Making a wavelet minimum-phase

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Problem 9.15

The wavelet $ [-0.9505,\;-0.0120,\;0.9915] $ is not minimum-phase. How would you change it to make it mimimum-phase without changing it any more than necessary? Give two methods.

Background

Minimum-phase wavelets are discussed in Sheriff and Geldart, 1995, section 9.4 and section 15.5.6a.

Solution

The transform of the wavelet is $ W(z)=-0.9505-0.0120z+0.9915z^{2} $. The roots of $ W(z)=0 $ are 0.9652 and $ -0.9531 $, so to make the wavelet minimum-phase, we must change it so that the magnitudes of both roots are greater than unity. To increase both roots by the same factor, we multiply both roots by a number slightly greater than $ 1/0.9531=1.0492 $. If we use the multiplier 1.0500 the roots become 1.0135 and $ -1.00076 $ and the revised equation is

$ {\begin{aligned}(z-1.0135)(z+1.0008)=-1.0143-0.0127z+z^{2}.\end{aligned}} $

To obtain the same value at $ t=0 $ as we had before, we multiply by $ 0.9505/1.0143=0.9371 $ and the transform becomes $ 0.9505-0.0119z+0.9371z^{2} $ which gives the wavelet [$ -0.9505 $, $ -0.0119 $, 0.9371]. The modified wavelet is nearly the same as the original except that its “tail” has been decreased from 0.9915 to 0.9371, a 5.5% decrease.

A second method is to use a taper to reduce the tail of the wavelet. A commonly used taper is $ k^{t} $ where $ t $ is in milliseconds and $ k $ is slightly less than unity, for example, 0.9950. Using this value for $ k $ and assuming that the sampling interval is 2 ms, the values of the taper are 1.0000, $ 0.9950^{2} $, $ 0.9950^{4} $, that is, 1.0000, 0.9900, 0.9801 and we get $ W(z)=-0.9505-0.9900\times 0.0120z+0.9801\times 0.9915z^{2}=-0.9505-0.0119z+0.9718z^{2} $. The roots of $ W(z)=0 $ are now 0.9904, $ -0.9783 $, both less than unity so we did not apply enough taper. We next try using $ 0.9850^{t} $, which reduces the wavelet to $ [-0.9505,\;-0.0116,\;0.9333] $. The roots are now 1.0154 and $ -1.0030 $ and $ W(z)=(z-1.0154)(z+1.0030)=-1.0184-0.0124z+z^{2} $. Multiplying by $ 0.9505/1.0184=0.9333 $ makes the wavelet [0.9505, $ -0.0116 $, 0.9333].

Comparing the two methods, we see that the second method has changed the wavelet slightly more than the first. In practice, a wavelet will have many more than three elements and the second method will generally be more practical. To verify that the taper is large enough to achieve its objective, we have to find all of the roots.

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