An efficient wavefield inversion: Using a modified source function in the wave equation

Geophysics ◽  
2019 ◽  
Vol 84 (6) ◽  
pp. R909-R922 ◽  
Author(s):  
Tariq Alkhalifah ◽  
Chao Song
Keyword(s):  
Wave Equation ◽  
Optimization Problem ◽  
Source Function ◽  
Waveform Inversion ◽  
Gradient Methods ◽  
Model Space ◽  
Velocity Model ◽  
Extended Model ◽  
Extended Source ◽  
Separate Step

The wavefield is often reconstructed through solving a wave equation corresponding to an active source and using our best knowledge of the medium to reproduce the data that we seek to fit the observed data as part of a process we call full-waveform inversion (FWI). Alternatively, the wavefield can be inverted and used to invert for the velocity model, in an extended optimization problem, in which we can relax the requirement that the wavefield satisfies the wave equation. In this case, the wavefield calculation, as in FWI, requires a matrix inversion (which depends on the velocity) at practically every iteration. Thus, we formulate a bilinear optimization problem with respect to the wavefield and a modified source function, as independent variables. We specifically recast the wave equation so that velocity perturbations are included in this modified source function (which includes secondary or contrast sources); thus, it represents the velocity perturbations implicitly. The optimization includes a measure of the wavefield’s fit to the data at the sensor locations and the wavefield, as well as the modified source function, compliance with a wave equation corresponding to the background model. This problem is, however, complex with an extended model space that is ill-posed, so we use an alternating-direction method to reduce the inversion to two subproblems for inverting each of the wavefields and extended source. On the other hand, the velocity perturbations can be extracted in a separate step via direct division (deconvolution). Because we avoid using gradient methods in extracting the velocity perturbations, we are less prone to crosstalk artifacts when we use simultaneous sources. We evaluate these features on a simple two-anomalies model and the modified Marmousi model.

Traveltime information-based wave-equation inversion

Geophysics ◽  
2009 ◽  
Vol 74 (6) ◽  
pp. WCC27-WCC36 ◽  
Author(s):  
Yu Zhang ◽  
Daoliu Wang
Keyword(s):  
Wave Equation ◽  
Conventional Method ◽  
Seismic Data ◽  
Waveform Inversion ◽  
Back Propagation ◽  
Velocity Model ◽  
Data Fitting ◽  
Inversion Method ◽  
New Wave ◽  
Seismic Record

We propose a new wave-equation inversion method that mainly depends on the traveltime information of the recorded seismic data. Unlike the conventional method, we first apply a [Formula: see text] transform to the seismic data to form the delayed-shot seismic record, back propagate the transformed data, and then invert the velocity model by maximizing the wavefield energy around the shooting time at the source locations. Data fitting is not enforced during the inversion, so the optimized velocity model is obtained by best focusing the source energy after a back propagation. Therefore, inversion accuracy depends only on the traveltime information embedded in the seismic data. This method may overcome some practical issues of waveform inversion; in particular, it relaxes the dependency of the seismic data amplitudes and the source wavelet.

Source-independent extended waveform inversion based on space-time source extension: Frequency-domain implementation

Geophysics ◽  
2018 ◽  
Vol 83 (5) ◽  
pp. R449-R461 ◽  
Author(s):  
Guanghui Huang ◽  
Rami Nammour ◽  
William W. Symes
Keyword(s):  
Source Function ◽  
Random Noise ◽  
A Priori ◽  
Waveform Inversion ◽  
Low Frequency ◽  
Inversion Method ◽  
Target Velocity ◽  
Extended Source ◽  
Full Waveform ◽  
Time Variables

Source signature estimation from seismic data is a crucial ingredient for successful application of seismic migration and full-waveform inversion (FWI). If the starting velocity deviates from the target velocity, FWI method with on-the-fly source estimation may fail due to the cycle-skipping problem. We have developed a source-based extended waveform inversion method, by introducing additional parameters in the source function, to solve the FWI problem without the source signature as a priori. Specifically, we allow the point source function to be dependent on spatial and time variables. In this way, we can easily construct an extended source function to fit the recorded data by solving a source matching subproblem; hence, it is less prone to cycle skipping. A novel source focusing annihilator, defined as the distance function from the real source position, is used for penalizing the defocused energy in the extended source function. A close data fit avoiding the cycle-skipping problem effectively makes the new method less likely to suffer from local minima, which does not require extreme low-frequency signals in the data. Numerical experiments confirm that our method can mitigate cycle skipping in FWI and is robust against random noise.

