Theory

The origin of the half-derivative filter lies in the simple operation of causal integration Claerbout (1993). With each pass of causal integration, we are actually convolving the signal in the time domain with a scaled ramp function which is equivalent to multiplication in the frequency domain with the inverse of frequency. This can be expressed as:

$$t^{n-1}\;\mathit{step(t)}\;=\;FT \left(\frac{1}{(-i \omega)^{n}}\right)$$ (1)

In two dimensions, the principal artifact that will affect our velocity transform occurs at $n\;=\;\frac{1}{2}$ Claerbout (1995). This leads to

$$\frac{1}{\sqrt{t}}\;\mathit{step(t)}\;=\;FT \left(\frac{1}{\sqrt{-i \omega}}\right)\;\;\mbox{\rm.}$$ (2)

To compensate,we need to apply $FT(\sqrt{-i \omega})$, which, recalling that:

$$\frac{d}{dt}\;=\;FT(-i \omega)\;\;\mbox{\rm ,}$$ (3)

we can obtain by the formula

$$\frac{d}{dt}\frac{1}{\sqrt{t}}\;\mathit{step(t)}\;=\;FT(\sqrt{-i \omega})\;\;\mbox{\rm.}$$ (4)

So, to repair the principal artifact of 2-D hyperbola summation, we need to apply this filter - the half-derivative filter.