Waveform acoustic impedance inversion with spectral shaping

Full waveform inversion applied to surface seismic data containing only reflection data generally gives an impedance map, the background velocity being assumed known. The first iteration update does not have a spectrum close to the Earth impedance spectrum because of source wavelet and wave propagation effects. To improve the convergence, these effects can be compensated by designing a spectral shaping filter that produces a gradient of the misfit function with a spectrum similar to the Earth spectrum. Based on an asymptotic analysis of the gradient of the misfit function, we rederive the theoretical spectral shaping filter. When the observed source wavelet is known or can be estimated from the data, we retrieve that, after source deconvolution/whitening of the data, the theoretical spectral shaping is in ω−β/2 for the data or in k −β for the gradient with ω the angular frequency and k the wavenumber. β is an exponent depending on acquisition and equal to 1 with areal (so-called 3-D) acquisition and to 2 with line (so-called 2-D) acquisition. Under acoustic assumption, this leads to a waveform acoustic impedance inversion approach. We test this approach with a small synthetic example and with a real data set. Since we did not use a priori impedance information to derive the spectral shaping, we validate the approach by comparing the spectrum of the inverted impedances with the one of the Earth impedance computed from well-log measurements. The results illustrate the relevance of the spectral shaping to improve the convergence of the waveform inversion of reflection data.