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

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.