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Abstract
This paper complements an earlier one that demonstrated the bias in the method-of-moments (MM) estimators frequently used to estimate parameters for drop size distribution (DSD) functions being “fitted” to observed raindrop size distributions. Here the authors consider both the bias and the errors in MM estimators applied to samples from known gamma DSDs (of which the exponential DSD treated in the earlier paper is a special case). The samples were generated using a similar Monte Carlo simulation procedure. The skewness in the sampling distributions of the DSD moments that causes this bias is less pronounced for narrower population DSDs, and therefore the bias problems (and also the errors) diminish as the gamma shape parameter increases. However, the bias still increases with the order of the moments used in the MM procedures; thus it is stronger when higher-order moments (such as the radar reflectivity) are used. The simulation results also show that the errors of the estimates of the DSD parameters are usually larger when higher-order moments are employed. As a consequence, only MM estimators using the lowest-order sample moments that are thought to be well determined should be used. The biases and the errors of most of the MM parameter estimates diminish as the sample size increases; with large samples the moment estimators may become sufficiently accurate for some purposes. Nevertheless, even with some fairly large samples, MM estimators involving high-order moments can yield parameter values that are physically implausible or are incompatible with the input observations. Correlations of the sample moments with the size of the largest drop in a sample (D max) are weaker than for the case of sampling from an exponential DSD, as are the correlations of the MM-estimated parameters with D max first noted in that case. However, correlations between the estimated parameters remain because functions of the same observations are correlated. These correlations generally strengthen as the sample size increases.
Abstract
This paper complements an earlier one that demonstrated the bias in the method-of-moments (MM) estimators frequently used to estimate parameters for drop size distribution (DSD) functions being “fitted” to observed raindrop size distributions. Here the authors consider both the bias and the errors in MM estimators applied to samples from known gamma DSDs (of which the exponential DSD treated in the earlier paper is a special case). The samples were generated using a similar Monte Carlo simulation procedure. The skewness in the sampling distributions of the DSD moments that causes this bias is less pronounced for narrower population DSDs, and therefore the bias problems (and also the errors) diminish as the gamma shape parameter increases. However, the bias still increases with the order of the moments used in the MM procedures; thus it is stronger when higher-order moments (such as the radar reflectivity) are used. The simulation results also show that the errors of the estimates of the DSD parameters are usually larger when higher-order moments are employed. As a consequence, only MM estimators using the lowest-order sample moments that are thought to be well determined should be used. The biases and the errors of most of the MM parameter estimates diminish as the sample size increases; with large samples the moment estimators may become sufficiently accurate for some purposes. Nevertheless, even with some fairly large samples, MM estimators involving high-order moments can yield parameter values that are physically implausible or are incompatible with the input observations. Correlations of the sample moments with the size of the largest drop in a sample (D max) are weaker than for the case of sampling from an exponential DSD, as are the correlations of the MM-estimated parameters with D max first noted in that case. However, correlations between the estimated parameters remain because functions of the same observations are correlated. These correlations generally strengthen as the sample size increases.
Abstract
Several multifrequency techniques for passive microwave estimation of precipitation based on the absorption and scattering properties of hydrometeors have been proposed in the literature. In the present study, plane-parallel limitations are overcome by using a model based on the discrete-ordinates method to solve the radiative transfer equation in three-dimensional rectangular domains. This effectively accounts for the complexity and variety of radiation problems encountered in the atmosphere. This investigation presents results for plane-parallel and three-dimensional radiative transfer for a precipitating system, discusses differences between these results, and suggests possible explanations for these differences.
Microphysical properties were obtained from the Colorado State University Regional Atmospheric Modeling System and represent a hailstorm observed during the 1986 Cooperative Huntsville Meteorological Experiment. These properties are used as input to a three-dimensional radiative transfer model in order to simulate satellite observation of the storm. The model output consists of upwelling brightness temperatures at several of the frequencies on the Special Sensor Microwave/Imager. The radiative transfer model accounts for scattering and emission of atmospheric gases and hydrometeors in liquid and ice phases.
Brightness temperatures obtained from the three-dimensional model of this investigation indicate that horizontal inhomogeneities give rise to brightness temperature fields that can be quite different from fields obtained using plane-parallel radiative transfer theory. These differences are examined for various resolutions of the satellite sensor field of view. In addition, the issue of boundary conditions for three-dimensional atmospheric radiative transfer is addressed.
