The missing data problem is probably the simplest to understand and interpret results. We begin by binning our data onto a regular mesh. For in fitting goals (2) we will use a selector matrix ,which is `1' at locations where we have data and `0' at unknown locations. As an example, let's try to interpolate a day's worth of data collected by SeaBeam (Figure ), which measures water depth under and to the side of a ship Claerbout (1999).
Figure 1 Depth of the ocean under ship tracks.
Figure shows the result of estimating a PEF from the known data locations and then using it to interpolate the entire mesh. Note how the solution has a lower spatial frequency as we move away from the recorded data. In addition, the original tracks of the ship are still clearly visible.
Figure 2 Result of using a PEF to interpolate Figure , taken from GEE Claerbout (1999).
If we look at a histograms of the known data and our estimated data we can see the effect of the PEF. The histogram of the known data has a nice Gaussian shape. The predicted data is much less Gaussian with a much lower variance. We want estimated data to have the same statistical properties as the known data (for a Gaussian distribution this means matching the mean and variance).
Figure 3 Histogram for the known data (solid lines) and the estimated data (`*'). Note the dissimilar shapes.
Geostatisticians are confronted with the same problem. They can produce smooth, low frequency models through kriging, but must add a little twist to get model with the statistical properties as the data. To understand how, a brief review of kriging is necessary. Kriging estimates each model point by a linear combination of nearby data points. For simplicity lets assume that the data has a standard normal distribution. The geostatistician find all of the points m1 .... mn around the point they are trying to estimate m0. The vector distance between all data points and each data point and the estimation point are then computed. Using the predefined covariance function estimate C, a covariance value is then extracted between all known point pairs Cij and between known points and estimation point Ci0 at the given distances and (Figure ). They compute the weights (w1 ... wn) by solving the set of equations implied by
The smooth models provided by kriging often prove to be poor representations of earth properties. A classic example is fluid flow where kriged models tend to give inaccurate predictions. The geostatistical solution is to perform Gaussian stochastic simulation, rather than kriging, to estimate the field Deutsch and Journel (1992). There are two major differences between kriging and simulation. The primary difference is that a random component is introduced into the estimation process. Stochastic simulation, or sequential Gaussian simulation, begins with a random point being selected in the model space. They then perform kriging, obtaining a kriged value m0 and a kriging variance .Instead of using m0 for the model value we select a random number from a normal distribution. We use as our model point estimate mi,
The difference between kriging and simulation has a corollary in our least squares estimation problem. To see how let's write our fitting goals in a slightly different format,
We can get an estimate of , or in the case of the missing data problem , by applying fitting goals (6). If we look at the variance of the model residual and we can get a good estimate of ,
Figure 6 Histogram of the known data (solid line) and the four different realizations of Figure .
Figure shows eight different realizations with a random noise level calculated through equation (8). Note how we have done a good job emulating the distribution of the known data. Each image shows some similar features but also significant differences (especially note within the `V' portion of the known data).
A potentially attractive feature of setting up the problem in this manner is that it easy to have both a space-varying covariance function (a steering filter or non-stationary PEF) along with a non-stationary variance. Figure shows the SeaBeam example again with the variance increasing from left to right.
Figure 8 Realization where the variance added to the image increases from left to right.