226 Geophysics in Geothermal Exploration of this variable during seismic inversion under this hypothesis is called acoustic inversion or post-stack inversion. When working on gathers, or on partial angle-stacked seismic data, the Zoeppritz equation can be applied. In practice, the Aki-Richards equation (Aki and Richards, 1980), a simplification, is preferred during elastic inversion or pre-stack inversion, whose partial derivatives are linear and offering a valid approximation up to an incidence angle of 45°. These elements allow good behavior during numerical optimization (inverse problem) and good optimization of P-impedances and S-impedances. Density, however, is not well optimized by this process; other techniques allow improving this result. As we will describe later, dealing with not only one but two variables to explain the characteristics of the reservoir allow capturing combined changes (lithology and fluid, or lithology and porosity, for instance). Figure 7.3 summarizes the two techniques in terms of inputs and outputs, respectively for acoustic inversion (left) and elastic inversion (right). Figure 7.3 Results obtained according to the full-stack (acoustic) or angle-stack (elastic) assumption. 7.1.3 Convolution and resolution Considering the medium as a series of reflection coefficients, the seismic response, in the two-way time domain, results from the convolution of this with an impulse response, called a wavelet. This operator has its own characteristics: (1) shape, (2) frequency spectrum, and (3) phase spectrum. It represents the impulse signal of the
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