12.747 Lecture 22: Section 5:

2-D Section Inversions

File last modified 1 December 1998

22.5 Additional Issues

There are a few additional, left-over, issues:

Resolution: the resolution matrix (VV^T) is I_M if the rank(A)=N, otherwise the elements of VV^T are < 1 and possibly off-diagonal.

Noise: add to equation 22.3.5 noise as in , then

and is the ratio of noise variance to solution variance.

Tapered SVD: use

in a "ridge analysis".

Inequalities: in addition to the commonly held belief that concentrations and eddy diffusivities must always be positive, there are also times when one would like to specify a direction, but not a magnitude. Could use something like a simplex, but the methods need improvement.

Assimilation: in numerical weather prediction data is "assimilated" into the model while it is running to improve the forecasts. In these simulations what works, works. Most techniques involve some variation of the Kalman Filter, a type of predictor-corrector technique.

There are a lot of steps in inverse theory applied to oceanographic problems. But the steps are actually composed of a short series of stages. Here is a quick list to get you started:

1. Choose initial reference level(s), but don't do it blindly, use what you know.
2. Choose a coordinate system, take you pick but be consistent.
3. Develop the statistics of the noise (residuals) as best you can (this is the n mention directly above.
4. Write down the equations for X and the statistics of X.
5. Add any other physics/chemistry/biology you think appropriate and begin.

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