A Theory for the Retrievals of Virtual Temperature from Remote Measurements of Horizontal Winds and Thermal Radiation

Abstract

This paper develops a theory for the estimation of virtual temperature from remote measurements of (i) emitted thermal radiation by microwave and infrared radiometers and (ii) horizontal winds by Doppler radars (or lidars). The problem of obtaining a temperature profile from measured radiances and an inversion of the radiative transfer equation is known to be ill posed/conditioned. For a meaningful inversion additional information is needed. In this work we utilize two sources of information and two physical relationships which are: 1) measured radiances at various frequencies; 2) the radiative transfer equation at various frequencies; 3) the measured horizontal wind, and 4) the horizontal equations of motions. We then look for a virtual temperature profile that will (in the least-square sense) satisfy both the horizontal equations of motions and the radiative transfer equations. The resulting calculus of variations problem, after suitable manipulations, is then reduced to solving at each horizontal level a Poisson equation for the virtual temperature. The input parameters are the measured horizontal winds from which, using objective analysis techniques, one can derive gradients and various nonlinear products of the wind, the measured radiances and its derived gradients and the kernels (at various frequencies) of the radiative transfer equation. The sensitivity of the solution to uncertainties in the vertical velocities and to objective analysis errors is also examined. The inclusion of fluid dynamical considerations also allows for the allows of components of the virtual temperature profile which make no contribution to the measured radiances.