Pentad SST Climatology

This climatology was interpolated from the monthly climatology by inverting the averaging operator using a .5 month correlation scale. This means we found a smooth solution that averaged to the monthly climatology. It is only an interpolation.

Suppose we use the following notation:

|m>

vector of monthly data

[est>

vector of pentad data

|A]

averaging operator going from pentads to months.

[M]

model-model covariance matrix, constructed from a correlation function exp(-T/L), where T is the time difference between the two points and L is about .5 month. T is periodic with period 1 year (365 days or more importantly 73 pentads). Covariances below 10^-5 were set to zero.

|A] is the averaging operator, so by construction

|A][est> = |m>

We invert this using a data-space inverse with the data-space error convariance set to zero (i.e. the average of our pentad estimates will be exactly equal to the monthly data).

[est> = [M][A|{|A][M][A|}^-1|m>