IMPROVING THE CONVECTIVE FORECASTS OF THE FEDERAL AVIATION ADMINISTRATION

Transcription

1 The Pennsylvania State University The Graduate School College of Earth and Mineral Sciences IMPROVING THE CONVECTIVE FORECASTS OF THE FEDERAL AVIATION ADMINISTRATION A Thesis in Meteorology by Marikate Lee Ellis Copyright 2010 Marikate Lee Ellis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science May 2010

2 The thesis of Marikate Lee Ellis was reviewed and approved* by the following: George S. Young Professor of Meteorology Thesis Adviser Andrew Kleit Professor of Energy and Environmental Economics Eugene E. Clothiaux Associate Professor of Meteorology Johannes Verlinde Associate Professor of Meteorology Associate Head, Graduate Program in Meteorology *Signatures are on file in the Graduate School. ii

3 ABSTRACT The effectiveness of Federal Aviation Administration (FAA) air traffic control is improved via a radar image generational algorithm developed to improve FAA convective forecasts by generating an ensemble of radar image realizations capable of being integrated into air traffic flow management software. The algorithm has the potential to result in a considerable cost savings to airlines and consumers. The algorithm not only accounts for a forecasted convective probability, but also incorporates a likely two-dimensional convective storm pattern, a characteristic currently absent in the FAA convective forecast. The ensemble forecast will provide the FAA with information on the fractional horizontal area the convective storm will cover, the two-dimensional shape of the convective storm, and the relative uncertainty associated with the forecast obtained by comparing the forecast radar images within the ensemble to each other. This information will be used to drive an existing air traffic flow model to produce an ensemble prediction of the air traffic capacity on routes into heavily utilized airports. At present, the algorithm successfully produces a single radar image realization containing the correct convective probability and a similar convective pattern. iii

4 TABLE OF CONTENTS List of Tables v List of Figures vi List of Abbreviations and Nomenclature xi Acknowledgements xii Chapter 1: Introduction Summary of Radar Image Generational Algorithm...3 Chapter 2: Data Chapter 3: Procedures Radar Image Generational Algorithm Inputs The Current Iterative Radar Image Spatial Conditional Probability The Swap Test Statistical Analysis Methods Chapter 4: Results Radar Image Generational Algorithm Stipulations Radar Image Generational Algorithm Strengths and Weaknesses Statistical Analysis Results...46 Chapter 5: Conclusions Future Work.. 50 References iv

5 LIST OF TABLES Table 1: Accompanies Figure 1. The table above describes the Q scales and the associated area sizes in relation to the center convective pixel (shaded in green in Figure 1)...16 Table 2: Bandwidth, the width of the narrowest axis of the convective pattern in the pattern training radar image, must be slightly less than or equal to the size of the center box of Q. 27 v

6 LIST OF FIGURES Figure 1: Accompanies Table 1. The figure above depicts three separate local Q matrices all centered on the same convective pixel (shaded in green). To compute the value of each box of each Q matrix, all the cells that fall into that box must be averaged.16 Figure 2: Accompanies Figure 3. Displays the 9 box values in each of the small, intermediate, and large scale local Q matrices surrounding the example green convective cell in Figure Figure 3: Accompanies Figure 2. The figure above depicts a 30 pixel by 30 pixel subsection of a radar image, where convective cells have a value of 1 and are highlighted in yellow and green. The green convective cell is the center of an example of three local Q matrices at three different scales. The colors of the Q matrices are the same colors as described in Table 1. Based on this example, the value of each of the 9 boxes in each of the three local Q matrices is calculated and displayed in Figure Figure 4-1: Front - Original National Radar Image Figure 4-2: Front - Processed National Radar Image Figure 4-3: Front - Pattern Training Radar Image Figure 4-4: Front - Initial Current Iterative Radar Image Figure 4-5: Front - Final Current Iterative Radar Image Figure 4-6: Front - Target Small Scale Q Figure 4-7: Front - Initial Small Scale Q Figure 4-8: Front - Final Small Scale Q Figure 4-9: Front - Small Scale Error..34 vi

7 LIST OF FIGURES Figure 4-10: Front - Target Intermediate Scale Q Figure 4-11: Front - Initial Intermediate Scale Q Figure 4-12: Front - Final Intermediate Scale Q Figure 4-13: Front - Intermediate Scale Error Figure 4-14: Front - Target Large Scale Q Figure 4-15: Front - Initial Large Scale Q...34 Figure 4-16: Front - Final Large Scale Q Figure 4-17: Front - Large Scale Error Figure 5-1: Squall Line - Original National Radar Image...35 Figure 5-2: Squall Line - Processed National Radar Image Figure 5-3: Squall Line - Pattern Training Radar Image...35 Figure 5-4: Squall Line - Initial Current Iterative Radar Image Figure 5-5: Squall Line - Final Current Iterative Radar Image Figure 5-6: Squall Line - Target Small Scale Q Figure 5-7: Squall Line - Initial Small Scale Q Figure 5-8: Squall Line - Final Small Scale Q Figure 5-9: Squall Line - Small Scale Error Figure 5-10: Squall Line - Target Intermediate Scale Q Figure 5-11: Squall Line - Initial Intermediate Scale Q Figure 5-12: Squall Line - Final Intermediate Scale Q...36 Figure 5-13: Squall Line - Intermediate Scale Error vii

8 LIST OF FIGURES Figure 5-14: Squall Line - Target Large Scale Q Figure 5-15: Squall Line - Initial Large Scale Q Figure 5-16: Squall Line - Final Large Scale Q Figure 5-17: Squall Line - Large Scale Error Figure 6-1: Isolated - Original National Radar Image Figure 6-2: Isolated - Processed National Radar Image..37 Figure 6-3: Isolated - Pattern Training Radar Image Figure 6-4: Isolated - Initial Current Iterative Radar Image Figure 6-5: Isolated - Final Current Iterative Radar Image Figure 6-6: Isolated - Target Small Scale Q Figure 6-7: Isolated - Initial Small Scale Q Figure 6-8: Isolated - Final Small Scale Q Figure 6-9: Isolated - Small Scale Error Figure 6-10: Isolated - Target Intermediate Scale Q...38 Figure 6-11: Isolated - Initial Intermediate Scale Q Figure 6-12: Isolated - Final Intermediate Scale Q Figure 6-13: Isolated - Intermediate Scale Error Figure 6-14: Isolated - Target Large Scale Q Figure 6-15: Isolated - Initial Large Scale Q Figure 6-16: Isolated - Final Large Scale Q Figure 6-17: Isolated - Large Scale Error viii

