Transmission and Spreading

Two Ways a Deep Reflection Loses Amplitude

Convolution hands every reflector the amplitude of its own reflection coefficient, regardless of depth. A real wave does not, and this section isolates the two reasons why. A stack of identical reflectors makes it vivid: in the ground they have the same reflection coefficient, yet the deep ones come back weaker.

• Geometric spreading. A wavefront expands as it travels, spreading its fixed energy over an ever-larger area, so its amplitude falls. In two dimensions this goes as one over the square root of the path length, and therefore of traveltime.
• Transmission loss. At every interface the wave crosses, part of the energy reflects away and less continues down. Crossing an interface down and back up multiplies the amplitude by $1 - R^2$, so a deep reflection, having passed many interfaces twice, is diminished before and after it even reflects.

Correcting for What the Wave Loses

The chart tells the story: convolution (dashed) keeps every reflection at full strength, while the wave equation (solid) fades them with depth, and the cyan curve shows how much of that fade is spreading alone versus transmission on top. The deepest reflector can return a small fraction of what convolution would give it.

Because these losses are real, raw seismic amplitudes are corrected before they are interpreted: a spherical-divergence gain undoes spreading, and true-amplitude processing accounts for transmission, so that a deep reflection can be compared fairly with a shallow one. A convolutional synthetic starts already corrected, having never applied the losses, which is convenient but means it over-reads deep amplitudes relative to raw field data. The wave equation applies them for free, because losing energy as it travels is simply what a wave does. Understanding this is what lets you match a synthetic to real data on amplitude, not just on time.