An improved formula for the complete data fusion

The complete data fusion is a method that combines independent measurements of atmospheric vertical profiles. Recently a new formula for the complete data fusion, which does not contain matrices that can be singular and overcomes the generalized inverse approximation used when singular matrices have to be inverted, has been proposed. We show that the new formula is a generalization of the original one and analyze the analytical relationship between the two formulas when generalized inverse matrices are used for the inversion of singular matrices. We extend the new formula to include interpolation and coincidence errors, which must be considered when the profiles to be fused are measured on different vertical grids and at different times and/or locations. Finally, we use a real measurement of the Infrared Atmospheric Sounding Interferometer (IASI) instrument to show the improved performances of the new formula with respect to the original one.