Calculation of altitude-dependent Tikhonov constraints for TES nadir retrievals

A key component in the regularization of vertical atmospheric trace gas retrievals is the construction of constraint matrices. We introduce a novel method for developing a constraint matrix based on altitude-varying combinations of zeroth-, first-, and second-order derivatives of the trace gas profile. This constraint matrix can be optimized to minimize the diagonal a posteriori error covariance and can also consider other factors such as degrees of freedom. This approach is applied to the calculation of constraint matrices for Tropospheric Emission Spectrometer nadir retrievals of atmospheric temperature, H2O, O-3, CO, and CH4. The retrieval error achieved with these constraints is comparable to the error achieved with the classical Bayesian constraint. Furthermore, these constraints are shown to be robust under uncertainty in the climatological conditions.