Offset to Angle Converter for AVO Analysis

Amplitude versus offset analysis is really amplitude versus angle analysis wearing an offset shaped disguise. Every AVO equation, from Shuey to Aki and Richards, is written in terms of incidence angle, not offset, so before any of that math can be applied to a real gather, each trace offset has to be converted into the angle at which it actually struck the reflector. Use the calculator above to convert a set of offsets into incidence angles for a given depth and velocity.

From Offset to Angle

For a flat reflector and a straight ray approximation, the geometry is simple triangle trigonometry. The source and receiver sit on the surface separated by the offset, and the reflection point sits directly below their midpoint at the reflector depth. Each leg of the ray path covers half the offset horizontally and the full depth vertically, so the incidence angle from vertical is:

theta = atan( offset / (2 x depth) )

Depth itself is estimated from the zero offset two way time and the average velocity above the reflector, using depth equals velocity times one way time. The calculator also reports the ray parameter for each offset, sin(theta) divided by velocity, which stays constant along a given ray path and is useful for more advanced layered velocity work.

Worked Example

With a zero offset two way time of 2000 ms and an average velocity of 2800 m/s, the reflector sits at about 2800 meters depth. An offset of 1000 meters then converts to an incidence angle of about 10.1 degrees, while a far offset of 3000 meters converts to about 28.2 degrees. This is exactly the kind of spread you want in a gather built for AVO analysis, since a real amplitude versus angle trend needs a wide enough angle range to be visible above noise.

Why This Step Matters

• AVO intercept and gradient attributes are defined in terms of angle, not offset
• The same offset means a very different angle at a shallow reflector than at a deep one
• Angle limited stacks and angle gathers used in modern interpretation depend on this conversion being correct
• A straight ray estimate is a reasonable starting point, though real velocity gradients bend rays and shift the true angle slightly

Related Calculator and Reference

Once you have angles for each trace, see the AVO Intercept Gradient Calculator to fit the amplitude versus angle trend. The offset to angle relationship used here follows the standard treatment in the SEG Wiki entry on AVO analysis.