Modeling 3-D anisotropic fractal media

Modeling seismic impedances

Seismic impedances have frequently been modeled as a Markov process. Godfrey et. al. (1980) modeled impedance as a special type of Markov chain, one that is constrained to have a purely exponential correlation function. They tested their method on three actual logs and compared the autocorrelation function to a best fit exponential curve. Apart from a small geologic noise component at the origin, their results showed excellent agreement between the theoretical exponential and the actual autocorrelation on two of the well logs they considered. For large lags, the actual correlation function had exponential behavior similar to that of the theoretical curve, but all data points fell below the synthetic curve showing a faster decay rate. The behavior of the autocorrelation could very well be interpreted as related to a rougher distribution than that predicted by the exponential correlation. A von Karman correlation function with a Hurst number smaller than 0.5 would have given a better fit to the autocorrelation of the impedance series. The autocorrelation of the impedance function provides information on the depositional pattern in the sedimentary column i.e, cyclic or transitional (O'Doherty and Anstey, 1971).

Figure 5 shows a comparison of one-dimensional random sequences that simulate synthetic impedances with von Karman correlation functions of varying fractal dimensions (i.e., Hurst number $\nu$). Again the smaller the value of $\nu$, the rougher the sequence. The impedance with exponential correlation seems smooth compared to the ones generated from autocorrelation functions with values of $\nu$ lower than 0.5.

imped
Figure 5. Synthetic random sequences simulating acoustic impedances with von Karman autocorrelation for varying Hurst number $\nu$.

Modeling 3-D anisotropic fractal media