In this cumulative dissertation, different features and methods are presented to assess and process multi-sensor derived radar data for climatological analysis. The overall objectives were to appraise the limitations of an hourly radar-based quantitative precipitation estimate (QPE) product and to develop and apply reasonable approaches to process these data. Hence the spatial and temporal limitations of radar-derived precipitation rates are discussed in the context of climatological applications, and two types of climatologies are obtained, first a climatology of daily precipitation fields and second a long term precipitation climatology. These relate to questions concerning the methodologies rather than climatological significance or assessment of precipitation and its role in the water balance. Current radar data availability limits such a hydro-climatic analysis. The thesis consists of three peer-reviewed publications. All investigations in this thesis are based on the RADOLAN rw-product of the German Weather Service (DWD) for an extended study region including the Free State of Saxony, Germany, for the period from April 2004 to November 2011. The first publication is dedicated to the classification of daily precipitation fields by unsupervised neural networks. In the presented work, the quality of the radar-derived precipitation rates is analysed by a temporal comparison between recording and non-recording gauges and the corresponding pixels of the RADOLAN rw-product on hourly and daily bases. The analysis shows that a temporal aggregation of the original product should be limited to a temporal scale up to 24 h because of the processing algorithms and the reappearance of previously suppressed errors. Nevertheless, an unsupervised neural network was successfully used for the classification of daily patterns. The derived daily precipitation classes and corresponding precipitation patterns could be assigned to properties of the associated weather patterns and seasonal dependencies. Hence, it could be shown that the classified patterns not only occurred by chance but by statistically proven properties of the atmosphere and of the season. The second publication is primarily concerned with two tasks: first, the pixel-wise fitting of mixture distributions on the bases of the obtained patterns from the first publication, and second, the analysis of spatial consistency of the radar-derived precipitation data set. The fitted parametric distribution functions were analysed in terms of Akaike\'s information criterion and the Kolmogorov-Smirnov test. These benchmarks showed, that the performances are best for mixture distributions derived by an initial classification by an unsupervised neural network and cluster analysis, and by gamma distributions. These results underline the significance of the derived precipitation classes obtained in the first publication. Furthermore, the Kolmogorov-Smirnov test indicates that independent of the distribution function, the radar-derived daily precipitation rates under the assumption of the deployed parametric distribution function has the best or most natural order of precipitation rates at spatial scales from 2 to 4 km for daily precipitation fields. Thus, it is recommended to use the original radar product at these scales rather than at 1 km resolution for daily precipitation sums. In the last publication, the focus shifts from daily to long-term precipitation climatology. The work introduces a rapid and simple approach for processing radar-derived precipitation rates for long-term climatologies. The method could successfully be applied to the radar-derived precipitation rates by excluding or correcting the errors that reappear due to temporal aggregation. Despite the fact that the approach is empirical, the introduced parameters could almost be objectively derived by means of simulation and optimisation. This could be achieved by utilising the reasonable relationship between elevation and precipitation rates for longer periods. Finally, the obtained results are compared to two independently derived precipitation data sets. The comparison shows good agreement of the precipitation fields and illustrates a reasonable application of the introduced procedure. The presented results support the application of the approach for precipitation aggregates of, at least, annual or longer periods. However the derivation of climatologies led to satisfactory results at the respective temporal scales, though the influence of radar-specific errors can only be minimized to a certain degree. Further studies have to prove if an application independent processing of radar-derived precipitation rates leads to higher qualities and validities of the derived data in time and space.

Scatterometers, such as SeaWinds, can provide wide coverage of ocean surface winds. They estimate near-surface wind vectors by relating measured radar backscatter to a geophysical model function. However, SeaWinds measurements are also sensitive to rain, and conventional wind retrieval degrades in rainy conditions. An algorithm that exploits SeaWinds' sensitivity to both wind and rain has be developed. This algorithm, termed simultaneous wind/rain retrieval, retrieves both wind vectors and rain rates for a given ocean area. Instantaneous results of simultaneous wind/rain retrieval in Hurricane events is analyzed through comparison with the NEXRAD ground-based radar system. This comparison allows validation of retrieved rains. Additionally, conditions that affect the accuracy of SeaWinds wind/rain observations are evaluated. It is shown that, when thresholded, the rains retrieved by SeaWinds give an adequate rain flag. The comparisons of SeaWinds and NEXRAD rain estimates facilitate construction of a model to simulate variability in the SeaWinds rain estimates. The model is used to show that rain estimates are unbiased, though with significant variability. The variability is likely to be primarily driven by the noise inherent to the SeaWinds system.

