# Shuey approximation

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## Introduction

The AVO equations are very important to hydrocarbon exploration and production because they help to estimate some rock properties, reservoir fluids and also enable to mitigate the drilling operations risks. Zoeppritz expressions give the refraction and reflection coefficients for different offsets considering a plane interface that separates two homogeneous and isotropic layers. Although, as commented by Yilmaz (2001)[1], Avseth et al. (2005)[2] and Aleardi et al. (2017)[3], the Zoeppritz equations are complex. The expressions comprehend various factors and the recognition of the practical meaning for each parameter is not intuitive, as reported in Sheriff and Geldart (1995)[4] and Rosa (2018)[5].  So, in many cases is useful to work with some simplifications such as Shuey approximation.

## Equations

Shuey (1985) [6] proposed a new arrangement for the Aki-Richards equation, which is already an linear approximation of Zoeppritz expressions. Gholami & Farshad (2019)[7] paper highlighted the relevance of Shuey (1985)[6] arrangement in AVO studies, that is widely applied in the industry. Shuey (1985)[6] paper shows the angle-dependent reflection amplitude with the following formula known as Shuey’s three-term AVO equation:

Shuey’s three-term AVO equation. Formulas from Shuey (1985).

In the formulas above, R is the P-wave reflection amplitude, θ is the average of incidence and transmission angles and VP1 and VP2 are the P-wave velocities above and below a given surface. Similarly, VS1 and VS2 are the S-wave velocities and ρ1 and ρ2 are densities. A0 specifies the gradual decrease of amplitude with offset and 𝜎 is the Poisson's ratio.

The first term (R0) is related to the zero offset reflectivity (normal incidence) and its value depends on P-wave velocity and density contrasts. The second term represents the intermediate offsets and it comprises one important parameter: Poisson’ ratio (𝜎). The third term is related to high angles (normally over 30 degrees), and according to Sheriff and Geldart (1995)[4], it assumes small values for the angles normally observed in reflection seismology. The relevant change proposed by Shuey (1985)[6] was the transformation of S-wave velocity to Poisson´s ratio with the following relationship:

Relation between S-wave velocity and Poisson's ratio. From Shuey (1985).

According to Shuey (1985)[6], the Poisson’s ratio is a key elastic parameter to describe the connection between the reflection coefficient and the incidence angles.  Others publications such as Yilmaz (2001)[1], Rosa (2018) [5] and Sheriff and Geldart (1995)[4] also emphasize this relationship between the two variables proposed by Shuey.

As described in Shuey (1985),[6] for studies that consider incidence angles where 0 < θ < 30 degrees, the third term can be omitted (exception when A<0 and B>0) and the Shuey’s equation can have only two-terms to get

Shuey approximation with two-terms.

It is important to mention that the Aki-Richards and Shuey’s approximation works under certain circumstances. First, the density and velocities contrasts needs to be small. Second, θ1 < 80° if VP2 < VP1 or θ1 about 10 degrees less than critical angle if VP2 > VP1. These restrictions do not represent a huge limitation because the conditions are satisfied for the majority of sedimentary situations where the reflection coefficient at normal incidence is smaller than 0.2.

The simplification proposed by Shuey (1985) can be applied for diffrerent purposes. Castanha and Smith (1994)[8], based on Shuey approximation formulas, highlighted that the product of the normal incidence P-wave reflection coefficient and the AVO gradient (slope) can be used for bright spots detection (AVO Class III). Aleardi et al. (2017)[9] paper describes how Shuey`s approximation can applied to help to predict overpressure intervals. Gholami & Farshad (2019)[7] study uses Shuey`s approximation in association with the concepts of Radon transform to model P-P reflections with AVO effects. On the other hand, Avseth et al. (2005)[2] describes situations (e.g., major faults and diapirs) where Shuey formulas should not be used due to the nonlinear moveouts and abrupt changes in the offset-dependent reflectivity.

Essentially, Shuey approximation is strongly associated with the Poisson’s ratio and that parameter is crucial to evaluate petrophysical properties.

## Reference

1. Yilmaz, Ö. (2001). Seismic data analysis: Processing, inversion, and interpretation of seismic data. Society of exploration geophysicists.
2. Avseth, P., Mukerji, T., & Mavko, G. (2010). Quantitative seismic interpretation: Applying rock physics tools to reduce interpretation risk. Cambridge university press.
3. Aleardi, M., Mapelli, L., & Mazzotti, A. (2017). A new AVA attribute based on P-wave and S-wave reflectivities for overpressure prediction. Journal of Applied Geophysics, 140, 11-23.
4. Sheriff, R. E., & Geldart, L. P. (1995). Exploration seismology. Cambridge university press.
5. Rosa, A.L.R. (2018). Análise do sinal sísmico. Sociedade Brasileira de Geofísica, Rio de Janeiro, 713 p.
6. Shuey, R. T. (1985). A simplification of the Zoeppritz equations. Geophysics, 50(4), 609-614.
7. Gholami, A., & Farshad, M. (2019). The Shuey-Radon transform. GEOPHYSICS, 84(3), V197-V206.
8. Castagna, J., & Smith, S. (1994). Comparison of AVO indicators: A modeling study. GEOPHYSICS, 59(12), 1849-1855.
9. Aleardi, M., Mapelli, L., & Mazzotti, A. (2017). A new AVA attribute based on P-wave and S-wave reflectivities for overpressure prediction. Journal of Applied Geophysics, 140, 11-23.