scholarly journals Intrinsic non-uniqueness of the acoustic full waveform inverse problem

2020 ◽  
Author(s):  
Yann Capdeville ◽  
Chao Lyu ◽  
David Al-Attar ◽  
Liang Zhao
Keyword(s):  
Inverse Problem ◽  
Frequency Band ◽  
Waveform Inversion ◽  
Real Data ◽  
Homogenization Method ◽  
Acoustic Medium ◽  
Fine Scale ◽  
Smooth Mapping ◽  
Full Waveform ◽  
Limited Frequency

<p>In the context of seismic imaging, the full waveform inversion (FWI) is more and more popular. Because of its lower numerical cost, the acoustic approximation is often used, especially at the exploration geophysics scale, both for tests and for real data. Moreover, some research domains such as helioseismology face true acoustic medium for which FWI can be useful. In this work, we show that the general acoustic inverse problem based on limited frequency band data is intrinsically non-unique, making any general acoustic FWI impossible. Our work is based on two tools: particle relabelling and homogenization. On the one hand, the particle relabelling method shows it is possible to deform a true medium based on a smooth mapping into a new one without changing the signal recorded at seismic stations. This is a potentially strong source of non-uniqueness for an inverse problem based a seismic data. Nevertheless, in the elastic case, the deformed medium loses the elastic tensor minor symmetries and, in the acoustic case, it implies density anisotropy. It is therefore not a source of non-uniqueness for elastic or isotropic acoustic inverse problems, but it is for the anisotropic acoustic case. On the other hand, the homogenization method shows that any fine-scale medium can be up-scaled into an effective medium without changing the waveforms in a limited frequency band. The effective media are in general anisotropic, both in the elastic and acoustic cases, even if the true media are isotropic at a fine scale. It implies that anisotropy is in general present and needs to be inverted. Therefore, acoustic anisotropy can not be avoided in general. We conclude, based on a particle relabelling and homogenization arguments, that the acoustic FWI solution is in general non-unique. We show, in 2-D numerical FWI examples based on the Gauss-Newton iterative scheme, the effects of this non-uniqueness in the local optimization context. We numerically confirm that the acoustic FWI is in general non-unique and that finding a physical solution is not possible.</p>

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.

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.

Targeted stacking of weak seismic phases for improving mantle discontinuity imaging using full-waveform inversion

2021 ◽  
Author(s):  
Maria Koroni ◽  
Andreas Fichtner
Keyword(s):  
Seismic Waves ◽  
Waveform Inversion ◽  
Real Data ◽  
Full Waveform Inversion ◽  
Finite Frequency ◽  
Adjoint Methods ◽  
Frequency Sensitivity ◽  
Full Waveform ◽  
Discontinuity Structure ◽  
Mantle Discontinuity

<p>This study is a continuation of our efforts to connect adjoint methods and full-waveform inversion to common beamforming techniques, widely used and developed for signal enhancement. Our approach is focusing on seismic waves traveling in the Earth's mantle, which are phases commonly used to image internal boundaries, being however quite difficult to observe in real data. The main goal is to accentuate precursor waves arriving in well-known times before some major phase. These waves generate from interactions with global discontinuities in the mantle, thus being the most sensitive seismic phases and therefore most suitable for better understanding of discontinuity seismic structure. </p><p>Our work is based on spectral-element wave propagation which allows us to compute exact synthetic waveforms and adjoint methods for the calculation of sensitivity kernels. These tools are the core of full-waveform inversion and by our efforts we aim to incorporate more parts of the waveform in such inversion schemes. We have shown that targeted stacking of good quality waveforms arriving from various directions highlights the weak precursor waves. It additionally makes their traveltime finite frequency sensitivity prominent. This shows that we can benefit from using these techniques and exploit rather difficult parts of the seismogram.  It was also shown that wave interference is not easily avoided, but coherent phases arriving before the main phase also stack well and show on the sensitivity kernels. This does not hamper the evaluation of waveforms, as in a misfit measurement process one can exploit more phases on the body wave parts of seismograms.</p><p>In this study, we go a step forward and present recent developments of the approach relating to the effects of noise and a real data experiment. Realistic noise is added to synthetic waveforms in order to assess the methodology in a more pragmatic scenario. The addition of noise shows that stacking of coherent seismic phases is still possible and the sensitivity kernels of their traveltimes are not largely distorted, the precursor waves contribute sufficiently to their traveltime finite-frequency sensitivity kernels.<br>Using a well-located seismic array, we apply the method to real data and try to examine the possibilities of using non-ideal waveforms to perform imaging of the mantle discontinuity structure on the specific areas. In order to make the most out of the dense array configuration, we try subgroups of receivers for the targeted stacking and by moving along the array we aim at creating a cluster of stacks. The main idea is to use the subgroups as single receivers and create an evaluation of seismic discontinuity structure using information from each stack belonging to a subgroup. <br>Ideally, we aim at improving the tomographic images of discontinuities of selected regions by exploiting weaker seismic waves, which are nonetheless very informative.</p>

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.

