# Microwave Integrated Retrieval System (MIRS) - Validity Space

Note that the same solutions (eq. 19 and eq. 20) could be obtained using different techniques such as the optimal estimation theory or the minimum variance solution (MVS). Both assume a local linearity of the forward model. These methods are therefore all mathematically equivalent and make the same assumptions that we summarize below:

- The probability density function of the geophysical vector X is assumed Gaussian with a mean background and a representative covariance matrix.
- The forward operator Y is able to simulate measurements-like radiances
- The errors of the models and the instrumental noise combined are assumed non-biased and normally distributed.
- The forward model is assumed to be locally linear at each iteration.

A legitimate question would be the following: What would happen if any of the assumptions above is not satisfied or if the mean and covariance information are not accurate enough? The solution that would be obtained under those conditions would likely be non-optimal. The term non-optimal here refers to the fact that the cost function would not necessarily be the smallest possible. The formulation of the cost function above (eq. 9 and the resulting solutions 19 and 20) were possible because of the simplifications introduced with those assumptions. In theory it is possible to derive another cost function based on non-Gaussian assumptions although it will likely be complicated. The corresponding solution could as well be determined by solving for the same equation 10 and resulting in a different formulation of the optimal solution.

## 1DVAR vs Regression Technique

Advantages of Physical Retrieval with Respect to Regression Technique

It is important to note that if (1) the same dataset used to generate the covariance matrix and background mean vector were also used to generate the coefficients of an algorithm based on multivariate regression technique, and if (2) the distributions of the geophysical vector and the modeling/instrument errors are both Gaussian and (3) the forward model is purely linear, then the application of the regression algorithm or the first iteration of the physical algorithm would mathematically result in the same optimal solution.

If however the problem is moderately non-linear or if the distributions non-Gaussian, the iterative-based numerical model is superior to the multivariate regression technique because it allows the use of the previous linear inversion locally and move, step by step, to the minimum-penalty solution. It is the iterative nature of the numerical method that is providing the added benefit with respect to the simple regression technique, by accommodating non-linearities and non- Normal distributions.