# Angle stacks

Angle stacks provide a means of accessing the AVO information in seismic data.[1] They are usually designed to measure the reflectivity at a given incidence angle, ${\displaystyle \theta }$, but the term angle-stack can also apply to a general combination of intercept and gradient.

There are a number of ways an angle stack can be constructed. Perhaps the most common is to stack the data from moveout corrected common reflection point gathers within constant angle mutes. Incidence angles to define the mutes must be estimated from a velocity field. A simple formula for incidence angle can be derived from the normal moveout equation.

${\displaystyle \sin ^{2}\theta ={\frac {x^{2}V_{i}^{2}}{V_{r}^{2}(V_{r}^{2}t_{0}^{2}+x^{2})}}}$, where:

${\displaystyle x}$ = offset

${\displaystyle V_{i}}$ = internal velocity

${\displaystyle V_{r}}$ = RMS velocity

${\displaystyle t_{0}}$ = zero offset TWT

This is applicable as far as the NMO equation is applicable; layer-cake overburden and moderate offsets. For more complex overburden or higher offsets raytracing will provide a more general solution. Incidence angle estimates will dependent on the quality of the velocity model and hence will be the cause of some uncertainty throughout all subsequent AVO analysis.

The figure shows a typical RMS and associated Dix interval velocity profile on the left. To the right are 15⁰ and 30⁰ angle mutes derived using the equation above. Stacking between the two would result in approximately a 22.5⁰ angle stack. In practice the mutes should be smoothed to ensure no discontinuities in the resultant stack.

An alternative way to derive angle stacks is by combining AVO intercept and gradient measurements using the simple two-term AVO equation:

${\displaystyle R(\theta )\approx A+Bsin^{2}\theta }$

Where A is the intercept and B the gradient. This should be a superior way to derive angle stacks as it makes full use of the available data and uses a theoretical model to fit the data, however it does make more demands of the processing, in particular ensuring the gathers are flat. Any number of theta angle stacks can be derived from a seismic dataset; 2, 3, or 4 is typical.

These two alternative ways of estimating angle dependent reflectivity provide a way to QC intercept and gradient estimates by using them to derive an angle stack and compare it with the equivalent derived by stacking between angle mutes.

Intercepts and gradients can be combined using a more general equation;

${\displaystyle R(\chi )=A\cos(\chi )+B\sin(\chi )}$

This is a coordinate rotation in AB space with rotation angle ${\displaystyle \chi }$ (chi). ${\displaystyle R(\chi )}$ is referred to as ‘scaled reflectivity’, ${\displaystyle R_{s}}$, in Whitcombe et al (2002).[2]

If the intercept and gradient have been estimated using simple linear regression then A, B and any ${\displaystyle R(\theta )}$ and ${\displaystyle R(\chi )}$ can be expressed as weighted stacks.

The impedance equivalent of ${\displaystyle R(\theta )}$ is elastic impedance and the impedance equivalent of ${\displaystyle R(\chi )}$ is extended elastic impedance. These can be used as the basis for well ties, rock property analysis and inversion of angle stacks.

## References

1. Castagna, J. P., and Backus, M. M., 1993, Oﬀset dependent reﬂectivity: Theory and practice of avo analysis: Society of Exploration Geophysicists
2. Whitcombe, D. N., P. A. Connolly, R. L. Reagan, and T. C. Redshaw, 2002, Extended elastic impedance for fluid and lithology prediction: Geophysics, 67, 63–67, http://dx.doi.org/10.1190/1.1451337.