Investigation of HWIND Simulation Wind Speeds and a Methodology for Simulating Distributions of Extreme Winds in Hurricane Environments
by
Joseph B. Dannemiller
A Dissertation
In
Wind Science & Engineering
Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of
DOCTOR OF PHILOSOPHY
Approved
Douglas A. Smith Co-Chair of Committee
Stephen M. Morse Co-Chair of Committee
John L. Schroeder
Kishor C. Mehta
Mark Sheridan Dean of the Graduate School
May 2019
Copyright 2018, Joseph Dannemiller
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ACKNOWLEDGEMENTS My thanks to Dr. Douglas Smith and Dr. Stephen Morse for chairing my committee, and Drs. John Schroeder and Kishor Mehta for their help and assistance. Also, thank you to Dr. Audra Morse, Dr. Anna Young, and Dr. Rich Krupar for their guidance, friendship and support in matters pertaining to wind, research and life. I would also like to thank all of the people associated with the Wind Science & Engineering Research Center, and the National Wind Institute, for their help and assistance along the way. Having been an IGERT Fellow I would like to thank the National Science Foundation for the IGERT program for funding me during this process and allowing me the freedom to pursue this research. I would like to thank, most of all, my wife Sandra, my children Mark, Alexandra and James, my family including Mom, Dad, Katherine and Christopher, and all of my friends for supporting me because without their love and support I would never have found the fortitude to complete this journey. During this process I have met, interacted, and made friends with so many people that I am not mentioning here specifically, so to everyone that helped me, in any way, thank you.
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TABLE OF CONTENTS ACKNOWLEDGEMENTS ...... iii ABSTRACT ...... xiii LIST OF TABLES ...... xiv LIST OF FIGURES ...... xvi DEFINITION OF VARIABLES ...... xxxii I. INTRODUCTION ...... 1 II. BACKGROUND ...... 3 Major Hurricane Damage ...... 3 Loads and Resistances ...... 6 Wind Loads ...... 9 Observational Systems ...... 17 Texas Tech University Hurricane Research Team ...... 19 Wind Speed Standardization ...... 28 Gust Factors ...... 33 Density of Surface Observation Deployments ...... 51 HWIND Hurricane Wind Field Model ...... 52 HWIND in SPA ...... 58 III. HWIND AND TTUHRT DATA PROCESSING ...... 60 Data Selection ...... 60 HWIND Data Processing ...... 60 Texas Tech University Hurricane Research Team Data Processing Methods ...... 71 Processed Texas Tech University Hurricane Research Team Data ...... 85 IV. TTUHRT VERSUS HWIND COMPARISON ...... 95 Differences Between TTUHRT and HWIND as a Function of Distance From Storm Center ...... 102 Comparison of TTUHRT and HWIND Gust Factors ...... 109
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Comparison of TTUHRT and HWIND Mean Values ...... 113 Comparison of Recorded Peak Wind Speeds and Maximum Sustained Wind Speeds from HWIND Reconstruction ...... 115 V. ESTIMATING PARAMETERS FOR DISTRIBUTIONS OF EXTREME WIND SPEED IN 600s WINDOWS USING TTUHRT DATA ...... 118 Distribution of Extreme Winds Location and Scale Parameters ...... 118 Parent Wind Distribution Mean Parameters Versus Extreme Wind Distribution Location Parameters...... 122 Estimating Scale Parameters for Distributions of Extreme Winds ...... 130 Extreme Value Location Parameters Separated into Smoother than Open, Open and Rougher than Open Exposure Categories ...... 142 Extreme Value Scale Parameters Separated into Smooth, Open and Rough Exposure Categories ...... 167 Estimating Values of Extreme Wind Speed Distribution Location Parameters ...... 176 Estimating Values of Extreme Wind Speed Distribution of Scale Parameters ...... 180 Iterative Procedure to Generate Distributions of Extreme Winds ...... 186 VI. CASE STUDIES ...... 193
Case Study One – Smoother Than Open Exposure (z0<0.03m) ...... 193
Case Study Two –Open Exposure (0.3m≤z0≤0.07m) ...... 209 Case Study Three –Rougher Than Open Exposure (z0>0.7m) ...... 226 Case Study Results ...... 243 VII. FUTURE POSSIBILITIES FOR SIMULATING DISTRIBUTIONS OF EXTREME WIND SPEEDS ...... 245 VIII. CONCLUSIONS ...... 253 Future Work ...... 256 BIBLIOGRAPHY ...... 258
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APPENDICES ...... 263 A1 HWIND ASSIGNED SURFACE ROUGHNESS VALUES FOR 30° SECTORS AT ALL TTHURT DEPLOYMENTS ...... 263 A2 HWIND SIMULATION OUTPUT DATA FIGURES ...... 295 A3 SUMMARY STATISTICS RECORDED BY TTUHRT WEMITE #1 DURING THE 1998 LANDFALL OF HURRICANE BONNIE BROKEN INTO SEQUENTIAL 600s WINDOWS ...... 349 A4 TEXAS TECH UNIVERSITY HURRICANE RESEARCH TEAM DEPLOYMENT SUMMARY STATISTICS ...... 455 A5 HWIND SIMULATION OUTPUT DATA TABLES ...... 480 A6 TTUHRT VERSUS HWIND COMPARISON PLOTS ...... 514 A7 CONDITIONAL DISTRIBUTIONS OF EXTREME VALUE LOCATION PARAMETERS USING RAW DATA FOR ALL 600S WINDOWS ...... 539 A8 CONDITIONAL DISTRIBUTIONS OF EXTREME VALUE LOCATION PARAMETERS USING 3s MOVING AVERAGE DATA FOR ALL 600S WINDOWS ...... 550 A9 CONDITIONAL DISTRIBUTIONS OF EXTREME VALUE LOCATION PARAMETERS USING 60s MOVING AVERAGE DATA FOR ALL 600S WINDOWS ...... 561 A10 CONDITIONAL DISTRIBUTIONS OF EXTREME VALUE LOCATION PARAMETERS USING RAW DATA FOR 600S WINDOWS WITH A MEAN ABOVE 15m/s ...... 572 A11 CONDITIONAL DISTRIBUTIONS OF EXTREME VALUE LOCATION PARAMETERS USING 3s MOVING AVERAGE DATA FOR 600s WINDOWS WITH A MEAN ABOVE 15m/s ...... 583 A12 CONDITIONAL DISTRIBUTIONS OF EXTREME VALUE PARAMETERS USING 60s MOVING AVERAGE DATA FOR 600s WINDOWS WITH A MEAN ABOVE 15m/s ...... 594 A13 CONDITIONAL DISTRIBUTIONS OF EXTREME VALUE LOCATION PARAMETERS USING RAW DATA FOR ALL 600s WINDOWS WITH A SURFACE ROUGHNESS VALUE z0 LESS THAN 0.03m ...... 605
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A14 CONDITIONAL DISTRIBUTIONS OF EXTREME VALUE LOCATION PARAMETERS USING RAW DATA FOR ALL 600s WINDOWS WITH A SURFACE ROUGHNESS VALUE z0 BETWEEN 0.03m AND 0.07m ...... 616 A15 CONDITIONAL DISTRIBUTIONS OF EXTREME VALUE LOCATION PARAMETERS USING RAW DATA FOR ALL 600s WINDOWS WITH A SURFACE ROUGHNESS VALUE z0 GREATER THAN 0.07m ...... 624 A16 CONDITIONAL DISTRIBUTIONS OF EXTREME VALUE LOCATION PARAMETERS USING RAW DATA FOR ALL 600s WINDOWS SEPARATED INTO SMOOTH (z0<0.03M) OPEN (0.03M
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A23 CONDITIONAL DISTRIBUTIONS OF EXTREME VALUE PARAMETERS USING 3s MOVING AVERAGE DATA FOR 600s WINDOWS WITH A MEAN ABOVE 15m/s ...... 728 A24 CONDITIONAL DISTRIBUTIONS OF EXTREME VALUE PARAMETERS USING 60s MOVING AVERAGE DATA FOR 600s WINDOWS WITH A MEAN ABOVE 15m/s ...... 743 A25 CONDITIONAL DISTRIBUTIONS OF EXTREME VALUE SCALE PARAMETERS USING RAW DATA FOR ALL 600s WINDOWS WITH A SURFACE ROUGHNESS VALUE z0 LESS THAN 0.03m ...... 755 A26 CONDITIONAL DISTRIBUTIONS OF EXTREME VALUE SCALE PARAMETERS USING RAW DATA FOR ALL 600s WINDOWS WITH A SURFACE ROUGHNESS VALUE z0 BETWEEN 0.03m AND 0.07m ...... 770 A27 CONDITIONAL DISTRIBUTIONS OF EXTREME VALUE SCALE PARAMETERS USING RAW DATA FOR ALL 600s WINDOWS WITH A SURFACE ROUGHNESS VALUE z0 GREATER THAN 0.07m ...... 787 A28 CONDITIONAL DISTRIBUTIONS OF EXTREME VALUE PARAMETERS USING RAW DATA FOR ALL 600s WINDOWS SEPARATED INTO SMOOTH (z0<0.03M) OPEN (0.03M
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A32 SUMMARY STATISTICS RECORDED BY TTUHRT WEMITE #2 DURING THE 2002 LANDFALL OF HURRICANE LILI BROKEN INTO SEQUENTIAL 600s WINDOWS ...... 947 A33 SUMMARY STATISTICS RECORDED BY TTUHRT WEMITE #1 DURING THE 2003 LANDFALL OF HURRICANE ISABEL BROKEN INTO SEQUENTIAL 600s WINDOWS ...... 1047 A34 SUMMARY STATISTICS RECORDED BY TTUHRT WEMITE #2 DURING THE 2003 LANDFALL OF HURRICANE ISABEL BROKEN INTO SEQUENTIAL 600s WINDOWS ...... 1190 A35 SUMMARY STATISTICS RECORDED BY TTUHRT PMT BLACK DURING THE 2003 LANDFALL OF HURRICANE ISABEL BROKEN INTO SEQUENTIAL 600s WINDOWS ...... 1385 A36 SUMMARY STATISTICS RECORDED BY TTUHRT PMT CLEAR DURING THE 2003 LANDFALL OF HURRICANE ISABEL BROKEN INTO SEQUENTIAL 600s WINDOWS ...... 1519 A37 SUMMARY STATISTICS RECORDED BY TTUHRT PMT WHITE DURING THE 2003 LANDFALL OF HURRICANE ISABEL BROKEN INTO SEQUENTIAL 600s WINDOWS ...... 1660 A38 SUMMARY STATISTICS RECORDED BY TTUHRT WEMITE #1 DURING THE 2004 LANDFALL OF HURRICANE FRANCES BROKEN INTO SEQUENTIAL 600s WINDOWS ...... 1801 A39 SUMMARY STATISTICS RECORDED BY TTUHRT WEMITE #2 DURING THE 2004 LANDFALL OF HURRICANE FRANCES BROKEN INTO SEQUENTIAL 600s WINDOWS ...... 2090 A40 SUMMARY STATISTICS RECORDED BY TTUHRT PMT BLACK DURING THE 2004 LANDFALL OF HURRICANE FRANCES BROKEN INTO SEQUENTIAL 600s WINDOWS ...... 2567 A41 SUMMARY STATISTICS RECORDED BY TTUHRT WEMITE #1 DURING THE 2005 LANDFALL OF HURRICANE DENNIS BROKEN INTO SEQUENTIAL 600s WINDOWS ...... 2769 A42 SUMMARY STATISTICS RECORDED BY TTUHRT WEMITE #1 DURING THE 2005 LANDFALL OF HURRICANE IVAN BROKEN INTO SEQUENTIAL 600s WINDOWS ...... 2877
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A43 SUMMARY STATISTICS RECORDED BY TTUHRT WEMITE #1 DURING THE 2005 LANDFALL OF HURRICANE KATRINA BROKEN INTO SEQUENTIAL 600s WINDOWS ...... 2930 A44 SUMMARY STATISTICS RECORDED BY TTUHRT WEMITE #2 DURING THE 2005 LANDFALL OF HURRICANE KATRINA BROKEN INTO SEQUENTIAL 600s WINDOWS ...... 3003 A45 SUMMARY STATISTICS RECORDED BY TTUHRT PMT BLACK DURING THE 2005 LANDFALL OF HURRICANE KATRINA BROKEN INTO SEQUENTIAL 600s WINDOWS ...... 3190 A46 SUMMARY STATISTICS RECORDED BY TTUHRT PMT CLEAR DURING THE 2005 LANDFALL OF HURRICANE KATRINA BROKEN INTO SEQUENTIAL 600s WINDOWS ...... 3311 A47 SUMMARY STATISTICS RECORDED BY TTUHRT PMT WHITE DURING THE 2005 LANDFALL OF HURRICANE KATRINA BROKEN INTO SEQUENTIAL 600s WINDOWS ...... 3454 A48 SUMMARY STATISTICS RECORDED BY TTUHRT WEMITE #1 DURING THE 2005 LANDFALL OF HURRICANE RITA BROKEN INTO SEQUENTIAL 600s WINDOWS ...... 3711 A49 SUMMARY STATISTICS RECORDED BY TTUHRT WEMITE #2 DURING THE 2005 LANDFALL OF HURRICANE RITA BROKEN INTO SEQUENTIAL 600s WINDOWS ...... 3877 A50 SUMMARY STATISTICS RECORDED BY TTUHRT PMT BLACK DURING THE 2005 LANDFALL OF HURRICANE RITA BROKEN INTO SEQUENTIAL 600s WINDOWS ...... 4026 A51 SUMMARY STATISTICS RECORDED BY TTUHRT PMT CLEAR DURING THE 2005 LANDFALL OF HURRICANE RITA BROKEN INTO SEQUENTIAL 600s WINDOWS ...... 4046 A52 SUMMARY STATISTICS RECORDED BY TTUHRT PMT WHITE DURING THE 2005 LANDFALL OF HURRICANE RITA BROKEN INTO SEQUENTIAL 600s WINDOWS ...... 4260 A53 SUMMARY STATISTICS RECORDED BY TTUHRT STICKNET 101A DURING THE 2008 LANDFALL OF HURRICANE IKE BROKEN INTO SEQUENTIAL 600s WINDOWS ...... 4332
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A54 SUMMARY STATISTICS RECORDED BY TTUHRT STICKNET 102B DURING THE 2008 LANDFALL OF HURRICANE IKE BROKEN INTO SEQUENTIAL 600s WINDOWS ...... 4385 A55 SUMMARY STATISTICS RECORDED BY TTUHRT STICKNET 103A DURING THE 2008 LANDFALL OF HURRICANE IKE BROKEN INTO SEQUENTIAL 600s WINDOWS ...... 4386 A56 SUMMARY STATISTICS RECORDED BY TTUHRT STICKNET 104B DURING THE 2008 LANDFALL OF HURRICANE IKE BROKEN INTO SEQUENTIAL 600s WINDOWS ...... 4424 A57 SUMMARY STATISTICS RECORDED BY TTUHRT STICKNET 105A DURING THE 2008 LANDFALL OF HURRICANE IKE BROKEN INTO SEQUENTIAL 600s WINDOWS ...... 4522 A58 SUMMARY STATISTICS RECORDED BY TTUHRT STICKNET 106B DURING THE 2008 LANDFALL OF HURRICANE IKE BROKEN INTO SEQUENTIAL 600s WINDOWS ...... 4676 A59 SUMMARY STATISTICS RECORDED BY TTUHRT STICKNET 107A DURING THE 2008 LANDFALL OF HURRICANE IKE BROKEN INTO SEQUENTIAL 600s WINDOWS ...... 4758 A60 SUMMARY STATISTICS RECORDED BY TTUHRT STICKNET 108B DURING THE 2008 LANDFALL OF HURRICANE IKE BROKEN INTO SEQUENTIAL 600s WINDOWS ...... 4845 A61 SUMMARY STATISTICS RECORDED BY TTUHRT STICKNET 109A DURING THE 2008 LANDFALL OF HURRICANE IKE BROKEN INTO SEQUENTIAL 600s WINDOWS ...... 4929 A62 SUMMARY STATISTICS RECORDED BY TTUHRT STICKNET 110A DURING THE 2008 LANDFALL OF
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HURRICANE IKE BROKEN INTO SEQUENTIAL 600s WINDOWS ...... 5059
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ABSTRACT The wind speeds simulated by the National Oceanic and Atmospheric Administration’s HWIND hurricane simulation model are compared to data gathered by Texas Tech University’s Hurricane Research Team. The errors associated with the HWIND reported wind speeds are quantified for data from 32 deployments, using 15 platforms, during the landfall of 10 hurricanes, from 1998 to 2005. The relationships between parent wind distribution parameters and extreme wind field parameters is explored and a methodology for simulating extreme wind speed distributions in hurricane environments is advanced. Utilizing the proposed methodology, hurricane wind field model data can be used to generate distributions of extreme winds for direct use in assessing structural performance during hurricane events.
