Frequency-power-law scattering by fractal inclusions
Abstract
Since the 1980s, the concept of fractals has been increasingly adopted in geophysics. Here, we study scattering by fractal inclusions which, under isothermal conditions, causes energy loss of the directly propagated field. If the medium follows a fractal model, then the effective sound velocity c averaged over the characteristic length is c(ω)proptoω1-α, where ω is the angular frequency and the parameter α characterizes the scale dependence of the bulk or shear modulus. This frequency-power-law (FPL) relation leads to a wave equation of hereditary type with a fractional-power derivative operator. The corresponding causal point-source solution has an analytic form. It is expressed in terms of the special function fα(t)=L-1srightarrow t\{exp(-sα)\}, where L-1srightarrow t represents the one-sided inverse Laplace transform supported on tin[0,propto]. Memory effects cause smoothing of the wavefield in the vicinity of the wavefront and a rapid amplitude decay far from the wavefront. In addition, a significant FPL pulse delay can be identified. In practice, this time delay can be measured during migration velocity analysis to estimate the fractal dimension D related to the dispersion parameter α. To examine the FPL dependence of direct body waves propagating in a homogeneous medium containing fractal inhomogeneities, we compute acoustic finite-difference snapshots in the frequency range f= 20-200 Hz. In contrast to regular fractals, the percolation model is constructed using the theory of percolation pertaining to stochastic geometry. This model is universal in the sense that it arises from randomly distributed particles provided their concentration is sufficiently high. This is why the use of percolation models allows quantitative characterization of real fracture/fault patterns and multi-scale interpretation of log data. The fractal structure of the model is very prominent because of frequency-dependent scattering with intrinsic correlation over a wide range of scales. We start by computing the F-K spectrum P for the scattered data. Next, the peak envelope of the k-averaged spectrum (P) is computed as a function of frequency f. The slope of a least-squares fit of the data (P) in the log-log domain gives an estimate of D. Results are important for inelastic depth imaging, inverse Q filtering, fracture detection, and integrated geophysical reservoir monitoring.
- Publication:
-
EGS - AGU - EUG Joint Assembly
- Pub Date:
- April 2003
- Bibcode:
- 2003EAEJA.....5034R