Beyond the basic case of fusing vertical profiles on the same grid, the CDF algorithm can be extended to more complex scenarios: fusing total-column measurements with profiles, and handling products with multiple atmospheric parameters.

Fusion of Profiles and Total Columns

Atmospheric remote sensing retrievals sometimes produce total-column quantities (vertically integrated amounts) instead of vertical profiles. The CDF method can fuse these columns with other profiles or columns, requiring only a slight modification to the standard formulation. For a total-column measurement, the averaging kernel \mathbf{A}_{i} becomes a row vector (1×N dimensions) representing the sensitivity of the retrieved column to the true vertical profile. The noise covariance becomes a scalar (variance). The cost function Eq. (2) must be reformulated as:
\mathbf{\alpha}_{i} = {\widehat{c}}_{i} – c_{ai} + \mathbf{A}_{i}\mathbf{x}_{ai}
(SC.1)
where {\widehat{c}}_{i} is the retrieved total column and c_{ai} is the total column corresponding to the a priori profile \mathbf{x}_{ai}. Once this substitution is made, the standard CDF(2015) equations (Configuration A from CDF Formulas page) can be applied directly. The constraint that the noise variance must be non-zero is typically satisfied in real measurements.

Converting Columns to Profiles

An alternative approach is to first transform the total column into a vertical profile using the CDF method itself, then fuse the resulting profile with other profiles using standard CDF equations (Tirelli et al., 2020). This transformation uses the column data as the only measurement with N=1:
\mathbf{x} = \left( \mathbf{A}_{i}^{T}\mathbf{S}_{ni}^{- 1}\mathbf{A}_{i} + \mathbf{S}_{a}^{- 1} \right)^{- 1}\left( \mathbf{A}_{i}^{T}\mathbf{S}_{ni}^{- 1}\mathbf{\alpha}_{i} + \mathbf{S}_{a}^{- 1}\mathbf{x}_{a} \right)
(SC.2)
The resulting profile and its characterisation matrices are:
\mathbf{A} = \left( \mathbf{A}_{i}^{T}\mathbf{S}_{ni}^{- 1}\mathbf{A}_{i} + \mathbf{S}_{a}^{- 1} \right)^{- 1}\mathbf{A}_{i}^{T}\mathbf{S}_{ni}^{- 1}\mathbf{A}_{i}
(SC.3)
\mathbf{S}_{n} = \left( \mathbf{A}_{i}^{T}\mathbf{S}_{ni}^{- 1}\mathbf{A}_{i} + \mathbf{S}_{a}^{- 1} \right)^{- 1}\mathbf{A}_{i}^{T}\mathbf{S}_{ni}^{- 1}\mathbf{A}_{i}\left( \mathbf{A}_{i}^{T}\mathbf{S}_{ni}^{- 1}\mathbf{A}_{i} + \mathbf{S}_{a}^{- 1} \right)^{- 1}
(SC.4)
\mathbf{S}_{s} = \left( \mathbf{A}_{i}^{T}\mathbf{S}_{ni}^{- 1}\mathbf{A}_{i} + \mathbf{S}_{a}^{- 1} \right)^{- 1}\mathbf{S}_{a}^{- 1}\left( \mathbf{A}_{i}^{T}\mathbf{S}_{ni}^{- 1}\mathbf{A}_{i} + \mathbf{S}_{a}^{- 1} \right)^{- 1}
(SC.5)
\mathbf{S} = \mathbf{S}_{n} + \mathbf{S}_{s} = \left( \mathbf{A}_{i}^{T}\mathbf{S}_{ni}^{- 1}\mathbf{A}_{i} + \mathbf{S}_{a}^{- 1} \right)^{- 1}
(SC.6)
The resulting noise covariance \mathbf{S}_{n} is a rank-1 matrix (singular). Because of this singularity, when fusing this transformed column-profile with other profiles, it is strongly recommended to use CDF(2022) instead of CDF(2015), as the latter requires inversion of singular matrices.

Multi-Target Retrievals (MTRs)

Some atmospheric remote sensing instruments retrieve multiple atmospheric parameters simultaneously (e.g., temperature, ozone, water vapor). This produces state vectors with components corresponding to different species. When fusing two MTR products that do not retrieve the same set of parameters, the standard CDF formulation must be adapted.

Example: Fusing Products with Different Parameters

Consider two instruments:
  • Instrument 1 retrieves parameters P1 and P2
  • Instrument 2 retrieves parameters P1 and P3
The individual state vectors are:
{\widehat{\mathbf{x}}}_{1} = \begin{pmatrix} \widehat{\mathbf{P}}\mathbf{1}_{1} \\ \widehat{\mathbf{P}}\mathbf{2}_{1} \end{pmatrix}\quad\quad{\widehat{\mathbf{x}}}_{2} = \begin{pmatrix} \widehat{\mathbf{P}}\mathbf{1}_{2} \\ \widehat{\mathbf{P}}\mathbf{3}_{2} \end{pmatrix}
(SC.7)
To fuse these products, we must extend both state vectors to include all parameters present in either product, padding with zeros where data are not available:
{\widehat{\mathbf{x}}}_{1}^{'} = \begin{pmatrix} \widehat{\mathbf{P}}\mathbf{1}_{1} \\ \widehat{\mathbf{P}}\mathbf{2}_{1} \\ \mathbf{0} \end{pmatrix}\quad\quad{\widehat{\mathbf{x}}}_{2}^{'} = \begin{pmatrix} \widehat{\mathbf{P}}\mathbf{1}_{2} \\ \mathbf{0} \\ \widehat{\mathbf{P}}\mathbf{3}_{2} \end{pmatrix}
(SC.8)
The averaging kernel matrices and noise covariance matrices must be extended accordingly, with zeros in rows and columns corresponding to non-retrieved parameters:
\mathbf{A}_{1}^{'} = \begin{pmatrix} \mathbf{A}_{\mathbf{11},1} & \mathbf{A}_{\mathbf{12},1} & \mathbf{0} \\ \mathbf{A}_{\mathbf{21},1} & \mathbf{A}_{\mathbf{22},1} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} \end{pmatrix}\quad\quad\mathbf{A}_{2}^{'} = \begin{pmatrix} \mathbf{A}_{\mathbf{11},2} & \mathbf{0} & \mathbf{A}_{\mathbf{13},2} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{A}_{\mathbf{31},2} & \mathbf{0} & \mathbf{A}_{\mathbf{33},2} \end{pmatrix}
(SC.9)
Once the matrices are extended to the common state vector format, standard CDF equations can be applied. The fused product contains:
  • Common parameters (P1): Improved estimates resulting from fusion of both instruments
  • Instrument-specific parameters (P2, P3): Also improved, but indirectly, through the correlation with P1
The quality improvement for instrument-specific parameters is directly proportional to the degree of correlation between that parameter and the common parameters. Reference: This extension for MTR fusion was described in Tirelli et al. (2021).