GEOSTATISTICS

Figures 21 and 22 show synthetic and real surveys containing regions where no data was recorded. The PEF method of this book produces the extrapolated image in the upper right. Notice that far from known data, the extrapolated data is weak. This is a result of a ``minimum variance'' calculation. Notice the weak interpolated data also has a different spectrum than the known data. It lacks the short wavelength fuzz. That's because the short wavelengths cannot be extrapolated long distances.

There is a simple way to acquire short wavelength fuzz at long distances by adding random fuzz of the proper spatial spectrum. This is what they do in Geostatistics. We can also do it within the usual geophysical estimation framework. Here is how we do it. Given the usual data fitting and model styling goals

		(19)
	(20)

We introduce a sample of random noise and fit instead these regressions

		(21)
	(22)

Of course you get a different solution for each different realization of the random noise. You also need to be a little careful to use noise of the correct variance. Bob Clapp (thesis) worked out the correct scalar variance. The wood image gives us the example in Figure 21.

 

manywood
Figure 21 Top left is wood with a hole punched out. Top right the hole filled with a PEF. The bottom two are realizations with different samples of noise . Notice they differ inside the hole.

Figure 22 shows the same concept with the seabeam data.

 

bobsea
Figure 22 Top left is binned data. Top right extends the data with a PEF. The bottom two panels add appropriately colored random noise in the regions of missing data. (Bob Clapp)