HR: 1340h
AN: S43B-1313    [Abstracts]
TI: Reconstructing impedance from noisy data using Bayesian inversion: preliminary results
AU: * Sanguinetti, R
EM: rrsangui@mail.uh.edu
AF: University of Houston, 312 Science & Research Building I, Houston, TX 77204, United States
AB: Acoustic impedance inversion is a technique for estimating acoustic impedance of subsurface layers from stacked seismic data. If we consider the earth as a series of discrete impedance layers, then each interface has associated with it a reflection coefficient. A seismic trace is just this series of reflection coefficients convolved with a seismic wavelet. Inversion attempts remove the seismic wavelet from the reflection series. Once we have the reflection coefficients it is relatively simple process to reconstruct the impedance earth layers. The task is the recovery of the acoustic impedance from a band-limited reflection seismogram. In terms of inverse theory, it assumed that impedance constraints are provided at different time levels. Also, noise in the trace is modeled by means of the usual Gaussian assumption. The uncertainties of the constraints are also assumed Gaussian. Thus, Bayes's rule is used to combine the data likelihood (by data I mean trace and constraints) with the prior probability of input reflectivity to construct the posteriori distribution of the output reflectivity. Thus, the maximum a posteriori (MAP) solution will be computed by minimizing the associated cost function. On the other hand, a reference model is used. For reaching it, it is necessary to perform the deconvolution of the desired reflectivity, from the observed data. This simple model must be modified to include the presence of random noise, where it emphasizes that r represents the primary reflectivity, i.e., the reflectivity of the layered earth assuming that multiple events have been attenuated and true amplitudes have been recovered (as well as possible) by an initial processing. This term is included into the cost function, and used as a reference model. Once the reference model, wavelet and impedance constraints, and the cost function are set, the inverse modeling is performed: the hyper-parameter is set, and the Cauchy prior is calculated, and added to the cost function. The minimization algorithm is based on the Conjugate Gradient (CG) extension for nonlinear functionals: the CG strategy can find the minimum of the cost function in a reasonable number of iterations. However, as well known, an incorrect selection of the hyper-parameter may yield a solution that is unreasonable. Technically speaking, if it is too small the data misfit is small, which indicates that over-fitting occurred. On the other hand, if it is too large, the regularization term will dominate the cost function, and the problem reduces to find the minimum. From the inversion, it is possible to see recovered models with (or without) using the reference model. However, these show some distortion around the spike, which could be telling us that the solution is pretty unstable. After that, an impedance estimate is reached: this solution is affected by the remaining noise which is seen as smooth distortions of the recovered reflectivity function in both sides of the spike model. So, the use of a reference model improves the solution. In our case, the use of this reference model helped to improve the performance of the inversion algorithm, giving us better results.
DE: 7290 Computational seismology
SC: Seismology [S]
MN: 2007 Fall Meeting