ABSTRACT

The article titled “Wavelengths of earth structures that can be resolved from seismic reflection data” by Jannane, et al., published in 1989 in Geophysics, followed Albert Tarantola’s idea and requested for proofing through the use of waveform least squares criteria, the brilliant intuition from J. Claerbout about the earth structures wavelengths that could be resolved from seismic reflection data. This intuition was described and summarized by a simple but meaningful graph into 1985 Claerbout’s book, “Imaging the Earth’s Interior” (p47, Fig.1.4-3, Reliability of information obtained from surface seismic measurements). By the experiments shown in this article, it clarified the resolution potential from LS waveform inversion, whatever the inversion technique which is used could be. It resulted into an experimental graph, extremely similar to Claerbout’s one. The methodology and techniques used are described here below. The aim of inverting seismic waveforms is to obtain the “best” earth model. The best model is defined as the one producing seismograms that best match (usually under a least-squares criterion) those recorded. Our approach is nonlinear in the sense that we synthesize seismograms without using any linearization of the elastic wave equation. Since we use rather complete data sets without any spatial aliasing, we do not have the problem of secondary minima (Tarantola, 1986, Geophysics. 51, 1893-1903). Nevertheless, our gradient methods fail to converge if the starting earth model is far from the true earth (Mora, 1987, Geophysics. 52, 121 l-1228; Kolb et al.. 1986, Proc. IEEE, 74, 498-508; Pica et al.. 1989, Geophysics, 55, 284-292). Our original motivation was to clear up this convergence problem; however, our investigation has provided a determination of which parameters of the earth model are resolved by a typical data set. To determine those parameters, we perturb some reference earth model with quasisinusoidal perturbations and measure the least-squares misfit between the data set generated (nonlinearly) from the reference model and the data sets generated from the perturbed models. Clearly, if the data are identical, the least-squares measure of misfit is zero: the perturbation belongs to the model null space, where a priori information may permit a choice, but data-based inferences may not. That is, the data are insufficient to resolve subsurface structures for certain ranges of wavelengths. Later on, Neves and Singh (1996, Geophys. J. Int. 126, 470–476) demonstrated that the model null space can be filled within some specific conditions, in particular by increasing the offset range, up to the critical angle at least. Claerbout’s intuition, Tarantola’s demonstration and further investigations from Neves and Singh, allowed for a better understanding of the resolution that can be expected from waveform least squares inversion. These past works still allow nowadays for a better understanding of any of modern waveform inversion experiment of seismic reflection data made in the industry or in the academic environment.

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