The phase spectrum is usually calculated by taking the arctangent of the ratio of imaginary to real parts of the Fourier transform. Because the arctangent function is a multivalue function, its principal value has to be determined as being, for example, between and . However, the phase spectrum calculated from the principal-value range is not a continuous function. It does not satisfy equation (5). One way to solve this problem is to add appropriate multiples of to the samples of the principal value. The appropriate multiple of can be determined by assuming that the sampling interval is sufficiently small so that the discontinuities between adjacent samples of the phase spectrum correspond to phase-wrap-around. If the phase spectrum varies rapidly, zero padding should be done in the time domain to ensure that there is a sufficiently small sampling interval in frequency.

swhfactm
The top panel shows a synthetic, minimum phase wavelet delayed for 80 ms.
The middle and bottom panels show the minimum phase and maximum phase
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.
Figure 1 |

swhfactz
The top panel shows a synthetic, zero phase wavelet delayed for 80 ms.
The middle and bottom panels show the minimum phase and maximum phase
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.
Figure 2 |

vwhfact
The top panel shows a trace recorded in a marine survey.
The middle and bottom panels show the minimum phase and maximum phase 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
Figure 3 |

Now let us look at some examples. Figure shows a minimum wavelet
with a delay of 0.08 second. After the spectrum factorization, the minimum
phase part is the original wavelet and the maximum phase part is a spike.
We see that the algorithm picks the first break. Figure
shows the results in the case of a zero phase wavelet.
As expected, the central peak of the zero phase wavelet is picked.
In both cases, the correct delays are found.
Figure shows the picking result of a trace recorded in a
marine survey. The wavelet generated by an air-gun propagates through water
and is recorded by a hydrophone. It is apparent that the picking position is
not the first break. The minimum phase part contains mainly low frequency
components while the maximum phase part contains only high frequency components.
The non-overlapping spectra of two parts indicate that the factorization
algorithm may become numerically unstable. However, the instability
of the spectrum factorization does not affect the results of traveltime
picking because the formula for calculating *n _{0}* uses the total phase
response. The spectrum factorization is only used to explain the
principle of picking.

12/18/1997