Which parameterization is suitable for acoustic vertical transverse isotropic full waveform inversion? Part 2: Synthetic and real data case studies from Valhall

Geophysics ◽  
2013 ◽  
Vol 78 (2) ◽  
pp. R107-R124 ◽  
Author(s):  
Yaser Gholami ◽  
Romain Brossier ◽  
Stéphane Operto ◽  
Vincent Prieux ◽  
Alessandra Ribodetti ◽  
...  
Keyword(s):  
Waveform Inversion ◽  
Real Data ◽  
Velocity Model ◽  
Oil Field ◽  
Full Waveform Inversion ◽  
Trade Off ◽  
Wave Speeds ◽  
Wide Aperture ◽  
Full Waveform ◽  
Transverse Isotropic

It is necessary to account for anisotropy in full waveform inversion (FWI) of wide-azimuth and wide-aperture seismic data in most geologic environments, for correct depth positioning of reflectors, and for reliable estimations of wave speeds as a function of the direction of propagation. In this framework, choosing a suitable anisotropic subsurface parameterization is a central issue in monoparameter and multiparameter FWI. This is because this parameterization defines the influence of each physical parameter class on the data as a function of the scattering angle, and hence the resolution of the parameter reconstruction, and on the potential trade-off between different parameter classes. We apply monoparameter and multiparameter frequency-domain acoustic vertical transverse isotropic FWI to synthetic and real wide-aperture data, representative of the Valhall oil field. We first show that reliable monoparameter FWI can be performed to build a high-resolution velocity model (for the vertical, the horizontal, or normal move-out velocity), provided that the background models of two Thomsen parameters describe the long wavelengths of the subsurface sufficiently accurately. Alternatively, we show the feasibility of the joint reconstruction of two wave speeds (e.g., the vertical and horizontal wave speeds) with limited trade-off effects, while Thomsen parameter [Formula: see text] is kept fixed during the inversion. The influence of the wave speeds on the data for a limited range of scattering angles when combined each other can, however, significantly hamper the resolution with which the two wave speeds are imaged. These conclusions inferred from the application to the real data are fully consistent with those inferred from the theoretical parameterization analysis of acoustic vertical transverse isotropic FWI performed in the companion report.

scholarly journals Time-domain full-waveform inversion of exponentially damped wavefield using the deconvolution-based objective function

Geophysics ◽  
2018 ◽  
Vol 83 (2) ◽  
pp. R77-R88 ◽  
Author(s):  
Yunseok Choi ◽  
Tariq Alkhalifah
Keyword(s):  
Objective Function ◽  
Frequency Band ◽  
Waveform Inversion ◽  
Low Frequency ◽  
Full Waveform Inversion ◽  
Low Frequencies ◽  
Full Waveform ◽  
Adjoint State ◽  
Recorded Data ◽  
Adjoint State Method

Full-waveform inversion (FWI) suffers from the cycle-skipping problem when the available frequency-band of data is not low enough. We have applied an exponential damping to the data to generate artificial low frequencies, which helps FWI to avoid cycle skipping. In this case, the least-squares misfit function does not properly deal with the exponentially damped wavefield in FWI because the amplitude of traces decays almost exponentially with increasing offset in a damped wavefield. Thus, we use a deconvolution-based objective function for FWI of the exponentially damped wavefield. The deconvolution filter includes inherently a normalization between the modeled and observed data; thus, it can address the unbalanced amplitude of a damped wavefield. We specifically normalize the modeled data with the observed data in the frequency-domain to estimate the deconvolution filter and selectively choose a frequency-band for normalization that mainly includes the artificial low frequencies. We calculate the gradient of the objective function using the adjoint-state method. The synthetic and benchmark data examples indicate that our FWI algorithm generates a convergent long-wavelength structure without low-frequency information in the recorded data.

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