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LIST OF TABLES
2.1 Gust factor computed in (Durst, 1970) ...... 42 2.2 Gust factors computed in (Krayer and Marshal, 1992) ...... 43 2.3 Gust factors, Cg3600,2, computed in (Vickery and Skerlj, 2005) ...... 45 2.4 Gust factors, Cg600,60, computed using (Vickery and Skerlj, 2005) data ...... 47 2.5 Gust factors, Cg600,2, computed using (Vickery and Skerlj, 2005) data ...... 48 2.6 Percent errors between simulated and observed peak wind speeds as reported by (Vickery et al., 2000) ...... 58 3.1 HWIND reconstruction data at the location of the TTUHRT WEMITE #1 during the 2005 landfall of Hurricane Katrina ...... 70 3.2 Breakdown of 600s windows from TTUHRT time histories ...... 82 3.3 Data for assessing the surface roughness of the upwind wind field for the TTUHRT StickNet 101A deployment during the 2008 landfall of Hurricane Ike ...... 84 4.1 Magnitude and percent differences between TTUHRT and HWIND maximum sustained wind speeds broken down by year...... 108 5.1 Frequency Table for μj60s, 600s vs σj60s, 600s histogram ...... 137 5.2 Expected Frequency Table for μj60s, 600s vs σj60s, 600s histogram ...... 139 5.3 Chi-Squared values for μj60s, 600s vs σj60s, 600s histogram ...... 140 5.4 GEV parameters for conditional location parameter distributions computed from TTUHRT Raw Data ...... 177 5.5 GEV parameters for conditional location parameter distributions computed from TTUHRT 60s MA Data...... 178 5.6 GEV parameters for conditional location parameter distributions computed from TTUHRT 3s MA Data ...... 179 5.7 GEV parameters for conditional scale parameter distributions computed from TTUHRT Raw data ...... 181 5.8 GEV parameters for conditional scale parameter distributions computed from TTUHRT 3s Data ...... 183
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5.9 GEV parameters for conditional scale parameter distributions computed from TTUHRT 60s MA Data...... 185 5.10 Mean and standard deviation of differences between the location and scale parameters recorded by TTUHRT raw data and the means of simulated distributions using only 600s windows with smoother than open exposure ...... 188 5.11 Mean and standard deviation of differences between the location and scale parameters recorded by TTUHRT raw data and the means of simulated distributions using only 600s windows with open exposure ...... 188 5.12 Mean and standard deviation of differences between the location and scale parameters recorded by TTUHRT raw data and the means of simulated distributions using only 600s windows with rougher than open exposure ...... 188 6.1 Assigned surface roughness values, surface roughness values upwind and downwind of the last roughness change, and distances to the last change in surface roughness at the TTUHRT WEMITE #1 deployment location during the 1998 landfall of Hurricane Bonnie in each 30 sector ...... 195 6.2 Assigned surface roughness values, surface roughness values upwind and downwind of the last roughness change, and distances to the last change in surface roughness at the TTUHRT PMT Black deployment location during the 2004 landfall of Hurricane Francis in each 30 sector ...... 211 6.3 Assigned surface roughness values, surface roughness values upwind and downwind of the last roughness change, and distances to the last change in surface roughness at the TTUHRT WEMITE #2 deployment location during the 2003 landfall of Hurricane Isabel in each 30 sector ...... 228
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LIST OF FIGURES 2.1 Bolivar Peninsula before Hurricane Ike Landfall, September 9, 2008 (FEMA 2009) ...... 5 2.2 Bolivar Peninsula after Hurricane Ike Landfall, September 15, 2008 (FEMA 2009) ...... 5 2.3 Normal distribution probability density function (PDF) ...... 15 2.4 Relationship between the mean and mode for positively skewed distributions ...... 16 2.5 Comparison of normal and skewed distributions ...... 17 2.6 (a) Automated Surface Observation System (ASOS) located in Austin, TX (www.ncdc.noaa.gov, 2017), (b) Weather buoy stationed in the Gulf of Mexico (www.ndbc.noaa.gov, 2017), (c) Geostationary Operational Environmental Satellite R Series (GOES-R) (www.lockheedmartin.com, 2017), (d) National Oceanic and Atmospheric Administration (NOAA) WP-3D Orion weather research and hurricane intercept aircraft (www.esrl.noaa.gov, 2017), (e) National Center for Atmospheric Research (NCAR) dropsonde device in flight (www2.ucar.edu, 2017), (f) ship mounted weather stations ...... 18 2.7 WEMITE #1 near Vacherie, LA prior to the landfall of Hurricane Katrina, 2005 (TTUHRT, 2006) ...... 20 2.8 WEMITE #2 at Stennis International Airport following the landfall of Hurricane Katrina, 2005 (TTUHRT, 2006) ...... 21 2.9 PMT Clear at Slidell Municipal Airport prior to the landfall of Hurricane Katrina, 2005 (TTUHRT, 2006) ...... 22 2.10 StickNet 101, A probe, deployed prior to the landfall of Hurricane Sandy ...... 23 2.11 StickNet 102, B Probe, deployed for testing at Reese Technology Center, Lubbock, TX ...... 24 2.12 Wind speed time history, recorded at 10m by WEMITE #1 during Hurricane Bonnie (1998) ...... 26 2.13 Wind direction time history, recorded at 10m by WEMITE #1 during Hurricane Bonnie (1998) ...... 27 2.14 Hurricane Bonnie, WEMITE #1 deployment site ...... 29 2.15 Comparison of boundary layer velocities over rough and smooth terrain ...... 30
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2.16 The boundary layer transition past a change in roughness, rough to smooth ...... 31 2.17 Probability of exceedance relationship in relation to parent wind speed parameters ...... 36 2.18 Comparison of gust factor curves from ESDU (1983), Durst (1960) and Krayer and Marshal (1992), reproduced from (Vickery and Skerlj, 2005)...... 44 2.19 Cg600,2 gust factors computed using TTUHRT platform data where z0 is between 0.03m and 0.07m, reproduced from (Paulsen and Schroeder, 2005) ...... 48 2.20 Cg600,3 gust factors computed using TTUHRT and FCMP platform data, reproduced from (Giammanco et.al., 2012) ...... 49 2.21 Locations of TTUHRT, Florid Coastal Monitoring Project and Louisiana Monroe atmospheric measurement platforms during the 2006 landfall of Hurricane Katrina (Giammanco et al, 2006) ...... 51 2.22 Locations of Automated Surface Observation Stations (ASOS) and Automated Weather Observation Stations (AWOS) ...... 52 2.23 Observations aggregated into the HWIND wind field simulation of Hurricane Katrina at 6:00 UTC on August 29, 2005, NOAA ...... 54 2.24 NOAA HRD created figure showing the HWIND reconstruction for Hurricane Katrina at 06:00 UTC on August 29, 2005 ...... 56 3.1 NOAA HRD created figure showing the HWIND reconstruction for Hurricane Katrina at 00:00 UTC on August 29, 2005 ...... 62 3.2 Grid points employed in HWIND reconstruction ...... 63 3.3 First enlarged view of the grid points employed in HWIND reconstruction...... 64 3.4 Second enlarged view of the grid points employed in HWIND simulation ...... 64 3.5 Wind speeds in m/s from the NOAA HRD HWIND reconstruction of Hurricane Katrina on 00:00UTC August 29, 2005 ...... 65 3.6 Storm relative quadrants in a hurricane ...... 67
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3.7 Wind speeds in m/s from the NOAA HRD HWIND reconstruction of Hurricane Katrina on 12:00UTC August 26, 2005 ...... 68 3.8 Wind speed time history for the raw data (blue) and the 60s MA data (red) recorded from 09:15UTC to 09:25UTC by TTUHRT WEMITE #1 during the 1998 landfall of hurricane Bonnie ...... 73 3.9 Peak wind speed values for 60-sec segments ...... 76 3.10 Aerial image of the deployment site for TTUHRT StickNet 101A captured January 2008 (© Google) ...... 83 3.11 Parent and extreme wind distribution parameters for the raw data recorded by TTUHRT WEMITE #1 during the 2005 landfall of Hurricane Katrina ...... 86 3.12 Parent and extreme wind distribution parameters for the 3s MA data recorded by TTUHRT WEMITE #1 during the 2005 landfall of Hurricane Katrina ...... 88 3.13 Parent and extreme wind distribution parameters for the 60s MA data recorded by TTUHRT WEMITE #1 during the 2005 landfall of Hurricane Katrina ...... 89 3.14 Parent and extreme wind distribution parameters for the raw data recorded by TTUHRT PMT Black during the 2005 landfall of Hurricane Katrina ...... 91 3.15 Parent and extreme wind distribution parameters for the 3s MA data recorded by TTUHRT PMT Black during the 2005 landfall of Hurricane Katrina ...... 92 3.16 Parent and extreme wind distribution parameters for the 60s MA data recorded by TTUHRT PMT Black during the 2005 landfall of Hurricane Katrina ...... 93 4.1 TTUHRT and HWIND maximum sustained wind speeds, differences between TTUHRT and HWIND maximum sustained wind speeds, and the distances from platform to center of the storm, for the deployment of TTUHRT PMT Clear during the 2005 landfall of Hurricane Katrina ...... 97 4.2 TTUHRT and HWIND maximum sustained wind speeds, differences between TTUHRT and HWIND mean wind speeds, and the distances from platform to center of the storm, for the deployment of TTUHRT WEMITE #1 during the 1998 landfall of Hurricane Bonnie ...... 99
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4.3 TTUHRT and HWIND maximum sustained wind speeds, differences between TTUHRT and HWIND mean wind speeds, and the distances from platform to center of the storm, for the deployment of TTUHRT WEMITE #2 during the 2003 landfall of Hurricane Isabel ...... 100 4.4 TTUHRT and HWIND maximum sustained wind speeds, differences between TTUHRT and HWIND mean wind speeds, and the distances from platform to center of the storm, for the deployment of TTUHRT StickNet 108B during the 2008 landfall of Hurricane Ike ...... 101 4.5 (a) Differences (m/s) between observed TTUHRT xj,k t,Tw and HWIND U 60s, 600s data versus the radial distance from storm center to the location of the TTUHRT platform Rp, (b) %-differences between observed TTUHRT xj, k t, Tw xjraw, 600s and HWIND U 60s, 600s U 600s data versus the radial distance from storm center to the location of the TTUHRT platform, Rp ...... 103 4.6 (a) Distribution of differences between TTUHRT xj, k t, Tw and HWIND U 60s, 600s data, (b) Distribution of %- differences between TTUHRT xj,k t,Tw and HWIND U 60s, 600s data ...... 105 4.7 (a) Differences between TTUHRT xj,k t,Tw and HWIND U 60s, 600s data with respect to storm year, (b) Percent differences between TTUHRT xj,k t,Tw and HWIND U 60s, 600s data with respect to storm year...... 107 4.8 Comparison between TTUHRT gust factors (blue histogram) and the single gust factor employed by HWIND (red) using only TTUHRT gust factors with smoother than open exposure (z0<0.03m) upwind during the recording of 600s time histories ...... 110 4.9 Comparison between TTUHRT (blue histogram) gust factors and the single gust factor employed by HWIND (red line) using only TTUHRT gust factors with open exposure (0.03m≤z0≤0.07m) upwind during the recording of 600s time histories ...... 111 4.10 Comparison between TTUHRT (blue histogram) gust factors and the single gust factor employed by HWIND (red line) using only TTUHRT gust factors with rougher than open exposure (z0>0.07m) upwind during the recording of 600s time histories ...... 112
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4.11 (a) Distribution of differences between TTUHRT xjraw, 600s and HWIND U 600s data, (b) Distribution of %-differences between TTUHRT xjraw, 600s and HWIND U 600s data ...... 114 4.12 (a) Distribution of differences between TTUHRT xj60s, 600s and HWIND U 60s, 600s , (b) Distribution of %-differences between TTUHRT xj60s, 600s and HWIND U 60s, 600s data ...... 116 5.1 Distribution of EV location parameters from 60s MA windows ...... 119 5.2 Distribution of EV location parameters from 3s MA windows ...... 120 5.3 Distribution of EV scale parameters from 60s MA windows ...... 121 5.4 Distribution of EV scale parameters from 3s MA windows ...... 122 5.5 Linear relationship between mean and location parameters from TTUHRT 60s MA data ...... 123 5.6 Magnitudes of differences between the TTUHRT μj60s, 600s and estimates using Equation 5.2 as a function of xj60s, 600s ...... 124 5.7 Percent of differences between the TTUHRT μj60s, 600s and estimates using Equation 5.2 as a function of xj60s, 600s ...... 125 5.8 Linear relationship to estimate μj60s, 600s parameters ...... 126 5.9 Magnitudes of differences between the TTUHRT μj60s, 600s and estimates using Equation 5.4 as a function of xjraw, 600s ...... 127 5.10 Percent of differences between the TTUHRT μj60s, 600s and estimates using Equation 5.4 as a function of xjraw, 600s ...... 127 5.11 Linear relationship to estimate μj3s, 600s parameters ...... 128 5.12 Magnitudes of differences between the TTUHRT μj60s, 600s and estimates using Equation 5.4 as a function of xjraw, 600s ...... 129 5.13 Percent of differences between the TTUHRT μj3s, 600s and estimates using Equation 5.5 as a function of xjraw, 600s ...... 129 5.14 Relationship between mean and scale parameters from 60s MA windows ...... 131
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5.15 Relationship between mean and scale parameters from 3s MA windows ...... 131 5.16 Relationship between EV location and scale parameters from 60s MA windows ...... 132 5.17 Relationship between EV location and scale parameters from ...... 132 5.18 Two parameter histogram plotting the correlated frequencies of occurrence for μj60s, 600s and σj60s, 600s values, view 1 ...... 134 5.19 Two parameter histogram plotting the correlated frequencies of occurrence for μj60s, 600s and σj60s, 600s values, view 2 ...... 134 5.20 Two parameter histogram plotting the correlated frequencies of occurrence for μj60s, 600s and σj60s, 600s values, view 3 ...... 135 5.21 Two parameter histogram plotting the correlated frequencies of occurrence for μj60s, 600s and σj60s, 600s values, view 4 ...... 135 5.22 Two parameter histogram plotting the correlated frequencies of occurrence for μj60s, 600s and σj60s, 600s values, heat map...... 136 5.23 Relationship between parent mean and EV location using TTUHRT 60s MA data ...... 143 5.24 Magnitudes of differences between the TTUHRT μj60s, 600s and estimates using 5.8 for smoother than open exposure data as a function of xj60s, 600s...... 144 5.25 Percent differences between the TTUHRT μj60s, 600s and estimates using 5.8 for smoother than open exposure data as a function of xj60s, 600s ...... 145 5.26 Magnitudes of differences between the TTUHRT μj60s, 600s and estimates using 5.8 for open exposure data as a function of xj60s, 600s ...... 146 5.27 Percent differences between the TTUHRT μj60s, 600s and estimates using 5.8 for open exposure data as a function of xj60s, 600s ...... 147 5.28 Magnitudes of differences between the TTUHRT μj60s, 600s and estimates using 5.8 for rougher than open exposure data as a function of xj60s, 600s...... 148
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5.29 Percent differences between the TTUHRT μj60s, 600s and estimates using 5.8 for rougher than open exposure data as a function of xj60s, 600s ...... 149 5.30 Relationship between parent mean and EV location using TTUHRT 3s MA data ...... 150 5.31 Magnitudes of differences between the TTUHRT μj3s, 600s and estimates using Equation 5.9 for smoother than open exposure data as a function of xj3s, 600s ...... 151 5.32 Percent differences between the TTUHRT μj3s, 600s and estimates using Equation 5.9 for smoother than open exposure data as a function of xj3s, 600s ...... 152 5.33 Magnitudes of differences between the TTUHRT μj3s, 600s and estimates using Equation 5.9 for open exposure data as a function of xj3s, 600s ...... 153 5.34 Percent differences between the TTUHRT μj3s, 600s and estimates using Equation 5.9 for open exposure data as a function of xj3s, 600s ...... 154 5.35 Magnitudes of differences between the TTUHRT μj3s, 600s and estimates using Equation 5.9 for rougher open exposure data as a function of xj3s, 600s ...... 155 5.36 Percent differences between the TTUHRT μj3s, 600s and estimates using Equation 5.9 for rougher than open exposure data as a function of xj3s, 600s ...... 156 5.37 Relationship between parent mean and EV location using TTUHRT raw data ...... 157 5.38 Magnitudes of differences between the TTUHRT μjraw, 600s and estimates using Equation 5.10 for smoother than open exposure data as a function of xjraw, 600s ...... 158 5.39 Percent differences between the TTUHRT μjraw, 600s and estimates using Equation 5.10 for smoother than open exposure data as a function of xjraw, 600s ...... 159 5.40 Magnitudes of differences between the TTUHRT μjraw, 600s and estimates using Equation 5.10 for open exposure data as a function of xjraw, 600s ...... 160 5.41 Percent differences between the TTUHRT μjraw, 600s and estimates using Equation 5.10 for open exposure data as a function of xjraw, 600s ...... 161
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5.42 Magnitudes of differences between the TTUHRT μjraw, 600s and estimates using Equation 5.10 for rougher open exposure data as a function of xjraw, 600s ...... 162 5.43 Percent differences between the TTUHRT μjraw, 600s and estimates using Equation 5.10 for rougher than open exposure data as a function of xjraw, 600s ...... 163 5.44 Two parameter histogram plotting the correlated frequencies of occurrence for xjraw, 600s and μjraw, 600s values for multiple roughness regimes, view 1 ...... 164 5.45 Two parameter histogram plotting the correlated frequencies of occurrence for xjraw, 600s and μjraw, 600s values for multiple roughness regimes, view 2 ...... 165 5.46 Two parameter histogram plotting the correlated frequencies of occurrence for xjraw, 600s and μjraw, 600s values for multiple roughness regimes, view 3 ...... 165 5.47 Two parameter histogram plotting the correlated frequencies of occurrence for xjraw, 600s and μjraw, 600s values for multiple roughness regimes, view 4 ...... 166 5.48 Two parameter histogram plotting the correlated frequencies of occurrence for xjraw, 600s and μjraw, 600s values, heat map...... 166 5.49 Relationship between EV location and scale parameters using TTUHRT 60s MA data ...... 168 5.50 Relationship between EV location and scale parameters using TTUHRT 3s MA data ...... 169 5.51 Relationship between EV location and scale parameters using TTUHRT raw data ...... 169 5.52 Two parameter histogram plotting the correlated frequencies of occurrence for μjraw, 600s and σjraw, 600s values for multiple roughness regimes, view 1 ...... 170 5.53 Two parameter histogram plotting the correlated frequencies of occurrence for μjraw, 600s and σjraw, 600s values for multiple roughness regimes, view 2 ...... 171 5.54 Two parameter histogram plotting the correlated frequencies of occurrence for μjraw, 600s and σjraw, 600s values for multiple roughness regimes, view 3 ...... 171
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5.55 Two parameter histogram plotting the correlated frequencies of occurrence for μjraw, 600s and σjraw, 600s values for multiple roughness regimes, view 4 ...... 172 5.56 Two parameter histogram plotting the correlated frequencies of occurrence for μjraw, 600s and σjraw, 600s values, heat map...... 172 5.57 Histograms for smooth, open and rough exposure σjraw, 600s values ...... 173 5.58 Coefficients of variation for the GEV distributions of σjraw, 600s values in all μjraw, 600s bins ...... 174 5.59 Differences in magnitude between the peak 3s wind speed recorded by the TTUHRT platforms and peak 3s wind speeds computed using the mean observed by the TTUHRT platforms and a gust factor equal to 1.44 ...... 189 5.60 Differences in percent between the peak 3s wind speed recorded by the TTUHRT platforms and peak 3s wind speeds computed using the mean observed by the TTUHRT platforms and a gust factor equal to 1.44 ...... 190 5.61 Differences in magnitude between the peak 60s wind speed recorded by the TTUHRT platforms and peak 60s wind speeds computed using the mean observed by the TTUHRT platforms and a gust factor equal to 1.18 ...... 191 5.62 Differences in percent between the peak 60s wind speed recorded by the TTUHRT platforms and peak 60s wind speeds computed using the mean observed by the TTUHRT platforms and a gust factor equal to 1.18 ...... 191 6.1 Deployment location for the TTUHRT WEMITE #1 platform during the 1998 landfall of Hurricane Bonnie ...... 194 6.2 Wind speed time history recorded by the TTUHRT WEMITE #1 platform during the 1998 landfall of Hurricane Bonnie ...... 195 6.3 Simulated distribution of location parameters for the smooth roughness case study using TTUHRT Hurricane Bonnie raw data for a single 600s window ...... 197 6.4 Simulated distribution of scale parameters for the smooth roughness case study using TTUHRT Hurricane Bonnie raw data for a single 600s window ...... 197 6.5 Differences in magnitude between the location parameter recorded by TTUHRT WEMITE #1 and the values in the
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simulated distribution of location parameters for the smooth roughness case study using TTUHRT Hurricane Bonnie raw data for a single 600s window ...... 198 6.6 Differences in percent between the location parameter recorded by TTUHRT WEMITE #1 and the values in the simulated distribution of location parameters for the smooth roughness case study using TTUHRT Hurricane Bonnie raw data for a single 600s window ...... 199 6.7 Differences in magnitude between the scale parameter recorded by TTUHRT WEMITE #1 and the values in the simulated distribution of scale parameters for the smooth roughness case study using TTUHRT Hurricane Bonnie raw data for a single 600s window ...... 200 6.8 Differences in magnitude between the scale parameter recorded by TTUHRT WEMITE #1 and the values in the simulated distribution of scale parameters for the smooth roughness case study using TTUHRT Hurricane Bonnie raw data for a single 600s window ...... 201 6.9 Distribution of mean differences in magnitude between the location parameter recorded by TTUHRT WEMITE #1 and the values in the simulated distributions of location parameters for the smooth roughness case study using TTUHRT Hurricane Bonnie raw data for a single 600s window...... 202 6.10 Distribution of mean differences in percent between the location parameter recorded by TTUHRT WEMITE #1 and the values in the simulated distributions of location parameters for the smooth roughness case study using TTUHRT Hurricane Bonnie raw data for a single 600s window...... 203 6.11 Distribution of mean differences in magnitude between the scale parameter recorded by TTUHRT WEMITE #1 and the values in the simulated distributions of scale parameters for the smooth roughness case study using TTUHRT Hurricane Bonnie raw data for a single 600s window ...... 204 6.12 Distribution of mean differences in percent between the scale parameter recorded by TTUHRT WEMITE #1 and the values in the simulated distributions of scale parameters for the smooth roughness case study using TTUHRT Hurricane Bonnie raw data for a single 600s window ...... 205
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6.13 Differences in magnitude between the peak 3s wind speed recorded by the TTUHRT WEMITE #1 during the 1998 landfall of Hurricane Bonnie and peak 3s wind speeds computed using the mean observed by the TTUHRT platform and a gust factor equal to 1.44 using all 600s windows in the Hurricane Bonnie time history ...... 207 6.14 Differences in percent between the peak 3s wind speed recorded by the TTUHRT WEMITE #1 during the 1998 landfall of Hurricane Bonnie and peak 3s wind speeds computed using the mean observed by the TTUHRT platform and a gust factor equal to 1.44 using all 600s windows in the Hurricane Bonnie time history ...... 207 6.15 Differences in magnitude between the peak 60s wind speed recorded by the TTUHRT WEMITE #1 during the 1998 landfall of Hurricane Bonnie and peak 60s wind speeds computed using the mean observed by the TTUHRT platform and a gust factor equal to 1.18 using all 600s windows in the Hurricane Bonnie time history ...... 208 6.16 Differences in percent between the peak 60s wind speed recorded by the TTUHRT WEMITE #1 during the 1998 landfall of Hurricane Bonnie and peak 60s wind speeds computed using the mean observed by the TTUHRT platform and a gust factor equal to 1.18 using all 600s windows in the Hurricane Bonnie time history ...... 209 6.17 Deployment location for the TTUHRT PMT Black platform during the 2004 landfall of Hurricane Francis ...... 210 6.18 Simulated distribution of location parameters for the smooth roughness case study using TTUHRT Hurricane Francis raw data for a single 600s window ...... 212 6.19 Simulated distribution of scale parameters for the smooth roughness case study using TTUHRT Hurricane Francis raw data for a single 600s window ...... 213 6.20 Differences in magnitude between the location parameter recorded by TTUHRT PMT Black and the values in the simulated distribution of location parameters for the smooth roughness case study using TTUHRT Hurricane Francis raw data for a single 600s window ...... 214 6.21 Differences in percent between the location parameter recorded by TTUHRT PMT Black and the values in the simulated distribution of location parameters for the smooth
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roughness case study using TTUHRT Hurricane Francis raw data for a single 600s window ...... 215 6.22 Differences in magnitude between the scale parameter recorded by TTUHRT PMT Black and the values in the simulated distribution of scale parameters for the smooth roughness case study using TTUHRT Hurricane Francis raw data for a single 600s window ...... 216 6.23 Differences in magnitude between the scale parameter recorded by TTUHRT PMT Black and the values in the simulated distribution of scale parameters for the smooth roughness case study using TTUHRT Hurricane Francis raw data for a single 600s window ...... 217 6.24 Distribution of mean differences in magnitude between the location parameter recorded by TTUHRT PMT Black and the values in the simulated distributions of location parameters for the smooth roughness case study using TTUHRT Hurricane Francis raw data for a single 600s window ...... 218 6.25 Distribution of mean differences in percent between the location parameter recorded by TTUHRT PMT Black and the values in the simulated distributions of location parameters for the smooth roughness case study using TTUHRT Hurricane Francis raw data for a single 600s window ...... 219 6.26 Distribution of mean differences in magnitude between the scale parameter recorded by TTUHRT PMT Black and the values in the simulated distributions of scale parameters for the smooth roughness case study using TTUHRT Hurricane Francis raw data for a single 600s window ...... 220 6.27 Distribution of mean differences in percent between the scale parameter recorded by TTUHRT PMT Black and the values in the simulated distributions of scale parameters for the smooth roughness case study using TTUHRT Hurricane Francis raw data for a single 600s window ...... 221 6.28 Differences in magnitude between the peak 3s wind speed recorded by the TTUHRT PMT Black during the 2004 landfall of Hurricane Francis and peak 3s wind speeds computed using the mean observed by the TTUHRT platform and a gust factor equal to 1.44 using all 600s windows in the Hurricane Francis time history ...... 223 6.29 Differences in percent between the peak 3s wind speed recorded by the TTUHRT PMT Black during the 2004
