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Penetration depth

Waves propagating in disordered media are exponentially attenuated by scattering (O'Doherty and Anstey, 1971; White et al., 1990). We define penetration depth $d(f)$ to be the skin depth (van der Baan et al., 2007) for low-frequency waves propagating in the heterogeneous medium:

$\displaystyle \frac{1}{d(f)}$ $\textstyle =$ $\displaystyle \frac{k_0}{2Q} = Im[k]\,,$ (33)
$\displaystyle d(f)$ $\textstyle =$ $\displaystyle \frac{b~(1+4\,b^2k_0^2)^{H+\frac{1}{2}}}
{\sigma^2\,(k_0b)^2\,C^{(1)}_{H}\left[(1+4\,b^2k_0^2)^{H+\frac{1}{2}}-1\right]}\,,$ (34)

where $f$ is frequency. Penetration depth $d(f)$ corresponds to a decrease of wave amplitude by $1/e$. For two-way traveltime, recorded amplitude is 14 % of the initial one. Figure 6 shows the frequency dependence of the penetration depth for different values of parameters $b$ and $H$ and the scattering attenuation for acoustic P- and S- waves. Seismic background velocities are $V_P=2700$ m/s and $V_S=1230$ m/s.

depthlfb05M025 depthlfb25025 depthlfb05025 depthlfb10025 depthlfb05075 llqfb05025
Figure 6.
Penetration depth for P (solid) and S (dashed) scalar waves in heterogeneous media described by the von Kármán model with $\sigma=30~\%$. For $b=5$m, a higher exponent $H$ decreases the penetration (a,c,e). The value of $b$ strongly influences the penetration of waves (b,d). The slope break in the attenuation curve (f) determines the frequency below which our scattering model is valid.
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Scattering attenuation $1/Q$ is proportional to $1/\lambda^3$ at low frequencies and corresponds to the Rayleigh diffusion regime. Attenuation increases at higher frequencies, where $1/Q$ is proportional to $1/\lambda$ and wavelength is comparable to the size of the heterogeneities. Nevertheless, the validity of our scattering theory is constrained to the low-frequency bandwidth until the attenuation curves in Figure 6(f) reach the change of slope at 45 Hz for S-waves and 70 Hz for P-waves. For a conventional seismic survey and for correlation length $b=5$m, previously estimated, wavelengths of P- and S-waves and ratio $\lambda/b$ are indicated in Table 5.

Wave Frequency (Hz) $\lambda $ (m) $\lambda/b$
P 10 270 54
  90 30 6
S 10 123 24
  50 25 5

Table 5. Ratio of the wavelength $\lambda $ over the size of heterogeneities $b=5$m for realistic seismic frequencies, when $V_P=2700$ m/s, $V_S=1230$ m/s.

Scattering is more important for seismic wavelengths with a dimension similar to that of heterogeneities: high frequencies and S-waves, because their wavelengths are shorter than P-waves, are more attenuated. Penetration depth is close to infinity at very low frequencies but decreases drastically in a narrow frequency band, depending on parameters $H$ and $b$ (see Figure 6). This steep descent shifts to higher frequencies when the fractal exponent decreases, corresponding to a stronger cyclicity of the layers. A shorter correlation length of heterogeneities highly improves penetration of high frequencies for both types of wave (see Figures 6(b) and 6(d)). For large-size heterogeneities, i.e. $b>20$ m, the scattering theory we use is not valid because the seismic frequencies statisfy $k_0b\geq
1$. Scattering regimes and the suggested description of heterogeneities are summarized in Figure 7. Our results therefore ignore the effects of large cycles in sediments. We refer the reader to Stovas and Ursin (2007) for more information on the effects of cycles on wave progagation.

Figure 7.
Schematic representation, including the suggested description of heterogeneities, of the different scattering regimes depending on the ratio between the seismic wavelength $\lambda $ and the size of heterogeneities $b$. The scale is for indication only and depends on the frequency band of the survey.
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Next: Dominant frequency versus depth Up: Attenuation in 3D fractal Previous: Attenuation in 3D fractal