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Jean-dominique Creutin - One of the best experts on this subject based on the ideXlab platform.
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analytical solutions to sampling effects in Drop Size Distribution measurements during stationary rainfall estimation of bulk rainfall variables
Journal of Hydrology, 2006Co-Authors: Remko Uijlenhoet, Josep M Porra, Daniel Sempere Torres, Jean-dominique CreutinAbstract:A stochastic model of the microstructure of rainfall is used to derive explicit expressions for the magnitude of the sampling fluctuations in rainfall properties estimated from rainDrop Size measurements in stationary rainfall. The model is a marked point process, in which the points represent the Drop centers, assumed to be uniformly distributed in space. This assumption, which is supported both by theoretical and by empirical evidence, implies that during periods of stationary rainfall the number of Drops in a sample volume follows a Poisson Distribution. The marks represent the Drop Sizes, assumed to be distributed independent of their positions according to some general Drop Size Distribution. Within this framework, it is shown analytically how the sampling Distribution of the estimator of any bulk rainfall variable (such as liquid water content, rain rate, or radar reflectivity) in stationary rainfall converges from a strongly skewed Distribution to a (symmetrical) Gaussian Distribution with increasing sample Size. The relevant parameter controlling this evolution is the average number of Drops in the sample ns. For a given sample Size, the skewness of the sampling Distribution is found to be more pronounced for higher order moments of the Drop Size Distribution. For instance, the sampling Distribution of the normalized mean diameter becomes nearly Gaussian for ns > 10, while the sampling Distribution of the normalized rain rate remains skewed for ns 500. Additionally, it is shown analytically that, as a result of the mentioned skewness, the median Q50 as an estimator of a bulk rainfall variable always underestimates its population value Qp in stationary rainfall. The ratio of the former to the latter is found to be , where b is a constant depending on the Drop Size Distribution. For bulk rainfall variables this constant is positive and therefore the median always underestimates the population value. This provides a theoretical confirmation and explanation of previously published simulation results. Finally, relationships between the expected number of rainDrops in the sample ns and the rain rate are established for different parametric forms of the rainDrop Size Distribution. These relationships are first compared to experimental results and then used to provide examples of sampling Distributions of bulk rainfall variables (in this case rain rate) for different values of the average rain rate and different integration times of the disdrometric device involved (in this case a Joss?Waldvogel disdrometer). The practical relevance of these results is (1) that they provide exact solutions to the sampling problem during (relatively rare) periods of stationary rainfall (e.g., drizzle), and (2) that they provide a lower bound to the magnitude of the sampling problem in the general situation where sampling fluctuations and natural variability co-exist.
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Analytical solutions to sampling effects in Drop Size Distribution measurements during stationary rainfall: Estimation of bulk rainfall variables
Journal of Hydrology, 2006Co-Authors: Remko Uijlenhoet, Josep M Porra, Daniel Sempere Torres, Jean-dominique CreutinAbstract:A stochastic model of the microstructure of rainfall is used to derive explicit expressions for the magnitude of the sampling fluctuations in rainfall properties estimated from rainDrop Size measurements in stationary rainfall. The model is a marked point process, in which the points represent the Drop centers, assumed to be uniformly distributed in space. This assumption, which is supported both by theoretical and by empirical evidence, implies that during periods of stationary rainfall the number of Drops in a sample volume follows a Poisson Distribution. The marks represent the Drop Sizes, assumed to be distributed independent of their positions according to some general Drop Size Distribution. Within this framework, it is shown analytically how the sampling Distribution of the estimator of any bulk rainfall variable (such as liquid water content, rain rate, or radar reflectivity) in stationary rainfall converges from a strongly skewed Distribution to a (symmetrical) Gaussian Distribution with increasing sample Size. The relevant parameter controlling this evolution is the average number of Drops in the sample ns. For a given sample Size, the skewness of the sampling Distribution is found to be more pronounced for higher order moments of the Drop Size Distribution. For instance, the sampling Distribution of the normalized mean diameter becomes nearly Gaussian for ns > 10, while the sampling Distribution of the normalized rain rate remains skewed for ns not, vert, similar 500. Additionally, it is shown analytically that, as a result of the mentioned skewness, the median Q50 as an estimator of a bulk rainfall variable always underestimates its population value Qp in stationary rainfall. The ratio of the former to the latter is found to be View the MathML source, where b is a constant depending on the Drop Size Distribution. For bulk rainfall variables this constant is positive and therefore the median always underestimates the population value. This provides a theoretical confirmation and explanation of previously published simulation results. Finally, relationships between the expected number of rainDrops in the sample ns and the rain rate are established for different parametric forms of the rainDrop Size Distribution. These relationships are first compared to experimental results and then used to provide examples of sampling Distributions of bulk rainfall variables (in this case rain rate) for different values of the average rain rate and different integration times of the disdrometric device involved (in this case a Joss–Waldvogel disdrometer). The practical relevance of these results is (1) that they provide exact solutions to the sampling problem during (relatively rare) periods of stationary rainfall (e.g., drizzle), and (2) that they provide a lower bound to the magnitude of the sampling problem in the general situation where sampling fluctuations and natural variability co-exist.