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landfall of Hurricane Francis and peak 3s wind speeds computed using the mean observed by the TTUHRT platform and a gust factor equal to 1.44 using all 600s windows in the Hurricane Francis time history ...... 223 6.30 Differences in magnitude between the peak 60s wind speed recorded by the TTUHRT PMT Black during the 2004 landfall of Hurricane Francis and peak 60s wind speeds computed using the mean observed by the TTUHRT platform and a gust factor equal to 1.18 using all 600s windows in the Hurricane Francis time history ...... 224 6.31 Differences in percent between the peak 60s wind speed recorded by the TTUHRT PMT Black during the 2004 landfall of Hurricane Francis and peak 60s wind speeds computed using the mean observed by the TTUHRT platform and a gust factor equal to 1.18 using all 600s windows in the Hurricane Francis time history ...... 225 6.32 Deployment location for the TTUHRT WEMITE #2 platform during the 2003 landfall of Hurricane Isabel ...... 227 6.33 Simulated distribution of location parameters for the smooth roughness case study using TTUHRT Hurricane Isabel raw data ...... 229 6.34 Simulated distribution of scale parameters for the smooth roughness case study using TTUHRT Hurricane Isabel raw data ...... 230 6.35 Differences in magnitude between the location parameter recorded by TTUHRT WEMITE #2 and the values in the simulated distribution of location parameters for the smooth roughness case study using TTUHRT Hurricane Isabel raw data ...... 231 6.36 Differences in percent between the location parameter recorded by TTUHRT WEMITE #2 and the values in the simulated distribution of location parameters for the smooth roughness case study using TTUHRT Hurricane Isabel raw data ...... 232 6.37 Differences in magnitude between the scale parameter recorded by TTUHRT WEMITE #2 and the values in the simulated distribution of scale parameters for the smooth roughness case study using TTUHRT Hurricane Isabel raw data ...... 233
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6.38 Differences in percent between the scale parameter recorded by TTUHRT WEMITE #2 and the values in the simulated distribution of scale parameters for the smooth roughness case study using TTUHRT Hurricane Isabel raw data ...... 234 6.39 Distribution of mean differences in magnitude between the location parameter recorded by TTUHRT WEMITE #2 and the values in the simulated distributions of location parameters for the smooth roughness case study using TTUHRT Hurricane Isabel raw data ...... 235 6.40 Distribution of mean differences in percent between the location parameter recorded by TTUHRT WEMITE #2 and the values in the simulated distributions of location parameters for the smooth roughness case study using TTUHRT Hurricane Isabel raw data ...... 236 6.41 Distribution of mean differences in magnitude between the scale parameter recorded by TTUHRT WEMITE #2 and the values in the simulated distributions of scale parameters for the smooth roughness case study using TTUHRT Hurricane Isabel raw data ...... 237 6.42 Distribution of mean differences in percent between the scale parameter recorded by TTUHRT WEMITE #2 and the values in the simulated distributions of scale parameters for the smooth roughness case study using TTUHRT Hurricane Isabel raw data ...... 238 6.43 Differences in magnitude between the peak 3s wind speed recorded by the TTUHRT WEMITE #2 during the 2003 landfall of Hurricane Isabel and peak 3s wind speeds computed using the mean observed by the TTUHRT platform and a gust factor equal to 1.44 using all 600s windows in the Hurricane Bonnie time history ...... 240 6.44 Differences in percent between the peak 3s wind speed recorded by the TTUHRT WEMITE #2 during the 2003 landfall of Hurricane Isabel and peak 3s wind speeds computed using the mean observed by the TTUHRT platform and a gust factor equal to 1.44 using all 600s windows in the Hurricane Bonnie time history ...... 241 6.45 Differences in magnitude between the peak 60s wind speed recorded by the TTUHRT WEMITE #2 during the 2003 landfall of Hurricane Isabel and peak 60s wind speeds computed using the mean observed by the TTUHRT platform
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and a gust factor equal to 1.18 using all 600s windows in the Hurricane Bonnie time history ...... 242 6.46 Differences in percent between the peak 60s wind speed recorded by the TTUHRT WEMITE #2 during the 2003 landfall of Hurricane Isabel and peak 60s wind speeds computed using the mean observed by the TTUHRT platform and a gust factor equal to 1.18 using all 600s windows in the Hurricane Bonnie time history ...... 243 7.1 Relationship between parent mean and turbulence intensity using TTUHRT data from all 600s windows...... 245 7.2 Relationship between parent mean and EV TI using TTUHRT data from all 600s windows, broken into roughness regimes (smooth, open and rough) ...... 246 7.3 Relationship between parent mean and EV TI using only TTUHRT data from 600s windows with a mean wind speed greater than 15m/s, broken into roughness regimes (smooth, open and rough) ...... 247 7.4 Relationship between parent mean and EV COV using TTUHRT data from all 600s windows, broken into roughness regimes (smooth, open and rough) ...... 248 7.5 Relationship between parent mean and EV COV using only TTUHRT data from 600s windows with a mean wind speed greater than 15m/s, broken into roughness regimes (smooth, open and rough) ...... 249 7.6 Relationship between TI and EV TI using TTUHRT data from all 600s windows, broken into roughness regimes (smooth, open and rough) ...... 250 7.7 Relationship between TI and EV COV using TTUHRT data from all 600s windows, broken into roughness regimes (smooth, open and rough) ...... 250 7.8 Relationship between TI and EV TI using only TTUHRT data from 600s windows with a mean wind speed greater than 15m/s, broken into roughness regimes (smooth, open and rough) ...... 251 7.9 Relationship between TI and EV COV using only TTUHRT data from 600s windows with a mean wind speed greater than 15m/s, broken into roughness regimes (smooth, open and rough) ...... 251
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DEFINITION OF VARIABLES
𝐶 – Expected gust factor 𝑓𝑟𝑒𝑞𝑢 – Sampling frequency for TTUHRT instrument platform, Hz 𝑔 𝑡 – Peak factor for computing gust factors
𝑔 , 𝑡, T – Segment gust factor of duration t seconds in segment k, from T second duration window 𝑗 of a time history 𝑖 – Index for time history data, from 1 to 𝑛
𝑗 – Index for windows in time history records, from 1 to 𝑤
𝑘 – Index for segmenting windows, from 1 to 𝑠 𝑙 – Largest lateral, longitudinal or diagonal distance for a structural component used in computing wind gust averaging times 𝑛 – Total number of data points per time history record
𝑛 – Total number of data points in a 𝑇 second segment
𝑛 – Total number of data points in a 𝑇 second window
𝑠 – Total number of segments in a time history window 𝑠𝑑 – Standard deviation 𝑠𝑑 𝑡, 𝑇 – Standard deviation of t second duration gusts wind speeds in T second time history
𝑠𝑑 𝑡, T – Standard deviation of t second gust wind speeds in the T second window 𝑗 of a time history
𝑠𝑑 𝑟𝑎𝑤, 600𝑠 – Standard deviation of wind speeds from window 𝑗 in a TTUHRT raw data time history, where the gust duration depends on the frequency of capture, and the record duration is equal to 600s
𝑠𝑑 3𝑠, 600𝑠 – Standard deviation of wind speeds from window 𝑗 in a TTUHRT 3s MA data time history, where the gust duration equals 3s, and the record duration is equal to 600s
𝑠𝑑 60𝑠, 600𝑠 – Standard deviation of wind speeds from window 𝑗 in a TTUHRT 60s MA data time history, where the gust duration equals 60s, and the record duration is equal to 600s 𝑡 – Time duration of gust wind speeds, second
𝑤 – Total number of complete 𝑇 second windows in a time history record
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𝑥 𝑡 – Single t second gust wind speed value 𝑥̅ – Distribution mean value 𝑥̅ 𝑡, 𝑇 – Mean of t second duration gusts wind speeds in a T second time history
𝑥 ̅ 𝑡, T – Mean wind speed of t second duration gusts in the T second window 𝑗 of a time history
𝑥 ̅ 𝑟𝑎𝑤, 600𝑠 – Mean wind speed from window 𝑗 in a TTUHRT raw data time history, where the gust duration depends on the frequency of capture, and the record duration is equal to 600s
𝑥 ̅ 3𝑠, 600𝑠 – Mean wind speed from window 𝑗 in a TTUHRT 3s MA data time history, where the gust duration equals 3s, and the record duration is equal to 600s
𝑥 ̅ 60𝑠, 600𝑠 – Mean wind speed from window 𝑗 in a TTUHRT 60s MA data time history, where the gust duration equals 60s, and the record duration is equal to 600s
𝑥̅ 𝑧 𝑎𝑛𝑑̅ 𝑥 𝑧 – mean wind speeds used when standardizing time history data 𝑥 𝑡, 𝑇 – Peak t second gust wind speed in a T second time history
𝑥 𝑡, T – Peak t second gust wind speed in the T second window 𝑗 of a time history
𝑥 , 𝑡, T – Peak t second gust wind speed, in the T second segment,k, from the j-th T second window of a time history
⏞𝑥 , 𝑡, 𝑇 – Estimate of the peak t second gust wind speed in the T second window 𝑗 of a time history computed using 𝑥 ̅ 𝑡, T multiplied by the HWIND gust factor equal to 1.18
𝑧 – Surface roughness, meters
𝑧 and 𝑧 – Heights used when standardizing time history data, meters
𝑧 and 𝑧 – Surface roughness values used when standardizing time history data, meters
𝑧 , – Surface roughness upwind of a roughness change, meters
𝑧 , – Surface roughness downwind of a roughness change, meters
𝑧 – Surface roughness where data is recorded, in a region containing a transitional boundary layer above the site, meters
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𝑧 – Height data is recorded by an observational platform, meters 𝐴 – Area over which wind induced pressures act 𝐶 – Product of coefficients accounting for phenomena affecting the wind field when computing wind induced pressures
𝐷 – Difference between x raw, 600s and U 600s , reported as both [m/s] and [%] 𝐺 𝑡,𝑇 – Gust factor for t second gust in a T second duration record 𝐺 60𝑠, 600𝑠 – Gust factor used in HWIND to compute the peak 60𝑠 of wind in a 600𝑠 window
𝐺 𝑡, T – Gust factor of t second duration gusts in the T second window 𝑗 of a time history
𝐻 – Height of an atmospheric observational system, used in HWIND wind speed standardization, meters
𝑆𝐷𝐸𝑉 𝑡, T – Standard deviation of extreme t second duration gusts wind speeds in a T second time history
𝑀𝐸𝑉 𝑡, T – Mean of extreme t second duration gusts wind speeds in a T second time history 𝑂 – Outcome of structural performance assessment, pass/fail or computed probability 𝑄 – Load acting on a component/system used in structural performance assessment 𝑅 – Resistance of a component/system used in structural performance assessment
𝑅 – Distance from observational system to the center of a hurricane
𝑅 – Radius of maximum winds, distance from the location of the maximum wind in a storm relative quadrant to the center of the hurricane
𝑇 – Time duration of time history record, seconds
𝑇 – Time duration of segment, seconds
𝑇 – Time duration of window, seconds
𝑇𝐻 – TTUHRT measurement station time history data where gust duration time depends on the frequency at which data was captured
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𝑇𝐻 – TTUHRT measurement station 3s moving average time history data
𝑇𝐻 – TTUHRT measurement station 60s moving average time history data
𝑇𝐼 𝑡, T – Turbulence intensity ratio of t second duration gusts in the T second window 𝑗 of a time history 𝑈 600𝑠 – HWIND computed 600s mean wind speed at 10-m in open exposure 𝑈 60𝑠, 600𝑠 – HWIND computed peak 60s wind speed in a 600s window, at 10-m over open exposure 𝑉 – Wind speed used to compute wind induced pressures 𝑉 – Mean wind speed used in the Time, Velocity, Length model for computing wind gust averaging times 𝑋 – Distance downwind from a change in roughness to where data is recorded, meters
𝑎 – Estimate of extreme value Type-I location parameter 𝜇 – Distribution location parameter 𝜇 𝑡, 𝑇 – Extreme Value Type-I location parameter for t second duration gusts wind speeds in a T second time history
𝜇 𝑡, T – Extreme Value Type-I location parameter for t second duration gusts wind speeds in a t second duration time history, unadjusted back to the windows length
𝑢 𝑡, T –Location parameter of t second duration gusts in the T second window 𝑗 of a time history, adjusted to the window length
𝜇 𝑟𝑎𝑤, 600𝑠 – Extreme Value Type-I location parameter from window 𝑗 in a TTUHRT raw data time history, where the gust duration depends on the frequency of capture, and the record duration is equal to 600s
𝜇 3𝑠, 600𝑠 – Extreme Value Type-I location parameter from window 𝑗 in a TTUHRT raw data time history, where the gust duration depends on the frequency of capture, and the record duration is equal to 600s
𝜇 60𝑠, 600𝑠 – Extreme Value Type-I location parameter from window 𝑗 in a TTUHRT 60s MA data time history, where the
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gust duration equals 60s, and the record duration is equal to 600s 𝜌 – Air density
𝛿 – Maximum height of a well established boundary layer after an upwind change in roughness
𝛿 – Maximum height of the transition boundary layer after an upwind change in roughness 𝜎 – Distribution scale parameter
𝜎 𝑡, T – Extreme Value Type-I scale parameter for t second duration gusts wind speeds in a T second time history, unadjusted back to the windows length
𝜎 𝑡, T – Extreme Value Type-I scale parameter for t second duration gusts wind speeds in a T second time history, adjusted back to the windows length
𝜎 𝑟𝑎𝑤, 600𝑠 – Extreme Value Type-I scale parameter from window 𝑗 in a TTUHRT raw data time history, where the gust duration depends on the frequency of capture, and the record duration is equal to 600s
𝜎 3𝑠, 600𝑠 – Extreme Value Type-I scale parameter from window 𝑗 in a TTUHRT 3s MA data time history, where the gust duration equals 3s, and the record duration is equal to 600s
𝜎 60𝑠, 600𝑠 – Extreme Value Type-I scale parameter from window 𝑗 in a TTUHRT 60s MA data time history, where the gust duration equals 60s, and the record duration is equal to 600s
𝜎 – Log-Logistic standard deviation parameter 𝛽 – Distribution shape parameter 𝛤 – Gamma function
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CHAPTER 1
INTRODUCTION
Hurricanes are responsible for billions of dollars in damages and place at risk over 6 million homes in the United States each year. In some cases, the devastation following hurricane landfall is so severe little-to-nothing of the built environment remains in entire regions. In cases like these, on site forensic investigation is no longer viable and alternative methods, like modeling, must be employed to assess the cause, sequence and probabilities associated with structural failures. Such an analysis can be done using deterministic or stochastic methods, but stochastic methods offer the advantage of accounting for the variability of random variables affecting both the loads and resistances factoring into the outcome of a failure assessment. Of course, such an analysis can only be employed if the true variability of each random variable can be accounted for. This requires understanding the true nature of each random variable and, in the case of hurricane wind loads, modeling the distribution of extreme winds becomes paramount. To correctly model the distribution of extreme winds at a site where failure assessment is sought, high resolution, high fidelity, high precision wind records are ideal as they provide data applicable directly to the site of interest. Unfortunately, while many groups do gather, high resolution, high fidelity, high precision wind data, the network of observation stations is sparse leaving a majority of the geography affect by hurricane winds without suitable data useful in structural performance analysis (SPA). As an alternative, hurricane wind field models can be used to provide wind data for any location in the region affected by a hurricane. Hurricane wind models are most useful if two things can be done: (1) the values produced by the model can be verified using high resolution, high fidelity, high precision wind data, and (2) a methodology is advanced to take hurricane wind field modeled data and generate distributions of extreme winds suitable for stochastic SPA.
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The objectives of this work are to investigate the hurricane wind field model developed by the National Oceanic and Atmospheric Administration Hurricane Research Division’s HWIND. To accomplish this investigation, HWIND reconstructions are compared against wind data gathered by Texas Tech University’s Hurricane Research Team (TTUHRT) collected for 32 total deployments, with 15 different platforms during 10 separate hurricane landfall events. The same TTUHRT data is then used to form relationships between parent and extreme wind fields. The established relationships and hurricane model wind field data can then be used together to simulate distributions of extreme winds for use in SPA to assess the cause, sequence and probabilities associated with structural failures leading to total structural collapse.
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CHAPTER 2
BACKGROUND
Major Hurricane Damage Of the 314 million people living in the United States in 2013, 85 million, or 27%, live in hurricane prone coastal counties between Maine and Texas (NOAA, 2013). In these areas, around 6.6 million homes are at risk of being damaged or destroyed during a hurricane’s landfall (Botts et al, 2015). Of the 6.6 million, 2.8 million homes are on the Gulf coast, Texas to Florida, with the remaining 3.8 million on the Atlantic coast, between Florida and Maine (Botts et al., 2015). Major hurricanes cause widespread devastation, disrupt communities and result in injury and death to many people in the affected regions. Examples of costly, major hurricanes to hit the Gulf coast include the Galveston Hurricane of 1900, Hurricane Camille in 1969, Hurricane Katrina in 2005 and Hurricane Ike in 2008. The most recent major hurricane to devastate the Atlantic coast is the 2012 landfall of Hurricane Sandy and in decreasing order, Hurricanes Katrina, Sandy, and Ike are the first, second and fifth costliest natural disasters in United States history. Together, these three hurricanes have resulted in a combined total of $110.4 billion in insured losses (FEMA, 2006), (FEMA, 2009), (FEMA, 2013). Between the Galveston Hurricane of 1900 and Hurricane Katrina in 2000, 10,000 people lost their lives making these two storms two of the deadliest natural disasters in United States history. Economically, the cost to rebuild all 6.6 million wood frame homes at risk on the Gulf and Atlantic coast is estimated by (Botts et al., 2015) to be over $1.5 trillion. The homes used by Botts et al. (2015) for this estimate are only those at risk of experiencing damage from hurricane surge, and does not include homes at higher elevations that are still vulnerable to damages caused by hurricane winds. It is unlikely the full $1.5 trillion estimate will ever be realized as a hurricane making landfall in North Carolina is unlikely to also affect homes on the Gulf coast of Texas. However, the economics associated with the structures at risk (the 6.6 million wood
3 Texas Tech University, Joseph Dannemiller, May 2019 frame houses plus engineered structures and infrastructure) makes assessing how and why (wind or surge) structures fail during hurricane landfall an important issue for policy makers, home and business owners, insurers, researchers, tax payers, and specifically any member of the public with at least one asset insured against such losses. Structural damages due to hurricane winds span from minor shingle and siding loss to being completely wiped off their foundations. Situations where structures are completely wiped off their foundations are often referred to as “slab-claims”, since the only remnant of the structure post event is the concrete foundation slab itself. For perspicuity, the term slab-claim will henceforth refer to a situation where only a foundation remains. In slab-claim cases it can be difficult to determine what structural component failed first, the sequence of structural failures, or make a conclusive assessment regarding whether hurricane winds or storm surge caused any one specific failure, where both hazards are present. In fortuitous cases, similarly constructed structures near a slab-claim survive providing examples of damage patterns that can be extrapolated to the analysis of a slab-claim. However, in the case of a slab-claim, where no similar structures in proximity survive the same event, observational analysis methods are no longer conclusive. Other methods, such as simulation, must be utilized to determine the most likely first component failure, the sequence of component failures, and whether wind or surge led to any, or all, of the component failures. Bolivar Peninsula, Texas is an example of a region devastated by hurricane winds and/or surge, leaving an entire region of slab-claims after the 2008 landfall of Hurricane Ike. Before and after aerial photographs are shown in Figure 2.1 and Figure 2.2 to illustrate the magnitude of the destruction.
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Figure 2.1 Bolivar Peninsula before Hurricane Ike Landfall, September 9, 2008 (FEMA 2009)
Figure 2.2 Bolivar Peninsula after Hurricane Ike Landfall, September 15, 2008 (FEMA 2009)
The arrows in the before and after photographs in Figure 2.1 and Figure 2.2 identify the locations of two wood frame structures for reference. Most of the other
5 Texas Tech University, Joseph Dannemiller, May 2019 structures between the referenced structures and the coastline are gone. These sites represent slab-claims and onsite forensic investigation is no longer conclusive for determining the hazard leading to failure, the progression of failures, or what ultimately led to total structural collapse. Attributing individual component failures to wind or surge and determining the order of component failures becomes a complex issue.
Loads and Resistances Assessing failures of structural components requires relating loads and strengths with deterministic, or stochastic, analysis. Both load and resistance values are random phenomena, each a function of other random variables, making assessments of failure, or non-failure, a random variable. For failure to occur, the loads acting on a component must exceed the strength of the component. In the opposite case, where the strength of a component exceeds the load, the component does not fail. Deterministic analysis is the process of using a single value for each random variable contributing to the production of a single outcome. The single values used in deterministic analysis are computed leveraging knowledge of the underlying distributions governing each random phenomenon. Stochastic analysis involves generating distributions for each random variable and then repeatedly sampling to compute a distribution of outcomes. With such a distribution, the probabilities associated with individual component failures can be assessed. Deterministic analysis is preferable where representative cases are sought to illustrate a common, or likely, result for a random outcome. In deterministic analysis, it is common for strength values to be underestimates of the mean strength of a sample of similar components, while the values of load are commonly overestimates of the mean of the load acting on a component resulting in a “worst case” condition i.e., a weak structure meets a strong load (Salmon et al., 2008). Structures survive this “worst-case” i.e., the structure can
6 Texas Tech University, Joseph Dannemiller, May 2019 still perform its intended function, when actual loads are lower than the overestimate, or when actual strengths are stronger than the underestimate. The alternative to deterministic analysis is stochastic analysis where distributions of random variables are simulated, and then sampled to compute instances of loads and strengths related together to compute probabilities of failure. The distributions of all random variables associated with the strengths and loads must be known to properly assess whether failure has occurred. The process of independently assessing whether failure has occurred is similar to deterministic analysis, except that in stochastic analysis the process is repeated many times using values sampled from each of the underlying distributions. The number of times this process is repeated, or repetitions, is governed by how many realizations are necessary to converge on a statistically significant solution, or one with an acceptable level of confidence. Once the independent assessments have been completed, the probability that failure, under the utilized loading condition, can be computed. An in-depth discussion of stochastic analysis specifically as it pertains to structures, is given in (Nowak and Collins, 2000). As stated above, the process of assessing whether failure has occurred in a structural system requires understanding and relating two phenomena, loads and strengths. The strength of a system, henceforth referred to as resistance, is governed by the properties of the materials used, member geometries and the type of structural system. The loads acting on the system are a function of the environment, and during hurricane landfall the two of principle concern are loads due to storm surge and hurricane winds. Storm surge is left for another discussion and only hurricane winds are discussed herein. Where loads and resistances are known, they can be related to determine if a failure has occurred using Equation 2.1.
O R Q 2.1
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Where O denotes the outcome, R denotes the resistance of the system, and Q denotes the load acting on the system. If the outcome is less than zero, the load exceeds the system’s ability to resist and failure occurs. Instead of referring to outcomes less than or equal to zero as failures, some literature refers to these scenarios as limit states. Failure of a limit state is the point where a structural component/system stops performing its intended function. Limit states fall into two main categories, ultimate limit states (ULS) and serviceability limit states (SLS). When ULS are reached, the properties of a system physically change. In most cases, a structural system can no longer resist the loading environment and the structure is no longer sound. In other words, the strength of a system can be reduced to a fraction of its original capacity, or all the way to zero. Serviceability limit states refer to vibration, deflection and localized deformations. When serviceability limit states have been surpassed, a structure can still be deemed sound, but may not be fit due to human perception. As an example, excessive vibrations and deflections can make structures undesirable to some human occupants. Occupant perceptions can lead to illness, or to some occupants losing faith in a structure as some equate deflections with failures. When a structure is wiped off its foundation, ULS have been exceeded and SLS are no longer a governing concern. The first ULS reached (first failure), as well as the sequence in which ULS occurred (damage sequence), is difficult to impossible to identify. Where damage sequences are difficult to ascertain, the systems comprising an entire structure can be analyzed, one at a time, to assess the conditional probabilities associated with all possible damage sequences. A scientist that understands the engineering and physics applicable to a specific assessment can reduce the number of possible damage sequences investigated when assessing causal failure. This can make the assessment more manageable in scope, but even analyses that are less than holistic can be long and cumbersome. The process of assessing the conditional probabilities, for all possible ULS damage sequences, for a single structure, is henceforth referred to as Structural Performance Analysis (SPA).
