For more complex optimization problems, further reduction of resolution may be needed to make objective functions convex. The severe loss of information may cause an erroneous global minimum of the objective function.
:
A real seismic trace and its duplication with an unknown shift.
:
The mean-squared error functions for two real seismic traces shown in
Figure 
.
Here, I show another example of residual statics correction problem for a trace with complex waveforms and unknown noise. Figure 12 shows a trace taken from a field seismic record, and its duplication with an unknown shift. We repeat the process discussed in the previous section on these two traces. Figure 13 shows the objective function for this optimization. Due the oscillatory nature of the seismic field data and unknown noise, the objective function shows complicated local and global structure. The basin of attraction leading to the global minimum point is extremely narrow and steep, which makes it almost impossible for any gradient searching methods to find the correct solution.
Again, the auto-correlation shell of the Daubechies basis is used to decompose the traces to coarse resolution levels. Figure 14 shows objective functions when applying various level of decomposition to traces in Figure 12. As expected, the complexity of the objective function is greatly reduced after the data being decomposed to coarse levels.
:
The mean-squared error functions for two real seismic traces shown in
Figure at various resolution levels. The traces are
decomposed with the auto-correlation shell of the Daubechies basis
with 
vanishing moments.
However, it is worth noticing the global minimum point are slightly shifted in Figure 14(d), though the objective function shows a nice, convexity shape. This problem may be caused by the loss of information when too much resolution was discarded from the data. In this case, an iterative process similar to a multi-grid iteration can be used to enhance the resolution progressively; i.e. the solution of a coarse-level optimization is used as the initial model to the following optimization at a finer level [3].
