A physical temperature profiling method using gradient flows

We present a new mathematical technique for retrieving the temperature profile from a ground based microwave profiler and ancillary measurements. It is based on a gradient flow approach. We are able to solve the inverse problem of radiative transfer and determine the temperature profile from ten simultaneous brightness temperature measurements in the range from 50.8 to 58.8 GHz. The approach uses no additional statistical information. A physical consistent solution for a temperature profile can be found within five seconds. First results of this technique show promising results with regard to absolute accuracy and inversion height determination, even if the brightness temperature measurements are assumed noisy to 0.5 K. 1 The inverse problem of radiative transfer A major desire for numerical weather prediction is the availability of continuously measured profiles of temperature ( ), humidity ( ) and pressure ( ). Such measurements are of extreme importance for operational data assimilation in the sense of short term weather forecasting. However, still today operational costand man-power extensive radiosondes launches (two to four times a day) state the main source for atmospheric profiles used to initialize weather forecast models. Recent technical advances in microwave technology have brought forth microwave profilers, which can simultaneously measure brightness temperatures at a multiple number of frequencies. Operation of these profilers in zenith and also at other elevation angles has the potential of deriving continuous atmospheric profiles of temperature and humidity. Standard methods for inferring these atmospheric profiles from brightness temperature measurements are mostly on a statistical basis, however some methods with physical constraints have also been developed. A standard approach [4] uses simultaneously measured microwave and infrared brightness temperatures and radiosonde profiles of and to create a representative data set and invert the radiative transfer equation via multiple regression and neural networks. In [8] retrieval methods are proposed based on a Bayesian approach. Here, the optimal estimation equations in the formulation of Newtonian Iteration can combine measurements and statistical information in an optimal way, such that certain physical constraints are fulfilled. Such an approach to infer , , and cloud liquid water profiles has been successfully applied to a multiple sensor combination in [6]. We present a novel and fast numerical technique to derive temperature profiles from microwave frequencies located at the wing of the 60 GHz oxygen absorption complex. Here we describe a new mathematical approach for a retrieval, which in contrast to many other retrieval methods, does not rely on statistical data sets, thus can be characterized as ”purely physical”. The advantage of such a method is that it may be applied virtually to any arbitrary site. The mathematical key tool for the solution of the problem consists in so called regularized gradient flows. For a theoretical background and applications to inverse problems we refer to [7, 9, 2, 1] 2 The gradient flow method In our measurement configuration the multi-channel microwave radiometer is located at the ground, the atmosphere is horizontally stratified and no clouds are present. We assume that the microwave radiation measured at the ground depends only on , , , elevation angle , and frequency . To compute the temperature profile we take into account brightness temperature measurements ( , ... , ) at frequencies to . In our special case these frequencies are chosen to be 50.8, 51.8, 52.8, 53.8, 54.8, 55.8, 56.8, 57.8, and 58.8 GHz, corresponding to one band of the 22 -channel microwave