Abstract
Several multifrequency techniques for passive microwave estimation of precipitation based on the absorption and scattering properties of hydrometeors have been proposed in the literature. In the present study, plane-parallel limitations are overcome by using a model based on the discrete-ordinates method to solve the radiative transfer equation in three-dimensional rectangular domains. This effectively accounts for the complexity and variety of radiation problems encountered in the atmosphere. This investigation presents results for plane-parallel and three-dimensional radiative transfer for a precipitating system, discusses differences between these results, and suggests possible explanations for these differences.
Microphysical properties were obtained from the Colorado State University Regional Atmospheric Modeling System and represent a hailstorm observed during the 1986 Cooperative Huntsville Meteorological Experiment. These properties are used as input to a three-dimensional radiative transfer model in order to simulate satellite observation of the storm. The model output consists of upwelling brightness temperatures at several of the frequencies on the Special Sensor Microwave/Imager. The radiative transfer model accounts for scattering and emission of atmospheric gases and hydrometeors in liquid and ice phases.
Brightness temperatures obtained from the three-dimensional model of this investigation indicate that horizontal inhomogeneities give rise to brightness temperature fields that can be quite different from fields obtained using plane-parallel radiative transfer theory. These differences are examined for various resolutions of the satellite sensor field of view. In addition, the issue of boundary conditions for three-dimensional atmospheric radiative transfer is addressed.
Improved Severe Weather Forecasts Using LEO and GEO Satellite SoundingsAbstract
It is shown here that improvements in numerical weather prediction (NWP) model forecasts of hazardous weather can be obtained by assimilating profile retrievals obtained in real time from combined direct broadcast system (DBS) polar satellite hyperspectral and geostationary satellite multispectral radiance data. Results of NWP model forecasts are shown for two recent tornado outbreak cases: 1) the 3 March 2019 tornado outbreak over the southeast United States and 2) the tornado outbreak that occurred across Illinois, Indiana, and Ohio during the night of 27 May and the morning of 28 May 2019, and 3) the 4 March 2019 severe precipitation event that occurred in southeast China. Improvements in both quantitative precipitation forecasts (QPFs) and predictions of the location of tornado occurrence are obtained. It is also shown that geostationary satellite hyperspectral soundings [i.e., Fengyun-4A (FY-4A) Geosynchronous Interferometric Infrared Sounder (GIIRS)] further improve hazardous precipitation forecasts when used, in addition to the combined polar hyperspectral and geostationary multispectral satellite profile data, to initialize the numerical forecast model. The lowest false alarm rate (FAR) and the highest probability of detection (POD) and critical success index (CSI) scores are achieved when assimilating atmospheric profile retrievals obtained by combining all the available satellite high-vertical-resolution hyperspectral radiance measurements with geostationary satellite high-spatial-resolution and high-temporal-resolution multispectral radiance measurements.
Abstract
It is shown here that improvements in numerical weather prediction (NWP) model forecasts of hazardous weather can be obtained by assimilating profile retrievals obtained in real time from combined direct broadcast system (DBS) polar satellite hyperspectral and geostationary satellite multispectral radiance data. Results of NWP model forecasts are shown for two recent tornado outbreak cases: 1) the 3 March 2019 tornado outbreak over the southeast United States and 2) the tornado outbreak that occurred across Illinois, Indiana, and Ohio during the night of 27 May and the morning of 28 May 2019, and 3) the 4 March 2019 severe precipitation event that occurred in southeast China. Improvements in both quantitative precipitation forecasts (QPFs) and predictions of the location of tornado occurrence are obtained. It is also shown that geostationary satellite hyperspectral soundings [i.e., Fengyun-4A (FY-4A) Geosynchronous Interferometric Infrared Sounder (GIIRS)] further improve hazardous precipitation forecasts when used, in addition to the combined polar hyperspectral and geostationary multispectral satellite profile data, to initialize the numerical forecast model. The lowest false alarm rate (FAR) and the highest probability of detection (POD) and critical success index (CSI) scores are achieved when assimilating atmospheric profile retrievals obtained by combining all the available satellite high-vertical-resolution hyperspectral radiance measurements with geostationary satellite high-spatial-resolution and high-temporal-resolution multispectral radiance measurements.
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Abstract
It is shown that the partial interferogram measurement technique, originally developed to separate the trace gas emissions from a spectral signal dominated by background radiation (from the earth's surface) and emissions from major constituents (H2O and CO2), has application to the vertical sounding problem. The interferometric technique will enable relatively high vertical temperature profile resolution to be achieved and will provide absolute accuracies of temperature approaching, and at same levels exceeding, 1°C.
