Seismic dip estimation based on the two-dimensional Hilbert transform and its application in random noise attenuation

Derivation of the FIR transfer function for the frequency response of digital differentiators

First, to characterize the FIR for signal differentiators, we transform the Leibniz series $$\frac{2{\rm {arcsin}}x}{\sqrt{1-x^2}}$$ to power series (Lehmer, 1985)

$$\frac{2{\rm {arcsin}}x}{\sqrt{1-x^2}}=2x \left[1+\sum_{m=1}^{\infty}\frac{(2{\rm {m}})!!} {(2{\rm {m}}+1)!!}x^{2m}\right].$$ (11)

We substitute sin $$(\frac{\omega}{2})$$ for $x$ , and after rearrangement and truncation of the first $M$ terms, we obtain

$$\begin{split}& \frac{\omega}{\sqrt{1-{\rm {sin}}^{2}\displays... ...m}+ o((\frac{1-{\rm {cos}}\omega}{2})^{M+1})\right] \end{split}$$ (12)

and after manipulation

$$\begin{split}\omega & =2{\rm {sin}}\frac{\omega}{2}{\rm {cos}... ...}+ o((\frac{1-{\rm {cos}}\omega}{2})^{M+1})\right]. \end{split}$$ (13)

We ignore the higher order terms and we obtain the $(2M+2)$ th-order causal transfer function of the derivative operator as

$$\hat{F}_{DD}(z)\approx-\frac{1-z^{-2}}{2}\left\{z^{-M}+ \sum_{m=1... ...\rm {m}}+ 1)!!}{\cdot}z^{-(M-m)}\left[-\frac{(1-z^{-1})^2}{4}\right]^m\right\}.$$ (14)

Seismic dip estimation based on the two-dimensional Hilbert transform and its application in random noise attenuation