Section \[sec5\] provides results of numerical experiments on simulated data. These results show that the basic method can work well for moderate values of $N$, even sometimes when the condition numbers of the covariance matrices do not stay bounded as the number of observations increases. Furthermore, the algorithm with the $U_j$’s chosen as in Section \[sec3\] can lead to substantially more accurate approximations for a given $N$. A large-scale numerical experiment shows that for observations on a partially occluded grid, the algorithm scales nearly linearly in the sample size. Section \[sec6\] applies the methods to OMI (Ozone Monitoring Instrument) Level 3 (gridded) total column ozone measurements for April 2012 in the latitude band $40^\circ$–$50^\circ$N.

![Demeaned ozone data (Dobson units) plotted using a heat color map. Missing data is colored white.[]{data-label="figozone"}](627f01.eps)

The data are given on a $1^\circ\times1^\circ$ grid, so if the data were complete, there would be a total of $360\times10\times30 = 108\mbox{,}000$ observations. However, as Figure \[figozone\] shows, there are missing observations, mostly due to a lack of overlap in data from different orbits taken by OMI, but also due to nearly a full day of missing data on April 29–30, so that there are 84,942 observations. By acting 