I've been learning about scattering from here and here. It seems most scattering equations are left in an arbitrary form of , and simply measure seismic noise then use it as in the following FFT:
If the sine wave were perfectly sinusoidal, say , our FFT would yield delta functions at . However, scattering is rarely so clean.
If where lambda is the laser wavelength, then our sine wave is approximately linear, and we get a clean spectrum at frequencies above the seismic noise.
When , we get "upconversion" of scatter noise, i.e. the higher order modes of the sine wave start to matter, and this extends the scatter shelf into higher frequencies.
I fit a scattering shelf of the functional form , where is a scatter coupling coefficient in units of hertz, and is a half width half maximum (HWHM) of an underlying Lorentzian .
I found and .
We can think of the HWHM as a function of overall scattering displacement and velocity: .
If , this gives