Debris flows occur frequently in remote areas of Great Smoky Mountains National Park, Tennessee. Rainfall gauges are not adequate for modeling infiltration required for triggering debris flows. Weather radar, providing frequently updated, continuous coverage, is a valuable tool for estimating rainfall intensity, duration, runoff, and infiltration. Daily rainfall from a sole gauge was compared with hourly rainfall from the Digital Precipitation Array weather radar product to model infiltration on August 5, 2012, the day before a debris flow was known to have occurred in the 91-km2West Prong Little Pigeon River watershed. Additionally, both gauge and radar data were used for rainfall-runoff-infiltration modeling for a 42-day period in July and August 2012. Runoff and infiltration were simulated using the conventional semi-distributed hydrological model HEC-HMS. A local bias correction of radar rainfall at the gauge location improved correlation between the radar rainfall and the gauge data. Peak daily rainfall for the August 5 storm was 93 mm (gauge) and 98 mm (radar), whereas average daily rainfall for the 42-day period was 10 mm and 7.75 mm, respectively. Over the study period, simulated daily infiltration declined from 28 mm to 0.5 mm for the gauge and from 15 mm to 0.14 mm for radar, indicating essentially saturated conditions on the day of the debris flow.

Complex engineered systems evolve, with a tendency toward self-organization, which can, paradoxically, frustrate the aims of those seeking to develop them. The systems engineer, seeking to promote the development in the context of changing and uncertain requirements, is challenged by conceptual gaps that emerge within engineering projects, particularly as they scale up, that inhibit communication among the various stakeholders. Overall optimization, involving multiple criterion, is often expressed in the language of the individual parties, increasing the complexity of the overall situation, subsuming the participants within the evolution of the complex engineered system, containing the objective and subjective in counterproductive or inefficient ways that can arrest healthy development. The conventional pragmatic systems engineering approach to the resolution of such situations is to introduce architectural discipline by way of separation of concerns. In complex engineered systems projects, the crucial interface, at any level of abstraction, is between the technical domain experts and higher level decision makers. Bridging the ensuing conceptual gap requires models and methods that provide communication tools promoting a convergence of the conversation between these parties on a common "common sense" of the underlying reality of the evolving engineered system. In the interest of conceptual clarity, we confine our investigation to a restricted, but important general class of evolving engineered system, information gathering and utilizing systems. Such systems naturally resolve the underlying domain specific measures by reduction into common plausible information measures aimed at an overall sense of informativeness. For concreteness, we further restrict the investigation and the demonstration to a species that is well documented in the open literature: weather radar networks, and in particular to the case of the currently emerging system referred to as CASA. The multiobjective problem of objectively exploring the high dimensionality of the decision space is done using multiobjective genetic algorithms (MOGA), specifically the John Eddy genetic algorithms (JEGA), resulting in well-formed Pareto fronts and sets containing Pareto optimal points within 20% of the ideal point. A visualization technique ensures a clear separation of the subjective criterion provided by the decision makers by superficially adding preferences to the objective optimal solutions. To identify the integrative objective functions and test patterns utilized in the MOGA analysis, explorations of networked weather radar technologies and configuration are completed. The explorations identify trends within and between network topologies, and captures both the robustness and fragility of network based measurements. The information oriented measures of fusion accuracy and precision are used to evaluate pairs of networked weather radars against a standardized low order vortex test pattern, resulting in a metrics for characterizing the performance of dual-Doppler weather radar pairs. To define integrative measures, information oriented measures abstracting over sensor estimators and parameters used to estimate the radial velocity and returned signal from distributed targets, specifically precipitation, are shown to capture the single radar predicted performance against standardized test patterns. The methodology bridges the conceptual gap, based on plausible information oriented measures, standardized with test patterns, and objectively applied to a concrete case with high dimensionality, allowed the conversation to converge between the systems engineer, decision makers, and domain experts. The method is an informative objective process that can be generalized to enable expansion within the technology and to other information gathering and utilizing systems and sensor technologies.

No description available.