Constrained full-waveform inversion by model reparameterization

Geophysics ◽  
2012 ◽  
Vol 77 (2) ◽  
pp. R117-R127 ◽  
Author(s):  
Antoine Guitton ◽  
Gboyega Ayeni ◽  
Esteban Díaz
Keyword(s):  
Waveform Inversion ◽  
Model Space ◽  
Real Data ◽  
Velocity Model ◽  
Full Waveform Inversion ◽  
Coherent Noise ◽  
Low Frequencies ◽  
Full Waveform ◽  
Geological Information ◽  
Ill Posed

The waveform inversion problem is inherently ill-posed. Traditionally, regularization schemes are used to address this issue. For waveform inversion, where the model is expected to have many details reflecting the physical properties of the Earth, regularization and data fitting can work in opposite directions: the former smoothing and the latter adding details to the model. We propose constraining estimated velocity fields by reparameterizing the model. This technique, also called model-space preconditioning, is based on directional Laplacian filters: It preserves most of the details of the velocity model while smoothing the solution along known geological dips. Preconditioning also yields faster convergence at early iterations. The Laplacian filters have the property to smooth or kill local planar events according to a local dip field. By construction, these filters can be inverted and used in a preconditioned waveform inversion strategy to yield geologically meaningful models. We illustrate with 2D synthetic and field data examples how preconditioning with nonstationary directional Laplacian filters outperforms traditional waveform inversion when sparse data are inverted and when sharp velocity contrasts are present. Adding geological information with preconditioning could benefit full-waveform inversion of real data whenever irregular geometry, coherent noise and lack of low frequencies are present.

Matrix-free anisotropic slope tomography: Theory and application

Geophysics ◽  
2019 ◽  
Vol 84 (1) ◽  
pp. R21-R43 ◽  
Author(s):  
Borhan Tavakoli F. ◽  
Stéphane Operto ◽  
Alessandra Ribodetti ◽  
Jean Virieux
Keyword(s):  
Inverse Problem ◽  
Waveform Inversion ◽  
Model Space ◽  
Velocity Model ◽  
Sensitivity Matrix ◽  
Model Quality ◽  
Depth Migration ◽  
Velocity Models ◽  
New Formulation ◽  
Matrix Free

Slope tomography uses traveltimes, source, and receiver slopes of locally coherent events to build subsurface velocity models. Locally coherent events by opposition to continuous reflections are suitable for semiautomatic and dense picking, which is conducive to better resolved tomographic models. These models can be further used as background/initial models for depth migration or full-waveform inversion. Slope tomography conventionally relies on ray tracing for traveltimes and slopes computation, where rays are traced from scatterers in depth to sources and receivers. The inverse problem relies on the explicit building of the sensitivity matrix to update the velocity model by local optimization. Alternatively, slope tomography can be implemented with eikonal solvers, which compute efficiently finely sampled traveltime maps from the sources and receivers, whereas slopes are estimated by finite differences of the traveltime maps. Moreover, a matrix-free inverse problem can be implemented with the adjoint-state method for the estimation of the data-misfit gradient. This new formulation of slope tomography is extended to tilted transverse isotropic (TTI) acoustic media, in which the model space is parameterized by four anisotropic parameters (e.g., vertical wavespeed, Thomson’s parameter [Formula: see text], [Formula: see text], and tilt angle) and the coordinates of the scatterers. A toy synthetic example allows for a first assessment of the crosstalk between anisotropic parameters and scatterer coordinates. A more realistic synthetic example indicates the feasibility of the joint update of the vertical wavespeed and [Formula: see text]. The slope tomography is finally applied to real broadband towed-streamer data to build the vertical velocity and the scatterers, while anisotropic parameters [Formula: see text] and [Formula: see text] are used as background parameters. The velocity model quality is assessed through common-image gathers computed by TTI Kirchhoff prestack-depth migration.

