ALBERT

Description: 〈span〉〈div〉Summary〈/div〉Full waveform inversion (FWI) is a nonlinear waveform matching procedure, which suffers from cycle skipping when the initial model is not kinematically-accurate enough. To mitigate cycle skipping, wavefield reconstruction inversion (WRI) extends the inversion search space by computing wavefields with a relaxation of the wave equation in order to fit the data from the first iteration. Then, the subsurface parameters are updated by minimizing the source residuals the relaxation generated. Capitalizing on the wave-equation bilinearity, performing wavefield reconstruction and parameter estimation in alternating mode decomposes WRI into two linear subproblems, which can solved efficiently with the alternating-direction method of multiplier (ADMM), leading to the so-called iteratively refined wavefield reconstruction inversion (IR-WRI). Moreover, ADMM provides a suitable framework to implement bound constraints and different types of regularizations and their mixture in IR-WRI. Here, IR-WRI is extended to multiparameter reconstruction for VTI acoustic media. To achieve this goal, we first propose different forms of bilinear VTI acoustic wave equation. We develop more specifically IR-WRI for the one that relies on a parametrisation involving vertical wavespeed and Thomsen’s parameters δ and ε. With a toy numerical example, we first show that the radiation patterns of the virtual sources generate similar wavenumber filtering and parameter cross-talks in classical FWI and IR-WRI. Bound constraints and TV regularization in IR-WRI fully remove these undesired effects for an idealized piecewise constant target. We show with a more realistic long-offset case study representative of the North Sea that anisotropic IR-WRI successfully reconstruct the vertical wavespeed starting from a laterally homogeneous model and update the long-wavelengths of the starting ε model, while a smooth δ model is used as a passive background model. VTI acoustic IR-WRI can be alternatively performed with subsurface parametrisations involving stiffness or compliance coefficients or normal moveout velocities and η parameter (or horizontal velocity).〈/span〉

Description: 〈span〉〈div〉SUMMARY〈/div〉Full-waveform inversion (FWI) is a waveform matching procedure, which can provide a subsurface model with a wavelength-scale resolution. However, this high resolution makes FWI prone to cycle skipping, which drives the inversion to a local minimum when the initial model is not accurate enough. Other sources of non-linearities and ill-posedness are noise, uneven illumination, approximate wave physics and parameter cross-talks. All these sources of error require robust and versatile regularized optimization approaches to mitigate their imprint on FWI while preserving its intrinsic resolution power. To achieve this goal, we implement bound constraints and total-variation (TV) regularization in the so-called frequency-domain wavefield reconstruction inversion (WRI) with the alternating direction method of multipliers (ADMM). In the ADMM framework, WRI relies on an augmented Lagrangian function, a combination of penalty and Lagrangian functions, to extend the FWI search space by relaxing the wave-equation constraint during early iterations. Moreover, ADMM breaks down the joint wavefield reconstruction plus parameter-estimation problem into a sequence of two linear subproblems, whose solutions are coordinated to provide the solution of the global problem. The decomposability of ADMM is further exploited to interface in a straightforward way bound constraints and TV regularization with WRI via variable splitting and proximal operators. The resilience of our regularized WRI formulation to cycle skipping and noise as well as its resolution power are illustrated with two targets of the large-contrast BP salt model. Starting from a 3Hz frequency and a crude initial model, the extended search space allows for the reconstruction of the salt and subsalt structures with a high fidelity. The TV regularization filters out the imprint of ambient noise and artefacts associated with multiscattering and Gibbs effects, while fostering large-contrast reconstruction. Compared to other TV-regularized WRI implementations, the proposed method is easy to tune due to its moderate sensitivity to penalty parameters and does not require a prior guess of the TV-norm ball.〈/span〉

Description: 〈span〉〈div〉Summary〈/div〉Full waveform inversion (FWI) is a waveform matching procedure, which can provide a subsurface model with a wavelength-scale resolution. However, this high resolution makes FWI prone to cycle skipping, which drives the inversion to a local minimum when the initial model is not accurate enough. Other sources of nonlinearities and ill-posedness are noise, uneven illumination, approximate wave physics and parameter cross-talks. All these sources of error require robust and versatile regularized optimization approaches to mitigate their imprint on FWI while preserving its intrinsic resolution power. To achieve this goal, we implement bound constraints and total variation (TV) regularization in the so-called frequency-domain wavefield-reconstruction inversion (WRI) with the alternating direction method of multipliers (ADMM). In the ADMM framework, WRI relies on an augmented Lagrangian function, a combination of penalty and Lagrangian functions, to extend the FWI search space by relaxing the wave-equation constraint during early iterations. Moreover, ADMM breaks down the joint wavefield reconstruction plus parameter estimation problem into a sequence of two linear subproblems, whose solutions are coordinated to provide the solution of the global problem. The decomposability of ADMM is further exploited to interface in a straightforward way bound constraints and TV regularization with WRI via variable splitting and proximal operators. The resilience of our regularized WRI formulation to cycle skipping and noise as well as its resolution power are illustrated with two targets of the large-contrast BP salt model. Starting from a 3Hz frequency and a crude initial model, the extended search space allows for the reconstruction of the salt and subsalt structures with a high fidelity. The TV regularization filters out the imprint of ambient noise and artifacts associated with multi-scattering and Gibbs effects, while fostering large-contrast reconstruction. Compared to other TV-regularized WRI implementations, the proposed method is easy to tune due to its moderate sensitivity to penalty parameters and does not require a prior guess of the TV-norm ball.〈/span〉

Description: 〈span〉In the original version of this paper, Algorithm 1 (BTV regularized IR-WRI algorithm based on the PRS algorithm) was omitted. This has now been corrected.〈/span〉