Time differentiation and integration of Fourier Transform

(Cand. Real. Knut Sørsdal) Differentiation amplifies the higher frequencies with respect to the lower because the original spectrum F(w) is multplied by iw. As w increases whatever the original value of f(w) it will be greater after multiplication by iw. Thus differentiation of a noisy seismic trace results in an even noisier seismic trace, especially if the noise is toward the higher end of the frequency spectrum.

Integration (divide with iw) is the opposite, thus time differentiation and time integration are examples of high-pass and low-pass filters respectively. The geology of the seismic structure can by means of certain velocity functions differentiate or integrate a seismic wavelet. Differentiation emphasizes the high frequencies over the low and integration the low over the high. In general the locations of anomalies in time or space are much more obvious in the dervative of the function than in the function itself.

Fig.1.gives an example of the Ricker wavelet 50 Hz centerfrequency - original (a), differentiated (b) and integrated(c).

Fig.1. a)original b)Differentiated c)Integrated wavelet

Fig.2. a)reflectivity convolved with original b)Differentiated

Differentiation is accomplished on the reflectivity series fig.2 convolved with the differentiated Ricker wavelet. It is an analog with using a high-pass filter on the series. In this way the geologic structure can act as a filter. Differentiation of a seismic trace is straightforward and easy to visualize. Integration is more complicated to visualize. In order to understand what division by iw means, suppose a seismic trace is composed of a time-invariant source wavelet that is convolved with a reflectivity function, a generally-accepted good first approximation. If this trace is convolved with a unit step function generated by:

UnitstepFuction = Table [1,Length[seismicTrace]]

Then the result is the same as you would get if an integrated source wavelet left the source and was reflected from those same reflection coefficients.

Fig.3. a)reflectivity convolved with original and the result convolved with unit step. b) reflectivity convolved with integrated wavelet. We got the same result as a).