scholarly journals Adding Prior Information in FWI through Relative Entropy

Entropy ◽  
2021 ◽  
Vol 23 (5) ◽  
pp. 599
Author(s):  
Danilo Cruz ◽  
João de Araújo ◽  
Carlos da Costa ◽  
Carlos da Silva
Keyword(s):  
Inverse Problem ◽  
Relative Entropy ◽  
Global Minimum ◽  
Prior Information ◽  
Waveform Inversion ◽  
Petroleum Industry ◽  
Synthetic Data ◽  
Full Waveform Inversion ◽  
Full Waveform

Full waveform inversion is an advantageous technique for obtaining high-resolution subsurface information. In the petroleum industry, mainly in reservoir characterisation, it is common to use information from wells as previous information to decrease the ambiguity of the obtained results. For this, we propose adding a relative entropy term to the formalism of the full waveform inversion. In this context, entropy will be just a nomenclature for regularisation and will have the role of helping the converge to the global minimum. The application of entropy in inverse problems usually involves formulating the problem, so that it is possible to use statistical concepts. To avoid this step, we propose a deterministic application to the full waveform inversion. We will discuss some aspects of relative entropy and show three different ways of using them to add prior information through entropy in the inverse problem. We use a dynamic weighting scheme to add prior information through entropy. The idea is that the prior information can help to find the path of the global minimum at the beginning of the inversion process. In all cases, the prior information can be incorporated very quickly into the full waveform inversion and lead the inversion to the desired solution. When we include the logarithmic weighting that constitutes entropy to the inverse problem, we will suppress the low-intensity ripples and sharpen the point events. Thus, the addition of entropy relative to full waveform inversion can provide a result with better resolution. In regions where salt is present in the BP 2004 model, we obtained a significant improvement by adding prior information through the relative entropy for synthetic data. We will show that the prior information added through entropy in full-waveform inversion formalism will prove to be a way to avoid local minimums.

scholarly journals Constrained Full Waveform Inversion for Borehole Multicomponent Seismic Data

Geosciences ◽  
2019 ◽  
Vol 9 (1) ◽  
pp. 45
Author(s):  
Marwan Charara ◽  
Christophe Barnes
Keyword(s):  
Inverse Problem ◽  
Spatial Correlation ◽  
Seismic Data ◽  
Waveform Inversion ◽  
Synthetic Data ◽  
Model Space ◽  
Elastic Model ◽  
Full Waveform Inversion ◽  
Spatial Correlations ◽  
Full Waveform

Full-waveform inversion for borehole seismic data is an ill-posed problem and constraining the problem is crucial. Constraints can be imposed on the data and model space through covariance matrices. Usually, they are set to a diagonal matrix. For the data space, signal polarization information can be used to evaluate the data uncertainties. The inversion forces the synthetic data to fit the polarization of observed data. A synthetic inversion for a 2D-2C data estimating a 1D elastic model shows a clear improvement, especially at the level of the receivers. For the model space, horizontal and vertical spatial correlations using a Laplace distribution can be used to fill the model space covariance matrix. This approach reduces the degree of freedom of the inverse problem, which can be quantitatively evaluated. Strong horizontal spatial correlation distances favor a tabular geological model whenever it does not contradict the data. The relaxation of the spatial correlation distances from large to small during the iterative inversion process allows the recovery of geological objects of the same size, which regularizes the inverse problem. Synthetic constrained and unconstrained inversions for 2D-2C crosswell data show the clear improvement of the inversion results when constraints are used.

Regularized seismic full waveform inversion with prior model information

Geophysics ◽  
2013 ◽  
Vol 78 (2) ◽  
pp. R25-R36 ◽  
Author(s):  
Amir Asnaashari ◽  
Romain Brossier ◽  
Stéphane Garambois ◽  
François Audebert ◽  
Pierre Thore ◽  
...  
Keyword(s):  
Prior Information ◽  
Initial Conditions ◽  
Waveform Inversion ◽  
Synthetic Data ◽  
Physical Property ◽  
Limited Range ◽  
Full Waveform Inversion ◽  
Prior Model ◽  
Full Waveform ◽  
The Impact

Full waveform inversion (FWI) delivers high-resolution quantitative images and is a promising technique to obtain macroscale physical property model of the subsurface. In most geophysical applications, prior information, such as that collected in wells, is available and should be used to increase the image reliability. For this, we propose to introduce three terms in the definition of the FWI misfit function: the data misfit itself, the first-order Tikhonov regularization term acting as a smoothing operator, and a prior model norm term. This last term is the way to smoothly introduce prior information into the FWI workflow. On a selected target of the Marmousi synthetic example, significant improvement was obtained when using the prior model term for noise-free and noisy synthetic data. The prior model term may significantly reduce the inversion sensitivity to incorrect initial conditions. The limited range of spatial wavenumber sampling by the acquisition may be compensated with the prior model information, for multiple-free and multiple-contaminated data. Prior and initial models play different roles in the inversion scheme. The starting model is used for wave propagation and therefore drives the data-misfit gradient, whereas the prior model is never explicitly used for solving the wave equation and only drives the optimization step as an additional constraint to minimize the total objective function. Thus, the prior model is not required to follow kinematic properties as precisely as the initial model, except in zones of poor illumination. In addition, we investigate the influence of a simple dynamic decreasing weighting of the prior model term. Once the cycle-skipping problem has been solved, the impact of the prior model term is gradually reduced within the misfit function to be driven by seismic-data only.

