4. Approximation of SQp and SQs attributes in the amplitude versus offset (AVO) domain

Derivations of SQp and SQs in the previous sub-chapter show that the parameters are taken from elastic properties extracted from seismic inversion results. To estimate the SQp and SQs directly from seismic data, the approximation of SQp and SQs through the AVO method is proposed.

To understand the approximation of SQp and SQs in the AVO domain, we started with the concept of AVO attributes: AVO intercept and AVO gradient methods. Castagna et al. [8] interprets the AVO intercept and AVO gradient using crossplot method (reader who is interested to get more details on this method can refer to Castagna et al. [8]). In AVO crossplot method, the diagonal line indicates the shale background, and potential hydrocarbon reservoirs will be identified as an AVO anomaly. Determination of the anomaly is measured from the shale background line (Figure 3).

The trend or background of Vp/Vs can be approximated with the formula

$$\mathbf{a}\_b = \mathbf{B}/\mathbf{A} \tag{11}$$

(12)

where B and A are AVO gradient and AVO intercept, respectively. Using Gardner equation [9], the relation between contrast density and contrast velocity is approximated with the formula

Δρ

#### Figure 4.

Comparison between conventional AVO crossplot and new SQp-SQs crossplot.

where Δ represents the different properties between the upper and lower media. Hence, Δρ ¼ ρ<sup>2</sup> � ρ1, and ρ ¼ ρ<sup>2</sup> þ ρ1, ΔVP ¼ VP<sup>2</sup> � VP1, and VP ¼ VP<sup>2</sup> þ VP1. The velocity contrast is approximated by

$$\frac{\Delta V\_P}{V\_P} \approx (\ $/\$ )A\tag{13}$$

Combining Eqs. (12) and (13) and substituting Eq. (10), SQs�<sup>1</sup> and SQp�<sup>1</sup> can be approximated with

$$\begin{aligned} \mathbf{SQp} &\approx \left(\frac{1}{3}\right) A \left(\frac{(-\mathbf{3} + \mathbf{2}(\mathbf{A} + \mathbf{B}))^{\mathbf{f}}}{2(1 - (\mathbf{A} + \mathbf{B}))}\right) \left( \\ \mathbf{SQs} &\approx \left(\frac{4}{3}\right) A \left(\frac{(\mathbf{2}(\mathbf{A} + \mathbf{B}) - 1)}{2(\mathbf{A} + \mathbf{B}) + 1}\right) \left( \end{aligned} \right)$$

where A and B are the intercept and gradient attributes. Eq. (14) shows that the SQp and SQs attributes can be approximated from the intercept and gradient of AVO.

The crossplot of both methods can be illustrated in Figure 4. The potential of hydrocarbon reservoir from oil (blue) to gas (red) is plotted close to each other in the conventional AVO crossplot. However, that anomaly is boosted in the SQp-SQs crossplot. The two different potential reservoirs are separated significantly.
