Stretching the traces

To realize transformation (2) each input trace, q(t,m,h), has to be stretched by the factor k/h:  

$${q_{1}(t_{1},m,h)} = {q(t_{1} {{h}\over{k}},m,h)}.$$ (4)

The differential time moveout $\Delta t = t - t_{1}$ of equation (1) is proportional to time:  

$${\Delta t} = {t_{1} ({{h}\over{k}}-1)}.$$ (5)

Logarithmic resampling is defined by:  

$${\tau_{1}} = \log({t_{1}\over{T_{c}}}),$$ (6)

where $\tau_{1}$ denotes the frequency variable after Fourier transformation of the input trace. T_c is the start of the unmuted frequency trace. Elsewhere in this report (), I show that After two Fourier transforms and the intermediate logarithmic resampling, each trace stretching amounts to a simple trace scaling:  

$${Q_{1}(\omega_{1},m,h)} = {Q(\omega_{1},m,h) \exp(-i ({{h}\over{k}}-1)\omega_{1})}$$ (7)

Resampling the Fourier transformed input trace rather than the trace itself takes advantage of the band limitation of typical seismic data (). The more compact data cube reduces the computation time and the amount of memory required.