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The second column of this table shows the different types of sub-grid variability in precipitation totals (Images show example radar-derived totals for cases that correspond to each of the example scenarios).

Based on clear-cut observational evidence from radar-derived totals, and from physical reasoning), one notes from the outset that within global ensemble member grid boxes (18km X 18km) very different geometries of sub-grid variability in precipitation totals can be observed. In principle we could have more (or indeed less) than three, though these serve as a useful illustration.

Let us assume that the number of meaningful different scenarios is given by n, and that each one is indexed with label i.

Scenario (i=) 1 could be said to be "zeroth order", i.e. point totals exhibit little sub-grid variability; scenario 2 is "first order", showing strong variability in one dimension; and scenario 3 is "second order" showing strong variability in two dimensions.

Accordingly, within the grid box the distribution of point totals (i.e. the pdf, or probability density function) is as follows: for scenario 1 it is roughly Gaussian, with a sharply defined peak, and so "confident"; For scenario 3 it is roughly exponential, with a high probability of small or zero totals, tapering down to a small probability of very high totals, and so not confident at all; and for scenario 2 it lies somewhere in between.

Clearly to anticipate point totals, one must recognise and understand these types of sub-grid variability.