Honeywell Technology Solutions Lab, India / Weather is a major contributing factor in aviation accidents, incidents and delays. Doppler weather radar has emerged as a potent tool to observe weather. Aircraft carry onboard radars but their range and angular resolution are limited. Networks of ground-based weather radars provide extensive coverage of weather over large geographic regions. It would be helpful if these data can be transmitted to the pilot. However, these data are highly voluminous and the bandwidth of the ground-air communication links is limited and expensive. Hence, these data have to be compressed to an extent where they are suitable for transmission over low-bandwidth links. Several methods have been developed to compress pictorial data. General-purpose schemes do not take into account the nature of data and hence do not yield high compression ratios. A scheme for extreme compression of weather radar data is developed in this thesis that does not significantly degrade the meteorological information contained in these data. The method is based on contour encoding. It approximates a contour by a set of systematically chosen ‘control points’ that preserve its fine structure up to a certain level. The contours may be obtained using a thresholding process based on NWS or custom reflectivity levels. This process may result in region and hole contours, enclosing `high' or `low' areas, which may be nested. A tag bit is used to label region and hole contours. The control point extraction method first obtains a smoothed reference contour by averaging the original contour. Then the points on the original contour with maximum deviation from the smoothed contour between the crossings of these contours are identified and are designated as control points. Additional control points are added midway between the control point and the crossing points on either side of it, if the length of the segment between the crossing points exceeds a certain length. The control points, referenced with respect to the top-left corner of each contour for compact quantification, are transmitted to the receiving end. The contour is retrieved from the control points at the receiving end using spline interpolation. The region and hole contours are identified using the tag bit. The pixels between the region and hole contours at a given threshold level are filled using the color corresponding to it. This method is repeated till all the contours for a given threshold level are exhausted, and the process is carried out for all other thresholds, thereby resulting in a composite picture of the reconstructed field. Extensive studies have been conducted by using metrics such as compression ratio, fidelity of reconstruction and visual perception. In particular the effect of the smoothing factor, the choice of the degree of spline interpolation and the choice of thresholds are studied. It has been shown that a smoothing percentage of about 10% is optimal for most data. A degree 2 of spline interpolation is found to be best suited for smooth contour reconstruction. Augmenting NWS thresholds has resulted in improved visual perception, but at the expense of a decrease in the compression ratio. Two enhancements to the basic method that include adjustments to the control points to achieve better reconstruction and bit manipulations on the control points to obtain higher compression are proposed. The spline interpolation inherently tends to move the reconstructed contour away from the control points. This has been somewhat compensated by stretching the control points away from the smoothed reference contour. The amount and direction of stretch are optimized with respect to actual data fields to yield better reconstruction. In the bit manipulation study, the effects of discarding the least significant bits of the control point addresses are analyzed in detail. Simple bit truncation introduces a bias in the contour description and reconstruction, which is removed to a great extent by employing a bias compensation mechanism. The results obtained are compared with other methods devised for encoding weather radar contours.

Para reduzir as perdas humanas e materiais durante as inundações, é possível realizar estudo conciso da previsão de chuva, etapa principal de um sistema de alerta antecipado de inundação. O uso de informações de radar de tempo, quando acopladas a modelos de previsão de precipitação baseados fisicamente, pode contribuir para o monitoramento e previsão de episódios de chuva intensa. Desta forma, a previsão de chuva, baseada no uso de informações de radar, juntamente com um modelo conceitual de previsão hidrometeorológica, foi descrita neste trabalho. Teve-se por objetivo aperfeiçoar as previsões de chuva de curtíssimo prazo (poucos minutos), que acopladas a um modelo chuva-vazão, podem ser usadas em sistemas de alerta antecipado. O modelo hidrometeorológico adotado, que considera uma nuvem hipotética unidimensional vertical, foi inicialmente desenvolvido por Georgakakos e Bras (1984a) e ampliado, neste trabalho. Para tal, adotou-se o uso das informações de Topo