Abstract
It is shown that the partial interferogram measurement technique, originally developed to separate the trace gas emissions from a spectral signal dominated by background radiation (from the earth's surface) and emissions from major constituents (H2O and CO2), has application to the vertical sounding problem. The interferometric technique will enable relatively high vertical temperature profile resolution to be achieved and will provide absolute accuracies of temperature approaching, and at same levels exceeding, 1°C.
Abstract
The traditional approach with experimental raindrop size data has been to use the method of moments in the fitting procedure to estimate the parameters for the raindrop size distribution function. However, the moment method is known to be biased and can have substantial errors. Therefore, the L-moment method, which is widely used by hydrologists, was investigated as an alternative. The L-moment method was applied, along with the moment and maximum likelihood methods, to samples taken from simulated gamma raindrop populations. A comparison of the bias and the errors involved in the L-moments, moments, and maximum likelihood procedures shows that, with samples covering the full range of drop sizes, L-moments and maximum likelihood outperform the method of moments. For small sample sizes the moment method gives a large bias and large error while the L-moment method gives results close to the true population values, outperforming even maximum likelihood results. Because the goal of this work is to understand the properties of the various fitting procedures, the investigation was expanded to include the effects of the absence of small drops in the samples (typical disdrometer minimum size thresholds are 0.3–0.5 mm). The results show that missing small drops (due to the instrumental constraint) can result in a large bias in the case of the L-moment and maximum likelihood fitting methods; this bias does not decrease much with increasing sample size. Because the very small drops have a negligible contribution to moments of order 2 or higher, the bias in the moment methods seems to be about the same as in the case of full samples. However, when moments of order less than 2 are needed (as in the case of modelers using moments 0 and 3), the moment method gives much larger bias. Therefore a modification of these methods is needed to handle the truncated-data situation.
Abstract
The traditional approach with experimental raindrop size data has been to use the method of moments in the fitting procedure to estimate the parameters for the raindrop size distribution function. However, the moment method is known to be biased and can have substantial errors. Therefore, the L-moment method, which is widely used by hydrologists, was investigated as an alternative. The L-moment method was applied, along with the moment and maximum likelihood methods, to samples taken from simulated gamma raindrop populations. A comparison of the bias and the errors involved in the L-moments, moments, and maximum likelihood procedures shows that, with samples covering the full range of drop sizes, L-moments and maximum likelihood outperform the method of moments. For small sample sizes the moment method gives a large bias and large error while the L-moment method gives results close to the true population values, outperforming even maximum likelihood results. Because the goal of this work is to understand the properties of the various fitting procedures, the investigation was expanded to include the effects of the absence of small drops in the samples (typical disdrometer minimum size thresholds are 0.3–0.5 mm). The results show that missing small drops (due to the instrumental constraint) can result in a large bias in the case of the L-moment and maximum likelihood fitting methods; this bias does not decrease much with increasing sample size. Because the very small drops have a negligible contribution to moments of order 2 or higher, the bias in the moment methods seems to be about the same as in the case of full samples. However, when moments of order less than 2 are needed (as in the case of modelers using moments 0 and 3), the moment method gives much larger bias. Therefore a modification of these methods is needed to handle the truncated-data situation.
Abstract
Use of the standard deviation σ m of the drop mass distribution as one of the three parameters of raindrop size distribution (DSD) functions was introduced for application to disdrometer data supporting the Global Precipitation Measurement dual-frequency radar system. The other two parameters are a normalized drop number concentration N w and the mass-weighted mean diameter D m . This paper presents an evaluation of that formulation of the DSD functions, in two parts. First is a mathematical analysis showing that the procedure for estimating σ m , along with the other DSD parameters, from disdrometer data is in essence another moment method. As such, it is subject to the biases and errors inherent in all moment methods. When the form of the DSD function is specified, it is inferior (like all moment methods) to the maximum likelihood technique for fitting parameters to sampled data. The second part is a series of sampling simulations illustrating the biases and errors involved in applying the formulation to the specific case of gamma DSDs. It leads to underestimates of σ m and consequently to overestimates of the gamma shape parameter—with large root-mean-square errors. Comparison with maximum likelihood estimates shows the degree of improvement that could be obtained in the estimates of the shape parameter. The propensity to underestimate σ m will be pervasive, and users of this DSD formulation should be cognizant of the biases and errors that can occur.
