Estimation of 2D random medium parameters from post-stack seismic data

To some extent, earth medium can be considered as random medium. The heterogeneity of random medium in true physical properties such as velocity may be mathematically descripted by using statistical parameters, including mean, standard deviation and auto-correlation function (ACF). The statistical parameters are very useful for the heterogeneity-constrained wave impedance inversion and reservoir characterization, which could give the results closer to the actual earth media. In the past, the heterogeneity research on earth medium is based on well logs so that can obtain the mean, standard deviation and only one-dimensional vertical ACF at well location. Here we propose a quantitative approach to estimate the 2D ACF of seismic velocity filed from post-stack seismic data. We assume the velocity field of medium is stochastic process in space and post-stack seismic data obey the Robinson convolutional model that the recorded 2D seismogram s(x, t) is the convolution of a reflectivity e(x, t) and a seismic wavelet w(t). In the wavenumber-frequency domain, the power spectrum of post-stack seismic data is equal to product of power spectra of reflectivity and wavelet. So according to Wiener-Khintchine theorem, we can find the ACF of reflectivity and furthermore get ACF of velocity field after we extract the wavelet and calculate its spectrum. With ACF of velocity field, the three parameters, ACF length a in horizontal direction and b in vertical direction and angle θ are then estimated. The numerical tests of 2D stationary and non-stationary random medium models show that the approach of estimating the statistical parameters of 2D random medium from post-stack seismic data is feasible and robust.