dos Ecos para determinação da altura das nuvens e considerou-se que a componente do modelo relativa à massa de água líquida no interior da nuvem corresponde à estimativa do conteúdo de água líquida integrado verticalmente (VIL) efetuada por radar. Para eventos de natureza frontal quente e convectiva, o modelo geralmente antecipou o início do processo de chuva, embora tenha conseguido simular seu comportamento e o instante em que o pico da precipitação ocorreu. Para os eventos frontais frios, o pico simulado registrou atraso. Para episódios de simulação com VIL maior que 3 Kg/\'M POT.2\', a chuva simulada acompanhou o comportamento temporal do VIL, mas com influências das variáveis meteorológicas pressão, temperatura e umidade relativa. As lâminas simuladas tornaram-se entrada para o modelo chuva-vazão do SCS, parcialmente calibrado com algoritmo genético. Para a escala de uma pequena bacia hidrográfica urbana brasileira, com estação fluviométrica experimental, preliminarmente, alguns resultados indicam vazões um pouco inferiores e com atrasos na previsão da vazão máxima. A antecedência proporcionada pelo modelo é de 15 minutos e, apesar da simplicidade, conseguiu-se prever o início do processo de precipitação que gerou inundação do córrego urbano. / To reduce human and material losses during floods it is feasible to concisely study the rainfall forecast as the main part of an early warning system. The use of weather radar information, when linked to physically-based forecast models, can contribute for monitoring and forecasting of intense rainfall episodes. Thus, the rainfall forecast, based on using of radar information along with a conceptual model of hydrometeorological forecast, was outlined in this dissertation. One goal of the study was to improve very short term rainfall forecasts, named as a \'nowcasting\' process throughout few minutes, coupled with a rainfall-runoff generation model to be used in early warning system. The hydrometeorological model, a hypothetical vertical 1-D cloud, was initially developed by Georgakakos and Bras (1984a) and further developed in this study to profit signals from radar tracking control volumes of moisture storage. Either \'Echo Top\' radar information was addressed to determinate the cloud height, and the vertically-integrated liquid (\'VIL\') water mass content inside the cloud, estimated by other radar signal, was considered into the model. On the one hand, during rainfall events of hot and convective front genesis monitored with on-field pluviometers, the model results generally anticipated the beginning of the rain process; however the model was capable of not only simulating well the behavior but also the time-step when rainfall monitored peak occurred. On the other hand, and for cold-front events, the simulating peaks depicted a time delay. For simulation episodes with VIL greater than 3 Kg/\'M POT.2\', the simulated rainfall accompanied the time behavior of VIL, but with influences from weather variables of pressure, temperature and relative humidity. These rainfall depths became inputs of a simple SCS rainfall-runoff model, partially calibrated with genetic algorithms. At the scale of a small brazilian urban river basin with experimental gauging station, some preliminary results modeled streamflows a bit lower than, and with some delay of maximum flow forecasts, observed discharges. The advanced lad time period used by model was of about 15 min and, besides its simplicity, it was quite able to anticipate the beggining of the rainfall process that generated the flood formation at the urban creek.

A rotina operacional de monitoramento e previsão de tempo tem mudado bastante nos últimos anos. Além de informações convencionais existentes, que são bem conhecidas nos centros operacionais, os dados obtidos por sensoriamento remoto através de satélites, radares meteorológicos e sensores de detecção de descargas atmosféricas fornecem informações vitais e em tempo real, sendo estas as principais ferramentas para a detecção e previsão de tempestades severas. Na America do Sul, em especial o nordeste da Argentina, Paraguai, Uruguai e o sul do Brasil são regiões particularmente sujeitas a ocorrência de eventos severos (precipitação intensa, granizo, enchentes e intensa atividade elétrica, além de vendavais e tornados). No sul do Brasil, a distribuição mensal de chuvas é bastante uniforme, porém com alta variabilidade diária associada, principalmente, à passagem das frentes frias pela região e aos Sistemas Convectivos de Mesoescala, que se formam nessa região. A principal atividade econômica nessa região do Brasil é a agroindústria, diretamente dependente da distribuição da precipitação para a produção, como também susceptível aos fenômenos meteorológicos adversos associados. Além desta atividade, a região sul é responsável pela produção de, aproximadamente, 35% de toda a energia elétrica utilizada no país. O objetivo principal desta pesquisa foi estudar os aspectos espaciais e temporais da atividade elétrica durante os eventos de Sistemas Convectivos de Mesoescala (SCM) e examinar as possíveis relações