9 LIST OF FIGURES Figure 7-1: Scattered - Original National Radar Image...39 Figure 7-2: Scattered - Processed National Radar Image Figure 7-3: Scattered - Pattern Training Radar Image Figure 7-4: Scattered - Initial Current Iterative Radar Image Figure 7-5: Scattered - Final Current Iterative Radar Image Figure 7-6: Scattered - Target Small Scale Q..40 Figure 7-7: Scattered - Initial Small Scale Q Figure 7-8: Scattered - Final Small Scale Q Figure 7-9: Scattered - Small Scale Error Figure 7-10: Scattered - Target Intermediate Scale Q Figure 7-11: Scattered - Initial Intermediate Scale Q..40 Figure 7-12: Scattered - Final Intermediate Scale Q Figure 7-13: Scattered - Intermediate Scale Error...40 Figure 7-14: Scattered - Target Large Scale Q Figure 7-15: Scattered - Initial Large Scale Q Figure 7-16: Scattered - Final Large Scale Q Figure 7-17: Scattered - Large Scale Error Figure 8-1: Cluster - Original National Radar Image..41 Figure 8-2: Cluster - Processed National Radar Image...41 Figure 8-3: Cluster - Pattern Training Radar Image Figure 8-4: Cluster - Initial Current Iterative Radar Image ix

10 LIST OF FIGURES Figure 8-5: Cluster - Final Current Iterative Radar Image Figure 8-6: Cluster - Target Small Scale Q Figure 8-7: Cluster - Initial Small Scale Q Figure 8-8: Cluster - Final Small Scale Q Figure 8-9: Cluster - Small Scale Error Figure 8-10: Cluster - Target Intermediate Scale Q Figure 8-11: Cluster - Initial Intermediate Scale Q Figure 8-12: Cluster - Final Intermediate Scale Q Figure 8-13: Cluster - Intermediate Scale Error Figure 8-14: Cluster - Target Large Scale Q Figure 8-15: Cluster - Initial Large Scale Q Figure 8-16: Cluster - Final Large Scale Q Figure 8-17: Cluster - Large Scale Error x

11 LIST OF ABBREVIATIONS AND NOMENCLATURE CAPE Convective Available Potential Energy CCFP Collaborative Convective Forecast Product CIN Convective Inhibition FAA Federal Aviation Administration GFS Global Forecast System LAMP Localized Aviation Model Output Statistics Product NEXRAD Next Generation Radar Q Spatial conditional probability map xi

12 ACKNOWLEDGEMENTS I d like to thank my advisor, George Young, and other committee members, Eugene Clothiaux and Andrew Kleit, for their support, encouragement, and patience. I d also like to thank my family and friends who have provided me with love, strength, and sanity at the most difficult times. This work was supported under FAA grant number PSU-0001-F800 ATP1 DTD 6/30/09. xii

13 CHAPTER 1: INTRODUCTION The Federal Aviation Administration (FAA) is responsible for the safety of civil aviation. FAA responsibilities include promulgating safety regulations, developing new aviation technologies, developing and overseeing air traffic control, and regulating commercial air traffic. 1 The traffic flow management performed by the FAA includes a number of tasks. 2 The FAA is responsible for balancing air traffic demand with national air space capacity, while maintaining maximum utilization of the air space. Air Route Traffic Control Centers work in conjunction with Terminal Radar Approach Control, which works in conjunction with the terminal tower, to see that aircraft safely and efficiently traverse routes. Certain rules of thumb are employed when making decisions regarding air space capacity. The threshold values for these rules are determined from experience and air traffic simulations using historical weather. In this effort to oversee air traffic, weather forecasts play a significant role. Snow, ice, wind, fog, and thunderstorms all create obstacles for pilots. Aircraft flights are often diverted, delayed, or cancelled due to the impact of weather. Thus, good weather forecasts are necessary to plan ahead and achieve a stable flow of air traffic through or around weather impediments. This research project is concerned with improving the convective storm or thunderstorm forecasts issued by the FAA for air flow management. The goal of the research is to create a radar image generational algorithm that generates an ensemble of forecast radar image realizations through combining a forecasted convective 1 What We Do. Federal Aviation Administration. 10 Mar Web. Feb < 2 Traffic Flow Management in the National Airspace System. Federal Aviation Administration Air Traffic Organization. Oct Web. Feb < 1

14 probability and quantitative descriptions of the expected two-dimensional convective storm pattern, referred to as the pattern training radar image spatial conditional probability maps. The ensemble forecast will provide the FAA with information on the fractional horizontal area the convective storm will cover, the two-dimensional shape of the convective storm, and the relative uncertainty associated with the forecast, obtained by comparing the forecast radar images within the ensemble to each other. The ensemble of forecast radar images will be used by the FAA as input to drive the Air Traffic Flow Management model. The model output will in turn drive air traffic flow management strategy. The radar image generational algorithm thus allows the FAA to have an air traffic flow management strategy prepared less than twenty-four hours in advance of an expected storm, in an effort to minimize flight delays, flight cancellations, airline costs, and consumer costs. Currently, the FAA uses the Collaborative Convective Forecast Product (CCFP) to forecast convective storms. The CCFP was developed in 1998 and is a product of collaboration between commercial airline weather offices, business aviation weather offices, and NWS Center Weather Service Units (CWSUs) located at 20 FAA field offices. 3 The collaboration results in a forecast comprised of three maps: 2 hour, 4 hour, and 6 hour advanced convective forecasts. It is important to note that the forecast is not necessarily a consensus of opinions. Each of the three maps depicts shaded areas that reflect convective forecasts containing the categorical expected probability of thunderstorm occurrence, thunderstorm growth, and top height of thunderstorm clouds. The disadvantage of this forecast is that it only forecasts a categorical range for the 3 The Collaborative Convective Forecast Product (CCFP). Collaborative Decision Making. Federal Aviation Administration. 15 Aug Web. Feb < 2

15 probability of thunderstorm occurrence. The shaded areas within the maps also do not necessarily reflect the shape of the convection. The shape of the convection is crucial for deciding whether or not an aircraft should be flown around the convection or if that is impossible and the aircraft must be flown over or through the convection. Another important disadvantage of this forecast is its collaborative formulation and unavoidable political implications of specific airline dominance and airline costs. Thus the CCFP may be purposely skewed by certain collaborating parties for the narrow interest of those parties. The radar image generational algorithm seeks to resolve those weaknesses posed by the current CCFP forecast. In the algorithm, the forecasted convective probability is extracted from the Localized Aviation Model Output Statistics Product (LAMP), which is derived from the operational Global Forecast System (GFS). The LAMP forecast provides point forecasts, which includes a forecast for the probability that convection will occur at a given location. This will provide the ensemble radar image forecast created by the algorithm with a more specific, quantitative convective probability rather than the categorical convective probability provided by the CCFP. Also, the very nature of the ensemble radar image forecast will provide information about the two-dimensional shape of a storm, a characteristic the CCFP lacks. Lastly, the radar image generational algorithm is purely objective in its creation of an ensemble of radar image realizations. The algorithm has no collaborative component prone to subjective influence. 1.1 Summary of Radar Image Generational Algorithm The radar image generational algorithm combines a spatially varying probabilistic convective forecast and quantitative descriptions of the expected two-dimensional convective storm pattern, referred to as the pattern training radar image spatial conditional probability maps, 3

