475 Passive Microwave Radiometer Channel Selection Based on Cloud and Precipitation Information Content Estimation Sabatino Di Michele and Peter Bauer Research Department Submitted to Q. J. Royal Meteor. Soc. July 2005 Series: ECMWF Technical Memoranda A full list of ECMWF Publications can be found on our web site under: http://www.ecmwf.int/publications/ Contact: [email protected] c Copyright 2005 European Centre for Medium-Range Weather Forecasts Shinfield Park, Reading, RG2 9AX, England Literary and scientific copyrights belong to ECMWF and are reserved in all countries. This publication is not to be reprinted or translated in whole or in part without the written permission of the Director. Appropriate non-commercial use will normally be granted under the condition that reference is made to ECMWF. The information within this publication is given in good faith and considered to be true, but ECMWF accepts no liability for error, omission and for loss or damage arising from its use. Microwave Channel Selection from Precipitation Information Content Abstract The information content of microwave frequencies between 5 and 200 GHz for rain, snow and cloud wa- ter retrievals over ocean and land surfaces was evaluated using optimal estimation theory. The study was based on large datasets representative of summer and winter meteorological conditions over North Amer- ica, Europe, Central Africa, South America and the Atlantic obtained from short-range forecasts with the operational ECMWF model. The information content was traded off against noise that is mainly produced by geophysical variables such as surface emissivity, land surface skin temperature, atmospheric temperature and moisture. The estimation of the required error statistics was based on ECMWF model forecast error statistics. The results suggest a number of frequency bands most suited for the retrieval of (1) rain over ¡ oceans: 17-18, 32-45, 80, 135-150, 118.75 ¡ 10-14 GHz; rain over land: 90, 135-140, 118.75 10-14 GHz; (2) snow over land and oceans: 135-155, 118.75 ¡ 10-14, 195 GHz, (3) clouds over oceans: 42, 82-83 GHz; clouds over land: 90-100, 135 GHz. For radiometers designed for global and multi-season applications, several channels in all of the above frequency ranges would be desirable for optimizing channel impact on hydrometeor retrievals depending on the observed situation. 1 Introduction Optimal estimation theory is widely used for the inversion of radiometric observations from space to retrieve physical state variables related to the Earth's atmosphere and surface. With the advent of infrared radiometer technology applying spectroscopic principles, instruments have been launched that provide a large number of channels over a wide spectral range. Several studies were carried out to optimize the usage of this data by reducing the number of channels and by maintaining the maximum possible information content at the same time (Rabier et al. 2002). Iterative 'selection' methods, first intro- duced by Rodgers (1996) turn out to be both practicable and efficient. Fourrie´ and Thepaut´ (2003) compared these methods against empirical techniques in the context of the Atmospheric Infrared Sounder (AIRS) con- cluding that a significant gain can be obtained when they are employed. Microwave frequencies are commonly used to remotely sense atmospheric temperature and moisture but also clouds and precipitation as well as sur- face properties. With increasing computer power and improvement in data assimilation techniques, cloud and precipitation microwave observations have been applied for Numerical Weather Prediction (NWP) showing en- couraging results (Moreau et al. 2003, Bauer et al. 2005a). Iterative methods loop over channels and sort them by decreasing information content given a priori information (from the model background) and that from the ones previously selected. Alternatively to 'selection' methods, which work top-down from highest to lowest information content, 'deselection' methods (that work bottom-up) also exist (Sofieva and Kyrol¨ a¨ 2003). The latter seem to be less vulnerable to non-uniqueness of solutions but are computationally more costly because the iteration must always be performed for the entire set of channels while selection methods may be truncated earlier. Two quantities are usually used as figure of merit for the information content, namely the degree of freedom for signal and the entropy reduction, both reflecting the improvement given by each channel to the retrieval error. The first determines the number of independent observations given an overdetermined inversion problem (Purser and Huang 1993), while the latter measures the reduction of entropy in a posteriori probability distribution functions (pd f , Shannon and Weaver 1949). In this paper these concepts are applied to define the optimized usage of channels in the microwave portion of the spectrum. The basics of optimal estimation and the information content methodology employed in this study are outlined in Section 