Seismic waveform inversion in the frequency domain, Part 1: Theory and verification in a physical scale model

Abstract

Seismic waveforms contain much information that is ignored under standard processing schemes; seismic waveform inversion seeks to use the full information content of the recorded wavefield. In this paper I present, apply, and evaluate a frequency-space domain approach to waveform inversion. The method is a local descent algorithm that proceeds from a starting model to refine the model in order to reduce the waveform misfit between observed and model data. The model data are computed using a full-wave equation, viscoacoustic, frequencydomain, finite-difference method. Ray asymptotics are avoided, and higher-order effects such as diffractions and multiple scattering are accounted for automatically. The theory of frequency-domain waveform/wavefield inversion can be expressed compactly using a matrix formalism that uses finite-difference/finite-element frequency-domain modeling equations. Expressions for fast, local descent inversion using back-propagation techniques then follow naturally. Implementation of these methods depends on efficient frequency-domain forward-modeling solutions; these are provided by recent developments in numerical forward modeling. The inversion approach resembles prestack, reverse-time migration but differs in that the problem is formulated in terms of velocity (not reflectivity), and the method is fully iterative. I illustrate the practical application of the frequencydomain waveform inversion approach using tomographic seismic data from a physical scale model. This allows a full evaluation and verification of the method; results with field data are presented in an accompanying paper. Several critical processes contribute to the success of the method: the estimation of a source signature, the matching of amplitudes between real and synthetic data, the selection of a time window, and the selection of suitable sequence of frequencies in the inversion. An initial model for the inversion of the scale model data is provided using standard traveltime tomographic methods, which provide a robust but low-resolution image. Twenty-five iterations of wavefield inversion are applied, using five discrete frequencies at each iteration, moving from low to high frequencies. The final results exhibit the features of the true model at subwavelength scale and account for many of the details of the observed arrivals in the data.

Acceleration strategies for elastic full waveform inversion workflows in 2D and 3D

“…1 is typically an irregular surface, and hence, finding a global minimum can be challenging if not impossible. In this context, multi-scale techniques [4,9,32,35,40,48], combined with gradient preconditioning and regularization methods, have been developed to sequentially incorporate higher wavenumber information into the inverted model. This results in a reduction of local minima effects by means of either selecting increasing individual frequency samples (in frequency domain) or broadening a low-pass filter by increasing its corner frequency (in time domain).…”

“…Nevertheless, and given the number of parametres to be inverted, statistical methods such as Monte Carlo are still not applicable for large 3D problems, especially when high-frequency contents are involved. In this sense, the centre piece of FWI is the so-called adjoint method, which allows us to obtain the gradient of the misfit function for the current model by cross-correlating both incident and back-propagated wavefield [15,16,31,32,44]. With the gradient at hand, an iterative optimization problem can be set in order to find the model that fits best the field data.…”

A parallel evolution strategy for an earth imaging problem in geophysics

“…Therefore, an additional regularization term is classically added to the inversion problem to make it well posed [50]. In addition to the velocity model, the source excitation is generally unknown and must be included as an unknown of the problem [35]. Provided that a good starting velocity model m s is available (good in the sense that smoothly represents the structure of the true velocity model), the minimization of the objective function (4) is in practice solved using a Newton type method (see [52] and references therein).…”

Full-waveform inversion in acoustic orthorhombic media and application to a North Sea data set

“…Over the past three decades, most FWI techniques have been designed to recover only a P-wave velocity model because of the high computational cost (e.g., Gauthier et al, 1986;Pratt, 1999). The recent progress in high-performance computing and the improvement in data acquisition resulted in the extension of FWI to 2D and 3D acoustic and elastic transverse isotropic media with a vertical axis of symmetry (VTI) (e.g., Warner et al, 2013;Operto et al, 2014;Wu and Alkhalifah, 2016;Kamath et al, 2017).…”