16 in order to generate an ensemble forecast of radar image realizations. The algorithm generates a single radar image realization forecast, the current iterative radar image, by iteratively swapping subsections of pixels within the current iterative radar image until its spatial conditional probability maps are effectively similar to the corresponding scale of pattern training radar image spatial conditional probability maps. Similar spatial conditional probability maps indicate that the radar images from which the spatial conditional probability maps were calculated have similar two-dimensional convective patterns. The forecasted convective probability is extracted from LAMP and is interpreted as the forecasted convective coverage percentage or the fractional area of pixels within a radar image that are classified as convective pixels. The convective coverage percentage is used to initialize the algorithm generated radar image realization, the current iterative radar image, by randomly classifying pixels in the image as convective until the convective coverage fractional number of pixels is achieved. Every realization within the ensemble of radar image realizations generated by the algorithm is characterized by having the forecasted convective coverage percentage as the basis for the number of convective pixels in the radar image. The second input required, the pattern training radar image spatial conditional probability maps, are based on an idealized convective pattern as depicted on a real radar image (i.e. front, squall line, clustered, scattered, or isolated convective pattern). This real radar image is referred to as the pattern training radar image and its spatial conditional probability maps, conditioned on a center convective pixel, are required for algorithm input. The pattern training radar image spatial conditional probability maps provide a quantitative way to compare the pattern training radar image to the current iterative radar image. The spatial conditional probability maps for a given radar image are differentiated by different scales or the size of the area of pixels in the 4

17 radar image that are quantitatively being described by each map. This quantitative comparison is crucial since the goal of the algorithm is to depict a pattern similar to that of the pattern training radar image on every radar image realization in the ensemble generated. The patterns on every realization will not be identical, however, since both the initial pixel positions and the swap locations are randomly selected. Spatial conditional probability maps are also calculated for the current iterative radar image and referred to as the current iterative radar image spatial conditional probability maps. Once the current iterative radar image spatial conditional probability maps and the pattern training radar image spatial conditional probability maps are defined, a single variable known as error is calculated for each identically scaled spatial conditional probability map pair. Error must be minimized in order to achieve a similar pattern on the current iterative radar image compared to the pattern training radar image. Error is minimized through the iterative process known as the swap test. The swap test randomly selects sections of pixels to swap within the current iterative radar image and recalculates the new error produced by the swap. If the swap effectively reduces the error, bringing the current iterative radar image pattern closer to the pattern on the pattern training radar image, then the swap is retained. This process is continued for millions of iterations. The resulting current iterative radar image contains a pattern similar to that of the pattern training radar image and is characterized by the forecasted convective coverage percentage. This process is repeated to generate each radar image in the ensemble of radar image realizations. Following the successful development of the algorithm, a statistical analysis is performed in an effort to automate the selection of the pattern training radar image spatial conditional probability maps, which is one of the two algorithm inputs. Successful automation is based on 5

18 finding a well correlated atmospheric variable with the convective pattern categories (i.e. front, squall line, clustered, scattered, and isolated convection). This is an area that will require future work. Once the radar image generational algorithm is successfully automated, several steps remain to achieve a version of the algorithm fit for FAA convective forecast purposes. This algorithm generated ensemble of radar image realizations can be used by the FAA to calculate the likelihood of a given air traffic situation occurring. Anticipating days requiring a reduced volume of air traffic reduces the strain on the air traffic management system by letting the FAA prepare in advance. 6

19 CHAPTER 2: DATA Observational data are required both as input for the radar image generational algorithm discussed above and for the statistical analysis used to automate the selection of the appropriate pattern training radar image spatial conditional probability map. For algorithm input, a convective coverage percentage and pattern training radar image are required, the latter being used to create spatial conditional probability maps. For the statistical analysis, sounding data and surface analysis data are used. In both the algorithm and statistical analysis datasets, actual observations are used rather than forecasts, making this a perfect prog method (Wilks 2006). The algorithm thus requires two pieces of input data: a convective coverage percentage and a pattern training radar image. When the algorithm is used to forecast operationally, the convective coverage percentage will be extracted from the LAMP forecast. In algorithm development, however, the convective coverage percentage is estimated from the fraction of convective pixels in the pattern training radar image. In contrast, observed radar images are used only during development where they are required to create the spatial conditional probability maps. These maps are then used in the operational forecast process. While we demonstrate the algorithm using a single radar image to generate each set of spatial conditional probability maps, one could increase the statistical robustness by averaging the results from several images to generate each set of spatial conditional probability maps. The algorithm s task will be easier in the forecast mode if multiple pattern training radar images with similar convective patterns and convective coverage percentages are used to generate each set of spatial conditional probability maps. In forecast mode it would be additionally useful to the algorithm if multiple subsets of pattern training radar images existed within a group of similar convective patterns (e.g. front 7

20 segments or clustered convection). Each subset of images would be characterized by similar convective patterns and convective coverage percentages but the subsets would be differentiated by the range of convective coverage percentages depicted in the images. The subset containing the most appropriate convective pattern and having the range of convective coverage percentages most similar to the convective coverage percentage from the LAMP forecast would be used to generate a single set of spatial conditional probability maps. In algorithm development, the pattern training radar images, used to create the spatial conditional probability maps, are extracted from archived national radar images of the United States. The archived images were extracted from the Pennsylvania State University Department of Meteorology e-wall as current radar images and archived. 4 The images were produced by Unidata utilizing 6 km resolution national reflectivity composites measured by the National Weather Service Next Generation Radar (NEXRAD) network. According to NOAA, the National Weather Service operates 159 NEXRAD radars, each with a maximum range of 250 nautical miles. 5 Each national radar image is composed of 870 pixels by 652 pixels for a total of 567,240 pixels. In order for a radar image case to be considered for processing in the radar image generational algorithm, each case is required to pass a manual screening. National radar images are manually grouped into convective pattern categories and are required to display radar reflectivities of at least 40 dbz. The value of 40 dbz was selected as a relative minimum value of radar reflectivity for classification of a convective storm. The higher the radar reflectivity, the 4 E-wall: The Electronic Map Wall. The Pennsylvania State University Department of Meteorology. N.d. Web < 5 About the Radar Operations Center. NOAA s National Weather Service Radar Operations Center. 9 Feb Web. Feb < 8

21 larger the size of precipitation particles and thus, the stronger the associated storm updraft, both of which can be dangerous to aircraft. The convective pattern categories are clustered convection, isolated convection, front segment, squall line, or scattered convection. Not all cases that fall into these categories are processed however. The number of cases processed from each category is relatively equivalent: 8 fronts, 8 clustered, 8 isolated, 8 scattered, and 9 squall lines. So only a select number of cases are processed. The 41 cases selected for processing in the radar image generational algorithm are predominantly from May through September 2006 but there are a few from May through September In order to complete the statistical analysis required to find a variable well correlated with the convective pattern categories and thus automate the selection of the appropriate set of spatial conditional probability maps, atmospheric sounding data and surface front analysis data associated with each case is used. The cases incorporated in the statistical analysis include those 41 cases previously used for processing by the algorithm in addition to 147 new cases. New cases are identified using the same national radar image and convective pattern categorization as described above. The only difference between the algorithm processed cases and the new cases is that the former are processed by the algorithm and a radar image forecast has been generated while the latter have not been processed and do not have a radar image forecast associated with them. This difference, however, is irrelevant for the purposes of the statistical analysis. The new cases are from May through September 2007 exclusively. For each of the new cases the associated sounding data and surface analyses are recorded. Thus a total of 188 possible cases are available for use in the statistical analysis. Not all these cases, however, are employed in the final statistical analysis presented in the results chapter. 9

