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:
In two dimensions, the principal artifact that will affect our velocity transform occurs at Claerbout (1995). This leads to
To compensate,we need to apply , which, recalling that:
we can obtain by the formula
So, to repair the principal artifact of 2-D hyperbola summation, we need to apply this filter - the half-derivative filter.