Remko Uijlenhoet - One of the best experts on this subject based on the ideXlab platform.
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analytical solutions to sampling effects in Drop Size Distribution measurements during stationary rainfall estimation of bulk rainfall variables
Journal of Hydrology, 2006Co-Authors: Remko Uijlenhoet, Josep M Porra, Daniel Sempere Torres, Jean-dominique CreutinAbstract:A stochastic model of the microstructure of rainfall is used to derive explicit expressions for the magnitude of the sampling fluctuations in rainfall properties estimated from rainDrop Size measurements in stationary rainfall. The model is a marked point process, in which the points represent the Drop centers, assumed to be uniformly distributed in space. This assumption, which is supported both by theoretical and by empirical evidence, implies that during periods of stationary rainfall the number of Drops in a sample volume follows a Poisson Distribution. The marks represent the Drop Sizes, assumed to be distributed independent of their positions according to some general Drop Size Distribution. Within this framework, it is shown analytically how the sampling Distribution of the estimator of any bulk rainfall variable (such as liquid water content, rain rate, or radar reflectivity) in stationary rainfall converges from a strongly skewed Distribution to a (symmetrical) Gaussian Distribution with increasing sample Size. The relevant parameter controlling this evolution is the average number of Drops in the sample ns. For a given sample Size, the skewness of the sampling Distribution is found to be more pronounced for higher order moments of the Drop Size Distribution. For instance, the sampling Distribution of the normalized mean diameter becomes nearly Gaussian for ns > 10, while the sampling Distribution of the normalized rain rate remains skewed for ns 500. Additionally, it is shown analytically that, as a result of the mentioned skewness, the median Q50 as an estimator of a bulk rainfall variable always underestimates its population value Qp in stationary rainfall. The ratio of the former to the latter is found to be , where b is a constant depending on the Drop Size Distribution. For bulk rainfall variables this constant is positive and therefore the median always underestimates the population value. This provides a theoretical confirmation and explanation of previously published simulation results. Finally, relationships between the expected number of rainDrops in the sample ns and the rain rate are established for different parametric forms of the rainDrop Size Distribution. These relationships are first compared to experimental results and then used to provide examples of sampling Distributions of bulk rainfall variables (in this case rain rate) for different values of the average rain rate and different integration times of the disdrometric device involved (in this case a Joss?Waldvogel disdrometer). The practical relevance of these results is (1) that they provide exact solutions to the sampling problem during (relatively rare) periods of stationary rainfall (e.g., drizzle), and (2) that they provide a lower bound to the magnitude of the sampling problem in the general situation where sampling fluctuations and natural variability co-exist.
