Full Waveform Inversion of Viscoelastic Media Based on P–S Separation

SeisElastic2D: An open-source package for multiparameter full-waveform inversion in isotropic-, anisotropic- and visco-elastic media

<p>The seismic full waveform inversion (FWI), as one of important ways to obtain the seismic wave velocity, has made rapid development in the last decade. In response to problems of cycle-skipping artifacts, dependence on the initial model, and low-frequency information in FWI, researchers have made many improvements, such as multi-scale envelope inversion and low-frequency extension. Recently, deep learning has been also adopted seismic data processing and interpretation, because of its strong nonlinear mapping ability. However, these works depend on labels used for training heavily, especially for the velocity model in the inversion, which prevents them from real application. Referring to these studies, this work combines low-frequency extension commonly as well as multiscale inversion with deep learning, and proposes a multi-scale FWI gradient optimization method based on CNN. CNN we designed is trained to predict the inversion gradient corresponding to the low-frequency band data in FWI, so that multi-scale gradient optimization can be directly used in multi-scale inversion, expanding the low-frequency information in the actual data and reducing the calculation in FWI. With a specially designed dataset, CNN is trained to optimize the gradients computed from the high-frequency band data by predicting the gradients corresponding to the low-frequency band data and the gradients corresponding to the mid-frequency band data, respectively. The predicted gradients are used in different stages of the multi-scale inversion. The low-frequency gradients are used to invert the initial structural construction so as not to rely on a good initial model, and the high-frequency gradients are used to improve the accuracy of the inversion results. In this way, low-frequency expansion and multiscale inversion can be achieved simultaneously. Our method achieves good results on the initial model for a given uniform wave velocity, effectively alleviating the reliance on the initial model in FWI. This study provides a new idea of combining deep learning and full waveform inversion, which will be effectively used in seismic data processing.</p>

In this paper, we study 3-D multiparameter full waveform inversion (FWI) in viscoelastic media based on the generalized Maxwell/Zener body including arbitrary number of attenuation mechanisms. We present a frequency-domain energy analysis to establish the stability condition of a full anisotropic viscoelastic system, according to zero-valued boundary condition and the elastic–viscoelastic correspondence principle: the real-valued stiffness matrix becomes a complex-valued one in Fourier domain when seismic attenuation is taken into account. We develop a least-squares optimization approach to linearly relate the quality factor with the anelastic coefficients by estimating a set of constants which are independent of the spatial coordinates, which supplies an explicit incorporation of the parameter <it>Q</it> in the general viscoelastic wave equation. By introducing the Lagrangian multipliers into the matrix expression of the wave equation with implicit time integration, we build a systematic formulation of multiparameter FWI for full anisotropic viscoelastic wave equation, while the equivalent form of the state and adjoint equation with explicit time integration is available to be resolved efficiently. In particular, this formulation lays the foundation for the inversion of the parameter <it>Q</it> in the time domain with full anisotropic viscoelastic properties. In the 3-D isotropic viscoelastic settings, the anelastic coefficients and the quality factors using bulk and shear moduli parametrization can be related to the counterparts using <it>P</it> and <it>S</it> velocity. Gradients with respect to any other parameter of interest can be found by chain rule. Pioneering numerical validations as well as the real applications of this most generic framework will be carried out to disclose the potential of viscoelastic FWI when adequate high-performance computing resources and the field data are available.

In viscoelastic media, reverse time migration (RTM) requires accurate Q value for energy compensation and phase correction. Full waveform inversion (FWI) has the potential to provide accurate Q value. In the wide frequency band seismic data, the quality factor Q almost does not change with frequency, which is called constant Q model, but its computational complexity is large. In the narrow frequency band data, when the memory variable order L is assumed to be 1 in the standard linear solid (SLS) model, the attenuation simulation effect of the subsurface Q value is acceptable, and the calculation amount is small. In this paper, we derive the back propagation wave field equation and gradient formula of full waveform inversion of viscoelastic medium based on the SLS model with memory variable order L = 1, and analyzes the effectiveness and feasibility of this method. In viscoelastic medium, low velocity body will produce low frequency noise in seismic record, which will lead to heterogeneous gradient energy. This low frequency noise is similar to the low frequency noise produced near the source of shallow strata. This will reduce the accuracy of the inversion results. The gradient preconditioning method using the pseudo-Hessian operator can make the gradient energy more balanced. Therefore, we introduce the pseudo-Hessian operator in gradient preconditioning and derives the gradient preconditioning formula for the full waveform inversion in viscoelastic media, which solves the problem of the non-uniform gradient energy. Examples show that the method can suppress the non-uniform gradient energy in the full waveform inversion of viscoelastic media.

