Tables of integrals

IMPROVING ACCURACY IN MICROWAVE RADIOMETRY VIA PROBABILITY AND INVERSE PROBLEM THEORY Derek L. Hudson Electrical and Computer Engineering Doctor of Philosophy Three problems at the forefront of microwave radiometry are solved using probability theory and inverse problem formulations which are heavily based in probability theory. Probability theory is able to capture information about random phenomena, while inverse problem theory processes that information. The use of these theories results in more accurate estimates and assessments of estimate error than is possible with previous, non-probabilistic approaches. The benefits of probabilistic approaches are expounded and demonstrated. The first problem to be solved is a derivation of the error that remains after using a method which corrects radiometric measurements for polarization rotation. Yueh [1] proposed a method of using the third Stokes parameter TU to correct brightness temperatures such as Tv and Th for polarization rotation. This work presents an extended error analysis of Yueh’s method. In order to carry out the analysis, a forward model of polarization rotation is developed which accounts for the random nature of thermal radiation, receiver noise, and (to first order) calibration. Analytic formulas are then derived and validated for bias, variance, and root-mean-square error (RMSE) as functions of scene and radiometer parameters. Examination of the formulas reveals that: 1) natural TU from planetary surface radiation, of the magnitude expected on Earth at L-band, has a negligible effect on correction for polarization rotation; 2) RMSE is a function of rotation angle Ω, but the value of Ω which minimizes RMSE is not known prior to instrument fabrication; and 3) if residual calibration errors can be sufficiently reduced via postlaunch calibration, then Yueh’s method reduces the error incurred by polarization rotation to negligibility. The second problem addressed in this dissertation is optimal estimation of calibration parameters in microwave radiometers. Algebraic methods for internal calibration of a certain class of polarimetric microwave radiometers are presented by Piepmeier [2]. This dissertation demonstrates that Bayesian estimation of the calibration parameters decreases the RMSE of the estimates by a factor of two as compared with algebraic estimation. This improvement is obtained by using knowledge of the noise structure of the measurements and by utilizing all of the information provided by the measurements. Furthermore, it is demonstrated that much significant information is contained in the covariance information between the calibration parameters. This information can be preserved and conveyed by reporting a multidimensional pdf for the parameters rather than merely the means and variances of those parameters. The proposed method is also extended to estimate several hardware parameters of interest in system calibration. The final portion of this dissertation demonstrates the advantages of a probabilistic approach in an empirical situation. A recent inverse problem formulation, sketched in [3], is founded on probability theory and is sufficiently general that it can be applied in empirical situations. This dissertation applies that formulation to the retrieval of Antarctic air temperature from satellite measurements of microwave brightness temperature. The new method is contrasted with the curvefitting approach which is the previous state-of-the-art. The adaptibility of the new method not only results in improved estimation but is also capable of producing useful estimates of air temperature in areas where the previous method fails due to the occurence of melt events.

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Derek Lavell Hudson

Papers

Improving accuracy in microwave radiometry via probability and inverse problem theory