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Separation

We can manipulate equation 6 and decompose it into two symmetric parts. First, we shift the Ricker wavelet to the center of the axes:

ricker ricker-freq
ricker,ricker-freq
Figure 1.
The fourth-order finite approximation of the Ricker wavelet: (a) in the time domain; (b) in the frequency domain.
[pdf] [pdf] [png] [png]


Ricker$\displaystyle = -\frac{(1-Z)^2}{Z}\frac{(1+Z)^{2N}}{Z^N}$     (7)

which rearranges to
Ricker$\displaystyle = [(1-\frac{1}{Z})(1+\frac{1}{Z})^N][(1-Z)(1+Z)^N] .$     (8)

We call one of these symmetric parts a ``half-Ricker wavelet'':

$\displaystyle H(Z)=(1-Z)(1+Z)^N$     (9)

Figures 2(a) and 2(b) show this finite approximation of the Ricker wavelet in the time and frequency domains.

We denote the inverse of the half-Ricker wavelet, which is our candidate as the initial guess for both filter $ f_a$ and filter $ f_b^r$ , as $ \hat f_a$ and $ \hat{f_b^r}$ in the time domain and $ \hat F_a$ and $ \hat{F_b^r}$ in the Z domain:

$\displaystyle \left\{ \begin{array}{l}
\hat F_a(Z)=\hat F_b(Z)=\frac{1}{H(Z)}=\...
...{H(\frac{1}{Z})}=\frac{1}{(1-\frac{1}{Z})(1+\frac{1}{Z})^N}
\end{array} \right.$     (10)

half-ricker half-ricker-freq
half-ricker,half-ricker-freq
Figure 2.
The fourth-order half-Ricker wavelet: (a) in the time domain; (b) in the frequency domain.
[pdf] [pdf] [png] [png]


next up previous [pdf]

Next: Avoiding the singularity Up: Approximation of the inverse Previous: Finite approximation

2011-05-24