22 The sounding data utilized in the statistical analysis to automate the selection of the set of spatial conditional probability maps, is downloaded from the University of Wyoming atmospheric sounding site. 6 Sounding data is measured through the National Weather Service Upper-Air Observations Program using radiosondes. As of January 2010, 102 radiosonde stations are operational in North America, the Pacific Islands, and the Caribbean. Soundings are conducted by the National Weather Service and normally are taken twice a day at 00 Z and 12 Z, 365 days a year. Radiosonde measurements include pressure, temperature, and relative humidity profiles, together with derived wind speed and direction from GPS tracking. All other variables listed on a sounding are calculated from these profiles. 7 The sounding recorded for each case is generally south to southeast of the specific convective storm, where the source of storm moisture is generally located. Since soundings are only available in 12 hour increments, the sounding recorded for each case is required to occur prior to the radar image time stamp. This was done in an effort to ensure that pre-storm atmospheric conditions or the atmospheric characteristics of the air feeding into the storm are recorded, not the post-storm atmospheric variables. Sounding data extracted for use in the statistical analysis includes the K-index, convective available potential energy (CAPE), convective inhibition (CIN), and wind speed and direction from the surface, 3000 m level, and 6000 m level. Note that these levels were used based on flight level standards in forecasts issued by the FAA. 6 Upper Air Sounding. University of Wyoming College of Engineering Department of Atmospheric Science. N.d. web. Feb < 7 What is a radiosonde? National Weather Service Radiosonde Observations. National Weather Service. N.d. web. Feb < 10

23 The statistical analysis required to automate the selection of the set of spatial conditional probability maps also incorporates surface analyses, containing information on the presence of fronts. Surface analyses are obtained from the Hydrometeorological Prediction Center (HPC). 8 Analyses are issued in 3 hour time intervals and depict surface synoptic and mesoscale features such as high and low pressure systems, fronts, troughs, outflow boundaries, squall lines and dry lines. The analyses include most of North America and its bordering oceans. 9 The surface analysis used in each case is the most recent surface analysis occurring prior to the radar image time stamp. In order for a stationary or cold front on a surface analysis to be associated with convection occurring in a case, the front must occur within 600 km of the convection, measured perpendicular to the front. This distance was judged to be a fair assessment of whether or not a front is in the vicinity of the convection. The radar image data in the radar image generational algorithm and sounding data and surface analyses in the statistical analysis are screened for completeness and errors. National radar images containing questionable reflectivity values are omitted and so are any associated cases. Atmospheric soundings containing incomplete information are omitted and replaced with substitute soundings from the same day and time but at an alternate sounding location. Surface analyses are examined for completeness but none are found deficient. 8 HPC s Surface Analysis Archive. Hydrometeorological Prediction Center. 1 Mar Web. Feb < 9 About the Surface Analysis. Hydrometeorological Prediction Center. 1 Mar Web. Feb < 11

24 CHAPTER 3: PROCEDURES As previously explained, the FAA convective forecast is concerned with how convective coverage translates into air space capacity within an air traffic sector. The goal of the radar image generational algorithm is to combine a convective coverage percentage forecast with a pattern training radar image to generate an ensemble of radar image realizations. This is achieved over a series of iterations, where the set of spatial conditional probability maps of the pattern training radar image will be compared to the set of spatial conditional probability maps of the current iterative radar image at each scale, as the latter is altered to appear similar to the former. The current iterative radar image is altered by randomly selecting sections of pixels and swapping them until the set number of iterations is achieved. Upon completion, the current iterative radar image will be characterized by the convective coverage percentage forecast and a convective pattern similar to that depicted on the pattern training radar image, as described by the similar spatial conditional probability map sets. If this iteration process is run repeatedly the algorithm output takes the form of an ensemble of radar image realizations, although currently the algorithm output is only a single radar realization. Operationally, the ensemble of radar realizations would then be used in conjunction with the air traffic modeling software to assess the impact of convective storms on the air traffic capacity. The availability of multiple radar realizations also will give an indication of the convective forecast uncertainty. 3.1 Radar Image Generational Algorithm Inputs The convective coverage percentage forecast, one of the two radar image generational algorithm inputs, will be derived from the LAMP forecast when the algorithm is forecasting 12

25 operationally. LAMP provides a forecast for the probability that convection will occur at given gridded locations. In order to use the LAMP probability grid, an assumption must be made. It is assumed that within a radar image forecast area, the average of the point forecast probabilities provided by LAMP is equivalent to the fractional amount of the land area that is covered with convection at the forecast time. For example, in a 5 pixel by 5 pixel radar image, an average LAMP probability of 20% would translate to 5 of the 25 pixels being defined as convective. The disadvantage of this assumption is that it is not completely accurate. In reality, converting the LAMP convective probability grid to a fractional area coverage will require recalibration, likely using linear regression. As previously explained, for developmental purposes, the convective coverage percentage is estimated from the fractional convective coverage area depicted in the pattern training radar image. The second algorithm input, the pattern training radar image, is derived from real radar images. Its purpose is to provide information about the size and shape of a convective storm, which is required to calculate the spatial conditional probability maps used as a goal standard of comparison throughout the algorithm iterations. The real national radar images are read in by the algorithm and all radar reflectivities 40 dbz and greater are interpreted as convective pixels. Reflectivities less than 40 dbz are interpreted as non-convective pixels. A two-dimensional matrix of ones and zeros is used to represent the original radar image depicting the pattern of the storm. Ones represent convective pixels in a radar image and zeros represent non-convective pixels. This two-dimensional matrix representing a convective radar pattern is known as the pattern training radar image. 13

26 3.2 The Current Iterative Radar Image In order to combine the convective coverage percentage and the pattern training radar image, a hypothetical radar map known as the current iterative radar image is created. The current iterative radar image, containing the forecasted convective coverage percentage, rearranges randomly located convective pixels into a pattern similar to the pattern on the pattern training radar image through many iterations of swapping subsections of pixels within the current iterative radar image. The final version of the current iterative radar image is a single radar realization among the multiple radar realizations necessary to create an ensemble of radar realizations. The current iterative radar image is initialized utilizing the convective coverage percentage. Using the assumption that a point probability of convection is equivalent to the fractional land coverage of convection, a two-dimensional matrix is created to represent the forecast area. This matrix is called the current iterative radar image. It is equivalent in size to the matrix used for the pattern training radar image. As in the case of the pattern training radar image, ones again represent convective pixels and zeros represent non-convective pixels. The cells or pixels in the current iterative radar image are randomly assigned a value of one until a fractional number of the pixels is equivalent to the convective coverage percentage. The remaining unassigned pixels are assigned a value of zero. 3.3 Spatial Conditional Probability In order to bring randomly assigned convective pixels in the current iterative radar image into a convective pattern resembling that in the pattern training radar image, a quantitative means of comparing the current iterative radar image to the pattern training radar image is required. 14