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Analytical solutions to sampling effects in Drop Size Distribution measurements during stationary rainfall: Estimation of bulk rainfall variables
Journal of Hydrology, 2006Co-Authors: Remko Uijlenhoet, Josep M Porra, Daniel Sempere Torres, Jean-dominique CreutinAbstract:A stochastic model of the microstructure of rainfall is used to derive explicit expressions for the magnitude of the sampling fluctuations in rainfall properties estimated from rainDrop Size measurements in stationary rainfall. The model is a marked point process, in which the points represent the Drop centers, assumed to be uniformly distributed in space. This assumption, which is supported both by theoretical and by empirical evidence, implies that during periods of stationary rainfall the number of Drops in a sample volume follows a Poisson Distribution. The marks represent the Drop Sizes, assumed to be distributed independent of their positions according to some general Drop Size Distribution. Within this framework, it is shown analytically how the sampling Distribution of the estimator of any bulk rainfall variable (such as liquid water content, rain rate, or radar reflectivity) in stationary rainfall converges from a strongly skewed Distribution to a (symmetrical) Gaussian Distribution with increasing sample Size. The relevant parameter controlling this evolution is the average number of Drops in the sample ns. For a given sample Size, the skewness of the sampling Distribution is found to be more pronounced for higher order moments of the Drop Size Distribution. For instance, the sampling Distribution of the normalized mean diameter becomes nearly Gaussian for ns > 10, while the sampling Distribution of the normalized rain rate remains skewed for ns not, vert, similar 500. Additionally, it is shown analytically that, as a result of the mentioned skewness, the median Q50 as an estimator of a bulk rainfall variable always underestimates its population value Qp in stationary rainfall. The ratio of the former to the latter is found to be View the MathML source, where b is a constant depending on the Drop Size Distribution. For bulk rainfall variables this constant is positive and therefore the median always underestimates the population value. This provides a theoretical confirmation and explanation of previously published simulation results. Finally, relationships between the expected number of rainDrops in the sample ns and the rain rate are established for different parametric forms of the rainDrop Size Distribution. These relationships are first compared to experimental results and then used to provide examples of sampling Distributions of bulk rainfall variables (in this case rain rate) for different values of the average rain rate and different integration times of the disdrometric device involved (in this case a Joss–Waldvogel disdrometer). The practical relevance of these results is (1) that they provide exact solutions to the sampling problem during (relatively rare) periods of stationary rainfall (e.g., drizzle), and (2) that they provide a lower bound to the magnitude of the sampling problem in the general situation where sampling fluctuations and natural variability co-exist.
Jaber Safdari - One of the best experts on this subject based on the ideXlab platform.
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using maximum entropy gamma inverse gaussian and weibull approach for prediction of Drop Size Distribution in a liquid liquid extraction column
Chemical Engineering Research & Design, 2017Co-Authors: Mehdi Asadollahzadeh, Meisam Torabmostaedi, Rezvan Torkaman, Jaber SafdariAbstract:Abstract In this study, Drop Size Distribution and Sauter mean Drop diameters were measured and correlated under operating variables and physical properties of the systems using a 113 mm diameter Kuhni column. Three systems including toluene-water, n -butyl acetate-water and n -butanol-water were experimented in this column. The countercurrent flow pattern of the liquid phases was characterized regarding the Sauter mean Drop diameter and Drop Size Distribution; a photographic method was used to assess Drop Sizes. The following operating variables were studied: rotor speed, flow rate of both liquid phases and interfacial tension. The Drop Size Distribution and Sauter mean Drop diameter were found to depend largely on the rotor speed and interfacial tension, albeit, only partially dependent on the phase velocities. The maximum entropy principle and the conventional probability Distribution functions (Gamma, Inverse Gaussian, Weibull) have already been applied to estimate the Drop Size Distribution. Experimental results show that the maximum entropy function describes the Drop Size Distribution better than the conventional probability Distribution functions for three systems in a Kuhni column extractor. An empirical correlation is proposed for the estimation of the Sauter mean Drop diameter. The acquired information would be useful in design of liquid–liquid extraction columns.
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a new model for prediction of Drop Size Distribution in a liquid liquid extraction column
RSC Advances, 2016Co-Authors: Mehdi Asadollahzadeh, Meisam Torabmostaedi, Rezvan Torkaman, Jaber SafdariAbstract:In this study, a new model for prediction of Drop Size Distribution is proposed in the asymmetric rotating disc pilot plant column (ARDC) by the maximum entropy density approximation technique. The liquid extraction systems including toluene–water, n-butyl acetate–water and n-butanol–water were used with this column. An image analysis technique was applied to determine the Drop Size Distribution as a function of operating parameters and physical properties. By applying abrupt changes of the operating parameters, the Drop behaviors in the column were investigated. The results show that the agitation speed has a main effect on the Drop Size Distribution in the column. However, the effects of phase flow rates are not significant. The empirical correlations are proposed to describe Lagrange multipliers in the maximum entropy function in terms of operating variables and physical properties of the systems. Except for these findings, an empirical correlation is proposed for estimation of the Sauter mean Drop diameter in terms of operating variables, column geometry and physical properties. The proposed correlations are evaluated based on the goodness of fit statistics, namely, χ2, R2 and RMSE. The fitting results by the maximum entropy principle method seem to be fairly accurate and reasonable on the basis of the experimental data. These completed sets of data could be used for modeling approaches in the liquid–liquid extraction columns.
