Attenuation

Our second observation is that all signals eventually attenuate, and this attenuation (Q) is a function of frequency and media properties. How much they attenuate is generally a function of the media, but a decent first approximation is that after a certain number of cycles, a given frequency attenuates a given percentageKjartansson (1979). Therefore, after a certain number of cycles, the wave components at that frequency will no longer be of usable strength so we can stop downward continuing it. This observation nicely compliments the wavelet sampling observation that higher frequencies will attenuate quicker in depth, reducing the distance we need to downward continue them.

We can combine these two ideas and replace our original cost equation with,

$$C \approx \sum_{f} * nz(f) * ( nh*nx log(nh) + nx*nh log(nx) + CEXP(nx,nh)) ,$$ (3)

where our number of depth steps nz(f) is a function of frequency.