27 The spatial conditional probability map (Q) is a quantitative description of the convective pattern observed on a radar image. Multiple Q maps exist for a given radar image, where each Q map is quantitatively describing the radar image convective pattern at a different scale. Keeping this in mind, the Q at each scale is a two-dimensional map conditioned on the presence of a convective center pixel as seen in Figure 1 and Table 1. Another way to interpret this: is that the Q at each scale is a two-dimensional matrix conditioned on the presence of a center cell with a value of one. In theory, the size of the Q matrix could be varied but in this application of the algorithm Q is a 3 box by 3 box two-dimensional matrix for all scales. Q maps exist at multiple scales to better capture the convective pattern within the radar images. In theory there could be any number of Q scales, as long as the pixel size of each Q map remains smaller than the radar image it describes. In most applications of the algorithm used here, Q is calculated at 3 scales: small, intermediate, and large. The reasoning behind this decision is related to maximizing algorithm performance, which is explained in the results chapter. Therefore, Q will generally be described as having only 3 scales in this text. The center pixel within the center box of each Q map is convective meaning that each of the 9 boxes of Q represents a portion of the radar image area surrounding a single convective pixel (i.e., upper left, lower center, etc.). As the size of the scale increases from small to large, the size of the radar image area Q is describing increases and more and more pixels fall into each of the 9 boxes of the Q matrix. 15

28 Scale Q Matrix Size cells x cells Specifics of Q at Three Scales Total No. of Cells in Q Matrix Shaded Color of Q Matrix Small 3x3 9 red green red Intermediate 9x9 81 blue red blue Large 27x gray blue black TABLE 1: Accompanies Figure 1. The table above describes the Q scales and the associated area sizes in relation to the center convective pixel (shaded in green in Figure 1). Q Matrices at Three Scales Shaded Color of Center Square Line Color of the 9 Q Matrix Boxes FIGURE 1: Accompanies Table 1. The figure above depicts three separate local Q matrices at three different scales all centered on the same convective pixel (shaded in green). To compute the value of each of the 9 boxes in each Q matrix, all the cells that fall into that box must be averaged. In order to quantitatively compare the Q matrices of the current iterative radar image to the Q matrices of the pattern training radar image, the global Q matrices of each image are utilized. A global Q matrix for a single scale is simply the average of all the local Q matrices at a single scale. A local Q is centered on a single convective pixel, indicating that a local Q at each scale exists for every convective pixel. In a given radar image, the number of local Q matrices at a single scale is equal to the number of convective pixels in the radar image. In order 16

29 to calculate the value of a single box in the local Q matrix at a given scale, all the pixels within a box are averaged, as seen in Figures 2 and 3. This is repeated for the 8 other boxes in the local Q matrix at that scale. As a reminder, the pixels are assigned a value of one or zero for convective and non-convective pixels respectively. Also, the number of pixels within a given box of the Q matrix is dependent upon the scale, with more pixels included in a box at larger scales. The global Q of a radar image at a single scale is equivalent to the sum of the local Q matrices at that scale divided by the total number of convective pixels within that radar image. For example, all the upper left boxes in the local Q matrices at a given scale are added together and divided by the total number of convective pixels in the radar image to reach the value of the upper left box in the global Q matrix at that scale. Note that the global Q is calculated for the same number of scales as the local Qs and therefore only the appropriately scaled local Qs are included in the calculation of each scale of the global Q. Also note that each Q is calculated here assuming periodic boundary conditions, though other approaches are possible. Small Scale Q Intermediate Scale Q Large Scale Q /9 0/9 0/9 1/81 1/81 1/ /9 1/9 0/9 0/81 2/81 1/ /9 0/9 0/9 2/81 1/81 0/81 FIGURE 2: Accompanies Figure 3. Displays the 9 box values in each of the small, intermediate, and large scale local Q matrices surrounding the example green convective cell in Figure 3. 17

30 FIGURE 3: Accompanies Figure 2. The figure above depicts a 30 pixel by 30 pixel subsection of a radar image, where convective cells have a value of 1 and are highlighted in yellow and green. The green convective cell is the center of an example of three local Q matrices at three different scales. The colors of the Q matrices are the same colors as described in Table 1. Based on this example, the value of each of the 9 boxes in each of the three local Q matrices is calculated and displayed in Figure 2. Once the global Q matrices at all scales are calculated for each the pattern training radar image and the current iterative radar image, a means of quantitatively comparing the two sets of matrices exists. The goal of the algorithm is to transform the current iterative radar image so that each of its global Qs become more similar to that of the pattern training radar image. This way the current iterative radar image, which already contains the correct convective coverage percentage, will develop an idealized pattern as portrayed in the pattern training radar image. 18

31 3.4 The Swap Test The swap test is the iterative portion of the algorithm. It repeatedly swaps subsections of pixels in the current iterative radar image with other subsections within the same image. The goal of the swap test is to bring the randomly assigned convection in the current iterative radar image into a coherent pattern resembling that in the pattern training radar image. It does this by comparing the global Q matrices of the two radar images after every swap to assess if at a given scale the Q matrix of the current iterative radar image is more similar to the target Q of the pattern training radar image. This verification process is the swap test. Error is the means of quantitatively comparing the global Q matrices of the current iterative radar image and the pattern training radar image at a single scale. As in the case of the global Q matrices, error also exists at multiple scales. The equation for error at a single scale is: Equation (1),, Equation 1 indicates that error is always non-negative. The larger the error value, the more dissimilar the Q matrices and associated radar images are. An error value of zero indicates that the two Qs are identical at that scale. In theory, if the error value at each and every scale is zero, the current iterative radar image and the pattern training radar image have identical patterns, but the feature location is free to vary due to the periodic boundary conditions used to calculate the Q matrices. However, an error value of zero is unlikely to ever occur on the scale of radar images with tens of thousands of pixels. Next, in order to iteratively swap subsections of the current iterative radar image, specifications concerning the swap must be designated. Specifically, the location and size of the 19