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A new model for prediction of Drop Size Distribution in a liquid–liquid extraction column
RSC Advances, 2016Co-Authors: Mehdi Asadollahzadeh, Meisam Torab-mostaedi, Rezvan Torkaman, Jaber SafdariAbstract:In this study, a new model for prediction of Drop Size Distribution is proposed in the asymmetric rotating disc pilot plant column (ARDC) by the maximum entropy density approximation technique. The liquid extraction systems including toluene–water, n-butyl acetate–water and n-butanol–water were used with this column. An image analysis technique was applied to determine the Drop Size Distribution as a function of operating parameters and physical properties. By applying abrupt changes of the operating parameters, the Drop behaviors in the column were investigated. The results show that the agitation speed has a main effect on the Drop Size Distribution in the column. However, the effects of phase flow rates are not significant. The empirical correlations are proposed to describe Lagrange multipliers in the maximum entropy function in terms of operating variables and physical properties of the systems. Except for these findings, an empirical correlation is proposed for estimation of the Sauter mean Drop diameter in terms of operating variables, column geometry and physical properties. The proposed correlations are evaluated based on the goodness of fit statistics, namely, χ2, R2 and RMSE. The fitting results by the maximum entropy principle method seem to be fairly accurate and reasonable on the basis of the experimental data. These completed sets of data could be used for modeling approaches in the liquid–liquid extraction columns.
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Prediction of Drop Size Distribution in a horizontal pulsed plate extraction column
Chemical Engineering and Processing: Process Intensification, 2015Co-Authors: M. Khajenoori, Jaber Safdari, A. Haghighi-asl, M.h. MallahAbstract:Abstract Mean Drop Size and Drop Size Distribution in a horizontal pulsed plate extraction column were investigated using different four binary systems. The effects of pulse intensity (af) and flow rates of both liquid phases have been investigated. The Drop Size decreased more rapidly with the increase of pulse intensities. It was observed that an increase in intensity of the pulses will lead to narrower ranges of Distribution for the Drop Size. Increasing the flow rate of dispersed phase tends to increase the Drop Size. The effect of continuous phase flow rate is weaker than the effect of the dispersed phase flow rate. By using results, a semi empirical correlation obtained for the estimation of mean Drop Size which proves to be in good agreement to the experimental data. The average absolute relative error (AARE) of this correlation is about 15.6%. In order to find a predictive correlation for Drop Size Distribution, four models of Distribution functions are tested. The normal probability density function is the only suitable way for representing the experimental Drop Size Distributions with an AARE of 13.7%.
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effect of structural parameters on Drop Size Distribution in pulsed packed columns
Chemical Engineering & Technology, 2014Co-Authors: Mahmoud Gholam Samani, Jaber Safdari, Ali Haghighi Asl, Meisam TorabmostaediAbstract:The effect of packing type on Drop Size Distribution in pulsed packed columns was investigated by means of different columns and three packing types with three liquid systems including n-butyl acetate, toluene, and kerosene with water. These liquid systems cover a wide range of interfacial tensions. Also the influence of operating variables in terms of pulse intensity and volumetric flow rates of dispersed and continuous phases was examined. Pulse intensity, interfacial tension, and packing shape were found as the main important factors for Drop Size Distribution while volumetric flow rates had no significant effect. Correlations are presented to predict Drop Distribution and mean Drop Size in pulsed packed columns.
Josep M Porra - One of the best experts on this subject based on the ideXlab platform.
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analytical solutions to sampling effects in Drop Size Distribution measurements during stationary rainfall estimation of bulk rainfall variables
Journal of Hydrology, 2006Co-Authors: Remko Uijlenhoet, Josep M Porra, Daniel Sempere Torres, Jean-dominique CreutinAbstract:A stochastic model of the microstructure of rainfall is used to derive explicit expressions for the magnitude of the sampling fluctuations in rainfall properties estimated from rainDrop Size measurements in stationary rainfall. The model is a marked point process, in which the points represent the Drop centers, assumed to be uniformly distributed in space. This assumption, which is supported both by theoretical and by empirical evidence, implies that during periods of stationary rainfall the number of Drops in a sample volume follows a Poisson Distribution. The marks represent the Drop Sizes, assumed to be distributed independent of their positions according to some general Drop Size Distribution. Within this framework, it is shown analytically how the sampling Distribution of the estimator of any bulk rainfall variable (such as liquid water content, rain rate, or radar reflectivity) in stationary rainfall converges from a strongly skewed Distribution to a (symmetrical) Gaussian Distribution with increasing sample Size. The relevant parameter controlling this evolution is the average number of Drops in the sample ns. For a given sample Size, the skewness of the sampling Distribution is found to be more pronounced for higher order moments of the Drop Size Distribution. For instance, the sampling Distribution of the normalized mean diameter becomes nearly Gaussian for ns > 10, while the sampling Distribution of the normalized rain rate remains skewed for ns 500. Additionally, it is shown analytically that, as a result of the mentioned skewness, the median Q50 as an estimator of a bulk rainfall variable always underestimates its population value Qp in stationary rainfall. The ratio of the former to the latter is found to be , where b is a constant depending on the Drop Size Distribution. For bulk rainfall variables this constant is positive and therefore the median always underestimates the population value. This provides a theoretical confirmation and explanation of previously published simulation results. Finally, relationships between the expected number of rainDrops in the sample ns and the rain rate are established for different parametric forms of the rainDrop Size Distribution. These relationships are first compared to experimental results and then used to provide examples of sampling Distributions of bulk rainfall variables (in this case rain rate) for different values of the average rain rate and different integration times of the disdrometric device involved (in this case a Joss?Waldvogel disdrometer). The practical relevance of these results is (1) that they provide exact solutions to the sampling problem during (relatively rare) periods of stationary rainfall (e.g., drizzle), and (2) that they provide a lower bound to the magnitude of the sampling problem in the general situation where sampling fluctuations and natural variability co-exist.