1. The definition of the candidate microwave channels and the description of the atmospheric profiles dataset used to perform the channels selection are given in Section 3. The conclusions from this new application for sensor optimization are finally presented in Section 4. Technical Memorandum No. 475 1 Microwave Channel Selection from Precipitation Information Content 2 Methodology The methodology employed for evaluating the information content of spaceborne passive microwave observa- tions of clouds and precipitation is based on the optimal estimation theory (e.g. Rodgers 2000). In this Section, first the basis of the forward and inverse problems will be briefly outlined, then the formulation used for the information content evaluation will be exposed. 2.1 Optimal estimation The problem of the inversion of observation from space is not fully constrained, in particular in the presence of clouds and precipitation. The application of statistical principles is therefore fundamental for solving the inverse problem. In the following, the state of the atmosphere (to be retrieved) is denoted as a vector x and the multiple-channel observations from a theoretical microwave radiometer are contained in a vector y. Their statistical link is expressed as conditional pd f 's, through the application of Bayes' theorem: £ ¤ ¢ ¤ P ¢ y x P x £ ¤¦¥ P ¢ x y (1) ¤ P ¢ y £ ¤ ¢ £ ¤ P ¢ x y is the posteriori probability of x when y is observed. P y x is the probability of making observation y ¤ ¢ ¤ ¢ ¤ when x is present, while P ¢ x and P y are the a priori probabilities of x and y, respectively. P x is assumed to fully describe the a priori distribution of x. The physical link between x and y is described by the observation operator, H, that may be non-linear. In our case, the observation operator only consists of a radiative transfer model. The modelled observation, y, is: ¢ ¤¨§ y ¥ H x ε (2) and ε summarizes observation errors (e.g. radiometer noise) and forward modeling errors (e.g. radiative transfer model uncertainties). £ ¤ The estimation of P ¢ x y is determined by the observations and an a priori estimate of the state, xb, as well as the error covariance matrices of that a priori state, B, and the errors originating from observation and modelling, § R ¥ E F: 1 T 1 1 T 1 £ ¤¦¥ © ¢ ¤ ¢ ¤ P ¢ exp H H (3) x y y x R y x x x B x x 2 2 b b Superscripts '-1' and 'T' denote inverse and transpose matrices, respectively. This only holds if the distributions ¢ ¤ y H x and x xb are uncorrelated as well as B and R have Gaussian characteristics and zero means. The information content estimation relies on the estimation of the additional contribution to retrieval accuracy provided by the observation. This means that the retrieval accuracy or the retrieval error have to be calculated. In general, this calculation is greatly facilitated by having linear observation operators. This is because the estimation of x requires finding the minimum of a cost-function that contains the right-hand side of Equation (3). The minimum is found using the first derivative of this term that contains the first derivative of the observation operator. This is rather simple because the linearity implies a quadratic cost function. For a non-linear H, the second derivative of H is needed whose actual computation may become difficult. Therefore, we will assume that H is only weakly non-linear and therefore that xb is rather close to x. Since we are not carrying out an actual retrieval this assumption is not critical. But even for retrieval studies this assumption may be valid if, for example, a statistically consistent perturbation approach is applied (Bauer et 2 Technical Memorandum No. 475 Microwave Channel Selection from Precipitation Information Content al. 2005b) that is based on small perturbations of the same order of magnitude as x xb differences. It can be shown (Rodgers, 2000) that in the linear case the optimal analysis state xa has the following expression: T 1 § ¢ ¤ xa ¥ xb AH R y H xb (4) with H and HT being the tangent-linear and adjoint of the observation operator, and A the analysis error covariance matrix: T T 1 ¢ § ¤ A ¥ B BH HBH R HB (5) 2.2 Channel selection The information content of an observation is a quantitative measure of the reduction of the estimation error pro- duced by the observation. The improvement of estimation error depends on the sensitivity of the observation to the state and on the accuracy of the measurement and modelling, i.e., on H and R, their respective magnitudes, and finally on the accuracy of the a priori information (through B). This makes information content similar to the signal-to-noise ratio. Historically, there have been several formulations of information content (Rodgers 2000 and references therein). As previously mentioned, in our application, the 'degrees of freedom for signal', DFS, and the 'entropy reduction', ER, are possible choices. The former estimates the number of independent pieces of information in a measurement vector that are related to the signal vs.