Factorizing for minimum phase and allpass wavelets

Using the Kolmogoroff spectrum factorization method (Claerbout, 1985), one can find the minimum phase wavelet that has the same amplitude spectrum as a given wavelet. Apply this algorithm to S(Z), we obtain its minimum phase equivalent W_min(Z)W_max(Z^-1). Then,

$$\begin{array} {lll} S(Z) & = & \alpha W_{min}(Z)W_{max}(Z)Z^... ...min}(Z)W_{max}(Z^{-1}) \\ \\ & = & \alpha P(Z)Q(Z),\end{array}$$ (7)

where Q(Z)=W_min(Z)W_max(Z^-1) is a new minimum phase wavelet and

$$P(Z)=Z^{n_0}{W_{max}(Z) \over W_{max}(Z^{-1})}$$

is an allpass wavelet. To estimate n₀, we can apply the method explained in the last section to P(Z). Figure shows an example with the same field data as in Figure . We see that two factors of this factorization process have similar frequency content. As expected, the traveltimes picked by two methods are equal. Because the phase spectrum of an allpass filter is always a nondecreasing function, the procedure for unwrapping the phase of P(Z) is easier than that for S(Z).

 

vwkfact Figure 4 The top panel shows the same trace as one in Figure . The middle and bottom panel show the minimum phase and allpass parts of this wavelet, respectively. The vertical dashed lines indicate the picking positions. The fat, dashed curve superposed on the curve of the top panel is the signal reconstructed from two factorized parts.