To apply the CDF to a generic level 2 atmospheric product (primarily vertical profiles of temperature or constituents), it must have specific characteristics, summarized below.
the product must be retrieved using the optimal estimation technique;
the full state vector {\widehat{\mathbf{x}}}_{\mathbf{i}} must be available;
the profiles that are part of the state vector {\widehat{\mathbf{x}}}_{\mathbf{i}}\mathbf{\ }can be defined in terms of concentrations or partial columns;
the vertical grid must be available (in pressure or in height, in levels or layers) on which the profiles belonging to the state vector are defined;
the complete a priori state vector \mathbf{x}_{\mathbf{ai}} must be available, where the parts representing profiles are defined on the same grid as the state vector;
two of the following three matrices have to be available:
the Averaging Kernels Matrix \mathbf{A}_{\mathbf{i}} referring to the full state vector must be available;
the regards Variance Covariance Matrix (VCM) referred to the total errors and to the full state vector \mathbf{S}_{\mathbf{i}};
the a priori VCM referred to the entire state vector (\mathbf{S}_{\mathbf{ai}}), used for product retrieval,
If only two of the matrices referenced at point 6 are available, the third one can be obtained using one of the following equations:
\mathbf{S}_{i} = \left( \mathbf{I} - \mathbf{A}_{i} \right) \cdot \mathbf{S}_{ai}
|
(P1) |
|---|---|
\mathbf{S}_{ai} = \left( \mathbf{I} - \mathbf{A}_{i} \right)^{- 1} \cdot \mathbf{S}_{i}
|
(P2) |
\mathbf{A}_{i} = \left( \mathbf{I} - \mathbf{S}_{i\ } \cdot {\mathbf{S}_{ai}}^{- 1} \right)
|
(P3) |
Both the inversions of equations (P2) (P3) refers to usually invertible matrices. The availability of the noise CM \mathbf{S}_{ni} instead of the total errors one \mathbf{S}_{i} is more critical because, while {\mathbf{S}_{ni}\mathbf{= \ A}}_{i\ } \cdot \mathbf{S}_{i} can be safely calculated, the inverse transformation entails the inversion of the AK matrix that in general is not invertible. This eventuality will be managed as a special case, if encountered.
For what regards the total columns products, the same considerations hold, considering that the CMs reduce to scalar values, the averaging kernels are row shaped and that the a priori profiles used in the retrieval of the total columns must be available. The averaging kernel of a total column represents the derivative of the retrieved column with respect to the true profile (VMR or cols).
The results of the CDF requirements tests will be summarized in tables like table 1, in which the colour code represents the test results and eventual notes give additional explanations.