A correlation-based misfit criterion for wave-equation traveltime tomography

Abstract

S U M M A R YWave-equation traveltime tomography tries to obtain a subsurface velocity model from seismic data, either passive or active, that explains their traveltimes. A key step is the extraction of traveltime differences, or relative phase shifts, between observed and modelled finite-frequency waveforms. A standard approach involves a correlation of the observed and measured waveforms. When the amplitude spectra of the waveforms are identical, the maximum of the correlation is indicative of the relative phase shift. When the amplitude spectra are not identical, however, this argument is no longer valid. We propose an alternative criterion to measure the relative phase shift. This misfit criterion is a weighted norm of the correlation and is less sensitive to differences in the amplitude spectra. For practical application it is important to use a sensitivity kernel that is consistent with the way the misfit is measured. We derive this sensitivity kernel and show how it differs from the standard banana-doughnut sensitivity kernel. We illustrate the approach on a cross-well data set.In ray-based tomography, the aim is to construct a subsurface velocity model that explains the picked traveltimes of the measured data. Such a model can be obtained in an iterative manner by back projecting the traveltime differences along rays in the current velocity model. This procedure will lead to satisfactory results when the wave propagation is sufficiently well approximated by ray theory. To extract more information from the data than just the traveltimes of a few selected arrivals, seismologists are moving towards full-waveform processing and inversion of all available data. This trend is driven by the availability of high-quality broad-band data (earthquake data from USArray, for example) and a need to incorporate finite-frequency effects to process data from geologically complex areas (sub-salt exploration for the detection of hydrocarbons, for example). Also, the computing resources needed to routinely model 3-D wave-propagation in complex media are becoming a commodity. There have been two major developments regarding finite-frequency or full-waveform inversion in the last decades. The first is least-squares inversion-or waveform tomography-of seismic data pioneered by Tarantola & Valette (1982) that was aimed originally at inversion of active seismic reflection data. The second is the development of a finite-frequency analogue of ray-based traveltime tomography by Luo & Schuster (1991) (see also TrompRe-use of this article is permitted in accordance with the Terms and Conditions set out at

An Auxiliary Bump Functional to Overcome Cycle Skipping in Waveform Inversion

“…Also, the surface located at z = 0 is assumed to be non-reflecting. We use a time-domain staggered-grid finite-difference code for the constant-density acoustic wave equation to model the data and to perform the adjoint wavefield computations required for calculating the gradients (Tarantola, 1984;Fichtner, 2010). The optimization was performed by the conjugate-gradient method.…”

“…Functionals that give more weight to the kinematic error than to the amplitude error between the seismic waveforms. They often involve the cross-correlation between the observed and modelled waveforms (Luo and Schuster, 1991;van Leeuwen and Mulder, 2010;van Leeuwen, 2010). After cross-correlation the arrival-time error can be picked by hand, which is tedious.…”

“…Full waveform inversion (FWI) is a non-linear optimization procedure that estimates the some of the Earth's parameters by least-squares fitting of the recorded arrivals in the seismic data (Tarantola, 1984;Virieux and Operto, 2009;Fichtner, 2010). Due to the computational cost of the wave-equation modelling, the optimization is usually performed with gradient-based techniques.…”

Adaptive waveform inversion: Theory

“…Although the genesis of our approach originally lay elsewhere, AWI is mathematically more closely related to the approaches of van Leeuwen and Mulder (2010) and Luo and Sava (2011); we discuss the relationship of AWI with these and other methods later in the paper.…”

Methods for source-independent Q estimation from microseismic and crosswell perforation shot data in a layered, isotropic viscoelastic medium

“…For traveltime tomography without picking, Van Leeuwen and Mulder (2010) proposed a weighted correlation-based misfit function that reduces sensitivity to phase shifts in the incorrect source wavelet. As a natural generalization, Luo and Sava (2011) proposed a deconvolution-based misfit function.…”