entre o ambiente no qual essas tempestades se desenvolvem e as características elétricas e hidrometeorológicas desses, conforme observados por um radar meteorológico Doppler, e uma rede de detecção de relâmpagos, principalmente, e também com informações de satélites meteorológicos, dados de superfície e análises de modelos numéricos. Os resultados deste trabalho mostraram a importância das características dinâmicas na região, em especial a presença dos jatos em baixos níveis com a convergência de umidade na região para a organização dos eventos de SCM, como também a distribuição dos regimes de precipitação com características distintas de estrutura de refletividade observada por radar e também de atividade elétrica durante os eventos analisados. Espera-se que os resultados deste trabalho ajudem a entender melhor a relação dos sistemas convectivos de mesoescala e sua estrutura e evolução, como observados e detectados pelos sistemas remotos de monitoramento hidrometeorológico, além de um melhor entendimento e aperfeiçoamento de nossas habilidades de análise e previsão de tempo relacionados a esses eventos severos com precipitação intensa. / The operational routine in weather monitoring and forecasting has changed a lot in the past years. Besides conventional information, well known in operational centers, data from remote sensing such as satellite, weather radars and lightning detection network provide vital information in real time, as the main tools for severe weather detection and forecasting In South America, specially northeastern Argentina, Paraguay, Uruguay and southern Brazil are regions prone to severe weather (intense precipitation, hail, floddings, lightning, tornadoes and gust winds). In the South of Brazil, monthly precipitation distribution is very uniform, but with daily variability associated, mostly, with the passage of cold fronts through the region and to mesoscale convective systems, forming in this area. The major economical activity in this region of Brazil is agroindustry, directly dependent of precipitation distribution for production and also susceptible to diverse meteorological events associated with it. Besides this activity, the south region is responsible for the production of, approximately, 35\\% of all electric energy used in the country. The main goal of this research was to study spatial and temporal aspects of the electrical activity during MCS events, as observed by a weather radar and a network of ligthning detection sensors in the south of Brazil, and to examine possible relations between the environment in which these storms develop and electrical characteristics of these weather systems, using weather radar, lightning, satellite and numerical model information. The results of this work showed the importance of the dynamic characteristics in the regial, specially the presence of low level jets and humidity convergence in the region to organize MCS events, as well as a distribution of precipitation regimes whith distinct characteristics of radar reflectivity and electrical activity during the analysed events. With this work we expect to contribute with the understanding of the relation of MCS structure and evolution as observed and detected by hydrometeorological monitoring systems and to improve the comprehension and ability to analyse and forecast such severe weather systems.

Para reduzir as perdas humanas e materiais durante as inundações, é possível realizar estudo conciso da previsão de chuva, etapa principal de um sistema de alerta antecipado de inundação. O uso de informações de radar de tempo, quando acopladas a modelos de previsão de precipitação baseados fisicamente, pode contribuir para o monitoramento e previsão de episódios de chuva intensa. Desta forma, a previsão de chuva, baseada no uso de informações de radar, juntamente com um modelo conceitual de previsão hidrometeorológica, foi descrita neste trabalho. Teve-se por objetivo aperfeiçoar as previsões de chuva de curtíssimo prazo (poucos minutos), que acopladas a um modelo chuva-vazão, podem ser usadas em sistemas de alerta antecipado. O modelo hidrometeorológico adotado, que considera uma nuvem hipotética unidimensional vertical, foi inicialmente desenvolvido por Georgakakos e Bras (1984a) e ampliado, neste trabalho. Para tal, adotou-se o uso das informações de Topo dos Ecos para determinação da altura das nuvens e considerou-se que a componente do modelo relativa à massa de água líquida no interior da nuvem corresponde à estimativa do conteúdo de água líquida integrado verticalmente (VIL) efetuada por radar. Para eventos de natureza frontal quente e convectiva, o modelo geralmente antecipou o início do processo de chuva, embora tenha conseguido simular seu comportamento e o instante em que o pico da precipitação ocorreu. Para os eventos frontais frios, o pico simulado registrou atraso. Para episódios de simulação com VIL maior que 3 Kg/\'M POT.2\', a chuva simulada acompanhou o comportamento temporal do VIL, mas com influências das variáveis meteorológicas pressão, temperatura e umidade relativa. As lâminas simuladas tornaram-se