Full waveform inversion (FWI) of 3-D data sets has recently been possible thanks to the development of high performance computing. However, FWI remains a computationally intensive task when high frequencies are injected in the inversion or more complex wave physics (viscoelastic) is accounted for. The highest computational cost results from the numerical solution of the wave equation for each seismic source. To reduce the computational burden, one well-known technique is to employ a random linear combination of the sources, rather that using each source independently. This technique, known as source encoding, has shown to successfully reduce the computational cost when applied to real data. Up to now, the inversion is normally carried out using gradient descent algorithms. With the idea of achieving a fast and robust frequency-domain FWI, we assess the performance of the random source encoding method when it is interfaced with second-order optimization methods (quasi-Newton l-BFGS, truncated Newton). Because of the additional seismic modelings required to compute the Newton descent direction, it is not clear beforehand if truncated Newton methods can indeed further reduce the computational cost compared to gradient algorithms. We design precise stopping criteria of iterations to fairly assess the computational cost and the speed-up provided by the source encoding method for each optimization method. We perform experiment on synthetic and real data sets. In both cases, we confirm that combining source encoding with second-order optimization methods reduces the computational cost compared to the case where source encoding is interfaced with gradient descent algorithms. For the synthetic data set, inspired from the geology of Gulf of Mexico, we show that the quasi-Newton l-BFGS algorithm requires the lowest computational cost. For the real data set application on the Valhall data, we show that the truncated Newton methods provide the most robust direction of descent.
We present some methodological aspects of full waveform inversion (FWI) in 3D elastic media. FWI is a local optimization scheme aiming to update the initial model with perturbations in order to minimize the misfit between the computed and observed data. The pertubations are found in the opposite direction of the gradient which can be evaluated efficiently with the adjoint state method. While this method is generally applied to a second order expression of the wave equation, we develop our inversion scheme on a first-order velocity-stress formulation which provides a great flexibility to recast the wave equation in pseudo-conservative form. We show how to take advantage of this formalism to develop the gradient of the misfit function with the adjoint-state method in a straightforward way. The inversion is implemented in the frequency domain, while the seismic modeling is performed in time. We propose also an abstraction concept between the forward and inverse problems that allows us to use different modeling engines in the inversion code and to perform target-oriented imaging.
We present an application of 2D acoustic frequency-domain Full Waveform Inversion (FWI) to the hydrophone component of 4-C ocean bottom cable (OBC) data recorded from the Valhall field in North sea. The starting model for FWI was built by reflection traveltime tomography (RTT). Although this starting model leads to flat common-image gathers (CIGs), it does not allow us to match first-arrival traveltimes of diving waves from above the gas layers. This mismatch between vertical and horizontal velocities is likely the footprint of anisotropy. We updated the RTT model by first-arrival traveltime tomography (FATT) to build a new starting model for FWI. The velocities above the gas layers of the updated model are significantly higher than velocities from in-well seismic (VSP) data. FWI models were computed from the two starting models just mentioned. More stable results were obtained with the starting model updated by FATT. The resulting FWI model shows a reasonable agreement with a former model developed by 3D FWI. A reasonable match of both short-aperture and wideaperture components of the data was obtained by isotropic FWI. This might indicate that layer-induced anisotropy was created by FWI in the gas layers to balance the increase of the shallow velocities created by the inversion of the wide-aperture data components.
Full waveform inversion (FWI) of seismic traces recorded at the free surface allows the reconstruction of the physical parameters structure on the underlying medium. For such a reconstruction, an optimization problem is defined, where synthetic traces, obtained through numerical techniques as finite-difference or finite-element methods in a given model of the subsurface, should match the observed traces. The number of data samples is routinely around 1 billion for 2D problems and 1 trillion for 3D problems while the number of parameters ranges from 1 million to 10 million degrees of freedom. Moreover, if one defines the mismatch as the standard least-squares norm between values sampled in time/frequency and space, the misfit function has a significant number of secondary minima related to the ill-posedness and the nonlinearity of the inversion problem linked to the so-called cycle skipping. Taking into account the size of the problem, we consider a local linearized method where gradient is computed using the adjoint formulation of the seismic wave propagation problem. Starting for an initial model, we consider a quasi-Newtonian method, which allows us to formulate the reconstruction of various parameters such as P and S waves velocities or density or attenuation factors. A hierarchical strategy based on the incremental increase of the data complexity starting from low-frequency content to high-frequency content, from initial wavelets to later phases in the data space from narrow azimuths to wide azimuths and from simple observables to more complex ones. Different synthetic examples on realistic structures illustrate the efficiency of this strategy based on the data manipulation. This strategy related to the data space has to be inserted into a more global framework where we could improve significantly the probability to converge to the global minimum. When considering the model space, we may rely on the construction of the initial model or add constraints such as smoothness of the searched model and/or prior informations collected by other means. An alternative strategy concerns the building of the objective function and various possibilities must be considered, which may increase the linearity of the inversion procedure.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.