scholarly journals Frequency Domain Wave Equation Inversion and Its Application on the Heterogeneous Reservoir Model Data

2016 ◽  
Vol 6 (1) ◽  
pp. 55
Author(s):  
Zhiyuan Tang
Keyword(s):  
Wave Equation ◽  
Velocity Field ◽  
Frequency Domain ◽  
Waveform Inversion ◽  
Gradient Methods ◽  
Carbonate Reservoir ◽  
Physical Models ◽  
Full Waveform Inversion ◽  
Matrix Formalism ◽  
Full Waveform

Seismic full waveform inversion seeks to make use of the full information based on full wave field modeling to extract quantitative information from seismograms. Its serious nonlinearity and high dependence on initial velocity model often results in unsatisfactory inversion results in paleo-karsts carbonate reservoir characterized by strong heterogeneity. The paper presents an improved strategy of multi-scale inversion to establish velocity field model of waveform tomography. the forward wave equation algorithm was derived in frequency domain, and then the Matrix formalism for the iterative inverse methods is derived by gradient methods to speed up calculation and to avoid convergence to local minimum value. After massive amount of frequencies tests, the appropriate bandwidth are extracted, and the velocity field calculated at low frequency is used as the input of the high frequency. After the iteration, the accurate velocity field is inverted. Finally, frequency domain wave equation full waveform inversion in mathematical and physical models is conducted in order to verify the inverse program. The method of selecting the inverse frequencies is proved to be effective.

scholarly journals Elastic Full-Waveform Inversion Using Migration‑Based Depth Reflector Representation in the Data Domain

2020 ◽  
Author(s):  
Vladimir Cheverda
Keyword(s):  
Seismic Data ◽  
Optimization Problem ◽  
Waveform Inversion ◽  
Nonlinear Least Squares ◽  
Model Space ◽  
Full Waveform Inversion ◽  
Brute Force ◽  
Data Inversion ◽  
Full Waveform ◽  
Least Squares Optimization

Full-waveform seismic data inversion has given rise to hope for the simultaneous and automated execution of tomography and imaging by solving a nonlinear least-squares optimization problem. As previously recognized, brute force minimization by classical methods is hopeless if the data lacks low temporal frequencies. The article developed a reliable numerical method for recovering smooth velocity using model space decomposition. We present realistic synthetic examples to test the presented algorithm.

scholarly journals Mitigating elastic effects in marine 3-D full-waveform inversion

2019 ◽  
Vol 220 (3) ◽  
pp. 2089-2104
Author(s):  
Òscar Calderón Agudo ◽  
Nuno Vieira da Silva ◽  
George Stronge ◽  
Michael Warner
Keyword(s):  
Wave Equation ◽  
Field Data ◽  
Waveform Inversion ◽  
Velocity Model ◽  
P Wave ◽  
Data Sets ◽  
Acoustic Wave Equation ◽  
Data Set ◽  
Velocity Models ◽  
Full Waveform

SUMMARY The potential of full-waveform inversion (FWI) to recover high-resolution velocity models of the subsurface has been demonstrated in the last decades with its application to field data. But in certain geological scenarios, conventional FWI using the acoustic wave equation fails in recovering accurate models due to the presence of strong elastic effects, as the acoustic wave equation only accounts for compressional waves. This becomes more critical when dealing with land data sets, in which elastic effects are generated at the source and recorded directly by the receivers. In marine settings, in which sources and receivers are typically within the water layer, elastic effects are weaker but can be observed most easily as double mode conversions and through their effect on P-wave amplitudes. Ignoring these elastic effects can have a detrimental impact on the accuracy of the recovered velocity models, even in marine data sets. Ideally, the elastic wave equation should be used to model wave propagation, and FWI should aim to recover anisotropic models of velocity for P waves (vp) and S waves (vs). However, routine three-dimensional elastic FWI is still commercially impractical due to the elevated computational cost of modelling elastic wave propagation in regions with low S-wave velocity near the seabed. Moreover, elastic FWI using local optimization methods suffers from cross-talk between different inverted parameters. This generally leads to incorrect estimation of subsurface models, requiring an estimate of vp/vs that is rarely known beforehand. Here we illustrate how neglecting elasticity during FWI for a marine field data set that contains especially strong elastic heterogeneities can lead to an incorrect estimation of the P-wave velocity model. We then demonstrate a practical approach to mitigate elastic effects in 3-D yielding improved estimates, consisting of using a global inversion algorithm to estimate a model of vp/vs, employing matching filters to remove elastic effects from the field data, and performing acoustic FWI of the resulting data set. The quality of the recovered models is assessed by exploring the continuity of the events in the migrated sections and the fit of the latter with the recovered velocity model.