entrada para o modelo chuva-vazão do SCS, parcialmente calibrado com algoritmo genético. Para a escala de uma pequena bacia hidrográfica urbana brasileira, com estação fluviométrica experimental, preliminarmente, alguns resultados indicam vazões um pouco inferiores e com atrasos na previsão da vazão máxima. A antecedência proporcionada pelo modelo é de 15 minutos e, apesar da simplicidade, conseguiu-se prever o início do processo de precipitação que gerou inundação do córrego urbano. / To reduce human and material losses during floods it is feasible to concisely study the rainfall forecast as the main part of an early warning system. The use of weather radar information, when linked to physically-based forecast models, can contribute for monitoring and forecasting of intense rainfall episodes. Thus, the rainfall forecast, based on using of radar information along with a conceptual model of hydrometeorological forecast, was outlined in this dissertation. One goal of the study was to improve very short term rainfall forecasts, named as a \'nowcasting\' process throughout few minutes, coupled with a rainfall-runoff generation model to be used in early warning system. The hydrometeorological model, a hypothetical vertical 1-D cloud, was initially developed by Georgakakos and Bras (1984a) and further developed in this study to profit signals from radar tracking control volumes of moisture storage. Either \'Echo Top\' radar information was addressed to determinate the cloud height, and the vertically-integrated liquid (\'VIL\') water mass content inside the cloud, estimated by other radar signal, was considered into the model. On the one hand, during rainfall events of hot and convective front genesis monitored with on-field pluviometers, the model results generally anticipated the beginning of the rain process; however the model was capable of not only simulating well the behavior but also the time-step when rainfall monitored peak occurred. On the other hand, and for cold-front events, the simulating peaks depicted a time delay. For simulation episodes with VIL greater than 3 Kg/\'M POT.2\', the simulated rainfall accompanied the time behavior of VIL, but with influences from weather variables of pressure, temperature and relative humidity. These rainfall depths became inputs of a simple SCS rainfall-runoff model, partially calibrated with genetic algorithms. At the scale of a small brazilian urban river basin with experimental gauging station, some preliminary results modeled streamflows a bit lower than, and with some delay of maximum flow forecasts, observed discharges. The advanced lad time period used by model was of about 15 min and, besides its simplicity, it was quite able to anticipate the beggining of the rainfall process that generated the flood formation at the urban creek.

Diese (Doktor)arbeit beschäftigt sich mit Radar-Polarimetrie, insbesondere mit der Untersuchung der Eigenschaften von polarimetrischen Variablen, die potenziellen Nutzen für die Radar-Meteorologie haben. Für den Einsatz in Dual-Polarisations-Radargeräten wird der Polarisationsgrad analysiert. Diese Variable wird in künftigen operationellen Radargeräten verfügbar sein. Der Polarisationsgrad hängt vom transmittierten Polarisationszustand und in weiterer Folge auch vom Betriebsmodus des Radargeräts ab. Der Hauptbetriebsmodus von operationellen Radargeräten sendet und empfängt gleichzeitig sowohl die horizontale als auch die vertikale Komponente. Der sekundäre Betriebsmodus sendet und empfängt simultan die horizontal polarisierte Komponente. In dieser Arbeit werden beide Polarisationsgrade untersucht. Da operationelle Systeme derzeit auf den Dual-Polarisationsmodus aufgerüstet werden, sollte künftig die Anwendungsmöglichkeiten von vollpolarimetrischen Wetterradarsystemen untersucht werden. Aus allen Variablen, die in diesem Betriebsmodus zur Verfügung stehen, wurde die Entropie (des gemessen Objektes) ausgewählt und wegen seiner engen Beziehung zum Polarisationsgrad näher untersucht. / The present doctoral thesis deals with radar polarimetry, namely with the investigation of properties of polarimetric variables potentially useful in radar meteorology. For use with dual-polarization radars, the degree of polarization is analyzed. This variable is available to planned operational radars. The degree of polarization is dependent on transmit polarization state and, consequently, it is dependent on the radar system operating mode. The primary operating mode of operational radars consists in simultaneous transmission and simultaneous receive of both horizontal and vertical components. The secondary operating mode consists of horizontal transmission and simultaneous receive. Both degrees of polarization are investigated in this thesis. Also, as operational systems are being updated to dual-polarization, research should start investigating the capabilities of fully polarimetric weather radar systems. Among the numerous variables available from this operating mode, the target entropy was chosen for investigation, also because of its close relation to the degree of polarization