# In inverse theory, how do I transform the averaging kernel matrix to a new grid?

Rodgers and Connor (2003) describe how measurements by remote sounders can be properly compared, taking into account differences in averaging kernels and error covariances. They make the assumption that the profiles are represented on the same vertical grid.

Rodgers (2000, section 3.1) describes how to transform the state vector ad the error covariance to get them on the same grid:

If we intend to compare MAP retrievals on different grids, we must ensure that we compare like with like. Not only must the state be properly transformed, but also the prior covariance. (...) A diagonal covariance with elements of say 100 K^2 on a grid of 1 km spacing is not equivalent to the same variance on a grid of 2 km spacing. (...)

...and proceeds to formulate in details how this can be done.

However, neither document describes how to properly transform the averaging kernel matrix.

What is a correct way to transform the averaging kernel matrix $\mathbf{A}$? The point Rodgers makes for the covariance matrix applies equally for the averaging kernel matrix; after all, the number of degrees of freedom ($tr(\mathbf{A})$), for example, must not change for the transformed matrix to remain consistent with the old one.

Rodgers, Clive D. Inverse methods for atmospheric sounding: theory and practice. Vol. 2. Singapore: World scientific, 2000.

Rodgers, C. D. and B. J. Connor (2003), Intercomparison of remote sounding instruments, J. Geophys. Res., 108(D3), 4116, doi:10.1029/2002JD002299.

This is considered in Calisesi et al. (2005). They derive that

$$\mathbf{A_{z_i}} = \mathbf{W_i^* A_x W_i} \, ,$$

where $\mathbf{A_{z_i}}$ is the averaging kernel for the new grid, $\mathbf{W_i}$ is the interpolation matrix with $\mathbf{W_i^*}$ its Moore-Penrose pseudo-inverse, and $\mathbf{A_x}$ is the averaging kernel matrix for the full state vector $\mathbf{x}$. For the inverse transformation,

$$\mathbf{A_x} = \mathbf{W_i A_{z_i} W_i^*} + \mathbf{\epsilon_{A_i}}\, ,$$

where $\mathbf{\epsilon_{A_i}} = \mathbf{A_x} - \mathbf{W_i W_i^* A_x W_i W_i^*}$. Across independent numerical grids,

$$\mathbf{A_{z_1}} = \mathbf{W_{12} A_{z_2} W_{21}} + \mathbf{W_1^* \epsilon_{A_2} W_1} \, .$$

For a full derevation, see Calisesi et al. (2005).

Calisesi, Y., V. T. Soebijanta and R. van Oss (2005), Regridding of remote soundings: Formulation and application to ozone profile comparison, J. Geophys. Res., 110, D23306, doi:10.1029/2005JD006122