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Wind Loads As hurricane wind loads are the primary concern herein, it is important to understand how winds impart loads on structures. Wind induced loads come from dynamic pressures exerted on structural systems, acting over the area of a system, or on a localized sub-system. An example of a sub-system representing the “weak link” in a structural system is a fastener used to attach metal roofing to a main structural element. The wind loads acting on the fastener are highly localized and are higher in magnitude per unit area than the wind loads acting on the panel, per unit area, that fastener attaches to the structure. The differentiation in the area used to compute wind pressures is important because wind loads do not act fully correlated, or can be lessened due to pressure coefficients applicable over differentiated areas. The area over which wind loads are of concern for any system can be computed using the Time, Velocity and Length (TVL) model from (Lawson, 1980). The TVL method is used to determine the size of wind gust that acts fully correlated over the area of a component. The size of gust is measured in time, t, and is computed using Equation 2.2 below.
tV 4.5l 2.2
Where the longest length between any two points bounding the area for a component is denoted as l, and the mean wind velocity is denoted as V . The wind gust of duration t is then used to compute the wind loads acting on a component based on its dimensions. As the mean wind speed for finite intervals changes, using the magnitude of the mean wind speed to determine the governing gust duration size is impractical in application. However, considerable work has been completed in developing structural codes which establish standard gust durations for: (1) entire structural systems, and (2) for smaller components of structures. For larger structural systems, (Cook, 1985b) uses a gust duration of 60s. In the United States, the gust durations established by ASCE 7-10 Design of Loads for Buildings and Other Structures (ASCE 7, 10) are 3s as 3s gust durations are recorded by the National Weather Service’s observational
9 Texas Tech University, Joseph Dannemiller, May 2019 systems (ASCE &, 10). The work herein will utilize raw wind data, as well as 60s and 3s gust wind data. Computing wind loads can vary based mostly on two things, how wind loads are assessed and how the phenomena affecting the wind field are considered. In terms of assessment, (Cook, 1985a) breaks the “problem” of assessing wind loads into categories that couple variations of three pieces: the wind climate, the atmospheric boundary layer, and a structure. These three pieces are separated because each represents a major factor governing when, and how, peak loads act on a structure. The wind climate refers to wind events that occur on the order of several days to years. Events occurring at this frequency that produce the highest wind speeds are thunderstorms, downbursts, hurricanes, tornados, and several other unique weather phenomena with winds higher than those produced by the daily diurnal cycle. The boundary layer is the second piece of how (Cook, 1985a) breaks apart the “problem” of assessing wind loads. Boundary layer considerations account for how the wind field is modeled near the surface of the earth. The shear caused by interactions with the surface of the earth itself, plus the man-made structures on the surface, slow wind speeds close to the surface. The last piece in assessing wind loads is the effects of a structure itself. Structural characteristics such as the height, geometry and material makeup affect what percentage of the dynamic pressures produced by the wind field translate into a structure, versus the percentage that breaks around a structure like water around a rock. Combining these three pieces together, a peak wind speed can be determined based on: (1) a type of wind event, and (2) our understanding of how such a wind field could be modeled near the surface. Once a wind speed has been selected, wind loads can be computed, for any assessment method (static, quasi-static, quasi- steady, pseudo-steady) described in (Cook, 1985a), using the same generic equation, Equation 2.3.
1 Q ∗ρ∗ V ∗C∗A 2.3 2
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Where the wind load, denoted as Q, is computed as a function of the air density, denoted as ρ, the square of a wind velocity, denoted as V, a set of coefficients accounting for the random variables affecting the wind velocity, denoted as C, and the area over which the wind pressure acts, denoted as A. Some phenomena included in computing values of C include localized pressure coefficients, upwind terrain roughness, terrain elevation changes, turbulence intensity, gust ratios, structural height, etc. Structures in the United States are governed by (ASCE 7,10). ASCE 7- 10, which uses both quasi-static theory, where fluctuations in loads are directly correlated to fluctuations in boundary layer winds; and the quasi-steady theory, where wind loads are a function of both boundary layer induced fluctuations in the wind field, and building induced fluctuations (Cook, 1985). Simulating wind loads in SPA requires knowledge of ρ, C, A, but most importantly V. The magnitude of the wind velocity is much larger than the other random variables, and then it is squared. This makes accurate estimates of wind speeds vital when computing wind loads. In stochastic SPA, it is important for the distribution of wind speeds to correctly model the highest wind speeds occurring in finite periods. The distribution that best models the highest wind speeds is an extreme value (EV) type-I distribution. But, before the EV distribution can be discussed further, the nature of the distribution of parent winds must be explored further The parent wind field contains every wind speed measurement in a time history record. The statistical distribution that best fits the parent wind field, in the hurricane environment, is the Weibull distribution (Hennessey, 1978), (Justus, 1978), (Simiu and Scanlan, 1996) and (Seguro and Lambert, 2000). The Weibull distribution is defined by location, scale and shapes parameters providing for translation of the distribution itself, and an adaptability in the shape of the distribution as the scale and shape parameters change. Some research has fit parent wind data with a Rayleigh distribution (Conradsen and Nielsen, 1984), where the Weibull shape parameter equals two, making this a specialized fit using a Weibull distribution by constraining the
11 Texas Tech University, Joseph Dannemiller, May 2019 shape parameter to a single value. The Probability Density Function (PDF) for the three parameter Weibull distribution is expressed in Equation 2.4.
β 𝑥 μ 𝑓 𝑥|μ,σ,β ∗ ∗𝑒 2.4 σ σ
Where 𝑓 𝑥|μ, σ, β denotes the probability, f(x), of a value x occurring given values of location, scale, and shape parameters, denoted as μ, σ, and β, respectively. The mean, denoted as 𝑥̅, and variance, denoted as 𝑣 , of the Weibull distribution are computed using numerical software and fit Equations 2.5 and 2.6.
1 𝑥̅ μ σ∗Γ 1 2.5 β
2 1 𝑣 σ ∗ Γ 1 Γ 1 2.6 β β
Where the gamma function, denoted as Γ, is defined in Equation 2.7 below, for any value symbolically represented by n.
Γ n n 1 ! 2.7
Where, for non-integer values, the gamma function is evaluated using the following integral (Mathworks, 2018).
Γ n 𝑒 𝑡 𝑑𝑡 2.8
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When assessing structural failures, it is important to consider that most failures occur when wind loads are high, almost always the result of higher wind speeds. Most often, failures do not occur from wind speeds in the low end of the distribution of the parent wind field. An exception to this statement would be if the mean wind speed was so high, that structural failure is all but certain for any wind speed in the distribution of parent winds. Therefore, when assessing failure deterministically, or when computing the probability of a failure stochastically, the frequency of occurrence for the highest wind speeds the distribution of parent winds becomes important to model. This is where the second distribution, the EV distribution, comes into play. While the distribution of parent winds excels in reasonably modeling the frequency of occurrence for wind speeds through the entire range of wind speeds in a time history, it loses its accuracy in the upper tail. The upper tail is where the highest wind speeds occur and these wind speeds are the most important wind speeds to model when assessing failure probabilities. Unlike the distribution of parent winds, the EV distribution does not accurately model the frequency of occurrence for all wind speeds through the entire range of wind speeds in a time history. It does, however, excel in modeling the frequency of occurrence for highest wind speeds during a defined duration. As such, the EV distribution, used to model extreme winds, serves as a companion to the distribution of parent winds, where each has benefits, and tradeoffs. The PDF for the EV Type- I distribution is expressed in Equation 2.9.
1 2.9 𝑓 𝑥|μ,σ ∗𝑒 ∗𝑒 σ
Where μ and σ denote the location and scale parameters, respectively. The location and scale parameters in Equation 2.4 are not the same as those in Equation 2.9 as the location and scale parameters are used to define the Weibull and EV distributions, respectively. The location parameter occurs at the mode of the EV distribution, corresponding to the value of x with the highest probability of occurrence. The scale
13 Texas Tech University, Joseph Dannemiller, May 2019 parameter quantifies the dispersion of data in the distribution, or the extent to which independent values are spread out away from the mean. A low scale parameter results in the wind data tightly organized around the mean. As the scale parameter increases, the distribution of wind speeds spreads out away from the location parameter, making the distribution wider and lower at the location of the mode. It is important to note that herein the scale parameter quantifies the dispersion of data, and this may be a source of some confusion. The term “dispersion” is sometimes used in statistics to describe an average distance between the individual data points and the mean of the dataset. In other instances, the term “dispersion” is used to define a parameter that is equal to 1/σ, a parameter some refer to as a rate parameter. In all applications, it is important to define exactly what is meant when using the term “dispersion”. Herein, any use of the term dispersion seeks to reference the concept of dispersion discussed earlier, the average distance from individual data points to the mean. Graphically, where the mean and mode occur for normally, and non-normally distributed, data are illustrated in Figure 2.3 through Figure 2.5. The mean and the mode are often introduced in statistics in a discussion focused on normally distributed data. One very important concept utilized in this work is the relationship between where the mean and mode occur in a distribution of non-normally distributed data. However, since most discussions begin discussing the locations of the mean and mode in a normally distributed dataset, a normal distribution is provided in Figure 2.3.
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0.06
0.05
0.04 Mean and Mode
0.03
f(x) 0.02
0.01
0 0 102030405060708090100
x = μ Wind Speed (x) [m/s] Figure 2.3 Normal distribution probability density function (PDF)
The PDF in Figure 2.3 is mean centered at x 45 𝑚/𝑠 with a standard deviation, a measure of dispersion, sd 6.24 𝑚/𝑠. The normal distribution has one advantage over most non-normal distributions in that it is symmetrical. Due to this symmetry, the mean, the value that divides the lower 50% of values from the upper 50%, is equal to the mode, the value with the highest probability of occurrence. The Weibull and EV distributions are both positively skewed distributions (not symmetrical) and therefore the mean and mode are not equal. The relationship between the mean and mode of a positively skewed distribution is illustrated below in Figure 2.4.
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0.06
0.05
0.04 Mode or Location, 𝜇
0.03 Mean, 𝑥̅ f(x)
0.02
0.01
0 0 102030405060708090100 Wind Speed (x) [m/s] Figure 2.4 Relationship between the mean and mode for positively skewed distributions
As shown in Figure 2.4, the mean is larger than the mode for a positively skewed distribution. The mode still represents the value with the highest probability of occurrence and occurs at the peak of the PDF. A comparison between the normal distribution in Figure 2.3 and the positively skewed distribution in Figure 2.4, is provided below in Figure 2.5.
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0.06 Normal Distribution Positively Skewed Distribution 0.05
0.04 Mean = 45 m/s
0.03 f(x)
0.02
0.01
0 0 102030405060708090100𝑥̅ Wind Speed (x) [m/s] Figure 2.5 Comparison of normal and skewed distributions
The mean values for the normal distribution, and the positively skewed distribution, in Figure 2.5 are both equal to 45 m/s. But, while the mode of the normal distribution is still 45 m/s, the mode of the skewed distribution is now 40.13 m/s. The skewing of the data is an important phenomenon to account for, as computing the most frequently occurring value in non-normally distributed data is no longer as simple as computing the mean. For clarity, it is also important to note that for positively skewed distributions, the mode is equal to the location parameter defined earlier. Going forward, the term location parameter will be used when referencing the value with the highest probability of occurrence when discussing both the Weibull and EV distributions.
Observational Systems Measurements of wind speed in proximity to a site are an important part of assessing the causality and probabilities of wind induced failures at the site itself. During hurricane landfall, wind speeds are measured using surface observation systems, aerial observation systems, and radar systems. The data collected by one or more observational system can be utilized in SPA if the data is gathered at or near the
17 Texas Tech University, Joseph Dannemiller, May 2019 site where it will be applied. Examples of data gathering systems are shown in Figure 2.6.
` (a) (b)
(c) (d)
(e) (f) Figure 2.6 (a) Automated Surface Observation System (ASOS) located in Austin, TX (www.ncdc.noaa.gov, 2017), (b) Weather buoy stationed in the Gulf of Mexico (www.ndbc.noaa.gov, 2017), (c) Geostationary Operational Environmental Satellite R Series (GOES-R) (www.lockheedmartin.com, 2017), (d) National Oceanic and Atmospheric Administration (NOAA) WP-3D Orion weather research and hurricane intercept aircraft (www.esrl.noaa.gov, 2017), (e) National Center for Atmospheric Research (NCAR) dropsonde device in flight (www2.ucar.edu, 2017), (f) ship mounted weather stations
Of the data gathering systems in Figure 2.6, surface observation systems are preferred as they record wind data close to the earth’s surface, where human occupied
18 Texas Tech University, Joseph Dannemiller, May 2019 structures exist, at frequencies necessary to match the temporal scales needed to analyze human occupied structures. In addition to ASOS/AWOS stations, many research universities build operate and maintain permanent, semi-permanent and portable systems that measure the hurricane wind field during landfall. One research university that has developed several generations of semi-permanent, and portable, surface observation systems is Texas Tech University’s Hurricane Research Team (TTUHRT).
Texas Tech University Hurricane Research Team (TTUHRT) TTUHRT deployed its first remote sensing field instrumentation during the 1998 landfall of Hurricane Bonnie. The first platform was the Wind Engineering Mobile Instrumented Tower Experiment (WEMITE), hereafter referred to as WEMITE #1. The WEMITE #1 platform was built on top of a structurally reinforced flatbed trailer and, when deployed, was stabilized against dynamic oscillation using outriggers and 6 guy wires. Horizontal wind speed and direction were collected at 3m, 6.1m and 10.7m using RM Young 05106MA Wind Monitors. Sometime between 1999 and 2004, the instruments at the 10.7m height were moved to 10m to collect data directly at the standard 10m height. The sampling rate for WEMITE #1 was 10Hz for all deployments except Hurricane Bonnie, 1998, where wind data was sampled at 5Hz. WEMITE #1 is shown in Figure 2.7 being deployed prior to the 2005 landfall of Hurricane Katrina.
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Figure 2.7 WEMITE #1 near Vacherie, LA prior to the landfall of Hurricane Katrina, 2005 (TTUHRT, 2006)
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One year after the construction of WEMITE #1, 1999, TTUHRT constructed a second platform, WEMITE #2. The sampling rate for WEMITE #2 was 10Hz for all deployments and like, WEMITE #1, WEMITE #2 was stabilized against dynamic oscillations via outriggers and guy wires. WEMITE #2 is shown in Figure 2.8 just following the 2005 landfall of Hurricane Katrina.
Figure 2.8 WEMITE #2 at Stennis International Airport following the landfall of Hurricane Katrina, 2005 (TTUHRT, 2006)
In 2002, TTUHRT constructed three Portable Mesonet Towers (PMT) named PMT Black, PMT White and PMT Clear. Each PMT tower collects data using a single RM Young 05106MA Wind Monitor placed atop a 10m, single pole mast, supported by six guy wires (TTUHRT, 2014). The sampling rate for the PMT towers was 10Hz for all deployments. PMT Clear is shown in Figure 2.9 deployed at the
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Slidell Municipal Airport in Slidell, LA prior to the landfall of Hurricane Katrina, 2005.
Figure 2.9 PMT Clear at Slidell Municipal Airport prior to the landfall of Hurricane Katrina, 2005 (TTUHRT, 2006)
More details on the design and construction of WEMITE #1 can be found in (Schroeder, 1999) (Schroeder and Smith, 2003). In 2005-2006, TTUHRT developed the StickNet Platform with two main objectives in mind: (1) develop a cheap, easy to deploy platform to facilitate greater spatial resolution in data capture and (2) maximize the number of platforms deployable in a short time frame. The StickNet platforms were first used to measure hurricane winds during the 2008 landfall of Hurricane Dolly. To date, the StickNet platforms have been used to gather data during the landfall of eight hurricanes. The
22 Texas Tech University, Joseph Dannemiller, May 2019 entire fleet consists of 2 different types of platforms, 12 of each, called A-probes and B-probes. The wind speed and direction are recorded on the A-probes, an example of which is shown in Figure 2.10, using RM Young 41382 wind monitor sampling at 1Hz, 5Hz, or 10 Hz depending on the StickNet platform and hurricane deployment. The wind speed and direction are recorded on the B-probes, an example of which is shown in Figure 2.11, using the Visalia WXT510 All-In-One instrument that also records temperature, relative humidity, barometric pressure, and total precipitation. The wind speed and direction data capture frequency for all the All-In-One platforms is 1 Hz.
Figure 2.10 StickNet 101, A probe, deployed prior to the landfall of Hurricane Sandy
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Figure 2.11 StickNet 102, B Probe, deployed for testing at Reese Technology Center, Lubbock, TX
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Between the WEMITE, PMT and StickNet platforms, there have been a total of 189 platform deployments during the landfall of 31 hurricanes. The WEMITE platforms comprise 35 total deployments, the PMT towers comprise 20 total deployments and the StickNet platforms comprise 134 total deployments. The data recorded by WEMITE, PMT and StickNet platforms is utilized by TTUHRT researchers to generate reports for storms both internally (e.g. Giammanco et al., 2006), and for outside entities (e.g. Schroeder and Edwards, 2005). The TTUHRT data is used by others in publishing reports (e.g. FEMA, 2009), as well as in technical publications in atmospheric science (e.g. Paulsen et al, 2005 and Hirth et al., 2012), wind engineering (e.g. Schroeder and Smith, 2003 and Letchford et al., 2001), economics (e.g. Ewing et al., 2006), and in applied structural engineering (e.g. Schroeder et al., 2009 and Giammanco et al., 2009). The data collection and deployments methods of the TTUHRT team have passed rigorous scrutiny. Routinely, TTUHRT wind speed and directions are plotted to show the time history of both. Examples of a wind speed and wind direction time histories are shown in Figure 2.12 and Figure 2.13, respectively, for the 1998 landfall of Hurricane Bonnie.
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Figure 2.12 Wind speed time history, recorded at 10m by WEMITE #1 during Hurricane Bonnie (1998)
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Figure 2.13 Wind direction time history, recorded at 10m by WEMITE #1 during Hurricane Bonnie (1998)
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Wind Speed Standardization To facilitate any comparison employing multiple datasets, mean values from each record must be standardized to ensure a direct “apples-to-apples” comparison. For such a comparison, all records need to be adjusted to represent values occurring at the same height, over the same surface terrain roughness, z and at a specified averaging time. Wind data can be standardized to 10m over open exposure using the log-law from (Holmes, 2001) shown in Equation 2.10.
𝑙𝑛 z ⁄z x z x z ∗ 2.10 𝑙𝑛 z ⁄z
Where the standardized mean wind speed, x z , is computed using the measured mean wind speed denoted as x z , measured at height z over terrain having a surface roughness z . The computed mean, x z , occurs at a height z over terrain with surface roughness z . The standard values of z and z are 10m and 0.03m, respectively, representing a mean value at 10m over open exposure. Figure 2.14 is provided to illustrate why standardization is important using the deployment site for WEMITE #1 during the 1998 landfall of Hurricane Bonnie.
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Figure 2.14 Hurricane Bonnie, WEMITE #1 deployment site
Arrows have been superimposed onto Figure 2.14 to reference two different wind directions for comparison. The first arrow shows the terrain over which wind travels when coming from 120º measured clockwise from true north. The second arrow shows the terrain over which wind ravels when coming from 300º measured clockwise from true north. The upwind terrain for the 120º wind direction contains flat, open exposure for 2000m. The 300º wind direction contains a forest less than 400m upwind. The passage of the wind field over the forested region results in more mechanically generated turbulence in the boundary layer, at lower levels, than is present in the 120º case. This disparity would make comparing mean values between two records, one from each of the 120º and 300º cases, inconsistent. Standardizing each record, using Equation 2.10 would make records from each of the two roughness regimes consistent, and therefore viable for comparison.
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If the surface conditions do not change in the last 1000m upwind of where data is recorded, then the z0 value for the upwind terrain can be used without modification (Melbourne, 1992) and (Weng et. al., 2010). If the terrain does change in the last
1000m, the z0 value can be estimated using methods discussed in (Deaves and Harris, 1981)). The idea behind the estimation is that after a change in roughness, the old established boundary layer begins to transition to a new boundary layer where velocity changes with height as a function of the new surface conditions. An example of this is shown below in Figure 2.15 where two established boundary layers are compared.
Figure 2.15 Comparison of boundary layer velocities over rough and smooth terrain
The boundary layer over rough terrain does not exhibit the same rapid increase in velocity with height, as the boundary layer profile over smooth terrain. This is due to a difference in shear at the surface over the two surface roughness conditions. While Figure 2.15 shows boundary layers before, and after, a change in roughness, the transition is not instantaneous. At times, an observational platform can gather wind data in a well-established boundary layer, or in a region where the boundary layer is
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Texas Tech University, Joseph Dannemiller, May 2019 transitioning from one well-established boundary layer to another well-established boundary layer. The properties of the upwind, downwind and transitional boundary layers, are necessary to determine z values utilized to standardize wind speed measurements. Transitional boundary layer theories have been studied extensively (Blackada et al., 1967), (Bietry et al, 1978), (Andreopoulos and Wood, 1982), (Wierenga, 1993) and (Garrett, 1990) to name a few. Many texts describe the process of adjusting wind speeds based on non-uniformity of surface roughnesses upwind of a location (Holmes, 2001), (Cook, 1985a), (Simiu and Scanlan, 1996), and (Dyrbe, 1997) to name a few. A good historical review of transitional boundary layer research is (Savelyev and Taylor, 2005). Herein, the z values for the upwind roughness regime, and at a deployment site, are assigned qualitatively using the definitions of z0 values for different roughness regimes in (Simiu and Scanlan, 1996). The assessment is completed for each 30º bin around the full 360º for each site. Figure 2.16 illustrates the heights through which the upwind, downwind, and transitional, boundary layers exist.
Figure 2.16 The boundary layer transition past a change in roughness, rough to smooth
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The value of z used to standardize data is a function of z , and z , , the surface roughness values upwind and downwind of the change in roughness, respectively. The heights for δ and δ , when moving from rougher to smoother terrain, are computed using Equations 2.11 and 2.12 from (Deaves and Harris, 1981).