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Analytical solutions to sampling effects in Drop Size Distribution measurements during stationary rainfall: Estimation of bulk rainfall variables
Journal of Hydrology, 2006Co-Authors: Remko Uijlenhoet, Josep M Porra, Daniel Sempere Torres, Jean-dominique CreutinAbstract:A stochastic model of the microstructure of rainfall is used to derive explicit expressions for the magnitude of the sampling fluctuations in rainfall properties estimated from rainDrop Size measurements in stationary rainfall. The model is a marked point process, in which the points represent the Drop centers, assumed to be uniformly distributed in space. This assumption, which is supported both by theoretical and by empirical evidence, implies that during periods of stationary rainfall the number of Drops in a sample volume follows a Poisson Distribution. The marks represent the Drop Sizes, assumed to be distributed independent of their positions according to some general Drop Size Distribution. Within this framework, it is shown analytically how the sampling Distribution of the estimator of any bulk rainfall variable (such as liquid water content, rain rate, or radar reflectivity) in stationary rainfall converges from a strongly skewed Distribution to a (symmetrical) Gaussian Distribution with increasing sample Size. The relevant parameter controlling this evolution is the average number of Drops in the sample ns. For a given sample Size, the skewness of the sampling Distribution is found to be more pronounced for higher order moments of the Drop Size Distribution. For instance, the sampling Distribution of the normalized mean diameter becomes nearly Gaussian for ns > 10, while the sampling Distribution of the normalized rain rate remains skewed for ns not, vert, similar 500. Additionally, it is shown analytically that, as a result of the mentioned skewness, the median Q50 as an estimator of a bulk rainfall variable always underestimates its population value Qp in stationary rainfall. The ratio of the former to the latter is found to be View the MathML source, where b is a constant depending on the Drop Size Distribution. For bulk rainfall variables this constant is positive and therefore the median always underestimates the population value. This provides a theoretical confirmation and explanation of previously published simulation results. Finally, relationships between the expected number of rainDrops in the sample ns and the rain rate are established for different parametric forms of the rainDrop Size Distribution. These relationships are first compared to experimental results and then used to provide examples of sampling Distributions of bulk rainfall variables (in this case rain rate) for different values of the average rain rate and different integration times of the disdrometric device involved (in this case a Joss–Waldvogel disdrometer). The practical relevance of these results is (1) that they provide exact solutions to the sampling problem during (relatively rare) periods of stationary rainfall (e.g., drizzle), and (2) that they provide a lower bound to the magnitude of the sampling problem in the general situation where sampling fluctuations and natural variability co-exist.
Daniel Sempere Torres - One of the best experts on this subject based on the ideXlab platform.