Time-domain seismic modeling in viscoelastic media for full waveform inversion on heterogeneous computing platforms with OpenCL

Summary “Processing of seismic data, for example for imaging, full waveform inversion and microseismic monitoring, can only be done accurately if seismic attenuation is accounted for. The basis for this is provided by a Q model. While inclusion of Q improves the processing results, it also requires an increase in computational resources as the computation of the waveforms involves now also the computation of a new quantity (typically, the complex valued travel time T*). We present two different viscoelastic applications of a GPU sum reduction scheme. In both cases the computational efficiency is drastically improved. The first application is the ray-Born integral, which forms the basis for the computation of viscoelastic full waveform inversion. The second application is the computation of seismic images using a viscoelastic background model. Our numerical results show both modelling and imaging in the overthrust model for various types of source mechanisms and attenuation models.”

plex shallow subsurface. This workflow incorporates dispersion curve inversion, traveltime tomography and full waveform inversion into a single framework to optimize the advantages of each. An accurate near-surface velocity is derived using the dispersion curve inversion technique that overcomes limitations of traditional traveltime tomography. This velocity is then incorporated into layer-stripping full waveform or RMO inversion schemes by “freezing” the ultra-shallow part of the model in order to retrieve the deeper velocity structure. We illustrate the workflow on a test data set from Inner Mongolia, China, whose long-offsets and low-frequency content provide the necessary input for both the dispersion curve and full waveform inversion techniques. Using this data set, we demonstrate the efficacy of the integrated methodology in determining both the macro-velocity model and near-surface velocities in structurally complex geological setting.

Summary We suggest a new method for the Laplace domain full waveform inversion (FWI) which use single damping constant. This method makes the Laplace domain FWI more efficient, because the proposed method uses only one damping constant during inversion process. To scale the gradient direction properly, we use simple depth window function to steepest descent direction. We demonstrate that the algorithm is working by numerical test using synthetic data set.

Summary Multi-component exploration seismic acquisition is widely used because it can obtain more detailed underground structure and attribute information. Multi-component seismic data is processed by elastic wave equations. The separation of P- and S-waves in the elastic wave equation is the key technology of multi-component elastic wavefield simulation, reverse time migration imaging, and full waveform inversion. Current elastic wave separation methods mainly include the curl-divergence operator based P- and S-waves separation, and the first-order velocity-stress wavefield decoupling separation. The curl-divergence operator based separation method will change the dynamic characteristics of the separated wavefield. The first-order velocity-stress wavefield decoupling method will cause separation artifacts in the separation wavefield. For the above problems, we analyze the causes of artifacts in the first-order velocity-stress equation, and propose a new P- and S-waves decoupling formulation based on the first-order velocity-strain equation. Through several numerical tests, we find that the separated P- and S-waves based on the first-order velocity-strain equation have no separation artifacts at the interface and have accurate separation results.

Viscous attenuation can have a strong impact on seismic wave propagation, but it is rarely taken into account in full waveform inversion (FWI). When viscoelasticity is considered in time domain FWI, the displacement formulation of the wave equation is usually used instead of the popular velocity–stress formulation. However, inversion schemes rely on the adjoint equations, which are quite different for the velocity–stress formulation than for the displacement formulation. In this paper, we apply the adjoint state method to the isotropic viscoelastic wave equation in the velocity–stress formulation based on the generalized standard linear solid rheology. By applying linear transformations to the wave equation before deriving the adjoint state equations, we obtain two symmetric sets of partial differential equations for the forward and adjoint variables. The resulting sets of equations only differ by a sign change and can be solved by the same numerical implementation. We also investigate the crosstalk between parameter classes (velocity and attenuation) of the viscoelastic equation. More specifically, we show that the attenuation levels can be used to recover the quality factors of P and S waves, but that they are very sensitive to velocity errors. Finally, we present a synthetic example of viscoelastic FWI in the context of monitoring CO₂ geological sequestration. We show that FWI based on our formulation can indeed recover P- and S-wave velocities and their attenuation levels when attenuation is high enough. Both changes in velocity and attenuation levels recovered with FWI can be used to track the CO₂ plume during and after injection. Further studies are required to evaluate the performance of viscoelastic FWI on real data.