. z , δ 0.07∗𝑋 ∗ 2.11 z ,
. . δ 10∗ z , ∗ 𝑋 2.12
Where X is the distance upwind past the change in roughness, illustrated in Figure
2.16. The height of δ , when moving from smoother to rougher terrain, is computed using Equation 2.13.
. . δ 0.36∗ z , ∗ 𝑋 2.13
With the heights of the transition region, 𝑧 used to standardize data is computed using Equation 2.14.
z , 𝑖𝑓 z δ ⎧ , ⎪ z , , 𝑖𝑓 z δ 𝑧 2.14 ⎨ 𝑧 𝛿 ∗ 𝑧 , 𝑧 , ⎪𝑧 , , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 ⎩ 𝛿 𝛿
Where z is the height at which data is recorded. Equation 2.14 shows that if z is below δ , the z value used to standardize data is equal to z , as the recording height is inside the internal boundary layer. If the height at which data is recorded is above δ , the z value used to standardize data is that of z , as the
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Texas Tech University, Joseph Dannemiller, May 2019 recording height is in the region where the new boundary layer is coherent and in equilibrium. If data is recorded at a height between δ and δ , the value of z used to standardize data is computed using linear interpolation between the heights δ and δ , and the z values from the upwind regime and deployment site. Utilizing the z value computed using Equation 2.14 as 𝑧 in Equation 2.10, while setting z 10𝑚, z 0.03𝑚, and z equal to the height at z , the mean wind speed x z can be standardized to a mean wind speed at 10𝑚 height over open exposure (z 0.03𝑚) using Equation 2.15.
𝑙𝑛 10𝑚⁄ 0.03𝑚 x z x z ∗ 2.15 𝑙𝑛 𝑧 ⁄𝑧
Because of the standardization completed using Equation 2.15, mean wind speeds can be compared against one another, or aggregated together to identify trends.
Gust factors While the distribution of extreme winds quantifies the probability of occurrence for the highest wind speeds in parent wind field, a gust factor is a common factor referenced to relate a peak wind speed to a mean wind speed in the parent wind field. Gust factors can be computed post-hoc using in-situ data, or an expected gust factor can be computed using assumed effects due to the turbulence mechanically generated by surface roughness. To keep these two concepts clear the notation used for all observed gust factors will be denoted using a G, while the gust factors estimated using models or methods will be denoted using a C or Cg. A gust factor can be easily computed from any observational record, like those recorded by TTUHRT, as the peak wind speed of some duration, t, divided by the mean of all winds occurring through a longer duration, T. For example, a 3𝑠 gust factor in a 600𝑠 window represents the fastest 3𝑠 of wind, divided by the mean of all wind in a 600𝑠 record.
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The equation for computing the gust factor, denoted as G t, T , is shown below in Equation 2.16.
x t, T G t, T 2.16 x t, T
Where x t,T denotes the peak t second duration wind speed, and x t, T denotes the mean of t second duration winds, both from a record of T seconds in length. Many different models exist for computing expected gust factors for non- hurricane environments, one example is (Durst, 1970), and for hurricane environments (Krayer and Marshal, 1992). Both of these studies use observations to form curves relating the duration of a t second gusts to a T second means. Due to stationarity durations and boundary layer behavior, it may not be possible to compute expected gust factor values for hurricane, and non-hurricane environments using the identical equations due to the underlying assumptions. Before this can be explained it is necessary to examine the statistical framework employed in many different methods to compute expected gust factor values, G t, T , using a form of Equation 2.17.
sd t, T C 1 g t ∗ 2.17 x t, T
Where g t is a peak factor from standard normal distribution tables, quantifying the distance from the mean to a value determined based on its probability of either being exceed, or not being exceeded. The use of Equation 2.17 is predicated on the assumption that deviations in the wind field are normally distributed. Research by (Ishizaka, 1983), specifically focused on Japanese typhoon wind field data, found computing average, or expected, gust factors to be possible using Equation 2.18.
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sd t, T 𝑇 C 1 g t ∗ ∗ln 2.18 x t, T 𝑡
It is important to note that values of C can be computed using Equation 2.16 only when in possession of time history data. Equations 2.17 and 2.18 are used to compute expected values of G t, T in the absence of time history data. Such expected C values can then be used to compute expected peak wind speeds, x t,T , by multiplying reported x t, T values by the expected value of C . A rearranged version of Equation 2.17 can be used to solve for x t,T , shown in Equation 2.19.
x t, T x t, T 𝑔 𝑡 ∗ sd t, T 2.19
Utilizing equation 2.19 to compute x t,T values requires establishing, a priori, a probability of exceedance so an associated value of g t can be selected from normal distribution tables. Figure 2.17 illustrates the graphical relationship between x t, T , sd t, T , x t,T and g t as a companion to Equation 2.17, to demonstrate the idea of probability of exceedance.
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x T g t ∗sd T Frequency f(x)
g t ∗sd T
x T Wind Speed (x) [m/s] Figure 2.17 Probability of exceedance relationship in relation to parent wind speed parameters
As discussed earlier, distributions of wind speeds are positively skewed, making the assumption of normally distributed deviations in wind speeds invalid. Therefore, Figure 2.17 is blending two concepts: (1) the distribution plotted exhibits a positive skew matching the expected shape of both a parent and EV distribution of wind speeds; and (2) the relationship between distribution parameters used when computing probabilities of exceedance are superimposed over the curve but are predicated on the assumption of normally distributed variations. The use of Equation 2.17 to compute G t, T , or Equation 2.19 to compute x t,T , are commonly used by modelers seeking an algorithmic methodology for computing both values. The wind speed value dividing the shaded area from the remaining unshaded area in Figure 2.17 is called an exceedance threshold. For the distribution plotted in Figure 2.17, the exceedance threshold is equal to 60 m/s. This threshold is a 95% exceedance threshold, where the non-shaded region under the curve contains 95% of the wind values recorded in the T second interval, and the shaded blue area under the curve contains 5% of the wind data recorded during the same T second interval. The term exceedance threshold means that 95% of the data does not exceed, and 5% of the
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Texas Tech University, Joseph Dannemiller, May 2019 data does exceed, the threshold. In practice, different statistical models use 90%, 95%, 99% and 99.9% exceedance thresholds. However, the 99% and the 99.9% exclusionary limits are difficult to show in illustration and therefore the 95% exclusionary limit is illustrated in Figure 2.17. Accurate estimates of C , computed using Equation 2.17, depend on a value of g t computed using large amounts of historical data. One such method for computing gust factors that has been verified using historical data is the Engineering Science Data Unit (ESDU) method (EDSU, 1983). The ESDU method computes gust factors as 2s peak wind speeds divided by the mean wind speed in a 3600s windows. This work focuses on 600s windows but the ESDU method has been used on 600s records before (ex: Krayer and Marshall, 1992) so subsequent discussion of the ESDU method will refer to C {2,600} values. The ESDU method computes a gust factor by dividing the recorded peak 2s wind speed by an assumed mean wind speed. The mean wind speed is computed using iterative analysis employing several relationships based on Prandtl’s Law of the Wall and the log law. To compute a gust factor for each 600s window using the ESDU method the following information is needed for each 600s window: the peak 2s wind speed, an assumed or calculated surface roughness value, the height at which data was recorded, and the latitude and longitude for each platform. The latitude is used to compute the Coriolis parameter, f using the Equation 2.38.
2𝜋 𝑓 2∗ ∗ 𝑠𝑖𝑛 𝑙𝑎𝑡 2.20 86400
Where 86400 is the time in seconds it takes the earth to rotate once and 𝑙𝑎𝑡 is the latitude at which measurements are recorded. The integral time scale, 𝑇 , is computed using Equation 2.21 as a function of the height at which observations are recorded, z.
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. 𝑇 3.13∗𝑧 2.21
The last set of equations are solved iteratively to compute a friction velocity, u*, that fits the observed data to the log law according to the assumed or computed z0 value for the upwind fetch. The first equation, Equation 2.22, computes the peak wind speed 𝑈 𝜏, 𝑧 as a function of the mean wind speed in a 3600s window is denoted as
𝑈 3600, 𝜏 , the peak factor 𝑔 𝜐, 𝜏, 𝑧 and the turbulence intensity 𝐼 𝑧 .
𝑈 𝜏, 𝑧 𝑈 𝑇 ,𝜏 ∗ 1 𝑔 𝜐, 𝜏, 𝑧 ∗𝐼 𝑧 2.22
The peak factor 𝑔 𝜐, 𝜏, 𝑧 is computed using Equation 2.23.
𝜎 𝑧, 𝜏 𝑔 𝜐, 𝜏, 𝑧 𝜎 𝑧 2.23 0.557 ∗ 2∗𝑙𝑛 𝑇 ∗ 2∗𝑙𝑛 𝑇 ∗𝜐 2∗𝑙𝑛 𝑇 ∗𝜐
Where 𝑇 denotes the window duration equal to 3600s. The cycling rate denoted as 𝜐 is computed using Equation 2.24.
0.007 0.213 ∗ 𝑇 ⁄ 𝜏 . 𝜐 2.24 𝑇
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The standard deviation of winds in a 𝑇 second window is denoted as 𝜎 𝑧, 𝜏 and is computed using Equation 2.25.
𝑢 ∗7.5∗𝜂∗ 0.538 0.09 ∗ 𝑙𝑛 𝑧𝑧⁄ 𝜎 𝑧 ∗ 𝑢 2.25 0 0.156 ∗ 𝑙𝑛 ∗ 𝑓 ∗𝑧
The standard deviation of winds in a 𝑇 second windows after the record has been passed through a low pass filter equal to 1/𝜏 Hz is computed using Equation 2.26.
. 𝜎 𝑧, 𝜏 𝜎 𝑧 ∗ 1 0.193 ∗ 𝑇 ⁄𝜏 0.1 2.26
The height scaling parameter denoted as 𝜂 is computed using Equation 2.27.
6∗𝑓 ∗𝑧 𝜂 1 2.27 𝑢∗
The iterations of Equations 2.22 through 2.27 are then used with the Equation 2.28 to map the mean profile according to the log-law and iteratively solve for 𝑢∗.
𝑧 𝑈 𝑇 ,𝑧 2.5∗𝑧∗ln 2.28 𝑧
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Once a value of 𝑢∗ has been computed, the gust factor 𝐶 𝑇 ,𝜏 is then computed using Equation 2.29.
𝑈 𝜏, 𝑧 𝐶 𝑇 ,𝜏 1 𝑔 𝜐, 𝜏, 𝑧 ∗𝐼 𝑧 2.29 𝑈 𝑇 ,𝜏
One issue with the ESDU method for computing gust factors is the use of the 2.5 value in Equation 2.28. This value is actually empirically computed to bring unity to a gust factor computed using a peak 3600s gust divided by the mean of the same 3600s window (Cook, 1985a). The use of the 2.5 value can be traced back in most literature to (Beljaars, 1987) which computes the value using data and methods from (Panofsky et. al., 1977). The method actually used an equivalence relating the standard deviation of wind speeds in a boundary layer bound by the log law of 𝜎 2.5𝑢∗. The product of multiplying 2.5 by the Von Karman constant (k = 0.4) is 2.5*0.4=1 and the two effectively cancel each other out over a 3600s window. The problem with the using
𝜎 2.5𝑢∗ is that (Beljaars, 1987) actually give a different equivalent relationship of
𝜎 2.2𝑢∗ for 600s windows, not 2.5. The 𝜎 2.2𝑢∗ relationship in (Beljaars, 1987) is also only for non-extreme wind events which would exclude hurricanes. A more recent study (Masters et. al., 2010) computed 𝜎/𝑢∗ ratios in 900s windows between 2.12 to 4.69 in nine platform deployments, through three hurricanes, using the Florida
Coastal Monitoring Project towers. Not only might the variability of the 𝜎/𝑢∗ explain the differences between gust factors, it also might call into question the use of the turbulence intensity (TI) method for computing surface roughness values in hurricane environments without adjusting the 𝜎 2.5𝑢∗ equivalence to a different value. This is an important note but beyond the scope of the work herein so it will be mentioned in the
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Texas Tech University, Joseph Dannemiller, May 2019 need for further work section but surface roughness values will be computed using both the TI method and a qualitative method when processing the TTUHRT data.
It is also important to note that not only does the ESDU method compute gust factors values using iterative analysis using Equations 2.20 through 2.28 employing the
𝜎 2.5𝑢∗ equivalence, but the ESDU method also assumes that deviations in wind speeds off of a 3600s mean value are normally distributed. This assumption of normally distributed deviations is stated by (Durst, 1970) and restated in further work (Krayer and Marshal, 1992) and (Powell et al., 1995b) to name two.
One common presentation of gust factor data is to produce curves delineating the value of C with respect to the 3600s mean. One of the most common curves is the
Durst curve (Durst, 1970) where C are computed using Equation 2.38.
𝑈 𝜏, 𝑧 𝑈 3600, 𝜏 ∗ 1 𝑆𝑈∗𝑆𝐷 3600, 𝜏 2.30
Where 𝑆𝑈 is the standard normal deviate for a value equal to 1 𝜏⁄ 𝑇 . The standard deviation of 𝜏 second winds about the 3600s mean is computed using Equation 2.31.
1 1 𝑆𝐷 3600, 𝜏 𝑈 𝜏, 𝑧 𝑈 3600, 𝑧 2.31 𝑈 3600, 𝑧 3600⁄ 𝜏
The first equation is reorganized, and gust factors are computed using Equation 2.32.
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𝐶 3600, 𝜏 1 𝑆𝑈∗𝑆𝐷 3600, 𝜏 2.32
The values of 𝑆𝐷 3600, 𝜏 , SU and 𝐶 3600, 𝜏 for several durations of 𝜏, compute in (Durst, 1970) are presented below in Table 2.1.
Table 2.1 Gust factor computed in (Durst, 1970)
𝝉 𝒔 𝑺𝑫 𝟑𝟔𝟎𝟎, 𝝉 𝝉⁄ 𝟑𝟔𝟎𝟎 SU 𝑪𝒈 𝟑𝟔𝟎𝟎, 𝝉 600 0.065 0.167 0.9 1.06 60 0.115 0.017 2.1 1.24 30 0.132 0.0085 2.4 1.32 20 0.140 0.0056 2.55 1.56 10 0.150 0.0028 2.8 1.42 5 0.159 0.0014 3.0 1.48 0.5 0.165 0.00014 3.6 1.59
Krayer and Marshall (1992) focused efforts on computing gust factors in hurricane wind fields using the same methods utilized in (Durst, 1970). The work of (Krayer and Marshal, 1992) stands as a compliment to the work done in (Durst, 1970). One reason a complimentary study was necessary is that the assumption of normally distributed deviance of wind speeds off a 3600s mean has been disproven by (Balderrama et. al., 2012) in hurricane wind fields. The 3600s duration used in ESDU and Durst is also improper for use in the hurricane environment as wind data gathered in 3600s records seldom meets the definition of time series stationary as outlined in (Bendat and Piersol, 1971). Bendat and Piersol (1971) stipulates that for a record to be stationary, the mean cannot change as a function of time. In most cases, the mean wind speed, at a single location, through a 3600s period, will increase or decrease and not remain constant. It is possible for a 3600s record to meet the definitions of stationarity, but it is very unlikely as the mean wind speeds change too frequently. Krayer and
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Marshal (1992) point out that a 3600s stationary record in non-extreme wind environments does not have a physical equivalent in the hurricane environment, particularly at or near the eyewall where winds are the highest and, for many applications, the most significant. For this reason (Krayer and Marshall, 1992) used 600s window durations to compute gust factors. The legitimacy of a 600s duration meeting stationarity is reiterated in many sources, (Vickery and Skerlj, 2005) and (Paulsen and Schroeder, 2005) to name a two. For ease in comparison, (Krayer and Marshal, 1992) also report gust factor values using 3600s windows as this is the duration most sources had used up to 1992. The gust factors computed in (Krayer and Marshal, 1992) are peak 2s winds in 600s windows and are presented below in the Table 2.2.
Table 2.2 Gust factors computed in (Krayer and Marshal, 1992) Number of Standard Mean C Dataset 600s deviation value windows of C values All 600s windows 1200 1.587 0.151 Actual exposure All 600s windows 1200 1.556 0.148 Standard exposure (z0 = 0.03m) Only 600s windows containing mean wind speeds above 15m/s 438 1.517 0.108 Standard exposure (z0 = 0.03m) Only 600s windows containing mean wind speeds above 25m/s 51 1.536 0.090 Standard exposure (z0 = 0.03m)
A comparison of the Durst Curve (Durst, 1970), the Krayer and Marshal Curve (Krayer and Marshal, 1992) and a curve delineating the upper and lower bounds of the ESDU method (ESDU, 1983) are presented below in Figure 2.18.
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Figure 2.18 Comparison of gust factor curves from ESDU (1983), Durst (1960) and Krayer and Marshal (1992), reproduced from (Vickery and Skerlj, 2005)
The gust factors employed in HWIND come from (Vickery and Skerlj, 2005) according to (Powell et al., 2010). Vickery and Skerlj sought to further quantify the gust factors in the hurricane environment, but in addition to open exposure gust factors, to also provide marine gust factor values as well. The gust factors presented in (Vickery and Skerlj, 2005) are provided below in Table 2.3.
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Table 2.3 Gust factors, 𝐶 3600,2 , computed in (Vickery and Skerlj, 2005)
Open exposure gust factors 𝑪𝒈 𝟑𝟔𝟎𝟎, 𝟐 Mean wind Number of Coefficient Mean speed (m/s) 3600s of 𝑪 𝟑𝟔𝟎𝟎, 𝟐 windows 𝒈 Variation 10 471 1.66 0.11 15 480 1.61 0.09 20 261 1.59 0.09 25 64 1.58 0.07 30 23 1.57 0.07 35 7 1.56 0.05 all 1294 1.62 0.10
The issue with (Powell et al., 2010) citing the gust factors presented in (Vickery and Skerlj, 2005) is that the gust factors presented in (Vickery and Skerlj, 2005) are given as peak 2s winds about 3600s mean wind speeds. Not only is the 3600s second window improper in the hurricane wind field, but HWIND reconstructions are computed every 600s. Therefore the 𝐶 3600,2 values are converted into 𝐶 600,60 used to compute the maximum sustained wind speeds reported in HWIND reconstructions. To compute the 𝐶 600,60 from the 𝐶 3600,2 , the 𝐶 3600, 𝜏 equation is reorganized to solve for 𝑆𝐷 3600,2 . This reorganized from is presented in Equation 2.33.
𝐶 3600,2 1 𝑆𝐷 3600,2 2.33 𝑆𝑈
Equation 2.33 produces an “equivalent” standard deviation using the observed mean
𝐶 3600,2 value. The 𝑆𝐷 3600,2 is then used with the value of 𝑆𝐷 3600,2 from
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(Durst, 1960) to compute a standard deviation of 2s gusts a 600s windows, 𝑆𝐷 600,2 using Equation 2.34.
𝑆𝐷 600,2 𝑆𝐷 3600,2 𝑆𝐷 3600,600 2.34
The gust factor for 2s gusts in 600s windows is then computed using Equation 2.35.
𝐶 600,2 1 2.7131 ∗ 𝑆𝐷 600,2 2.35
Where 𝑆𝑈 2.7131 for the deviate equal to 1 2⁄ 600 . Last, the gust factor for a 60s wind can be computed using Equation 2.36.
1.32 𝐶 600,60 𝐶 600,2 ∗ 2.36 1.69
Where 1.32 and 1.69 are the Krayer and Marshall gust factors for 60s and 2s gusts about
3600s mean wind speeds, respectively. This process is used to convert the 𝐶 3600,2 gust factors in each 5m/s mean wind speed bin in (Vickery and Skerlj, 2005) to
𝐶 600,60 values. Since the period from which peak gust has changed from 3600s to 600s, the mean wind speeds also need to be converted using the methods in (Simiu and Miyata, 2006). The 600s mean is computed using Equation 2.37.
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𝜂∗𝑐 𝜏 𝑈 600, 𝑧 𝑈 3600, 𝑧 2.37 2.5 ∗ 𝑙𝑛 𝑧𝑧⁄
Using tables provided in (Simiu and Miyata, 2006) the values of 𝜂 is equal to 2.45 and corresponds to open terrain where 𝑧 0.03𝑚; and 𝑐 𝜏 is equal to 1.284 corresponding to gust duration equal to 60s. The values of 𝐶 600,60 and 𝑆𝐷 600,2 computed from (Vickery and Skerlj, 2005) Table 4 are presented in the table below, centered at the converted 600s mean wind speeds.
Table 2.4 Gust factors, 𝐶 600,60 , computed using (Vickery and Skerlj, 2005) data
Open exposure gust factors 𝑪 𝟔𝟎𝟎, 𝟔𝟎 Mean wind 𝒈 speed (m/s) # of Standard Mean observations Deviation 10.607 471 1.187 0.057 15.911 480 1.153 0.047 21.215 261 1.139 0.043 26.518 64 1.132 0.041 31.822 23 1.125 0.038 37.126 7 1.118 0.036 all 1294 1.184 0.056
Using Equations 2.33 through 2.37 to compute 𝐶 600,60 gust factors using
𝐶 3600,2 gust factors can also be used to compute 𝐶 600,2 gust factors. The
𝐶 600,2 can be compared to previous works to verify whether these techniques produce values consistent with other referenced values. The converted 𝐶 600,2 gust factors are presented in Table 2.5.
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Table 2.5 Gust factors, 𝐶 600,2 , computed using (Vickery and Skerlj, 2005) data
Open exposure gust factors 𝑪 𝟔𝟎𝟎, 𝟐 Mean wind 𝒈 speed (m/s) # of Standard Mean observations Deviation 10.607 471 1.520 0.192 15.911 480 1.476 0.175 21.215 261 1.458 0.169 26.518 64 1.449 0.166 31.822 23 1.440 0.162 37.126 7 1.431 0.159 all 1294 1.485 0.179
The converted 𝐶 600,2 gust factors by bin range from 1.431 to 1.520 and the converted mean 𝐶 600,2 gust factor for all the data is 1.485. This matches well with the 𝐶 600,2 values in (Paulsen and Schroeder, 2005) presented in histogram form in Figure 2.19.
Figure 2.19 𝐶 600,2 gust factors computed using TTUHRT platform data where z0 is between 0.03m and 0.07m, reproduced from (Paulsen and Schroeder, 2005)
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The 𝐶 600,2 gust factors in Table 2.5 range from (1.431,1.520). This range is near the center of the 𝐶 600,2 histogram from (Paulsen and Schroeder, 2005). It is hard to tell exactly where the mean and mode of the tropical gust factors are but the values (1.431,1.520) do appear to be close to center showing good agreement with past
TTUHRT work. Additionally, (Giammanco et.al., 2012) tabulated 10m 𝐶 600,3 values using TTUHRT WEMITE and Florida Coastal Monitoring Project (FCMP) tower data gathered during hurricane landfalls from 2004 to 2008. The 𝐶 600,3 values presented in (Giammanco et.al., 2012) are illustrated in Figure 2.20.
Figure 2.20 𝐶 600,3 gust factors computed using TTUHRT and FCMP platform data, reproduced from (Giammanco et.al., 2012)
While Figure 2.20 illustrates 𝐶 600,3 data, that will be lower than the 𝐶 600,2 data, the differences should not be so large as to preclude comparing the two sets of
𝐶 600, 𝜏 values. Again, the 𝐶 600,2 gust factors in Table 2.5 show good agreement with the data in (Giammanco et.al., 2012). The range of (1.43127,1.52005) lies above and below the blue line mapping the mean trend of 𝐶 600,3 values versus mean wind speed.
