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][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>**