Elastic Wavefield Inversion of Reflection and Transmission Data , by Peter Mora

Elastic inversion of seismic data can be performed by finding the earth parameters (P- and S-wave velocities and density) that minimize the square error between the wavefield computed using this model and the observed wavefield (Tarantola (1984) and Mora (1986a)). The resultant model is a maximum probability solution provided the assumptions implied using least squares, namely Gaussian probability distributions of the data and model parameters, are valid. This is not a bad assumption considering the elastic inversion algorithm assumes the elastic wave equation and therefore accounts for S-waves, mode conversions, head waves and rayleigh waves etc. that are normally considered as coherent noise. As the number of shot profiles used in an inversion is increased, the signal to noise ratio increases and there are more illumination angles of the seismic waves on the subsurface. More illumination angles implies a more complete picture of the subsurface can be obtained. Even using a single shot profile, Mora (1986a) showed with some synthetic studies that a reasonable resolution can be achieved (both spatially and between the P- and S-wave velocities). When many shots are used to invert surface seismic data (reflection data), the result of the inversion is comparable to the expected result a prestack elastic shot profile depth migration, namely, a high frequency image of P- and S-wave velocities and density. However, compared to migration, artifacts are smaller, the P and S-wave velocity images are better resolved from one another, the magnitudes of the velocity and density perturbations have significance in an absolute sense and there is a slight increase in the lower frequency components of the image making this result easier to interpret (see also Mora (1986a)). Synthetic studies show that when signal to noise ratios are high, that an inversion of just a few shots profiles results in a \fIcomplete\fR image of the subsurface (whereas conventional processing (stack and migration) would have required at least an order of magnitude more data (Ronen (1985) comes to the same conclusion)). In comparison to the case of inversion of reflection data where only the high frequency part of the model can be resolved, inversion of transmission data yields a result containing \fIboth high and low frequencies\fR (and hence blockiness in the model due to layering and other gross features). Transmission data (such as offset VSP and/or well to well data) contains direct P- and S-waves that are most affected by the low frequency blocky velocity perturbations which cause significant delays to these direct waves. If multiple shots are used and hence the direct waves have illuminated the subsurface at many different angles, then there is enough redundancy to resolve these gross features (this is comparable to elastic diffraction tomography, see Devaney (1984) for the case of linearized acoustic diffraction tomography). Synthetic studies show that even using offset VSP's from only two wells located on either side of a region of interest, it is possible to obtain an inversion result that is almost identical to the true solution.


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