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## Synthetic realizations

 Steps Domain Operation for fGn Operation for fBm 1 Space Generate Gaussian white noise with zero mean and unit variance 2 Fourier Generate energy spectrum 3 Fourier  4 Space Obtain correlated fGn Obtain correlated fBm Table 2. Synthesis of correlated heterogeneous media by generating fractional Gaussian noise (fGn) or fractional Brownian motion (fBm) with the spectrum . The spatial Fourier transforms in dimensions of the distributions and are respectively and with , and .

Our method of synthesizing correlated random media is summarized in Table 2 for fractional Gaussian noise (fGn) and fractional Brownian motion (fBm). The Gaussian nature of the initial white noise is contained in the phase of its Fourier transform , whereas the amplitude is constant with frequency because the noise is white. The causal integration, to produce the fBm, is performed by a phase rotation in the Fourier domain, strictly equivalent to the Hilbert transform. The fractal property, below spatial scale , is imposed by the amplitude of the von Kármán model.    signal,f2dfile,cgaussb,fcgaussbp
Figure 1.
Variations of in a high-resolution reservoir model based on seismic and well data from a field in Canada (a) and its spectral energy density (b). Synthetic realization of 2D fGn using the von Kármán spectral amplitude with the exponent and elliptical anisotropy (c,d).         For dimension , the exponent in equation 7 is . Therefore, the energy spectrum of the 2D fGn with in Figure 1 is . Klimes (2002) in fact proposed using to synthesize geologically realistic 2D models with the von Kármán function. Because sediments are made up of layers, we consider the autocorrelation function to be vertical transverse isotropic. We use two different correlation lengths and for horizontal and vertical directions, and define the Riemannian relative distance (Goff and Jordan, 1988): (12)

Figure 1 shows, for comparison, the signal and associated energy spectrum of the synthetic fGn and of a 2D section from a high-resolution model of a clastic reservoir in Canada. The spectrum of the synthetic heterogeneous medium is similar to the one from the reservoir model, but the synthetic fGn, although exhibiting some comparable roughness in the space domain, does not contain coherent, large geological structures, i.e. folded beds.    Fractal heterogeneities in sonic logs and low-frequency scattering attenuation  Next: Nonlinear parameter estimation on Up: Statistical model of heterogeneities Previous: Fractal statistics

2013-07-26