:
The observed data in the first example. Two traces contains identical
waveforms of Ricker wavelet with a 
Hz peak frequency. The
relative time-shift is the unknown we are seeking.
:
The error-fitting function with respect to the relative time-shift
between two traces. The goal is to find the optimal point where the
mean-squared error is minimum.
:
The histogram of the obtained time-shifts of 
conjugate-gradient
optimization experiments starting from uniformly distributed random
initial models between 
s. The horizontal axis is the
number of shift-samples, where the sample interval is 
s, and
the grid size of the histogram is 
samples. The number of times
that found the true global minimum is 
out of .
Let us first consider a simple residual statics problem. Consider a trace containing one Ricker wavelet; duplicate the trace with an unknown shift. Figure 6 shows two traces as described above. Now, we look for the time-shift between the two traces by applying an optimization, that is, searching for the time-shift which maximally aligns the two traces. This is a simple residual statics estimation problem using the stacking power method; there is only one unknown in the optimization. The objective function is formulated as a least-squared error,
where 
and 
are the two data traces,
is the number of samples per trace, and 
is the unknown
time-shift. The goal is to find the time-shift that minimizes
the error function 
.
Figure 7 shows the error function as in
equation (24) for the fitting of these two traces. In
addition to possible problems caused by the local-minima, the
basin of attraction leading to the global-minimum is ``steep'' and
narrow, while the two areas to the sides are ``flat''. The global
structure of this objective function suggests that the global
minimum point may be hard to find by traditional gradient-based
searching methods. Assuming that we know a priori the time-shift
between the traces lies in the range of s,
the searching range is restricted to this interval.
Figure 8 shows the histogram of the obtained time-shift
for optimizations by using the Conjugate-Gradient and Cubic-Line-Search tools provided in the CWP Object-Oriented
Optimization Library [7];
initial models are randomly chosen
between s. As expected, the chances of finding the
correct global minimum is small. In the case of this test, there are
out of experiments that the correct time-shift was found.