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Both (Paulsen and Schroeder, 2005) and (Giammanco et.al., 2012) are past works completed by members of the TTUHRT team. Examples of work by researchers outside the TTUHRT team are also necessary to validate the 𝐶 600,2 gust factors in Table 2.5 for processing HWIND wind speeds. One work is (Balderrama et al., 2012) where 600s gust factor curves are presented for various ranges of TI. Through a range of TI from 0.15 to 0.3 (z0 between 0.0127 and 0.357), all with mean wind speeds above
20m/s, the 𝐶 600,60 spans from 1.12 to 1.23. These values are approximated visually from six figures presented in (Balderrama et al., 2012). The World Meteorological
Organization (WMO) computed a mean value of 𝐶 600,3 equal to 1.38 (WMO,
2010). Cao et al. (2015) computed a mean 𝐶 600,3 values from 1.351 to 1.416 for various ranges of mean wind speed. All of these show good agreement with the
𝐶 600,60 gust factors in Table 2.4, and the 𝐶 600,2 gust factors in Table 2.5.
The 𝐶 600,60 gust factors in Table 2.4 are for open (z0 = 0.03m) only. It is important to note that gust factors, and the standard deviation of gust factors, vary greatly based on roughness changes. Independent studies have been completed to quantify condition specific gust factors based on: a specific storm, a specific site and a specific observation system. These conditional studies attempt to tease out storm specific, and site specific, effects to quantify the effects of the upwind terrain on the boundary layer wind field. One such study recently quantified gust factors for permanent surface observation systems (particularly ASOS stations), reported in radial bins, so engineers and scientists can process data recorded at those sites after future hurricane events (Masters et. al., 2010). Data like this is very valuable. However, for locations where such a study has not been completed, statistical methods derived from historical data are still the preferred method to predict gust factor values. Therefore, the
𝐶 600,60 gust factors in Table 2.4 will be used to compute mean wind speeds using the maximum sustained wind speeds taken from HWIND reconstructions.
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Density of Surface Observation System Deployments Unfortunately, surface observation systems are expensive to build, operate and maintain. Due to the expensive nature of research and ASOS systems, observational data is not gathered in sufficient density to provide useable wind speed information for most of the geography affected by hurricane landfall. To demonstrate the geographic sparsity of surface observation systems, Figure 2.21 illustrates the locations of seven research grade surface observations systems deployed during the 2005 landfall of Hurricane Katrina. As a companion, the locations of ASOS/AWOS stations in Gulf Coast states is illustrated in Figure 2.22.
Figure 2.21 Locations of TTUHRT, Florid Coastal Monitoring Project and Louisiana Monroe atmospheric measurement platforms during the 2006 landfall of Hurricane Katrina (Giammanco et al, 2006)
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Figure 2.22 Locations of Automated Surface Observation Stations (ASOS) and Automated Weather Observation Stations (AWOS)
The observational systems in Figure 2.21 and Figure 2.22 are represented by markers that far exceed the size of the platforms themselves. If the size of each marker is assumed to be representative of the area over which the data captured is valid, large swaths of land remain where no recorded wind data is viable for analytical applications. For slab claim cases, alternative sources of wind data are necessary to provide useable wind data over the entire region affected during hurricane landfall. One such alternative is the use of a hurricane simulation model to identify wind speeds at specific site.
HWIND Hurricane Wind Field Model One of the largest hurricane models offering data to the public is the HWIND model from The National Oceanic and Atmospheric Administration’s (NOAA) Hurricane Research Division (HRD) (Powell, 1996; Powell, 1998). The HWIND model has been developed over the last 25 years and is now one of, if not the, foremost models for hurricane wind field reconstruction. The Federal Emergency Management Agency (FEMA, 2006) report outlining the devastation from Hurricane Katrina, stated HWIND is “…the best-known model for predicting wind variation” in
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Texas Tech University, Joseph Dannemiller, May 2019 hurricane wind fields. This assessment stems from NOAA HRD scientists’ stated objective of creating HWIND to produce hurricane wind fields in a manner sufficient to meet the needs of both the atmospheric science and wind engineering communities (Powell et al, 1996). HWIND assimilates as much atmospheric data as possible to seed wind field reconstruction, including data from all of the observation systems illustrated in Figure 2.6. The timing of each record is synchronized and each record is then split into sequential 10min windows. The reconstructions are completed on a window by window basis where all observational data gathered during the same window is aggregated to feed a single reconstruction. The mean recorded by each system, in a single window, is computed and then standardized to 10𝑚 over open
(z 0.03𝑚) and marine (z 0.001𝑚) exposures. Gust factors are used to compute maximum sustained wind speeds, the peak 60𝑠 of wind occurring in a 600s window, using the standardized means. The gust factor used for each observational system is a function of whether data was gathered over land or water and, for observations over water, the magnitude of the mean wind speed. For more information, the process is explained, in detail, in (Powell et al, 2010). Once all observational data has been standardized, the location of all observations is mapped according to latitude and longitude. An example of this is shown in Figure 2.23 for the reconstruction of the Hurricane Katrina wind field at 6:00 UTC on August 29, 2005.
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Figure 2.23 Observations aggregated into the HWIND wind field simulation of Hurricane Katrina at 6:00 UTC on August 29, 2005, NOAA
HWIND provides figures like Figure 2.23 to show the locations of all observations used in each HWIND wind field reconstruction. While it is difficult to see the coastline in Figure 2.23, the location and scale of both Figure 2.21 and Figure 2.23 are nearly identical to provide ease in comparison. The case in Figure 2.23 is chosen specifically to compare the number of observations used in HWIND versus the number of research grade surface observation systems gathering data during landfall, illustrated in Figure 2.21. As stated earlier, surface observations are preferred to many other observation systems like those shown in Figure 2.23 because the data gathered by non-surface observation systems can be difficult to use directly in SPA. However, HWIND uses these observations together to reconstruct a wind field which benefits from the volume of measurements and, in the aggregate, balances out any problems
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Texas Tech University, Joseph Dannemiller, May 2019 stemming from a device used to capture data, or the geography surrounding each observation. The wind speed reported in HWIND reconstructions are maximum sustained wind speeds over open (z 0.03𝑚) and marine(z 0.001𝑚) exposure. The gust factor used to compute maximum sustained wind speeds over open exposures comes from (Vickery and Skerlj, 2005) according to (Powell et. al., 2010) and was shown in the previous section to be 1.18 per Table 2.4. Using the maximum sustained wind speeds reported over land, the mean wind speed at any location in a reconstruction can be computed using a rearranged form of Equation 2.16, and setting G 60s, 600s 1.18. The resulting form is shown below in Equation 2.38.
U 60s, 600s U 600s 2.38 1.18
Where the HWIND reported maximum sustained speeds are denoted as U 60s, 600s and the mean wind speed from HWIND reconstructions, over open exposure, are denoted as U 600s . Care must be taken to ensure that the maximum sustained wind speeds used in Equation 2.38 occur over land and not over water. This is reasonably easy to do as the HWIND datafiles illustrate discontinuities in isotachs when plotted. An example of an HWIND reconstruction plot is provided in Figure 2.27 showing HWIND reconstruction of the Hurricane Katrina wind field 06:00 UTC on August 29, 2005. Figure 2.27 verifies wind speed values reported over land are adjusted back to open exposure as discontinuities in lines of equipotential wind speeds can be observed at the coastal boundary. One such case is highlighted in red.
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Figure 2.24 NOAA HRD created figure showing the HWIND reconstruction for Hurricane Katrina at 06:00 UTC on August 29, 2005
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The first HWIND reconstruction was completed for the 1992 landfall of Hurricane Andrew and according to (Powell et al, 1996), simulated wind speeds corresponding to marine observations were accurate within 7%, and simulated wind speeds corresponding to land-based observations were accurate within 10.5%. The largest reconstruction, Hurricane Katrina, including 189 land, coastal and marine observation platforms, 400 dropsondes from NOAA WP-3D and United States Air Force C-130 flights, ground based radar and satellite observations (Powel et al., 2010). Unlike the Andrew reconstruction, no percent error values were reported for the Katrina reconstruction. To date, the only independent investigation of HWIND reconstruction values was completed by (Dinapoli and Bourassa, 2012). The stated objective in (Dinapoli and Bourassa, 2012) was to quantify the level of uncertainty in HWIND computed by non-NOAA HRD scientists. The (Dinapoli and Bourassa, 2012) analysis focused on offshore HWIND data from six hurricanes: the 2004 landfalls of Hurricanes Charley and Ivan, the 2005 landfalls of Hurricanes Katrina, Rita and Wilma, and the 2007 landfall of Hurricane Felix. Four sources of errors were identified: “random observation errors, relative biases between observation types, temporal drift resulting from combining non-simultaneous measurements into a single analysis and smoothing and interpolation errors.” The sole purpose of their analysis was to quantify uncertainty for observations occurring over marine exposure and they found that HWIND simulation values are accurate within 7.5% to 13% depending on the measurement platform/system. Again, Dinapoli and Bourassa (2012) ignored all land-based observations. While it did not study the HWIND model, (Vickery et al., 2000) did quantify the uncertainties between the peak wind speeds computed by their hurricane wind field model and observed peak wind speeds and reported percent errors based on gust wind speed ranges. The reported statistics are reproduced in Table 2.6 below.
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Table 2.6 Percent errors between simulated and observed peak wind speeds as reported by (Vickery et al., 2000) Percent Error (%) for Percent Error (%) for Gust wind speed r/Rmax ≤ 2 2 < r/Rmax ≤ 4 ranges(m/s) Sample Standard Sample Standard Mean Mean Size Deviation Size Deviation 15≤U 60s, 600s <25 105 26.09 23.19 217 19.47 12.97
25≤U 60s, 600s <35 169 19.09 13.87 202 5.98 9.15
35≤U 60s, 600s <45 35 11.38 10.11 14 -5.82 3.18
U 60s, 600s 45 3 4.12 4.79 0 0 0
Where the percent errors are broken into two categories governed by the ratio of the distance from the storm center to the observed gust wind speed denoted as r, divided by the distance to the maximum wind speed denoted as Rmax. The percent differences are larger at lower wind speeds where more data is available than in the bins containing higher gust wind speeds. This data will be used a reference point for the TTUHRT vs HWIND comparisons.
HWIND Data in SPA As HWIND reconstructions, unlike observational data, provide wind data for the entire geographic region affected by hurricane landfall, HWIND stands as a potential solution for the main objective of this work: forwarding a model to simulate distributions of extreme winds for use in stochastic SPA at any site. Before the HWIND data can be used, it must first be validated over land. Such a validation stands as a companion to work completed by (Dinapoli and Bourassa, 2012) who validated HWIND data over marine exposure. Fortunately, the TTUHRT data used by HWIND scientists when reconstructing hurricane wind fields, is available, as are all HWIND reconstructions prior to 2015. This makes a direct comparison between TTUHRT Data and HWIND reconstructions possible. If such a comparison between
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TTUHRT and HWIND data shows good agreement, the HWIND wind data can be used directly in deterministic SPA or are as a starting point when estimating EV distribution parameters for extreme winds useable when computing probabilities in stochastic SPA.
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CHAPTER 3
HWIND AND TTUHRT DATA PROCESSING Data Selection The first step in using HWIND wind field reconstructions to simulate EV wind speed distributions is to verify HWIND values using TTUHRT data. TTUHRT data is used by HWIND scientists in computing HWIND wind fields so selecting hurricane data for the verification process requires the existence of both TTUHRT and HWIND data. HWIND data exists for every tropical storm and hurricane since 1998 so the limiting factor in selecting storms was the existence of data in the TTUHRT archives. Specifically, information pertaining to the deployment locations for TTUHRT platforms, the time at which each platform began recording data, the height at which data was recorded, and the frequency of data capture for each deployment. After looking through TTUHRT archives, all necessary data for the following storms was available: the 1998 landfall of Hurricane Bonnie, the 2003 landfall of Isabel, the 2004 landfall of Frances, the 2005 landfalls of Dennis, Katrina and Rita, and the 2008 landfall of Ike.
HWIND Data Processing All HWIND data processed herein comes from shapefiles provided by NOAA HRD’s historical archives. Each shapefile has over 20,000 points, each containing the latitude, longitude, wind speed and wind direction. The points create a rectangular grid, the bounds of which are set by HWIND scientists. The wind speeds, as discussed in Chapter 2, are maximum sustained wind speeds (peak 60𝑠 average wind speed in a 600𝑠 window), at 10𝑚 height, over open or marine exposure. The exposure characterization is determined by HWIND scientists where observations occurring over land are provided as maximum sustained wind speeds occurring over open exposure (𝑧 0.03𝑚), and observations occurring over water are provided as maximum sustained wind speeds occurring over marine exposure (𝑧 0.001𝑚). NOAA HRD also provides figures illustrating the magnitudes of the HWIND
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Texas Tech University, Joseph Dannemiller, May 2019 reconstructed wind fields for sequential 600𝑠 windows through the duration of a hurricane event. An example of such a figure is provided below, Figure 3.1, for the 00:00UTC August 29, 2005 reconstruction of Hurricane Katrina.
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Figure 3.1 NOAA HRD created figure showing the HWIND reconstruction for Hurricane Katrina at 00:00 UTC on August 29, 2005
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The smoothing of the wind field in Figure 3.1 is accomplished using a spline interpolation applied across the 20,000+ points for which wind speeds have been computed. Because 20,000+ points are difficult to show without a graphic becoming so dense the points appear as one solid color, the background grid used to seed a reconstruction is shown as a substitute on the right-hand side of Figure 3.2. The background grid is magenta, and the wind field (minus the header and footer information) from Figure 3.1 is provided side-by-side for context.
Figure 3.2 Grid points employed in HWIND reconstruction
The number of background wind field points illustrated in Figure 3.2 is less than 1,500. It is for this reason, that the background wind field is used to illustrate the gridding as the 20,000+ reconstructed is quite substantial. Figure 3.3 shows an enlargement, centered at the eye, of the pictures in Figure 3.2, to provide clear distinction between background grid points, and observations used to seed the reconstruction. Figure 3.4 shows an even closer enlargement, to add even more clarity.
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Figure 3.3 First enlarged view of the grid points employed in HWIND reconstruction
Figure 3.4 Second enlarged view of the grid points employed in HWIND simulation
The locations of all TTUHRT deployments were verified as being reported over the parts of HWIND reconstruction domain utilizing open exposure (z 0.03𝑚) terrain. This is verified for all HWIND reconstructions and as an example, the HWIND reconstruction data for the 00:00UTC August 29, 2005 reconstruction of Hurricane Katrina is plotted with the locations of the TTUHRT platforms superimposed using white shapes.
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Figure 3.5 Wind speeds in m/s from the NOAA HRD HWIND reconstruction of Hurricane Katrina on 00:00UTC August 29, 2005
The locations of the TTUHRT platforms occur inland according to the coastline plotted in Figure 3.5. Verification that the TTUHRT platforms are reported over open exposure is also accomplished by looking for discontinuities in equipotential wind speeds at the coastal boundary. Figure 3.5 shows the same wind field reconstruction shown in Figure 3.1 except the units used to create Figure 3.5 are m/s, while the units used to create Figure 3.1 are knots. Illustrating a verification of TTUHRT platforms
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Texas Tech University, Joseph Dannemiller, May 2019 deployments occurring on land, and not over water, is the focus of Figure 3.5. All future HWIND analysis herein is completed in m/s. Publicly available HWIND data, for 600𝑠 windows, are only available at 3- hour increments through the life of a hurricane event. The data for every 600𝑠 window through the life of an event is available for purchase, but was cost prohibitive for this analysis so only the publicly available data is used. Processing HWIND data involves computing the following values at the locations of TTUHRT platform deployments: (1) the HWIND reported maximum sustained wind speed, (2) the maximum wind speed recorded in storm relative quadrant containing the TTUHRT platform, (3) the distance from the center of the storm to the TTUHRT deployment location, and (4) the distance from the center of the storm to the maximum wind speed in the storm relative quadrant containing the TTUHRT platform. Storm relative quadrants are shown in Figure 3.6, and are important to separate as storm mechanics influence the wind field differently in each quadrant. By classifying each HWIND reported wind speed by storm relative quadrants, observations from different storms, can be compared together to identify trends.
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Figure 3.6 Storm relative quadrants in a hurricane
The center of a hurricane in a reconstruction is identified by finding the single grid point with a wind speed equal to zero. This point is used to measure both the distance from the center of the storm to each TTUHRT deployment, denoted as R , and the distance from the center of the storm to the radius of maximum winds in the quadrant containing each TTUHRT deployment, denoted as R . For many of the HWIND reconstructions, the locations of TTUHRT platforms do not fall in the reconstruction domain. An example of this is shown in Figure 3.7 for the 12:00UTC August 26, 2005 reconstruction of Hurricane Katrina, where the TTUHRT platforms are deployed in Louisiana, but the storm can be seen just passing the Florida peninsula.
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Figure 3.7 Wind speeds in m/s from the NOAA HRD HWIND reconstruction of Hurricane Katrina on 12:00UTC August 26, 2005
HWIND reconstructions, like the one shown in Figure 3.7, net no useable information to compare against TTUHRT data and are therefore discarded. In reconstructions where the TTUHRT platforms do fall in the simulation domain, the TTUHRT
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Texas Tech University, Joseph Dannemiller, May 2019 platforms themselves do not fall exactly at the location of a single HWIND reconstruction grid point. Therefore, the HWIND reconstruction maximum sustained wind speeds at the four grid points around the TTUHRT deployment are used to interpolate a value at the exact location of the TTUHRT deployment. The interpolated maximum sustained wind speed is denoted U 60s, 600s , and is used in Equation 2.38 to compute a mean wind speed, denoted as U 600s . A summary of data from HWIND reconstructions for the 2005 landfall of Hurricane Katrina, including the 00:00UTC August 29, 2005 reconstruction illustrated in Figure 3.5, is provided in Table 3.1.
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Table 3.1 HWIND reconstruction data at the location of the TTUHRT WEMITE #1 during the 2005 landfall of Hurricane Katrina Radius of Maximum Maximum Radial Distance Mean Wind Winds for Sustained of TTUHRT Speed Location of Quadrant Date Time Wind Speed Probe to the R ⁄R U 600s TTUHRT Probe Containing U 60s, 600s Storm Center [m/s] TTUHRT [m/s] (R ) [km] Platform R [km] 08/28/2005 18:00UTC 6.45 5.47 451.39 Quadrant II 21.62 20.88 08/28/2005 21:00UTC 7.46 6.32 392.02 Quadrant II 21.54 18.20 08/29/2005 00:00UTC 9.84 8.34 342.36 Quadrant II 24.13 14.19 08/29/2005 03:00UTC 13.60 11.53 290.13 Quadrant II 36.25 8.00 08/29/2005 06:00UTC 16.04 13.59 232.89 Quadrant II 36.25 6.43 08/29/2005 09:00UTC 16.95 14.36 171.91 Quadrant II 54.26 3.17 08/29/2005 12:00UTC 23.70 20.08 125.21 Quadrant II 60.38 2.07 08/29/2005 15:00UTC 24.54 20.80 112.08 Quadrant III 36.25 3.09 08/29/2005 18:00UTC 17.95 15.21 159.54 Quadrant III 36.25 4.40 08/29/2005 21:00UTC 15.33 12.99 240.92 Quadrant III 48.26 4.99 08/30/2005 00:00UTC 8.83 7.48 329.56 Quadrant III 71.72 4.60
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Table 3.1 shows several interesting things about the location of WEMITE #1 during Katrina’s landfall. Per the quadrant delineation illustrated in Figure 3.6, the WEMITE #1 was in quadrant II until 15:00UTC on 8/29/2005, after which WEMITE #1 was in quadrant III for the duration of the landfall event. At the same time, the WEMITE #1 was located in quadrant II, the distance R decreases as the storm moves on land. Once the of the eye makes it on land it then passes WEMITE #1 at which time the deployment location falls in quadrant III and R continually increases as the storm moves inland. Also, through the landfall, and up until the eye passes the coastal boundary between 12:00UTC and 15:00UTC, the distance R continually increases as the storm changes terrain environments from marine, to various land exposures. Figures plotted using HWIND data files, for each hurricane used herein, are provided in Appendix A1, and Tables containing HWIND reconstruction wind speed data and distance like Table 3.1 are provided in Appendix A2.
Texas Tech University Hurricane Research Team Data Processing Methods Processing TTUHRT data records involves computing parameters for both the parent wind speed distribution and the extreme value wind speed distribution. Each time history is split into in sequential windows, each with a duration T 600𝑠 to remain consistent with the window duration used in HWIND reconstructions. However, prior to splitting each record, two moving averages (MA), a 3𝑠 and a 60𝑠, are applied separately and separate time records are kept for each of the three gust durations: (1) t equal to the frequency at which data was captured, (2) t 3𝑠 and (3) t 60𝑠. Each record is then split into sequential 600s segments henceforth referred to as windows. HWIND reconstructions report maximum sustained wind speeds (the peak 60𝑠 of wind in a 600𝑠 window) in 600𝑠 windows. Applying a 60𝑠 MA to each time history, and then identifying the peak wind speed value in the time history, ensures the gust duration of the observed TTUHRT peak wind speed matches the gust duration used in HWIND reconstruction. Henceforth, the time history saved after a 60𝑠 MA is applied is referred to as the 60𝑠
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MA time history data, or the 60𝑠 MA data for short. A second MA is applied to the raw time history and the gust duration for this MA is 3𝑠. The 3𝑠 gust duration matches the gust duration identified in ASCE 7 used when computing localized wind pressures. As the goal of this work is to create a method for estimating extreme wind speeds for use in SPA, estimating EV distribution parameters for 3𝑠 gust duration wind speeds, is required to analyze structural components. Henceforth, the time history saved after a 3𝑠 MA is applied is referred to as the 3𝑠 MA time history data, or the 3𝑠 MA data for short. The original time history, and the parameters computed from it are referred to as the raw time history data, or raw data for short. Numerical software was used to apply both the 3𝑠 and 60𝑠 moving averages. Because of how the MA is applied, some of the data must be trimmed from each of the raw, 3𝑠 and 60𝑠 time histories. The data that is lost occurs at the beginning and the end of each time history. To explain why this happens, the case of applying the 60𝑠 MA can be explored. The software averages 60𝑠 of data and reports the new value at the last time stamp of the averaging period. Unfortunately, if 60𝑠 data is kept this way, the result would be the raw data and the 60𝑠 data being 30𝑠 out of phase in the temporal domain. Therefore, the 60𝑠 time history is adjusted 30𝑠 back in time ensuring consistent temporal reporting for sequential data points in each of the raw and 60𝑠 time histories. A similar sequence of steps is completed when applying the 3𝑠 MA with the data being adjusted 1.5𝑠 backwards. Lastly, for the 60𝑠 MA time history, the first 30𝑠, and the last 30𝑠, of data is lost as these two periods contain no wind information. To ensure that all three time histories, raw, 3𝑠 MA and 60𝑠 MA, remain in sync, the first and last 30𝑠 of data in each time history is removed. With this process complete, all three time histories begin at the exact same time, and end at the exact same time. Each time history is then divided into sequential 600𝑠 windows to compute both parent wind field, and extreme wind field parameters. An example of a raw time history in comparison to the time history after a 60s moving average has been applied is presented below in Figure 3.8.
