• Introduction
  
• Data Assimilation
  
• SSI
  
• Harmonics
• Variables
  
• Example
• Sources

Data assimilation

 Since there aren't enough observations at one time to figure out what's happening in the atmosphere all over the globe, a weather forecast is made for the same time as the observations, and then that forecast is changed just enough so that it matches the available observations.  This is called data assimilation.

What goes into data assimilation?

The forecast data is evenly spaced all over the globe, but the observations are not.  Besides that, the observations might have errors (from the instruments, record keeping, transmission of the data, etc). So the data assimilation has to move the forecast data location to the location of the observation, figure out the probable error, and then change the value of the data.

And since this all has to be done fore the entire globe, which that takes a lot of time even on a fast computer, any solutions to these problems have to be computationally fast.  There are a lot of different methods to do this, but this page only talks about one method: Spectral Statistical Interpolation (SSI).

What's good about spectral statistical interpolation?

• No need to initialize: One problem with a lot of models is that they have to do two big steps in their data assimilation cycle: bring the forecast to match the observations, then change it just enough to make sure all the mass terms are  balanced with gravity and buoyancy.  The reason the second step is necessary is that the model will over-compensate for unbalanced data and show a big wave expanding in time.  These waves aren't really there: a good assimilation process brings the forecast to match the observations without upsetting the hydrostatic balance.  SSI uses a linear balance equation during that first step, so the data is already balanced and no further change is necessary.
  

• Another problem is that just bringing the forecast to match the observations takes a long time.  But since all the dependent variables can be represented in spherical harmonics, this model can do that step in the spherical harmonic coordinate system, which is a lot faster than the latitude/longitude coordinate system.  Also, other methods (like optimal interpolation) do these adjustments at one location at a time, whereas SSI does every adjustment over the entire world at once.

What's bad about SSI?

• The main thing that needs to be calculated is the difference between the forecast and the best answer.  In SSI, this difference (called background error covariance) is very easy to find as long as it is the same in all directions, but in reality this is not the case.  Finding the slight directional differences in the background error covariance is difficult using this method.
  
• Faster is better: new innovations in data assimilation algorithms make newer methods faster than SSI.