frequency-domain versions of the wave equations for the generalized standard-linear-solid model are scattered throughout the literature. Here, we revisit the generalized standard-linear-solid model and seek to overcome the confusion concerning the expression for the relaxation function (or modulus). We present a unified approach to derive the viscoacoustic wave equations. We start with the time- and frequency-domain formulations separately to derive two sets of viscoacoustic wave equations. All these viscoacoustic wave equations are expressed in a simple and compact form. The two sets of viscoacoustic wave equations are equivalent to each other. The proposed method to derive the appropriate viscoacoustic wave equations can be extended to derive wave equations for other dissipative media.

scholarly journals Land seismic multiparameter full waveform inversion in elastic VTI media by simultaneously interpreting body waves and surface waves with an optimal transport based objective function

2019 ◽  
Vol 219 (3) ◽  
pp. 1970-1988 ◽  
Author(s):  
Weiguang He ◽  
Romain Brossier ◽  
Ludovic Métivier ◽  
René-Édouard Plessix
Keyword(s):  
Objective Function ◽  
Surface Waves ◽  
Least Squares ◽  
Optimal Transport ◽  
Waveform Inversion ◽  
Body Waves ◽  
Model Parameters ◽  
Full Waveform Inversion ◽  
Initial Model ◽  
Full Waveform

SUMMARY Land seismic multiparameter full waveform inversion in anisotropic media is challenging because of high medium contrasts and surface waves. With a data-residual least-squares objective function, the surface wave energy usually masks the body waves and the gradient of the objective function exhibits high values in the very shallow depths preventing from recovering the deeper part of the earth model parameters. The optimal transport objective function, coupled with a Gaussian time-windowing strategy, allows to overcome this issue by more focusing on phase shifts and by balancing the contributions of the different events in the adjoint-source and the gradients. We first illustrate the advantages of the optimal transport function with respect to the least-squares one, with two realistic examples. We then discuss a vertical transverse isotropic (VTI) example starting from a quasi 1-D isotropic initial model. Despite some cycle-skipping issues in the initial model, the inversion based on the windowed optimal transport approach converges. Both the near-surface complexities and the variations at depth are recovered.

Applied 3D salt body reconstruction using shape optimization with level sets

Geophysics ◽  
2020 ◽  
Vol 85 (5) ◽  
pp. R437-R446
Author(s):  
Taylor Dahlke ◽  
Biondo Biondi ◽  
Robert Clapp
Keyword(s):  
Objective Function ◽  
Level Sets ◽  
Oil And Gas ◽  
Waveform Inversion ◽  
Complex Salt ◽  
Model Parameters ◽  
Gradient System ◽  
Implicit Surface ◽  
Data Set ◽  
Velocity Models

As oil and gas extraction becomes more advanced, deep-water exploration becomes increasingly focused on imaging near or under complex salt geology, which necessitates detailed velocity models with strong contrast interfaces. These interfaces can be elegantly tracked using the level sets of an implicit surface. One can invert for the velocity model that best fits the recorded data in a full-waveform inversion (FWI) style objective function by reparameterizing the model in terms of an implicit surface representation of the salt interface. With this parameterization of the FWI objective function, we find the Hessian and solve a conjugate gradient system for the Newton step at every nonlinear iteration. We sparsify the representation of the implicit surface using radial basis functions, which can hasten convergence of the inner inversion by reducing the number of model parameters. We have developed a guided inversion approach that embeds information about the certainty of different salt boundary regions by the initialization of the implicit surface slope at the salt interface. This can help guide the inversion away from perceived local minima. The results of testing this inversion workflow on a 3D Gulf of Mexico data set show that it can be a useful tool for refining salt models because the initial and final seismic images show clearer and more consistent features below the updated salt area.

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.

scholarly journals Nearly constant Q models of the generalized standard linear solid type and the corresponding wave equations

Geophysics ◽  
2021 ◽  
pp. 1-125
Author(s):  
Qi Hao ◽  
Stewart Greenhalgh
Keyword(s):  
Quality Factor ◽  
Time Domain ◽  
Inverse Modeling ◽  
Wave Equations ◽  
New Models ◽  
Q Model ◽  
Standard Linear Solid ◽  
Standard Linear ◽  
Forward And Inverse Modeling ◽  
Solid Type