Full Waveform Inversion (FWI) is a seismic imaging technique that utilizes the entire content of seismic trace. This method aims to minimizes differences between real and synthetic seismograms. FWI consists of three main steps: forward modelling to simulate synthetic seismograms, adjoint simulation to calculate the gradient misfit, and optimization to obtain an appropriate model perturbation. FWI is an extremely non-linear inversion technique and computationally intensive since it involves many model parameters. In the context of geophysics, finding parameter models that describe the structure of the earth’s subsurface is the main goal. Geophysical measurements in the field only obtain data that is influenced by the parameter model, in this case seismic recording data, therefore mathematical techniques are studied and applied to determine the correct parameter model from the data obtained, also known as the inversion process. In this study, main research area is on optimization step which include mathematical approach to find perturbation model effectively. The Newton method can be used in the optimization step of FWI. However, its implementation is not promising due to high computational cost in calculating Hessian matrix. To reduce computation cost, an identity matrix is chosen to replace the Hessian matrix, as it usually done in the steepest descent method. Since FWI has a high level of non-linearity, steepest descent method requires many iterations to minimize the objective function. Therefore, this study uses a conjugate gradient method to make the data misfit minimized more quickly. Apart from that, the Line search technique is used to find an appropriate step-length at each iteration on both methods. The FWI workflow is applied to the synthetic acoustic velocity model. We conduct FWI using steepest descent and conjugate gradient methods. The results show that the conjugate gradient method requires fewer iterations than the steepest descent method to minimize the objective function. FWI model result using conjugate gradient method is closer with the true model compared to steepest descent method, showed on misfit result and model image. By comparing these methods we can see that conjugate gradient method can be chosen for application on FWI optimization step for better result on seismic imaging. On the industry scale, better results provide less uncertainty on the model parameters that useful for energy exploration or engineering.

P068 Full waveform inversion of multi-component OBC seismic data Abstract 1 Advances in computer technology have made full waveform inversion a more realistic and achievable process. In addition to overcoming the computational challenges involved the inverse problem itself has been helped through different techniques. The use of long-offset data and preliminary inversion of lower frequencies are seen to mitigate problems of nonlinearity. Medium wavelength features thought missing from near offset reflection data may then be resolved and convergence to a unique solution is made possible. However while the inversion of compressional (P) -wave velocity from marine streamer data is more

The quantitative reconstruction of subsurface earth properties from the propagation of waves follows an iterative minimization of a misfit functional. In marine seismic exploration, the observed data usually consist of measurements of the pressure field, but dual-sensor devices also provide the normal velocity. Consequently, a reciprocity-based misfit functional is specifically designed, and it defines the full reciprocity-gap waveform inversion (FRgWI) method. This misfit functional provides additional features compared to the more traditional least-squares approaches, in particular, in that the observational and computational acquisitions can be different. Therefore, the positions and wavelets of the sources from which the measurements are acquired are not needed in the reconstruction procedure and, in fact, the numerical acquisition (for the simulations) can be chosen arbitrarily. Based on 3D experiments, FRgWI is shown to behave better than full-waveform inversion in the same context. It allows for arbitrary numerical acquisitions in two ways: when few measurements are given, a dense numerical acquisition (compared to the observational one) can be used to compensate. However, with a dense observational acquisition, a sparse computational one is shown to be sufficient, for instance, with multiple-point sources, hence reducing the numerical cost. FRgWI displays accurate reconstructions in both situations and appears more robust with respect to crosstalk than least-squares shot stacking.

Summary Full Waveform Inversion (FWI) is now a well-established technology widely used to derive migration velocities in complex settings where conventional travel time tomography is inadequate. Most seismic contractors have developed their own flavours of FWI workflows which differ in several ways; they utilize different codes with their own strengths and weaknesses, utilize different parts of the recorded wavefield and follow different update-strategies. When running a production FWI project several complex choices must be made and their combined impact is difficult to predict. This paper investigates some of the impacts of different FWI approaches by comparing the results from a benchmark study where 3 seismic processing contractors used the same 2D data set to run FWI free of tomography. It was found that each of them have a distinctive piece of technology that helps to overcome specific challenges in a FWI production flow. Here we show and discuss their results, as well as their route map to get there.

Converted wave imaging is challenging, especially in the ultra-shallow waters offshore Abu Dhabi. We acquired the densest ever ocean bottom node (OBN) survey up to date, and demonstrated that additional field effort can significantly enhance the potential of converted waves as was proven mathematically prior. We used state-of-the-art processing techniques and utilizing auxiliary data, such as Surface Wave Inversion (SWI), Near field Hydrophone (NFH), Full Waveform Inversion (FWI) and joint non-linear PP-PS tomography with Common Conversion Point (CCP) criterion QC to achieve our objectives of velocity model building and, subsequently, reservoir characterization using Compressional-converted-Shear waves (PS). Physical limitations that are intrinsically related to wave conversion result in a PS image that is generally weaker than its PP counterpart, and that will always be the case. The novelty is in pushing the PS image quality through extending the usable bandwidth of the data encompassing a larger frequency range that was previously restricted to PP due to the natural signal to noise ratio profile relative to each wave type. Additionally, it has been theorized for a very long time that it is possible to derive anisotropic parameters epsilon (ε) and delta (δ) individually and not coupled in the form of eta (η=ε‒δ) as conventionally in PP processing and relying on drilled wells information. This can be done in a solely seismic driven manner without the need of well calibration relying instead on PS data and converted wave imaging, and with the use of the ratio between P wave velocity and S wave velocity (Vp/Vs), we end up having enough functions/equations to solve for the unknowns at hand.