Spherical Divergence Correction Calculator: Geometric Spreading Gain

Long before amplitude versus offset analysis or any other quantitative interpretation can happen, raw seismic amplitudes need one basic correction. As a wavefront travels away from its source, its energy spreads over a growing sphere, so amplitude falls off with distance even in a perfectly lossless earth. This has nothing to do with rock properties, it is pure geometry, and it has to be corrected before amplitude can be trusted as a reflectivity indicator. Use the calculator above to compute the gain needed to undo that spreading loss at any two way time.

Why Amplitude Falls Off with Distance

Picture a small explosion at the surface sending energy outward in all directions. Close to the source, that energy is packed into a small sphere and the amplitude is high. Farther away, the same total energy is spread across a much larger sphere, so the amplitude at any one point is smaller. In a constant velocity medium, amplitude falls off as one over the travel distance. In the real earth, where velocity generally increases with depth, the falloff follows a more involved but still predictable curve.

The Correction

A widely used practical approximation for spherical divergence gain in a medium where velocity varies with depth is:

Gain(t) = t x V(t) squared

where t is two way time and V(t) is the rms or average velocity at that time. Multiplying the recorded amplitude by this gain, normalized to some reference time, restores the amplitude to what it would have looked like without spherical spreading loss. The calculator above models velocity as increasing linearly with time from a reference point, then computes the gain multiplier and corrected amplitude relative to that reference.

Worked Example

Using a reference time of 1000 ms at 2500 m/s, with velocity increasing at 600 m/s per second of two way time, a query time of 2500 ms gives a velocity there of 3400 m/s. The gain multiplier works out to about 4.62, meaning an amplitude of 1.0 recorded at that later time actually represents about 4.62 units of true reflectivity strength once the spreading loss is restored.

Why This Correction Matters

• Deeper reflections would look artificially weak on a seismic section without it
• Amplitude versus offset and other quantitative amplitude analysis depend on correctly recovered relative amplitudes
• Getting the gain function wrong can create false bright spots or hide real ones
• It is normally applied early in processing, before deconvolution and other amplitude sensitive steps

Related Calculator and Reference

For the underlying velocity relationships used in this gain function, see the Seismic Velocity Converter. The t times V squared approximation for spherical divergence gain follows the treatment in Yilmaz, O. (2001), summarized in the SEG Wiki entry on spherical divergence.