Tackling cycle skipping in full-waveform inversion with intermediate data

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. R411-R427 ◽  
Author(s):  
Gang Yao ◽  
Nuno V. da Silva ◽  
Michael Warner ◽  
Di Wu ◽  
Chenhao Yang
Keyword(s):  
Objective Function ◽  
Waveform Inversion ◽  
Synthetic Data ◽  
Velocity Model ◽  
Full Waveform Inversion ◽  
Data Set ◽  
Intermediate Data ◽  
Full Waveform ◽  
L2 Norm ◽  
Recorded Data

Full-waveform inversion (FWI) is a promising technique for recovering the earth models for exploration geophysics and global seismology. FWI is generally formulated as the minimization of an objective function, defined as the L2-norm of the data residuals. The nonconvex nature of this objective function is one of the main obstacles for the successful application of FWI. A key manifestation of this nonconvexity is cycle skipping, which happens if the predicted data are more than half a cycle away from the recorded data. We have developed the concept of intermediate data for tackling cycle skipping. This intermediate data set is created to sit between predicted and recorded data, and it is less than half a cycle away from the predicted data. Inverting the intermediate data rather than the cycle-skipped recorded data can then circumvent cycle skipping. We applied this concept to invert cycle-skipped first arrivals. First, we picked up the first breaks of the predicted data and the recorded data. Second, we linearly scaled down the time difference between the two first breaks of each shot into a series of time shifts, the maximum of which was less than half a cycle, for each trace in this shot. Third, we moved the predicted data with the corresponding time shifts to create the intermediate data. Finally, we inverted the intermediate data rather than the recorded data. Because the intermediate data are not cycle-skipped and contain the traveltime information of the recorded data, FWI with intermediate data updates the background velocity model in the correct direction. Thus, it produces a background velocity model accurate enough for carrying out conventional FWI to rebuild the intermediate- and short-wavelength components of the velocity model. Our numerical examples using synthetic data validate the intermediate-data concept for tackling cycle skipping and demonstrate its effectiveness for the application to first arrivals.

scholarly journals ML-descent: An optimization algorithm for full-waveform inversion using machine learning

Geophysics ◽  
2020 ◽  
Vol 85 (6) ◽  
pp. R477-R492 ◽  
Author(s):  
Bingbing Sun ◽  
Tariq Alkhalifah
Keyword(s):  
Neural Network ◽  
Machine Learning ◽  
Inverse Problem ◽  
Optimization Algorithm ◽  
Waveform Inversion ◽  
Slight Modification ◽  
Descent Method ◽  
Quadratic Functions ◽  
Full Waveform Inversion ◽  
Full Waveform

Full-waveform inversion (FWI) is a nonlinear optimization problem, and a typical optimization algorithm such as the nonlinear conjugate gradient or limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) would iteratively update the model mainly along the gradient-descent direction of the misfit function or a slight modification of it. Based on the concept of meta-learning, rather than using a hand-designed optimization algorithm, we have trained the machine (represented by a neural network) to learn an optimization algorithm, entitled the “ML-descent,” and apply it in FWI. Using a recurrent neural network (RNN), we use the gradient of the misfit function as the input, and the hidden states in the RNN incorporate the history information of the gradient similar to an LBFGS algorithm. However, unlike the fixed form of the LBFGS algorithm, the machine-learning (ML) version evolves in response to the gradient. The loss function for training is formulated as a weighted summation of the L2 norm of the data residuals in the original inverse problem. As with any well-defined nonlinear inverse problem, the optimization can be locally approximated by a linear convex problem; thus, to accelerate the training, we train the neural network by minimizing randomly generated quadratic functions instead of performing time-consuming FWIs. To further improve the accuracy and robustness, we use a variational autoencoder that projects and represents the model in latent space. We use the Marmousi and the overthrust examples to demonstrate that the ML-descent method shows faster convergence and outperforms conventional optimization algorithms. The energy in the deeper part of the models can be recovered by the ML-descent even when the pseudoinverse of the Hessian is not incorporated in the FWI update.

