Fractal heterogeneities in sonic logs and low-frequency scattering attenuation

Dominant frequency versus depth

If seismic pulse is defined as a Ricker wavelet, a relation can be derived for modification of the frequency content of P and S acoustic waves by scattering attenuation. Dominant frequency $f_{{dom}}(z)$ with depth $z$ and initial spectrum $S_I(f)$ of the source are defined by

$$S_I(f) = \left(\frac{f}{f_0}\right)^2e^{-f^2/f_0^2},$$ (35)

$$f_{{dom}}(z) = \frac{d}{df}\left(S_I(f)\,e^{-z/d(f)}\right),$$ (36)

with initial condition $f_{{dom}}(0)=f_0$, and where $d(f)$ is the penetration depth defined in equation 34. Dispersion involves different traveltimes at different frequencies but does not modify the frequency content or amplitude. For convenience, we estimate dominant frequency as frequency expectation:

$$f_{{dom}}(z) = \frac{1}{\langle S_I \rangle_z}\int f\,S_I(f)\,e^{-z/d(f)}\,df\,,$$ (37)

$$\langle S_I \rangle_z = \int S_I(f)\,e^{-z/d(f)}\,df\,.$$ (38)

Figure 8 shows the evolution of the dominant frequency with depth in fractal media, with $V_P=2700$ m/s, $V_S=1230$ m/s, and standard deviation $\sigma=30\,\%$. The value of correlation length $b$ again has a very high impact, whereas the fractal exponent moderately influences results. For a multicomponent seismic survey in a clastic reservoir, evolution of the peak frequency should show a more important decrease with depth for PS data than for PP data.

fdomfb25025,fdomfb05M025,fdomfb05025,fdomfb0505,fdomfb10025,fdomfb05075
Figure 8. Evolution of the dominant frequency with depth for P (solid line) and S (dashed line) scalar waves modeled by a Ricker wavelet ($f_0=60$Hz) in heterogeneous media with $\sigma=30~\%$. For a constant exponent $H=0.25$, the dominant frequency shifts to lower frequencies faster for larger values of $b$ (a,c,e). The exponent $H$ weakly influences the evolution of the dominant frequency (b,d,f).

Fractal heterogeneities in sonic logs and low-frequency scattering attenuation