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10min window beginning 24-Sep-1998 09:15:00 20
15
10
5 09:15 09:16 09:17 09:18 09:19 09:20 09:21 09:22 09:23 09:24 Time
Figure 3.8 Wind speed time history for the raw data (blue) and the 60s MA data (red) recorded from 09:15UTC to 09:25UTC by TTUHRT WEMITE #1 during the 1998 landfall of hurricane Bonnie
Figures and tables illustrating and enumerating the summary statistics for the remaining 600s windows in the time history recorded by TTUHRT WEMITE #1 during the 1998 landfall of Hurricane Bonnie are presented in Appendix A3. The total number of sequential 600𝑠 windows in a time history, denoted as w , is computed using Equation 3.1.
T w 3.1 T
Where the duration of each window is denoted as T , and the total time for each
TTUHRT platform record is denoted as T . In this work, T will always equal 600𝑠. However, future work may investigate different window durations, and using a variable reference maintains consistency with the abstract concepts discussed in
Chapter 2, so henceforth the duration of each window is referred to as T . The total number of wind speed measurements in each window, denoted as n , after the truncation of the data record, is computed using Equation 3.2.
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𝑛 𝑓𝑟𝑒𝑞𝑢 ∗𝑇 3.2
Where the frequency at which each TTUHRT platform records data is denoted by frequ. The enumeration of windows in a TTUHRT time history is delineated using the j subscript, equal to one through n . The mean and standard deviations of wind speeds, for t second gusts, in T second windows, is computed using Equations 3.3 and 3.4, respectively, where the mean values are denoted as x t, T , and standard deviations are denoted as sd t, T .
1 x t, T x t 3.3 n
1 sd t, T x t x t, T 3.4 n
Each value in a time history is denoted as x t where the gust duration is specified in the curly brackets. The turbulence intensity, denoted as TI t, T , and the gust factor, denoted as G t, T are computed using Equations 3.5 and 3.6, respectively.
sd t, T TI t, T 3.5 x t, T
x t, T G t, T 3.6 x t, T
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Where TI t, T is computed by dividing sd t, T by x t, T and is sometimes referred to as the turbulence intensity ratio but herein, 𝑇𝐼 𝑡, 𝑇 will be referred to as simply, turbulence intensity. The turbulence intensity represents a coefficient of variation for t second winds in a T second period measuring the relative variance of deviations in wind speed about a center value. The gust factor, G t, T , for each window is computed by dividing the single maximum wind speed of t second duration, denoted as x t, T , by the mean of t second duration winds in a T second record. The formulation of Equation 3.6 follows the discussion of Equation 2.16 for recorded time history data. The distribution of extreme wind speeds is fit with an Extreme Value (EV) Type-I distribution. Prior to computing the EV location and scale parameters for each 600𝑠 window, each window must be subdivided into smaller, sequential segments. The segment time used herein is equal to 60𝑠, resulting in 10 equal duration segments in each window. The segments duration was selected by first plotting the autocorrelation function of each window and determining the magnitude of the first zero crossing. For all 600𝑠 windows used herein, the first zero crossing occurred before 60𝑠. Statistically, values in a random process cannot be accurately predicted past the time duration indicated by the first zero crossing, i.e. data points separated by more time than the first zero crossing are independent. The smallest, and largest, values computed for the first zero crossings were 42 and 58 seconds, respectively.
The segment duration, T , is set equal to 60s as it both exceeds the 58 second value, and evenly divides each 600𝑠 window into 10 equal segments. Another segment duration can be used in the future, so the number of segments in a window, denoted as s , is presented in abstraction and is computed using Equation 3.7.
T s 3.7 T
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The number of data points per segment, denoted as n , is computed using Equation 3.8.
n n 3.8 s
A peak wind speed in each segment k, in each window j, denoted as x , t, T , is recorded for each window. An illustration of all 10 denoted as x , t, T values is provided below in Figure 3.9 where the 10 peak wind speeds are circled in red. To compute the extreme value Type-I location and scale parameters for each window, denoted as u t, T and σ t, T , respectively, the 10 x , t, T values in a window are collected and maximum likelihood estimation (MLE) is used to compute u t, T and σ t, T . The MLE method is completed in the numerical software package. It is important to note that the algorithm is built to predict negative extremes, so the peak wind speed values extracted from the TTUHRT records,
𝑥 , 𝑡, 𝑇 , were multiplied by negative one to flip their signs prior to using the MLE algorithm to compute location and scale parameters. An example window record is illustrated in Figure 3.9 where the segments are separated by black vertical lines, and the peak wind speeds in each segment are circled in red.
Figure 3.9 Peak wind speed values for 60-sec segments
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The equations to compute u t, T and σ t, T , and the form of the likelihood function are shown in Equations 3.9 through 3.12. In the process of computing EV parameters applicable to a 600s window duration, the EV parameters are computed for the 60s segment duration and are then transformed. To start, Equation 3.9 computes the likelihood function for the peak wind speeds in all 10 segments, conditional upon values of EV location and scale parameters about the segment duration, denoted as μ t, T and σ t, T , respectively. As Equation 3.9 would become too long to communicate using the previous variable formats, the values of μ t, T and σ t, T have been truncated to 𝜇 and 𝜎 , and the values of
𝑥 , 𝑡, 𝑇 have been truncated to 𝑥 , .
𝐿 𝑥 , …𝑥 , |𝜇 ,𝛼 3.9
𝑃 𝑥 , |𝜇 ,𝛼 ∗𝑃 𝑥 , |𝜇 ,𝛼 ∗…∗𝑃 𝑥 , |𝜇 ,𝛼
Equation 3.9 converges upon values of 𝜇 and 𝜎 that maximize the likelihood function, L. To maximize L, the conditional probabilities of occurrence for 𝑥 , are multiplied together. The equation to compute the conditional probability for each 𝑥 , is computed using Equation 3.10.
∗ ∗ , , 3.10 𝑃 𝑥 , 𝜇 ,𝛼 𝛼 ∗𝑒 ∗𝑒
Lastly, the point estimates for location and scale parameters are computed by solving Equations 3.11 and 3.12 simultaneously.
, ∑ 𝑥 , ∗𝑒 0 𝑥 ̅ 𝛼 3.11 , ∑ 𝑒
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1 , 0 𝛼 ∗𝑙𝑜𝑔 𝑒 𝜇 3.12 𝑛
The EV location and scale parameters computed so far are, again, the parameters for t second gust winds about the segment duration, T . To compute the location and scale parameters about the window duration, denoted u t, T and σ t, T Equations 3.13and 3.14 are employed (Cook, 1985a).
T u t, T μ t, T 𝑙𝑛 ∗σ t, T 3.13 T
σ t, T σ t, T 3.14
The mean and standard deviations of the EV distribution, denoted as MEV t, T and
SDEV t, T , are computed using Equations 3.15 and 3.16, respectively (Simiu and Scanlan, 1996).
MEV t, T u t, T 0.5772 ∗ σ t, T 3.15
𝜋 SDEV t, T σ t, T 3.16 √6
After the parent and EV wind field parameters have been computed, the windows are checked for validity, stationarity and neutral stability. All three of these qualifications are discussed further but to start with, instrument error was identified in several records by: the presence of erroneous spikes in wind speeds (either hundreds of m/s or negative values), the absence of wind data for short time periods, and the absence of fluctuations in the recorded signal (usually manifesting in hours of recording wind speed equal to 0 m/s). If the errors only occurred during a portion of a
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Texas Tech University, Joseph Dannemiller, May 2019 platform time history, then the erroneous windows were removed. In some cases, entire time histories were problematic and had to be removed. Problems included: the 2005 landfall of Hurricane Katrina when the WEMITE #1 tower suffered a power failure and did not record data past the onset of peak winds, and during the 2005 landfall of Hurricane Rita the PMT Black tower was struck by debris several hours into the deployment. In the Rita case, the tower did not record valid data post impact and therefore data recorded post-impact was unusable. Also, during Hurricane Rita, the WEMITE #1 tower suffered a power failure like the failure it experienced during Hurricane Katrina. Past these, several TTUHRT StickNet towers recorded data days past the dissipation of hurricane wind fields and sporadic windows exhibited errors resulting in those windows being eliminated. Windows were then eliminated that did not meet the criterion of neutral stability. Neutral stability is a boundary layer characteristic occurring when all fluctuation in wind speed are the result of mechanical disturbances in the wind field. This means that all turbulence, or fluctuations in wind speed are a function of, and only of, the surface roughness at the base of the boundary layer. One source of fluctuations that is not mechanically generated is the rising of air pockets as the ground surface radiates latent heat absorbed from the sun. These rising air pockets disturb the wind field, but when the mean wind speed rises above a threshold, these effects become marginal to nonexistent. To ensure data meets the criteria of neutrally stability, a minimum x t, T threshold of 15 m/s is established. All windows having x t, T 15m/s are discarded. One additional benefit to discarding windows having x t, T 15m/s is an added focus on records possessing higher wind speeds. It was mentioned, in Chapter 2, that it is uncommon for structural failures to occur at lower wind speeds and one targeted objective herein is to forward a model for use in SPA. Therefore, focusing on records possessing higher wind speeds optimizes the data used to build such a model. The last condition for 600s windows is that each window kept for analysis must meet the criteria of stationarity. Stationarity has several classical definitions
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Texas Tech University, Joseph Dannemiller, May 2019 across different fields of statistics and signal analysis. Herein stationarity implies the properties of a window, containing a truly random signal, do not change inside smaller pieces of that window (Bendat and Piersol, 1986). More specifically, where parent distribution summary statistics (x t, T and sd t, T ) do not change with time, and where no discernable pattern exists in a record, the data is said to be stationary. Phenomena that lead to non-stationarity include: (1) the mean gradually rising, or falling, through the duration of a window; (2) the standard deviation increasing or decreasing through the duration of a window, and (3) the presence of a fluctuating/cyclical/sinusoidal pattern such that data predictably increases and then decreases repeatedly. In cases like these, data records may still meet the definition of random, in that a future instance of wind speed cannot be predicted exactly using a defined mathematical relationship, but the trend can be used to predict future wind speed values, to a quantified level of level of precision. Herein, only stationary windows are used in analysis so all windows failing to meet the criteria of stationary are discarded. The time history for each window used to check for stationarity is the raw wind data time history. Each window is checked for stationarity using the Run Test (RunT) and the Reverse Arrangement Test (RAT) (Bendat and Piersol, 1986). Both the RunT and the RAT are good at detecting monotonic trends (changes in 𝑥 ̅ 𝑡, 𝑇 and sometimes in 𝑠𝑑 𝑡, 𝑇 ), however only the RunT is good at detecting fluctuating patterns. Zero windows failed both the RunT, and the RAT. One reason for this could be that windows checked for stationarity have mean wind speeds above 15m/s. Cyclical trends are less common in records containing high mean wind speeds than in records with lower means. Ultimately, for any window failing either the RunT, or the RAT, the raw, 3s MA and 60s MA records were discarded. Table 3.2 breaks out the accounting of complete 600𝑠 windows recorded by TTUHRT platforms. The first column contains the name of the hurricane and TTUHRT platform for each TTUHRT record. The second column contains the number of complete 600s windows in each platform record after the time histories are
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Texas Tech University, Joseph Dannemiller, May 2019 split into sequential, complete, 600𝑠 windows. The third column contains the number of complete 600𝑠 windows that did not exhibit any problems when recording wind data. The fourth column contains the number of complete 600s windows, exhibiting zero problems when recording wind data, with x t, T 15𝑚/𝑠 thus meeting the criteria of neutral stability. The fifth column contains the number of complete 600𝑠 windows that met all prior criterion, but failed to meet the criteria for stationarity. And, the sixth column contains the number of complete 600𝑠 windows, exhibiting zero problems when recording wind data, meeting the criteria for stationarity and neutrally stability. The number of windows, from each TTUHRT record, used in future subsequent analyses are those appearing in the sixth column of Table 3.2.
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Table 3.2 Breakdown of 600s windows from TTUHRT time histories Number of Number Number windows Number of of Number of with stationary windows of windows x t, T > windows Hurricane exhibiting complete with with no errors 15m/s windows x t, T > x t, T > in data removed for capture 15m/s non- 15m/s stationarity Bonnie WEMITE #1 209 209 59 2 57 Francis WEMITE #1 576 576 83 0 83 Francis WEMITE #2 376 376 98 0 98 Francis PMT Black 402 175 175 2 173 Ike SN101A 210 105 0 0 0 Ike SN102B 984 196 63 0 63 Ike SN103A 151 75 35 1 34 Ike SN104B 978 195 71 0 71 Ike SN105A 614 307 53 0 53 Ike SN106B 808 161 69 1 68 Ike SN107A 344 172 35 0 35 Ike SN108B 830 166 44 0 44 Ike SN109A 519 259 58 0 58 Ike SN110A 1822 911 59 1 58 Isabel WEMITE #1 283 283 103 2 101 Isabel WEMITE #2 389 389 29 1 28 Isabel PMT Black 266 266 15 0 15 Isabel PMT Clear 282 282 14 0 14 Isabel PMT White 280 280 19 0 19 Ivan WEMITE #1 103 103 4 0 4 Dennis WEMITE #1 213 213 3 0 3 Katrina WEMITE #1 143 130 29 1 28 Katrina WEMITE #2 228 228 29 1 28 Katrina PMT Black 240 231 41 3 38 Katrina PMT Clear 283 283 40 2 38 Katrina PMT White 228 228 21 1 20 Lili WEMITE #1 197 197 34 0 34 Rita WEMITE #1 46 35 0 0 0 Rita WEMITE #2 296 296 118 0 118 Rita PMT Black 39 21 6 0 6 Rita PMT Clear 426 426 108 2 106 Rita MT White 141 141 98 2 96 Total 12906 7915 1613 22 1591
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Aerial imagery was then used to assess the upwind surface roughness regimes, and the distance to the last change in upwind surface roughness in accordance to the methods of standardization discussed in Chapter 2. These data were used in Equation 2.14 to standardize x t, T computed from TTUHRT records, to 10𝑚 height over open exposure (z 0.03𝑚). This standardization of x t, T values allows for direct comparison of values between different TTUHRT records, and between TTUHRT and HWIND records. An example of an aerial image used in this process is provided in Figure 3.10 showing the deployment location of the StickNet 101A platform during the 2008 landfall of Hurricane Ike.
Figure 3.10 Aerial image of the deployment site for TTUHRT StickNet 101A captured January 2008 (© Google)
Google Earth was used to acquire an aerial image for each TTUHRT deployment site, captured as close as possible to, but still before, the time data was recorded. By only using pre-storm imagery, the destruction of the topography, foliage and structures does
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not factor into the qualitative assessment of z . The case illustrated in Figure 3.10 is for the TTUHRT StickNet 101A deployment during the 2008 landfall of Hurricane Ike. Hurricane Ike made landfall over Texas in mid-September 2008. The image used in Figure 3.10 is from eight months prior and while seasonal variations in foliage may have changed the upwind roughness this image stands as the best approximation of the surface conditions to use when assessing the surface roughness prior to a change in exposure denoted as 𝑧 , , the surface roughness after a change in exposure denoted as
𝑧 , and the surface roughness computed using Equation 2.13 denoted as 𝑧 . The z values assigned to terrain regimes were referenced from (Holmes, 2001). Table 3.3 accounts for the X, 𝑧 , , 𝑧 , , δ , δ and 𝑧 values for each HWIND reconstruction reporting time for the 2008 landfall of Hurricane Ike.
Table 3.3 Data for assessing the surface roughness of the upwind wind field for the TTUHRT StickNet 101A deployment during the 2008 landfall of Hurricane Ike Internal Transition Surface Surface Distance Boundary Boundary Roughness Roughness from Layer Layer Computed Sector Upwind of downwind Roughness Height Height Surface (degrees) the of the Change to Over the Over the Roughness [º] Roughness Roughness Tower (X) TTUHRT TTUHRT (z ) [m] Change Change [m] Platform Platform (z , ) [m] (z , ) [m] (δ ) [m] (δ ) [m] 0‐30 1400 0.50 0.05 30.99 585.15 0.050 30‐60 800 0.75 0.03 11.20 491.90 0.030 60‐90 500 0.20 0.03 13.56 218.67 0.030 90‐120 400 0.20 0.03 10.84 191.27 0.030 120‐150 300 0.20 0.03 8.13 160.95 0.032 150‐180 500 0.05 0.03 27.11 125.59 0.027 180‐210 800 0.03 0.01 32.33 135.74 0.006 210‐240 800 0.03 0.01 32.33 135.74 0.006 240‐270 2000 0.05 0.05 140.00 288.54 0.050 270‐300 1000 0.03 0.50 285.77 155.18 0.500 300‐330 1200 0.10 0.10 84.00 280.23 0.100 330‐360 2000 0.10 0.10 140.00 380.73 0.100
Processed Texas Tech University Hurricane Research Team Data
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Figures are assembled for each TTUHRT deployment raw, 3s and 60s MA time histories. Each figure contains the mean, standard deviation, EV location and EV scale parameters plotted with respect to the numbered windows through the duration of the time history. For the raw data the mean, standard deviation, EV location and
EV scale parameters are denoted x raw, 600s , sd raw, 600s , μ raw, 600s and
σ raw, 600s , respectively. The parent and extreme wind field statistics from the raw data recorded by WEMITE #1 during Hurricane Katrina are plotted in Figure 3.10.
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raw, 600s [m/s] Mean wind speed wind speed Mean x 600s windows (j) [#]
] raw, 600s m/s Standard deviation of wind speeds sd [ 600s windows (j) [#]
raw, 600s EV location EV location parameters μ [m/s] 600s windows (j) [#]
raw, 600s EV scale EV scale parameters σ [m/s] 600s windows (j) [#] Figure 3.11 Parent and extreme wind distribution parameters for the raw data recorded by TTUHRT WEMITE #1 during the 2005 landfall of Hurricane Katrina
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While recording wind data during the 2005 landfall of Hurricane Katrina, WEMITE #1 suffered a power failure. Before the platform stopped recording data all together, several complete 600s windows exhibited erroneous data capture and were removed from further analysis. In total, the 13 600𝑠 windows at the end of the WEMITE #1 record were removed. These 13 windows can be seen in Figure 3.11 exhibiting inconsistent mean, EV location and scale values, and non-existent standard deviation values. The standard deviations not included in Figure 3.10 were either negative values, or very high (100𝑚/𝑠) as the WEMITE #1 platform had begun recording wind speed measurements that belied belief. These 13 windows are the 13 windows listed in Table 3.2 as having been removed from the 143 complete 600𝑠 windows recorded by WEMITE #1. As a result, only 130 windows were checked for stationarity and neutral stability. Additionally, a thick horizontal blue line has been superimposed in the plot of x raw, 600s values at the top of Figure 3.10 to indicate the 15m/s threshold used in identifying 600𝑠 windows meeting the criterion of neutral stability. The values of sd raw, 600s , μ raw, 600s and σ raw, 600s corresponding to windows containing a 𝑥 ̅ 𝑟𝑎𝑤, 600𝑠 15𝑚/𝑠 are also circled in red. For comprehensiveness, the figures for the 3s, and 60s MA parent and extreme wind field parameters, computed from WEMITE #1 3s and 60s data, are provided in Figure 3.11 and Figure 3.12, respectively, for comparison.
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3s, 600s [m/s] Mean wind Mean speed x 600s windows (j) [#]
] 3s, 600s m/s Standard deviation of wind speeds sd [ 600s windows (j) [#]
3s, 600s [m/s] EV location EV location parameters μ 600s windows (j) [#]
3s, 600s [m/s] EV scale EV scale parameters σ 600s windows (j) [#] Figure 3.12 Parent and extreme wind distribution parameters for the 3𝑠 MA data recorded by TTUHRT WEMITE #1 during the 2005 landfall of Hurricane Katrina
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60s, 600s Mean wind Mean speed x [m/s] 600s windows (j) [#]
] 60s, 600s m/s Standard deviation of wind speeds sd [ 600s windows (j) [#]
60s, 600s EV location EV location parameters μ [m/s] 600s windows (j) [#]
60s, 600s [m/s] EV scale EV scale parameters σ 600s windows (j) [#] Figure 3.13 Parent and extreme wind distribution parameters for the 60𝑠 MA data recorded by TTUHRT WEMITE #1 during the 2005 landfall of Hurricane Katrina
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Figure 3.10, Figure 3.11 and Figure 3.12 do an excellent job illustrating the manifestation of erroneous data capture at the end of the record, and the location of the neutral stability threshold. Due to the power failure suffered by the platform, wind data was not captured during the time of peak winds for Hurricane Katrina, stopping after recording data for 143 600𝑠 windows. Figure 3.13, Figure 3.14 and Figure 3.15 illustrate the raw, 3s and 60s data for the 2005 landfall of Hurricane Katrina. One item to note in the Katrina figures is that they exhibit the characteristic rise and fall of wind speed values indicative of a storm making landfall that was absent in Figure 3.10, Figure 3.11 and Figure 3.12.