32 area being swapped must be indicated. Two pixel locations within the current iterative radar map are randomly selected and identified as pixel 1 and pixel 2. The zero or one value of the pixels is irrelevant. A Q scale is also randomly selected from among small, intermediate, and large scales. The Q matrices have boxes containing a varying number of pixels with larger scaled boxes containing more pixels as seen in Table 1. The pixel size of a single box is important because it designates the size of the swap-able area surrounding the randomly chosen pixels. If specifically scaled Q matrix overlays are centered over each of the randomly selected pixel 1 and pixel 2 locations, all the pixels included in the center box of the Q matrix are included in the swap. More specifically, pixels contained in the center box of the Q matrix centered on pixel 1 are swapped with the pixels contained in the center box of the Q matrix centered on pixel 2. After a swap has been executed it is necessary to assess if the swap decreased the error measurement at the scale randomly designated for the swap. A decreased error value indicates that the change in the global Q matrix of the current iterative radar image was an adjustment that brought it closer to the global Q matrix of the pattern training radar image. Since this is the goal of the swap test, only those swaps that cause a decrease in the error value at the selected scale are retained. Swaps that do not meet the criteria are reversed and new random specifications are selected. For retained swaps, several variables are updated as well. The global Q matrix of the current iterative radar image must be calculated for the other scales that were not randomly chosen for comparison use in the swap test. Also, the associated errors for other scales must be calculated. The swap test is repeated for a set number of iterations as defined by the user. The number of iterations necessary depends upon the pixel grid size and convective coverage 20

33 percentage of the radar image being processed. An ideal number of iterations will illustrate an error value that asymptotes as the number of iterations increases. Of course squandering computer resources is undesirable and the number of iterations that can be used are limited. The number of iterations utilized in development is discussed in the results chapter. It is important to note that in order for a swap to be accepted and retained, only the error at a single scale, the randomly selected scale, is required to decrease. The errors at the other scales may increase or decrease. They are irrelevant to the acceptance of a swap. Although by randomly addressing all of the scales, it is hoped that all three errors are driven down by the end of the iterative process. It is known from conducted algorithm runs, that only requiring one of the scales of error to decrease, minimizes error more efficiently and avoids local minima of error, as opposed to requiring all three scales of error to decrease in order to retain a swap. 3.5 Statistical Analysis Methods The radar image generational algorithm, at this point, is not automated for the selection of the pattern training radar image and its associated spatial conditional probability maps. It is hypothesized, however, that certain atmospheric variables are likely indicative of the type of convective storm pattern. If the algorithm were to utilize forecasts of atmospheric variables well correlated with the convective storm pattern, it would allow for automated selection of the spatial conditional probability maps by the algorithm. A statistical analysis is performed to find atmospheric variables that are well correlated with specific convective pattern categories. The convective pattern categories are scattered convection, isolated convection, clustered convection, squall lines, and front segments. The atmospheric variables of interest are the K-index, Convective Available Potential Energy (CAPE), Convective Inhibition (CIN), the presence or 21

34 absence of a front in the vicinity, and wind shear from the surface to 3000 m, 3000 m to 6000 m, and the surface to 6000 m. These variables are indicators of storm moisture, energy, shear strength, and shear direction, which all influence a convective storm pattern. A total of 188 cases are included in the statistical analysis database. The first step of the analysis is to manually classify the convective pattern category of each of the 188 cases. Front segment: subsection of a larger front associated with a low pressure system, visible on a national or regional scale. Squall line: a pattern of short linear convection not adjacent to any apparent front. Isolated convection: one to a few convective cells in an area largely devoid of connective activity. Scattered convection: multiple cells of convection randomly distributed throughout a wide area in no apparent pattern. Clustered convection: multiple cells of convection in close proximity and lacking a linear structure to the convective pattern. Upon completion of the manual analysis, atmospheric variable data are recorded for each case as described in the data chapter. The K-index, CAPE, CIN, surface wind speed and direction, 3000 m wind speed and direction, and 6000 m wind speed and direction are collected from associated upper air soundings. The presence of a cold or stationary front in the vicinity of the convection of interest is recorded utilizing HPC surface analyses. Shear in the u and v wind components, wind shear magnitude, and wind shear direction are calculated at three levels: the surface to 3000 m, 3000 m to 6000 m, and the surface to 6000 m. 22

35 Two statistical approaches are utilized in an effort to find the atmospheric variable(s) that are needed to correctly classify the highest percentage of cases based on the five convective pattern category classifications. First, cluster analysis in Matlab is explored and then the regression analyses in Weka (Witten and Frank 2005) are explored. Also, instead of only calculating the percentage of correctly classified cases based on the five convective pattern category classifications, the convective pattern category classifications can be simplified. The convective pattern categories are divided into two groups of convective pattern types: linearly organized convective patterns and randomly organized convective patterns. Random convective patterns include isolated convection, scattered convection, and clustered convection. Linear convective patterns include front segments and squall lines. This simplistic approach is logical because the three classes of random convection are only differentiated by the fractional coverage of convection. Isolated convection covers the least area, scattered covers an intermediate area, and clustered convection covers the most area but all three are randomly organized. The linearly organized convection on the other hand is often indistinguishable on a 100 pixel by 100 pixel scale. In either case, if the algorithm were successful in selecting the random or linear type of pattern training radar image, it only requires the convective coverage percentage to narrow down which type of random classification the case falls into if the algorithm selects the random type. It doesn t need to distinguish squall lines and fronts at all for purposes of producing 100 pixel by 100 pixel radar images. So if the atmospheric variables do not correctly classify cases based on the five convective pattern categories, a simpler approach using the two convective pattern types can also be explored. The simpler approach has the possibility of achieving a higher percentage of correctly classified cases based on the two convective pattern types rather than the five 23

36 convective pattern categories. The higher the percentage of cases correctly classified, the more accurately the algorithm will be able to forecast the appropriate set of spatial conditional probability maps. This is crucial to creating a useful, autonomous algorithm. 24

37 CHAPTER 4: RESULTS The radar image generational algorithm successfully combines a convective coverage percentage forecast with a pattern training radar image to generate an ensemble of radar image realizations when both synthetic and real radar images are used to create the pattern training spatial conditional probability maps. The algorithm, however, displays both strengths and weaknesses in its ability to resolve certain convective pattern characteristics and successful use of the algorithm is stipulated by certain Q-scale size and swap test repetition requirements. Overall, the generated radar image realizations display the correct convective coverage percentage and a pattern similar to that displayed on the pattern training radar image. The statistical analysis necessary to automate the selection of the pattern training spatial conditional probability maps through correlating atmospheric variables with convective pattern categories, is less successful. Weka regression analysis became the preferred statistical analysis tool over Matlab cluster analysis but the percentage of correctly classified cases that the atmospheric variables achieve based on the five convective pattern categories is poor even in Weka. The simpler method of using two convective pattern types instead of five convective pattern categories is more successful but this is an area that will require future work. 4.1 Radar Image Generational Algorithm Stipulations In order to successfully utilize the radar image generational algorithm, stipulations concerning the minimum size of the largest Q scale and minimum number of swap test iterations must be met. The number of swap test iterations dictates the degree to which the Q matrices of the current iterative radar image can be altered to appear similar to the corresponding Q matrices 25