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analytical solutions to sampling effects in Drop Size Distribution measurements during stationary rainfall estimation of bulk rainfall variables
Journal of Hydrology, 2006Co-Authors: Remko Uijlenhoet, Josep M Porra, Daniel Sempere Torres, Jean-dominique CreutinAbstract:A stochastic model of the microstructure of rainfall is used to derive explicit expressions for the magnitude of the sampling fluctuations in rainfall properties estimated from rainDrop Size measurements in stationary rainfall. The model is a marked point process, in which the points represent the Drop centers, assumed to be uniformly distributed in space. This assumption, which is supported both by theoretical and by empirical evidence, implies that during periods of stationary rainfall the number of Drops in a sample volume follows a Poisson Distribution. The marks represent the Drop Sizes, assumed to be distributed independent of their positions according to some general Drop Size Distribution. Within this framework, it is shown analytically how the sampling Distribution of the estimator of any bulk rainfall variable (such as liquid water content, rain rate, or radar reflectivity) in stationary rainfall converges from a strongly skewed Distribution to a (symmetrical) Gaussian Distribution with increasing sample Size. The relevant parameter controlling this evolution is the average number of Drops in the sample ns. For a given sample Size, the skewness of the sampling Distribution is found to be more pronounced for higher order moments of the Drop Size Distribution. For instance, the sampling Distribution of the normalized mean diameter becomes nearly Gaussian for ns > 10, while the sampling Distribution of the normalized rain rate remains skewed for ns 500. Additionally, it is shown analytically that, as a result of the mentioned skewness, the median Q50 as an estimator of a bulk rainfall variable always underestimates its population value Qp in stationary rainfall. The ratio of the former to the latter is found to be , where b is a constant depending on the Drop Size Distribution. For bulk rainfall variables this constant is positive and therefore the median always underestimates the population value. This provides a theoretical confirmation and explanation of previously published simulation results. Finally, relationships between the expected number of rainDrops in the sample ns and the rain rate are established for different parametric forms of the rainDrop Size Distribution. These relationships are first compared to experimental results and then used to provide examples of sampling Distributions of bulk rainfall variables (in this case rain rate) for different values of the average rain rate and different integration times of the disdrometric device involved (in this case a Joss?Waldvogel disdrometer). The practical relevance of these results is (1) that they provide exact solutions to the sampling problem during (relatively rare) periods of stationary rainfall (e.g., drizzle), and (2) that they provide a lower bound to the magnitude of the sampling problem in the general situation where sampling fluctuations and natural variability co-exist.
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Analytical solutions to sampling effects in Drop Size Distribution measurements during stationary rainfall: Estimation of bulk rainfall variables
Journal of Hydrology, 2006Co-Authors: Remko Uijlenhoet, Josep M Porra, Daniel Sempere Torres, Jean-dominique CreutinAbstract:A stochastic model of the microstructure of rainfall is used to derive explicit expressions for the magnitude of the sampling fluctuations in rainfall properties estimated from rainDrop Size measurements in stationary rainfall. The model is a marked point process, in which the points represent the Drop centers, assumed to be uniformly distributed in space. This assumption, which is supported both by theoretical and by empirical evidence, implies that during periods of stationary rainfall the number of Drops in a sample volume follows a Poisson Distribution. The marks represent the Drop Sizes, assumed to be distributed independent of their positions according to some general Drop Size Distribution. Within this framework, it is shown analytically how the sampling Distribution of the estimator of any bulk rainfall variable (such as liquid water content, rain rate, or radar reflectivity) in stationary rainfall converges from a strongly skewed Distribution to a (symmetrical) Gaussian Distribution with increasing sample Size. The relevant parameter controlling this evolution is the average number of Drops in the sample ns. For a given sample Size, the skewness of the sampling Distribution is found to be more pronounced for higher order moments of the Drop Size Distribution. For instance, the sampling Distribution of the normalized mean diameter becomes nearly Gaussian for ns > 10, while the sampling Distribution of the normalized rain rate remains skewed for ns not, vert, similar 500. Additionally, it is shown analytically that, as a result of the mentioned skewness, the median Q50 as an estimator of a bulk rainfall variable always underestimates its population value Qp in stationary rainfall. The ratio of the former to the latter is found to be View the MathML source, where b is a constant depending on the Drop Size Distribution. For bulk rainfall variables this constant is positive and therefore the median always underestimates the population value. This provides a theoretical confirmation and explanation of previously published simulation results. Finally, relationships between the expected number of rainDrops in the sample ns and the rain rate are established for different parametric forms of the rainDrop Size Distribution. These relationships are first compared to experimental results and then used to provide examples of sampling Distributions of bulk rainfall variables (in this case rain rate) for different values of the average rain rate and different integration times of the disdrometric device involved (in this case a Joss–Waldvogel disdrometer). The practical relevance of these results is (1) that they provide exact solutions to the sampling problem during (relatively rare) periods of stationary rainfall (e.g., drizzle), and (2) that they provide a lower bound to the magnitude of the sampling problem in the general situation where sampling fluctuations and natural variability co-exist.
