Observation impact to the 4DVAR data assimilation systems

[arango]

A new option, OBS_IMPACT, has been coded to compute the observation impact to W4DPSAS_SENSITIVITY, W4DVAR_SENSITIVITY, and IS4DVAR_SENSITIVITY algorithms.

The difference between observation impact and observation sensitivity is a subtle but important one. Actually, in some cases they explain the same thing in two different ways:

• The observation impact allows us to attribute to each observation the change that occurs in I(dx) where dx is the analysis or forecast increment, and I is some scalar functional of the state (e.g. transport). In this case dx is the change in the state-vector x that occurred by assimilating data (i.e. dx=xa-xb, where xa is the analysis and xb is the background).

• The observation sensitivity answers the question: what would the change to I be if we changed the observations y by an amount dy? Now, it turns out that if we let dy=d (the innovation vector), then the change in I is approximately equal to I(dx) above so observation sensitivity gives the same answer as observation impact. However, the observation sensitivity is more general. For example, if we want to know what the impact on the assimilation would be if we remove all of the SSH data, then choose dy=-dSSH. We can estimate what the change in I would be without having to rerun the assimilation. So the observation sensitivity has great utility for obervation system design and monitoring.

Also corrected miscelaneous bugs in unused options of the spatial convolutions.

Many thanks to Andy Moore for his help coding and testing this new option.