38 of the pattern training radar image. This is important because the goal of the algorithm is to make these two sets of Q matrices as similar as possible at each scale. The size of the largest Q scale dictates the largest scale at which the algorithm can see the convective pattern. Wider convective patterns will need a larger size for the largest Q scale in order for the algorithm to perceive the large scale width and correctly resolve the pattern. As discussed in the procedures chapter, the number of iterations dictates how many iterations of the swap test are performed. The swap test is the process of comparing the corresponding Q matrices of the pattern training radar image and the current iterative radar image before and after swapping a section of pixels in the current iterative radar image. If the error, the quantitative difference between two Q matrices at the same scale, decreases, then the swap is retained. The minimum number of iterations must depict an error that asymptotes as the number of iterations increases. This illustrates that the pairs of Q matrices at the same scale have become as similar as they reasonably can, which is the goal of the swap test. When using 100 pixel by 100 pixel radar images, as is done in algorithm development, the number of iterations required is between 1 million and 10 million, depending on the convective coverage percentage. A higher convective coverage percentage will require a larger number of iterations to successfully resolve the current iterative radar image. A benchmark of 3 million iterations is used during algorithm development, in order to develop a standard of comparison. The algorithm is able to successfully resolve the majority of the 41 test cases at this level. The second stipulation for successful use of the radar image generational algorithm, is that the largest sized Q scale must meet a minimum threshold, so the algorithm can see the largest convective pattern. As explained in the procedures chapter, during the swap test, a 26

39 random scale of Q is selected. This random Q scale determines the size in pixels of the swappable areas in the current iterative radar image. The size of the center box of the random Q scale is the size of the two areas swapped. In order for the algorithm to accurately resolve the convection, it must see the largest scale of the convection. The bandwidth 10 is the width of the narrowest axis of the convective pattern depicted in the pattern training radar image. The algorithm must be able to perceive the width of the bandwidth, in order to correctly depict the convective pattern as an unbroken, continuous pattern, if that is the case. So the size of the center box at the largest Q scale, must be slightly less than or equal to the size of the bandwidth. This can be seen in Table 2. Table 2 lists 4 possible scale sizes and the size of the center box of Q at each of those scales. For each scale, 4 different algorithm runs are conducted, where different simulated convective patterns are used for the pattern training radar image. The simulated convective patterns are differentiated by having bandwidths of sizes 1, 3, 7, and 25. For a given scale, the algorithm is best able to resolve the pattern that has a bandwidth most similar to the size of the center box at that scale. Thus, the size of the center box of the largest Q scale must be slightly less than or equal to the size of the pattern bandwidth depicted in the pattern training radar image. Q Size Requirements of Correctly Calibrated Algorithm Scale Size Number (Small to Large) Size of Center Box (no. pixels x no. pixels) 1x1 3x3 9x9 27x27 Bandwidth Size With Best Algorithm Results (Possible Bandwidths: 1, 3, 7, 25 ) 1 1 & TABLE 2: Bandwidth, the width of the narrowest axis of the convective pattern in the pattern training radar image, must be slightly less than or equal to the size of the center box of Q. 10 This is the width of the convective band, not to be confused with the definition used in the signal processing literature. 27

40 In algorithm development, the largest scale size is set to 3, the large scale described in Table 1. This benchmark is set to allow for a standard of comparison. A scale of 3 is chosen because it is the largest scale necessary out of the 41 cases the algorithm processed from 100 pixel by 100 pixel pattern training radar images. 4.2 Radar Image Generational Algorithm Strengths and Weaknesses The radar image generational algorithm displays both strengths and weaknesses in its ability to evolve the pattern in the current iterative radar image into a pattern similar to that in the pattern training radar image. This is a direct reflection of the effectiveness of using the spatial conditional probability maps as a means to quantitatively compare the convective pattern on the pattern training radar image to the convective pattern on the current iterative radar image. The algorithm strengths, however, outweigh the algorithm weaknesses. The algorithm strengths and weaknesses are best illustrated with the assistance of example cases. The graphics from example cases are followed by a discussion of the algorithm strengths and weaknesses. Five example algorithm processed cases are displayed as follows, one from each convective pattern category: front segment, squall line, isolated convection, scattered convection, and clustered convection. The front segment case is from September 23, 2006, at 19:45 Z. The squall line case is from August 13, 2006, at 21:45 Z. The isolated convection case is from June 19, 2006, at 19:45 Z. The scattered convection case is from August 7, 2008, at 23:45 Z. The clustered convection case is from August 23, 2006, at 10:45 Z. The image panel for each case is composed of: 28

41 1. Original national radar image: An original 6 km national composite radar image from NEXRAD radars, from which the case was identified for inclusion in algorithm processing. 2. Processed national radar image: A processed version of the original national radar image depicting reflectivities 40 dbz and greater as convective pixels with a value of one, nonconvective pixels with a value of zero. The color bar indicates pixel values while the x and y axes indicate the number of pixels West to East and North to South. 3. Pattern training radar image: A 100 pixel by 100 pixel subsection of the processed national radar image, containing the convective area of interest. Convective pixels have a value of one and non-convective pixels have a value of zero. The color bar indicates pixel values while the x and y axes indicate the number of pixels West to East and North to South. 4. Initial current iterative radar image: The initial version of the current iterative radar image containing the correct convective coverage percentage as a fractional area of randomly assigned convective pixels. Convective pixels have a value of one and non-convective pixels have a value of zero. The color bar indicates pixel values while the x and y axes indicate the number of pixels West to East and North to South. 5. Final current iterative radar image: The final processed version of the current iterative radar image containing the correct convective coverage percentage and a pattern based off of the pattern in the pattern training radar image. Convective pixels have a value of one and nonconvective pixels have a value of zero. The color bar indicates pixel values while the x and y axes indicate the number of pixels West to East and North to South. 6. Target small scale Q: The small scale Q of the pattern training radar image. The shading indicates the value of the conditional probability in the nine boxes of the Q matrix, where magenta indicates a value of one (convective) and cyan indicates a value of zero 29

42 (nonconvective). The x and y axes indicate the box numbers of the Q matrix West to East and North to South. 7. Initial small scale Q: The small scale Q of the initial version of the current iterative radar image. The shading indicates the value of the conditional probability in the nine boxes of the Q matrix, where magenta indicates a value of one (convective) and cyan indicates a value of zero (nonconvective). The x and y axes indicate the box numbers of the Q matrix West to East and North to South. 8. Final small scale Q: The small scale Q of the final version of the current iterative radar image. The shading indicates the value of the conditional probability in the nine boxes of the Q matrix, where magenta indicates a value of one (convective) and cyan indicates a value of zero (nonconvective). The x and y axes indicate the box numbers of the Q matrix West to East and North to South. 9. Small scale error: The small Q scale error value as a function of 3 million iterations where the number of iterations is on the x axis and the value of the error is depicted on the y axis. 10. Target intermediate scale Q: The intermediate scale Q of the pattern training radar image. The shading indicates the value of the conditional probability in the nine boxes of the Q matrix, where magenta indicates a value of one (convective) and cyan indicates a value of zero (nonconvective). The x and y axes indicate the box numbers of the Q matrix West to East and North to South. 11. Initial intermediate scale Q: The intermediate scale Q of the initial version of the current iterative radar image. The shading indicates the value of the conditional probability in the nine boxes of the Q matrix, where magenta indicates a value of one (convective) and cyan 30

