Comparison of linear forms of the radiative transfer equation with analytic Jacobians

Determining the Jacobians of the radiative transfer equation (RTE) is important to the qualities of the simultaneous retrieval of geophysical parameters from satellite radiance observations and the assimilation of radiance data into a numerical weather prediction system. Two linear forms of the RTE with analytic Jacobians are formulated. The first linear form has approximate analytic Jacobians, which involves some monochromatic approximation applied to a fast transmittance model. Unlike previous research, which lacks the transmittance Jacobian with respect to the atmospheric temperature profile, this form is complete in the sense that the transmittance Jacobians with respect to atmospheric temperature and absorbing constituent profiles are both present. The second linear form has exact analytic Jacobians derived consistently from the same fast transmittance model without using any monochromatic approximation. By numerical comparison between the two linear forms for the NOAA-12 High-Resolution Infrared Sounder, we show significant errors in the linear form with approximate analytic Jacobians. The relative absolute linearization error from the linear form with approximate analytic Jacobians is shown to be 2-4 orders of magnitude larger than that from the linear form with exact analytic Jacobians, even for the case of a 0.1% perturbation of the U.S. Standard Atmosphere. The errors unnecessarily complicate the ill-posed retrieval problem of atmospheric remote sensing and can be avoided if the correct linear form of the RTE with exact analytic Jacobians is adopted.