Calibration Parameters and Scaling Function

In order to determine the theoretical seafloor AVO response, I need to assume elastic properties for the water layer and the near-seafloor sediments. The reflection amplitude is generated by an impedance contrast averaged one Fresnel zone above and one Fresnel zone below the reflector, in this case the seafloor. After migration, the Fresnel zone collapses to 1/4th of a seismic wavelength $\lambda\:=\:\rm v/\rm f$, where $\rm v$ is the rms velocity of the reflection event and $\rm f$ is the dominant frequency. In the case of the Blake Outer Ridge data, the dominant frequency is approximately 30 Hz and the velocity at the seafloor is 1.5 km/s. Therefore, the seafloor reflection amplitude is generated in a region approximately 12.5 m above and 12.5 m below the seafloor.

The water can be assumed to have a P-wave velocity of 1.5 km/s, zero S-wave velocity, and a density of 1.0 ${\rm g}/{\rm cm}^3$. The properties of the near-surface sediments are estimated based on results of the drilling at sites 994, 995 and 997 Matsumoto et al. (1996) and on observations by Hamilton Hamilton (1976). Since the marine sediments are highly unconsolidated, I chose a P-wave velocity of approximately 1.55 km/s, a S-wave velocity of about 0.1 km/s and a density of 1.3 ${\rm g}/{\rm cm}^3$ to represent the near-surface marine sediments. Based on these elastic parameters, I construct a theoretical seafloor AVO response using the Zoeppritz equations Aki and Richards (1980).

The Zoeppritz equations Aki and Richards (1980) describe the amplitudes of transmitted and reflected P- and S-waves in the case of plane waves incident on a reflector. Assuming small layer contrasts and angles coverage well within the pre-critical region, Aki and Richards 1980 linearized these equations. The resulting acoustic reflection coefficient can be described as  

$$\rm R(\theta)\:= {1\over{2 \cos^2(\theta)}}\:{\rm I_{\rm p}}... ...(2\:\gamma^2\:\sin^2(\theta)\:-\:0.5\:\tan^2(\theta))\: {\rm D}$$ (1)

where $\rm I_{\rm p},\rm I_{\rm s}$ and $\rm D$ are the relative contrasts in P-impedance, S-impedance and density, $\theta$ is the reflection angle and $\gamma$ is an estimate of the background shear to compressional velocity ratio $\rm v_{\rm s}/\rm v_{\rm p}$.

Using the estimated elastic parameter changes across the interface from water to near-surface sediment, I can thus determine the functional form of the seafloor AVO response using equation . The appropriate scaling function of the traces after they have been normalized using the maximum seafloor amplitudes is therefore $A(\theta) \:=\:{R(\theta)\over R(0)}$ and can be calculated from equation . The result is shown in Figure . It shows that the scaling function preserves the near-offset amplitudes and increases the far-offset amplitudes by less than 2%. This will result in nearly constant seafloor amplitudes with increasing offset.

 

seaavo
Figure 14 AVO scaling function.

 

avo-ann
Figure 15 Final CMP gather after amplitude calibration.