43 indicates a value of zero (nonconvective). The x and y axes indicate the box numbers of the Q matrix West to East and North to South. 12. Final intermediate scale Q: The intermediate scale Q of the final version of the current iterative radar image. The shading indicates the value of the conditional probability in the nine boxes of the Q matrix, where magenta indicates a value of one (convective) and cyan indicates a value of zero (nonconvective). The x and y axes indicate the box numbers of the Q matrix West to East and North to South. 13. Intermediate scale error: The intermediate Q scale error value as a function of 3 million iterations where the number of iterations is on the x axis and the value of the error is depicted on the y axis. 14. Target large scale Q: The large scale Q of the pattern training radar image. The shading indicates the value of the conditional probability in the nine boxes of the Q matrix, where magenta indicates a value of one (convective) and cyan indicates a value of zero (nonconvective). The x and y axes indicate the box numbers of the Q matrix West to East and North to South. 15. Initial large scale Q: The large scale Q of the initial version of the current iterative radar image. The shading indicates the value of the conditional probability in the nine boxes of the Q matrix, where magenta indicates a value of one (convective) and cyan indicates a value of zero (nonconvective). The x and y axes indicate the box numbers of the Q matrix West to East and North to South. 16. Final large scale Q: The large scale Q of the final version of the current iterative radar image. The shading indicates the value of the conditional probability in the nine boxes of the Q matrix, where magenta indicates a value of one (convective) and cyan indicates a value of 31

44 zero (nonconvective). The x and y axes indicate the box numbers of the Q matrix West to East and North to South. 17. Large scale error: The large Q scale error value as a function of 3 million iterations where the number of iterations is on the x axis and the value of the error is depicted on the y axis. Note: The images are organized according to case. Each image is assigned a figure number, which consists of a case identification number followed by the image number from the descriptive list above (i.e. case number-image number). 32

45 Front Segment at 19:45Z Figure 4-1: Front - Original National Radar Image Figure 4-2: Front Processed National Radar Image Figure 4-3: Front Pattern Training Radar Image Figure 4-4: Front Initial Current Iterative Radar Image Figure 4-5: Front Final Current Iterative Radar Image 33

46 Figure 4-6: Front Target Small Scale Q Figure 4-10: Front Target Intermediate Scale Q Figure 4-14: Front Target Large Scale Q Figure 4-7: Front Initial Small Scale Q Figure 4-11: Front Initial Intermediate Scale Q Figure 4-15: Front Initial Large Scale Q Figure 4-8: Front Final Small Scale Q Figure 4-12: Front Final Intermediate Scale Q Figure 4-16: Front Final Large Scale Q Figure 4-9: Front Small Scale Error Figure 4-13: Front Intermediate Scale Error Figure 4-17: Front Large Scale Error 34

47 Squall Line at 21:45Z Figure 5-1: Squall Line - Original National Radar Image Figure 5-2: Squall Line Processed National Radar Image Figure 5-3: Squall Line Pattern Training Radar Image Figure 5-4: Squall Line Initial Current Iterative Radar Image Figure 5-5: Squall Line Final Current Iterative Radar Image 35

48 Figure 5-6: Squall Line Target Small Scale Q Figure 5-10: Squall Line Target Intermediate Scale Q Figure 5-14: Squall Line Target Large Scale Q Figure 5-7: Squall Line Initial Small Scale Q Figure 5-11: Squall Line Initial Intermediate Scale Q Figure 5-15: Squall Line Initial Large Scale Q Figure 5-8: Squall Line Final Small Scale Q Figure 5-12: Squall Line Final Intermediate Scale Q Figure 5-16: Squall Line Final Large Scale Q Figure 5-9: Squall Line Small Scale Error Figure 5-13: Squall Line Intermediate Scale Error Figure 5-17: Squall Line Large Scale Error 36

49 Isolated at 19:45Z Figure 6-1: Isolated - Original National Radar Image Figure 6-2: Isolated Processed National Radar Image Figure 6-3: Isolated Pattern Training Radar Image Figure 6-4: Isolated Initial Current Iterative Radar Image Figure 6-5: Isolated Final Current Iterative Radar Image 37

50 Figure 6-6: Isolated Target Small Scale Q Figure 6-10: Isolated Target Intermediate Scale Q Figure 6-14: Isolated Target Large Scale Q Figure 6-7: Isolated Initial Small Scale Q Figure 6-11: Isolated Initial Intermediate Scale Q Figure 6-15: Isolated Initial Large Scale Q Figure 6-8: Isolated Final Small Scale Q Figure 6-12: Isolated Final Intermediate Scale Q Figure 6-16: Isolated Final Large Scale Q Figure 6-9: Isolated Small Scale Error Figure 6-13: Isolated Intermediate Scale Error Figure 6-17: Isolated Large Scale Error 38

51 Scattered at 23:45Z Figure 7-1: Scattered - Original National Radar Image Figure 7-2: Scattered Processed National Radar Image Figure 7-3: Scattered Pattern Training Radar Image Figure 7-4: Scattered Initial Current Iterative Radar Image Figure 7-5: Scattered Final Current Iterative Radar Image 39

52 Figure 7-6: Scattered Target Small Scale Q Figure 7-10: Scattered Target Intermediate Scale Q Figure 7-14: Scattered Target Large Scale Q Figure 7-7: Scattered Initial Small Scale Q Figure 7-11: Scattered Initial Intermediate Scale Q Figure 7-15: Scattered Initial Large Scale Q Figure 7-8: Scattered Final Small Scale Q Figure 7-12: Scattered Final Intermediate Scale Q Figure 7-16: Scattered Final Large Scale Q Figure 7-9: Scattered Small Scale Error Figure 7-13: Scattered Intermediate Scale Error Figure 7-17: Scattered Large Scale Error 40

53 Cluster at 10:45Z Figure 8-1: Cluster - Original National Radar Image Figure 8-2: Cluster Processed National Radar Image Figure 8-3: Cluster Pattern Training Radar Image Figure 8-4: Cluster Initial Current Iterative Radar Image Figure 8-5: Cluster Final Current Iterative Radar Image 41

54 Figure 8-6: Cluster Target Small Scale Q Figure 8-10: Cluster Target Intermediate Scale Q Figure 8-14: Cluster Target Large Scale Q Figure 8-7: Cluster Initial Small Scale Q Figure 8-11: Cluster Initial Intermediate Scale Q Figure 8-15: Cluster Initial Large Scale Q Figure 8-8: Cluster Final Small Scale Q Figure 8-12: Cluster Final Intermediate Scale Q Figure 8-16: Cluster Final Large Scale Q Figure 8-9: Cluster Small Scale Error Figure 8-13: Cluster Intermediate Scale Error Figure 8-17: Cluster Large Scale Error 42