Abstract
Use of the standard deviation σ m of the drop mass distribution as one of the three parameters of raindrop size distribution (DSD) functions was introduced for application to disdrometer data supporting the Global Precipitation Measurement dual-frequency radar system. The other two parameters are a normalized drop number concentration N w and the mass-weighted mean diameter D m . This paper presents an evaluation of that formulation of the DSD functions, in two parts. First is a mathematical analysis showing that the procedure for estimating σ m , along with the other DSD parameters, from disdrometer data is in essence another moment method. As such, it is subject to the biases and errors inherent in all moment methods. When the form of the DSD function is specified, it is inferior (like all moment methods) to the maximum likelihood technique for fitting parameters to sampled data. The second part is a series of sampling simulations illustrating the biases and errors involved in applying the formulation to the specific case of gamma DSDs. It leads to underestimates of σ m and consequently to overestimates of the gamma shape parameter—with large root-mean-square errors. Comparison with maximum likelihood estimates shows the degree of improvement that could be obtained in the estimates of the shape parameter. The propensity to underestimate σ m will be pervasive, and users of this DSD formulation should be cognizant of the biases and errors that can occur.
Abstract
The robust diffusion sensor provides a simple, inexpensive, accurate method to integrate temperature, relative humidity (or water activity), or salinity for time periods up to several years. It does not require power or maintenance. Depending on temperature, precision of about 0.1°C for temperature and a few percent for relative humidity or salinity can be achieved. Various combinations of water-filled and desiccant-filled cells can be used for simultaneously integrating temperature, relative humidity or salinity by simply weighing the cells before and after exposure to the environment.
Abstract
The robust diffusion sensor provides a simple, inexpensive, accurate method to integrate temperature, relative humidity (or water activity), or salinity for time periods up to several years. It does not require power or maintenance. Depending on temperature, precision of about 0.1°C for temperature and a few percent for relative humidity or salinity can be achieved. Various combinations of water-filled and desiccant-filled cells can be used for simultaneously integrating temperature, relative humidity or salinity by simply weighing the cells before and after exposure to the environment.
To test the impact of high-resolution Nimbus-6 sounding data on Australian region forecasts, two parallel analysis/forecast cycling experiments were carried out, using data for 14 days during August and September 1975. In one of these cycles, only conventional data and manual interpretation of satellite imagery were used as input, while the other cycle used conventional and Nimbus-6 sounding data. A manual mean sea level pressure analysis was used in each cycle to provide reference level information over the oceans.
Two series of 24 h limited area prognoses were prepared from these two sets of analyses, using the primitive equations prognosis model developed at the Australian Numerical Meteorology Research Centre. An average improvement in geopotential forecasts of more than 5 skill score points was achieved at all levels over the Australian continent when the Nimbus-6 data were included in the base analyses. Also, significant reductions were obtained in 24 h forecast root-mean-square (rms) temperature errors.
To test the impact of high-resolution Nimbus-6 sounding data on Australian region forecasts, two parallel analysis/forecast cycling experiments were carried out, using data for 14 days during August and September 1975. In one of these cycles, only conventional data and manual interpretation of satellite imagery were used as input, while the other cycle used conventional and Nimbus-6 sounding data. A manual mean sea level pressure analysis was used in each cycle to provide reference level information over the oceans.
Two series of 24 h limited area prognoses were prepared from these two sets of analyses, using the primitive equations prognosis model developed at the Australian Numerical Meteorology Research Centre. An average improvement in geopotential forecasts of more than 5 skill score points was achieved at all levels over the Australian continent when the Nimbus-6 data were included in the base analyses. Also, significant reductions were obtained in 24 h forecast root-mean-square (rms) temperature errors.
Abstract
A method to depict quasi-continuous surface temperature features is presented. Half-hourly GOES window channel brightness temperature determinations are employed to monitor time changes in the surface temperature field. TIROS-N water vapor channel measurements, within 6 h of the GOES measurements, are used to generate water vapor absorption corrections to the window channel brightness temperatures. Two case studies are presented that show the resulting surface radiating temperature estimates to be accurate close to 1 K. In regions where conventional ground based measurements are sparse, this method is demonstrated to be most useful. The potential for using time sequences of these surface temperature fields as a diagnostic aid for forecasting severe weather is exhibited in one of the case studies.
Abstract
A method to depict quasi-continuous surface temperature features is presented. Half-hourly GOES window channel brightness temperature determinations are employed to monitor time changes in the surface temperature field. TIROS-N water vapor channel measurements, within 6 h of the GOES measurements, are used to generate water vapor absorption corrections to the window channel brightness temperatures. Two case studies are presented that show the resulting surface radiating temperature estimates to be accurate close to 1 K. In regions where conventional ground based measurements are sparse, this method is demonstrated to be most useful. The potential for using time sequences of these surface temperature fields as a diagnostic aid for forecasting severe weather is exhibited in one of the case studies.
