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.