Building FWI starting model from wells with dynamic time warping and convolutional neural networks

Geophysics ◽  
2021 ◽  
pp. 1-35
Author(s):  
Jiashun Yao ◽  
Yanghua Wang
Keyword(s):  
Dynamic Time Warping ◽  
Waveform Inversion ◽  
Model Space ◽  
Velocity Model ◽  
Time Warping ◽  
Spatial Features ◽  
Training Samples ◽  
Structural Coherence ◽  
Dynamic Time ◽  
Starting Model

Full waveform inversion (FWI) needs a feasible starting model, because otherwise it might converge to a local minimum and the inversion result might suffer from detrimental artifacts. We built a feasible starting model from wells by applying dynamic time warping (DTW) localized rewarp and convolutional neural network (CNN) methods alternatively. We used the DTW localized rewarp method to extrapolate the velocities at well locations to the non-well locations in the model space. Rewarping is conducted based on the local structural coherence which is extracted from a migration image of an initial infeasible model. The extraction uses the DTW method. The purpose of velocity extrapolation is to provide sufficient training samples to train a CNN, which maps local spatial features on the migration image into the velocity quantities of each layer. We further designed an interactive workflow to reject inaccurate network predictions and to improve CNN prediction accuracy by incorporating the Monte Carlo dropout method. We demonstrated that the proposed method is robust against the kinematic incorrectness in the migration velocity model, and is capable to produce a feasible FWI starting model.

Wave-equation reflection traveltime inversion with dynamic warping and full-waveform inversion

Geophysics ◽  
2013 ◽  
Vol 78 (6) ◽  
pp. R223-R233 ◽  
Author(s):  
Yong Ma ◽  
Dave Hale
Keyword(s):  
Wave Equation ◽  
Waveform Inversion ◽  
Velocity Model ◽  
Full Waveform Inversion ◽  
Local Minima ◽  
Low Frequencies ◽  
Traveltime Inversion ◽  
Velocity Models ◽  
Full Waveform ◽  
Dynamic Warping

In reflection seismology, full-waveform inversion (FWI) can generate high-wavenumber subsurface velocity models but often suffers from an objective function with local minima caused mainly by the absence of low frequencies in seismograms. These local minima cause cycle skipping when the low-wavenumber component in the initial velocity model for FWI is far from the true model. To avoid cycle skipping, we discovered a new wave-equation reflection traveltime inversion (WERTI) to update the low-wavenumber component of the velocity model, while using FWI to only update high-wavenumber details of the model. We implemented the low- and high-wavenumber inversions in an alternating way. In WERTI, we used dynamic image warping (DIW) to estimate the time shifts between recorded data and synthetic data. When compared with correlation-based techniques often used in traveltime estimation, DIW can avoid cycle skipping and estimate the time shifts accurately, even when shifts vary rapidly. Hence, by minimizing traveltime shifts estimated by dynamic warping, WERTI reduces errors in reflection traveltime inversion. Then, conventional FWI uses the low-wavenumber component estimated by WERTI as a new initial model and thereby refines the model with high-wavenumber details. The alternating combination of WERTI and FWI mitigates the velocity-depth ambiguity and can recover subsurface velocities using only high-frequency reflection data.

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