scholarly journals Bayesian Full-waveform Inversion with Realistic Priors

Geophysics ◽  
2021 ◽  
pp. 1-20
Author(s):  
Xin Zhang ◽  
Andrew Curtis
Keyword(s):  
Prior Information ◽  
Velocity Structure ◽  
Waveform Inversion ◽  
Full Waveform Inversion ◽  
Reflection Seismic ◽  
Variational Bayesian Inference ◽  
Full Waveform ◽  
Highly Nonlinear ◽  
Seismic Acquisition ◽  
Using Data

Seismic full-waveform inversion (FWI) uses full seismic records to estimate the subsurface velocity structure. This requires a highly nonlinear and nonunique inverse problem to be solved, so Bayesian methods have been used to quantify uncertainties in the solution. Variational Bayesian inference uses optimization to provide solutions efficiently. However, previously the method has only been applied to a transmission FWI problem, and with strong prior information imposed on the velocity such as is never available in practice. We show that the method works well in a seismic reflection setting, and with realistically weak prior information, representing the type of problem that occurs in reality. We conclude that the method can produce high-resolution images and reliable uncertainties using data from standard reflection seismic acquisition geometry, realistic nonlinearity, and practically achievable prior information.

Seismic full-waveform inversion using minimization of virtual scattering sources

Geophysics ◽  
2020 ◽  
Vol 85 (3) ◽  
pp. R299-R311
Author(s):  
Donguk Lee ◽  
Sukjoon Pyun
Keyword(s):  
Newton Method ◽  
Scattering Problem ◽  
Waveform Inversion ◽  
Synthetic Data ◽  
Inverse Scattering Problem ◽  
Simulated Data ◽  
Model Space ◽  
Full Waveform Inversion ◽  
Velocity Inversion ◽  
Full Waveform

Full-waveform inversion (FWI) is a powerful tool for imaging underground structures with high resolution; however, this approach commonly suffers from the cycle-skipping issue. Recently, various FWI methods have been suggested to address this problem. Such methods are mainly classified into either data-space manipulation or model-space extension. We developed an alternative FWI method that belongs to the latter class. First, we define the virtual scattering source based on perturbation theory. The virtual scattering source is estimated by minimizing the differences between observed and simulated data with a regularization term penalizing the weighted virtual scattering source. The inverse problem for obtaining the virtual scattering source can be solved by the linear conjugate gradient method. The inverted virtual scattering source is used to update the wavefields; thus, it helps FWI to better approximate the nonlinearity of the inverse scattering problem. As the second step, the virtual scattering source is minimized to invert the velocity model. By assuming that the variation of the reconstructed wavefield is negligible, we can apply an approximated full Newton method to the velocity inversion with reasonable cost comparable to the Gauss-Newton method. From the numerical examples using synthetic data, we confirm that the proposed method performs better and more robust than the simple gradient-based FWI method. In addition, we show that our objective function has fewer local minima, which helps to mitigate the cycle-skipping problem.

Source-domain full waveform inversions

Geophysics ◽  
2020 ◽  
pp. 1-50
Author(s):  
Yulang Wu ◽  
George A. McMechan
Keyword(s):  
Differential Operators ◽  
Waveform Inversion ◽  
Synthetic Data ◽  
Velocity Model ◽  
Low Noise ◽  
Full Waveform Inversion ◽  
Virtual Source ◽  
Reflected Waves ◽  
Source Domain ◽  
Full Waveform

Conventional full waveform inversion (FWI) updates a velocity model by minimizing the data residuals between predicted and observed data, at the receiver positions. We propose a new full waveform inversion to update the velocity model by minimizing virtual source artifacts, at the receiver positions, in the source domain (SFWI). Virtual source artifacts are created by replacing the propagating source wavefield by the forward-time observed data at the receiver positions, as a data-residual constraint. Therefore, no matter whether the velocity model is correct or not, the data residuals, at the receiver positions, are always forced to be zero. If the velocity model is correct, this data-residual constraint has no effect on the wavefield, since the predicted data is the same as the observed data. However, if the estimated velocity model is incorrect, the mismatch between the replaced forward-time observed data and the incorrect predicted upgoing waves (e.g., reflected waves) at the receiver positions, will produce downgoing artifact waves. Thus, the data-residual constraint behaves as a virtual source to create artifact wavefields. By minimizing the virtual source artifacts (equivalent to producing the artifact wavefield), the velocity model can be iteratively updated toward the true velocity model. Similar to conventional FWI, SFWI can be implemented in either the frequency or the time domain, which is unlike previous source-domain solutions, which have to be implemented only in the frequency domain, to solve the normal equations. SFWI does more over-fitting of noisy observed data than conventional FWI does, because noise is amplified by the differential operators when calculating the virtual source artifacts. Tests on synthetic data show that the SFWI inverts for the velocity model more accurately than conventional FWI for noise-free or low-noise data.

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