Time-domain seismic forward and inverse modeling for a dissipative medium is a vital research topic to investigate the attenuation structure of the Earth. Constant Q, also called frequency independence of the quality factor, is a common assumption for seismic Q inversion. We propose the first- and second-order nearly constant Q dissipative models of the generalized standard linear solid type, using a novel Q-independent weighting function approach. The two new models, which originate from the Kolsky model (a nearly constant Q model) and the Kjartansson model (an exactly constant Q model), result in the corresponding wave equations in differential form. Even for extremely strong attenuation (e.g., Q = 5), the quality factor and phase velocity for the two new models are close to those for the Kolsky and Kjartansson models, in a frequency range of interest. The wave equations for the two new models involve explicitly a specified Q parameter and have compact and simple forms. We provide a novel perspective on how to build a nearly constant Q dissipative model which is beneficial for time-domain large scale wavefield forward and inverse modeling. This perspective could also help obtain other dissipative models with similar advantages. We also discuss the extension beyond viscoacousticity and other related issues, for example, extending the two new models to viscoelastic anisotropy.

scholarly journals Viscoacoustic anisotropic wave equations

Geophysics ◽  
2019 ◽  
Vol 84 (6) ◽  
pp. C323-C337 ◽  
Author(s):  
Qi Hao ◽  
Tariq Alkhalifah
Keyword(s):  
Wave Equation ◽  
Point Source ◽  
Wave Equations ◽  
Transverse Isotropy ◽  
Standard Linear Solid ◽  
Standard Linear ◽  
Forward And Inverse Modeling ◽  
Asymptotic Point ◽  
Attenuation Anisotropy ◽  
And Inversion

The wave equation plays a central role in seismic modeling, processing, imaging and inversion. Incorporating attenuation anisotropy into the acoustic anisotropic wave equations provides a choice for acoustic forward and inverse modeling in attenuating anisotropic media. However, the existing viscoacoustic anisotropic wave equations are obtained for a specified viscoacoustic model. We have developed a relatively general representation of the scalar and vector viscoacoustic wave equations for orthorhombic anisotropy. We also obtain the viscoacoustic wave equations for transverse isotropy as a special case. The viscoacoustic orthorhombic wave equations are flexible for multiple viscoacoustic models. We take into account the classic visocoacoustic models such as the Kelvin-Voigt, Maxwell, standard-linear-solid and Kjartansson models, and we derive the corresponding viscoacoustic wave equations in differential form. To analyze the wave propagation in viscoacoustic models, we derive the asymptotic point-source solution of the scalar wave equation. Numerical examples indicate a comparison of the acoustic waveforms excited by a point source in the viscoacoustic orthorhombic models and the corresponding nonattenuating model, and the effect of the attenuation anisotropy on the acoustic waveforms.

scholarly journals Full-waveform inversion in acoustic orthorhombic media and application to a North Sea data set

Geophysics ◽  
2018 ◽  
Vol 83 (5) ◽  
pp. C179-C193 ◽  
Author(s):  
Nabil Masmoudi ◽  
Tariq Alkhalifah
Keyword(s):  
North Sea ◽  
Waveform Inversion ◽  
Computational Cost ◽  
Model Parameters ◽  
Full Waveform Inversion ◽  
Data Set ◽  
Model Inversion ◽  
Multistage Model ◽  
Full Waveform ◽  
Trade Offs

Full-waveform inversion (FWI) in anisotropic media is challenging, mainly because of the large computational cost, especially in 3D, and the potential trade-offs between the model parameters needed to describe such media. By analyzing the trade-offs and understanding the resolution limits of the inversion, we can constrain FWI to focus on the main parameters the data are sensitive to and push the inversion toward more reliable models of the subsurface. Orthorhombic anisotropy is one of the most practical approximations of the earth subsurface that takes into account the natural horizontal layering and the vertical fracture network. We investigate the feasibility of a multiparameter FWI for an acoustic orthorhombic model described by six parameters. We rely on a suitable parameterization based on the horizontal velocity and five dimensionless anisotropy parameters. This particular parameterization allows a multistage model inversion strategy in which the isotropic, then, the vertical transverse isotropic, and finally the orthorhombic model can be successively updated. We applied our acoustic orthorhombic inversion on the SEG-EAGE overthrust synthetic model. The observed data used in the inversion are obtained from an elastic variable density version of the model. The quality of the inverted model suggests that we may recover only four parameters, with different resolution scales depending on the scattering potential of these parameters. Therefore, these results give useful insights on the expected resolution of the inverted parameters and the potential constraints that could be applied to an orthorhombic model inversion. We determine the efficiency of the inversion approach on real data from the North Sea. The inverted model is in agreement with the geologic structures and well-log information.

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