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raw, 600s [m/s] Mean wind Mean speed x 600s windows (j) [#]
] raw, 600s m/s Standard deviation of wind speeds sd [ 600s windows (j) [#]
raw, 600s EV location EV location parameters μ [m/s] 600s windows (j) [#]
raw, 600s EV scale EV scale parameters σ [m/s] 600s windows (j) [#] Figure 3.14 Parent and extreme wind distribution parameters for the raw data recorded by TTUHRT PMT Black during the 2005 landfall of Hurricane Katrina
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3s, 600s [m/s] Mean wind Mean speed x 600s windows (j) [#]
] 3s, 600s m/s Standard deviation of wind speeds sd [ 600s windows (j) [#]
3s, 600s [m/s] EV location EV location parameters μ 600s windows (j) [#]
3s, 600s [m/s] EV scale EV scale parameters σ 600s windows (j) [#] Figure 3.15 Parent and extreme wind distribution parameters for the 3𝑠 MA data recorded by TTUHRT PMT Black during the 2005 landfall of Hurricane Katrina 92
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60s, 600s Mean wind speed wind speed Mean x [m/s] 600s windows (j) [#]
] 60s, 600s m/s Standard deviation of wind speeds sd [ 600s windows (j) [#]
60s, 600s EV location EV location parameters μ [m/s] 600s windows (j) [#]
60s, 600s [m/s] EV scale EV scale parameters σ 600s windows (j) [#] Figure 3.16 Parent and extreme wind distribution parameters for the 60𝑠 MA data recorded by TTUHRT PMT Black during the 2005 landfall of Hurricane Katrina 93
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Like the WEMITE #1 data, the PMT Black time history contained 12 600𝑠 windows containing erroneous data between windows 22 and 37. These 12 windows were removed. Where the PMT Black figures excel is in showing the effect of the moving average on both the parent and extreme wind field parameters. The moving average does not affect as large a change in the mean values as it does the standard deviation,
EV location and scale parameters. Observe, the peak sd raw, 600s value in Figure
3.13 is equal to 6.4𝑚/𝑠, the peak sd 3s, 600s in Figure 3.14 is also equal to 6.4m/s, but the peak value of sd 60s, 600s in Figure 3.15 is equal to 2.8𝑚/𝑠, a difference of 3.6𝑚/𝑠 between the raw and 60s MA values. This means the 60𝑠 MA data shows less dispersion meaning wind speeds in the 60𝑠 MA time history are more closely centered around the mean of each 600𝑠 window. Similarly, the difference between peak μ raw, 600s and μ 60s, 600s is equal to 17.7𝑚/𝑠 and the difference between peak σ raw, 600s and σ 60s, 600s is equal to 2.3𝑚/𝑠. Figures showing the 600s statistics for the remaining TTUHRT platform deployments are presented in Appendix A4.
While sd t, T and σ t, T are measures of dispersion for the parent and extreme wind fields, respectively, their decrease in magnitudes after the application of a MA is not entirely a bad phenomenon. While the extent of the deviation between the peak wind speeds compared to a mean is decreased, the closer concentration of the wind data around the mean the more predictable the parent and extreme wind speed distributions become.
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CHAPTER 4
TTUHRT VERSUS HWIND COMPARISON The sensitivity of using HWIND model data to construct/estimate EV wind speed distribution parameters first requires understanding the errors in HWIND values. To assess these errors a direct comparison between HWIND reconstruction data and TTUHRT time history data is completed. To facilitate the best possible comparison, peak wind speeds are computed using TTUHRT records using the methods employed by HWIND scientists. The mean of the TTUHRT raw data record is standardized to open exposure and then multiplied by the HWIND utilized gust factor equal to 1.18. This process is shown below in Equation 4.1.
⏞x , t, T x t, T ∗ 1.18 4.1
Where ⏞x , t, T is a point estimate of the peak wind speed in a 600s window. The
⏞x , t, T values are the same peak 60s of wind in a 600s windows, or maximum sustained wind speeds reported by HWIND U 600s . This procedure results in an ability to assess the differences in every HWIND value whereas using the actual peak wind speeds from TTUHRT records would require selective disqualification of all wind speeds not occurring over open exposure. The maximum sustained wind speeds provided by HWIND at the locations of the TTUHRT platforms are denoted as U 60s, 600s . The difference, denoted as D is computed and reported in both m/s and %. Equation 4.2 quantifies how differences are computed when reported in m/s.
D 𝑚/𝑠 ⏞ x , t, T 𝑈 60𝑠, 600𝑠 4.2
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The HWIND values are subtracted from the TTUHRT values to net positive values of D when the actual values are larger, and negative values where the actual values are smaller. The difference is also reported as a percent difference using Equation 4.3.
⏞x , t, T 𝑈 60𝑠, 600𝑠 D % ∗ 100% 4.3 ⏞x , t, T
Figures are assembled for each TTUHRT deployment illustrating the TTUHRT and HWIND peak wind speed values, the radial distance from storm center to the location of the TTHURT platform, denoted as R , at the time of each HWIND reporting time, and D[m/s], all plotted with respect to coordinated universal time (UTC). The figure for the TTUHRT PMT Clear platform, from the 2005 landfall of Hurricane Katrina, is shown in Figure 4.1.
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80 TTUHRT data 60 HWIND data
40
20
0 2005-08-29 00:00:00 2005-08-30 00:00:00
Maximum sustained wind speed [m/s] sustained Maximum Time [UTC]
15 Observed TTUHRT mean windspeed - Computed HWIND mean windspeed
10
5
0
DIfference [m/s] DIfference -5
-10 2005-08-29 00:00:00 2005-08-30 00:00:00 Time [UTC]
400 ) [km]
p 300
200
100
Radial Distance (R 0 2005-08-29 00:00:00 2005-08-30 00:00:00 Time [UTC]
Figure 4.1 TTUHRT and HWIND maximum sustained wind speeds, differences between TTUHRT and HWIND maximum sustained wind speeds, and the distances from platform to center of the storm, for the deployment of TTUHRT PMT Clear during the 2005 landfall of Hurricane Katrina
Nearly all of the differences in Figure 4.1 are between +/-5m/s with the exception of a single value exhibiting a difference of a little more than 10m/s. The HWIND data does follow the wind speed time history recorded by the PMT Clear platform. Tables containing the results from processed HWIND datafiles are presented in table form in Appendix A5. More figures, like Figure 4.1 are provided for other TTUHRT
97 Texas Tech University, Joseph Dannemiller, May 2019 deployments in Appendix A6. A deployment from Hurricane Katrina was plotted first as several examples in previous chapters have utilized data from Hurricane Katrina. However, to show difference trends here, a chronological sequence is presented. First, the earliest HWIND reconstruction for which TTUHRT data is available, the 1998 landfall of Hurricane Bonnie. The figure for WEMITE #1 deployment on Hurricane Bonnie is illustrated in Figure 4.2.
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35 TTUHRT data 30 HWIND data
25
20
15
10 1998-08-27 00:00:00
Maximum sustained wind speed [m/s] sustained Maximum Time [UTC]
8
6
4
2
DIfference [m/s] DIfference 0 Observed TTUHRT mean windspeed - Computed HWIND mean windspeed -2 1998-08-27 00:00:00 Time [UTC]
200 ) [km]
p 150
100
50
Radial Distance (R 0 1998-08-27 00:00:00 Time [UTC]
Figure 4.2 TTUHRT and HWIND maximum sustained wind speeds, differences between TTUHRT and HWIND mean wind speeds, and the distances from platform to center of the storm, for the deployment of TTUHRT WEMITE #1 during the 1998 landfall of Hurricane Bonnie
The deployment of TTUHRT WEMITE #2 during the 2003 landfall of Hurricane Isabel is illustrated in Figure 4.3.
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40 TTUHRT data 30 HWIND data
20
10
0 2003-09-17 00:00:00 2003-09-18 00:00:00 2003-09-19 00:00:00 2003-09-20 00:00:00
Maximum sustained wind speed [m/s] sustained Maximum Time [UTC]
5 Observed TTUHRT mean windspeed - Computed HWIND mean windspeed
4
3
2 DIfference [m/s] DIfference
1 2003-09-17 00:00:00 2003-09-18 00:00:00 2003-09-19 00:00:00 2003-09-20 00:00:00 Time [UTC]
600
) [km] 500 p
400
300
200
Radial Distance (R 100 2003-09-17 00:00:00 2003-09-18 00:00:00 2003-09-19 00:00:00 2003-09-20 00:00:00 Time [UTC]
Figure 4.3 TTUHRT and HWIND maximum sustained wind speeds, differences between TTUHRT and HWIND mean wind speeds, and the distances from platform to center of the storm, for the deployment of TTUHRT WEMITE #2 during the 2003 landfall of Hurricane Isabel
The 2008 deployment of TTUHRT StickNet 108B during the 2008 landfall of Hurricane Ike is illustrated in Figure 4.4.
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40 TTUHRT data 30 HWIND data
20
10
0 2008-09-13 00:00:00 2008-09-14 00:00:00
Maximum sustained wind speed [m/s] Maximum Time [UTC] Observed TTUHRT mean windspeed - Computed HWIND mean windspeed 0
-2
-4
-6
DIfference [m/s] DIfference -8
-10 2008-09-13 00:00:00 2008-09-14 00:00:00 Time [UTC]
300
) [km] 250 p
200
150
100
Radial Distance (R 50 2008-09-13 00:00:00 2008-09-14 00:00:00 Time [UTC]
Figure 4.4 TTUHRT and HWIND maximum sustained wind speeds, differences between TTUHRT and HWIND mean wind speeds, and the distances from platform to center of the storm, for the deployment of TTUHRT StickNet 108B during the 2008 landfall of Hurricane Ike
The D[m/s] values do not show any consistency other than that most are between +/- 5m/s. Some HWIND values are above the TTUHRT values, and some below. No clear trend or pattern is evident. Originally, the differences were examined with respect to several distances or ratios of distances, but no clear patterns emerged in those
101 Texas Tech University, Joseph Dannemiller, May 2019 analyses. As no clear trends were present in the plots above, and the remaining figures provided in Appendix A6, are plotted with respect to time. The radial distance is plotted at the bottom of each array of plots to provide a relation showing the distance from the TTUHRT platforms and the center of the storm as reported by HWIND.
Differences Between TTUHRT and HWIND as a Function of Distance From Storm Center The D[m/s] and D[%] values are aggregated together for all TTUHRT deployments and plotted in Figure 4.5(a) and (b), respectively, with respect to radial distance from storm center, for all datapoints where both an HWIND wind speed and a TTUHRT wind speed are available.
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Difference [m/s]
(a) Percent Difference [%]
(b) Figure 4.5 (a) Differences (m/s) between observed TTUHRT ⏞x , t, T and HWIND U 60s, 600s data versus the radial distance from storm center to the location of the TTUHRT platform R , (b) %-differences between observed TTUHRT ⏞x , t, T x raw, 600s and HWIND U 60s, 600s U 600s data versus the radial distance from storm center to the location of the TTUHRT platform, R
The data plotted in Figure 4.5(a) and (b) includes all U 60s, 600s values from HWIND reconstructions where the TTUHRT platforms were in the HWIND domain. This includes 600s windows with mean wind speeds both above and below 15m/s.
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Figure 4.5(b) shows %-differences between HWIND and TTUHRT values of +/-50% with a few observations even exceeding that range. The large percent differences from the 2005 datasets were either: (1) from 600s windows containing low wind speeds, so while the percent difference was large, the difference in m/s was not more than 5m/s; or (2) from Hurricane Rita where the HWIND model did not accurately model the wind field at and around the eye. The percent differences that are particularly problematic are those from the TTUHRT StickNet platforms. The large differences in the StickNet data cannot currently be explained beyond accounting for the previously mentioned issue where using a TI derived z0 may not be correct for data measured 2.25m off the ground nominally. The histograms of differences, and percent differences for all TTUHRT vs HWIND comparisons are shown in Figure 4.6(a) and (b), respectively.
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Frequency f(x)
a) Difference between TTUHRT ⏞x , t, T and HWIND U 60s, 600s D[m/s] Frequency f(x)
b) Difference between TTUHRT ⏞x , t, T and HWIND U 60s, 600s D[%]
Figure 4.6 (a) Distribution of differences between TTUHRT ⏞x , t, T and HWIND 𝑈 60𝑠, 600𝑠 data, (b) Distribution of %-differences between TTUHRT ⏞𝑥 , 𝑡, 𝑇 and HWIND U 60s, 600s data
The differences in Figure 4.6(a) fit a normal distribution mean centered at 0.145 with a standard deviation of 4.303. Again, this data includes all HWIND simulation values including comparisons with TTUHRT windows with mean wind speeds above and below 15 m/s. Including all of this data provides more observations to attempt to boost the significance of distributions fit, or used to analyze, the difference. The implication of the fit in Figure 4.6(a) is that 95% of the differences between TTUHRT
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⏞x , t, T x raw, 600s and HWIND U 60s, 600s U 600s values are between - 8.461m/s and 8.751m/s. Figure 4.6(b) shows the percent differences between
TTUHRT x t, T x raw, 600s and HWIND U 60s, 600s . U 600s Assuming the percent differences are also normally distributed, the mean and standard deviation are -0.781% and 22.844%, respectively. With such large differences, and percent differences, the work herein does not agree with (Powell et al, 1996) that states HWIND U 60s, 600s values are within +/-12% of the land observations (including
TTUHRT x t, T x raw, 600s data). The trend in differences as years progress was also investigated and those results are plotted below in Figure 4.7(a) and (b) for the magnitudes and percent differences, respectively.
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Difference [m/s] Difference
a) Percent Error [%] Error Percent
b)
Figure 4.7 (a) Differences between TTUHRT ⏞x , t, T and HWIND 𝑈 60𝑠, 600𝑠 data with respect to storm year, (b) Percent differences between TTUHRT ⏞x , t, T and HWIND 𝑈 60𝑠, 600𝑠 data with respect to storm year
The means and standard deviations of the differences plotted in Figure 4.7(a) and (b) are given in Table 4.1 below, including the number of observations per year.
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Table 4.1 Magnitude and percent differences between TTUHRT and HWIND maximum sustained wind speeds broken down by year Magnitude (m/s) Percent (%) Number of Standard Standard Year Mean Mean Observations Deviation Deviation 1998 4 2.854 3.684 11.102 13.735 2003 34 1.484 2.026 8.753 12.900 2004 9 4.025 3.137 25.497 18.680 2005 59 2.429 4.7193 9.256 24.018 2008 67 -2.708 3.033 -15.168 18.288
There does not appear to be an improvement in the HWIND model with time in either the mean or standard deviation values. It is possible that more data may improve this analysis, but the dispersion of the differences is quite large, so the inclusion of more data may not be enough to bring these differences in line with the +/-12% quoted in (Powell et al, 1996). The percent differences do however compare quite well to the data presented in Table 2.6 showing the errors computed by (Vickery et al., 2000) for their hurricane wind field simulation model. Table 2.6 had mean percent errors as high as 26.09% for a dataset containing 105 observations, and as low as -5.82% for a dataset containing only 14 observations. The minimum standard deviations from Table 2.6 is 3.18% from the same dataset exhibiting the lowest mean which contained 14 observations. And the maximum standard deviation from Table 2.6 is 23.19% from the same dataset exhibiting the highest mean which had 105 observations. Compared to the lowest and highest mean percent differences in Table 4.1 of -15.168 and 25.497 for datasets containing 67 and 9 observations, respectively, the values match well with the data in Table 4.1. The lowest mean in Table 4.1 is lower than the lowest mean in Table 2.6, but the lowest mean in Table 4.1 comes from 2008 with the TTUHRT StickNet data, which as mentioned in Chapter 2, may not be standardized properly if
HWIND does in fact use TI derived z0 values from data gathered at a 2.25m height. Given the complexity of HWIND simulations, it will be necessary to reach out to HWIND scientists to explore this work further and especially prior to publishing
108 Texas Tech University, Joseph Dannemiller, May 2019 any of these results. Unfortunately, to date, several attempts have been made contact HWIND scientists, but no responses have been received.
Comparison of TTUHRT and HWIND Gust Factors An additional comparison of TTUHRT and HWIND datasets involves comparing the gust factor utilized by HWIND to the values recorded by TTUHRT platforms. The gust factor used by HWIND is 1.18 and the gust factors computed using TTUHRT data were computed by taking the peak 60s wind speed divided by the mean of the raw records for each 600s time history. The TTUHRT data was then segregated by roughness into three categories, smoother than open (z0<0.03m), open
(0.03m≤z0≤0.07m), and rougher than open (z0>0.07m) exposures to determine if the HWIND gust factor’s location differed amongst gust factors from differing roughness regimes. The z0 values used to segregate the TTUHRT values were the assigned z0 values discussed in Chapter 2 instead of TI derived z0 values. The TTUHRT gust factors are presented in histogram form with the HWIND gust factor superimposed in red. These are illustrated in Figure 4.8, Figure 4.9, and Figure 4.10, for smoother than open, open and rougher than open exposures, respectively.
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Figure 4.8 Comparison between TTUHRT gust factors (blue histogram) and the single gust factor employed by HWIND (red) using only TTUHRT gust factors with smoother than open exposure (z0<0.03m) upwind during the recording of 600s time histories
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Figure 4.9 Comparison between TTUHRT (blue histogram) gust factors and the single gust factor employed by HWIND (red line) using only TTUHRT gust factors with open exposure (0.03m≤z0≤0.07m) upwind during the recording of 600s time histories
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Figure 4.10 Comparison between TTUHRT (blue histogram) gust factors and the single gust factor employed by HWIND (red line) using only TTUHRT gust factors with rougher than open exposure (z0>0.07m) upwind during the recording of 600s time histories
The location of the HWIND gust factor in all three roughness distributions in Figure 4.8 through Figure 4.10 is at or near the highest quantile (near the mode) in all three histograms. Since the HWIND gust factor is an expected value, these results show excellent agreement between the mode of the distribution of gust factors computed using TTUHRT data for all three roughness regimes and the gust factor employed by HWIND. Unfortunately, this only reinforces the previous claim that differences between TTUHRT and HWIND maximum sustained wind speeds, presented in Table 4.1, are most likely due to something more complex in the HWIND algorithm that cannot be presently explained.
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Comparison of TTUHRT and HWIND Mean Values The previous section compared the gust factors observed by TTUHRT platforms to the single gust factor used in HWIND reconstructions. To compute maximum sustained wind speeds in HWIND reconstructions, the mean wind speed in a 600s record is also needed. The differences between TTUHRT observed mean wind speeds and mean wind speeds computed by dividing the HWIND reported maximum sustained wind speeds, at the locations of TTUHRT platforms, by the single gust factor equal to 1.18, denoted as U 600s . The differences between mean wind speeds are provided below for the magnitudes of the difference, in m/s, and as percent differences in Figure 4.11(a) and Figure 4.11(b), respectively.
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Number of Observations (n) [#]
a) Difference between TTUHRT x raw, 600s and HWIND U 600s D[m/s] Number of Observations (n) [#]
b) Difference between TTUHRT x raw, 600s and HWIND U 600s D[%]
Figure 4.11 (a) Distribution of differences between TTUHRT 𝑥 ̅ 𝑟𝑎𝑤, 600𝑠 and HWIND U 600s data, (b) Distribution of %-differences between TTUHRT 𝑥 ̅ 𝑟𝑎𝑤, 600𝑠 and HWIND U 600s data
The differences in the means are similar to the values illustrated in Figure 4.6 since the comparison between TTUHRT ⏞x , t, T and HWIND U 60s, 600s involved multiplying the TTUHRT 𝑥 ̅ 𝑟𝑎𝑤, 600𝑠 by 1.18 to compute maximum sustained
114 Texas Tech University, Joseph Dannemiller, May 2019 wind speeds estimates for comparison. The only difference in the values reported in Figure 4.6 and Figure 4.11 is the amplification of the magnitudes and percent differences in Figure 4.11 due to the gust factor.
Comparison of TTUHRT Recorded Peak Wind Speeds and Maximum Sustained Wind Speeds from HWIND Reconstructions The differences between the actual peak 60s of wind in 600s windows recorded by TTUHRT platforms, denoted as 𝑥 60𝑠, 600𝑠 and the maximum sustained wind speeds reported by HWIND, denoted as U 60s, 600s , are provided below in Figure 4.12(a) and Figure 4.12(b), respectively.
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Number of Observations (n) [#]
Difference between TTUHRT 𝑥 60𝑠, 600𝑠 and HWIND 𝑈 60𝑠, 600𝑠 a) D[m/s] Number of Observations (n) [#]
b) Difference between TTUHRT 𝑥 60𝑠, 600𝑠 and HWIND 𝑈 60𝑠, 600𝑠 D[%]
Figure 4.12 (a) Distribution of differences between TTUHRT 𝑥 60𝑠, 600𝑠 and HWIND 𝑈 60𝑠, 600𝑠 , (b) Distribution of %-differences between TTUHRT 𝑥 60𝑠, 600𝑠 and HWIND 𝑈 60𝑠, 600𝑠 data
These differences combine the errors in the gust factors recorded by TTUHRT platforms versus the single gust factor used in HWIND, presented in Figure 4.8 through Figure 4.10, with the differences between the mean wind speeds recorded by TTUHRT platforms and the mean wind speeds computed using the maximum sustained wind speeds from HWIND reconstructions. The magnitudes of the errors
116 Texas Tech University, Joseph Dannemiller, May 2019 presented in Figure 4.12 are close to those in Figure 4.11 but seem to have moved in the positive direction by a few m/s. The percent errors in Figure 4.12 are more concentrated around a mean value than those presented in Figure 4.11 but the maximum percent errors in Figure 4.12 are larger (but negative) than the greatest percent differences in Figure 4.11. One takeaway from the comparison of Figure 4.11 and Figure 4.12 is that the percent errors for the comparison of the maximum sustained wind speeds are less than the percent errors for the comparison of mean values. This means the maximum sustained wind speeds computed by HWIND are closer to the actual peak wind speeds recorded by TTUHRT, than are the mean wind speeds computed from HWIND reconstructions and the mean wind speeds recorded by TTUHRT platforms. If the percent errors in Figure 4.11 are moved to the right about 10% they would closely match the percent errors in Figure 4.12 without being magnified due to errors associated with variations in gust factors presented in Figure 4.8 through Figure 4.10. The conclusion here is there is more error in the mean wind speeds used in HWIND reconstructions than there is in the single gust factor used in HWIND reconstructions. In theory, the HWIND reported maximum sustained wind speeds should be at the top of the distribution of wind speeds recorded by TTUHRT platforms. In reality, the maximum sustained wind speeds from HWIND reconstructions did not occur consistently at, or around, any specific quantile, or range of quantiles, in the distribution of parent winds, or in the distribution of extreme winds.
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CHAPTER 5
ESTIMATING PARAMETERS FOR DISTRIBUTIONS OF EXTREME WIND SPEEDS IN 600s WINDOWS USING TTUHRT DATA To simulate distributions of extreme winds, relationships between the parent and extreme wind speed distribution parameters, and between EV location and scale parameters are formed using raw, 3s MA and 60s MA data. Some of these relationships are compared to previous works, but for some, there exists no prior work for comparison of wind data so statistical tests are run to validate certain properties, and to quantify errors and uncertainties.
Distributions of Extreme Winds Location and Scale Parameters One interesting property of both the location and scale parameters is that when the values of each, from all TTUHRT 600s records, are aggregated together they can be well fit with a distribution that one could use in estimation or simulation. The distributions of μ 60s, 600s , μ 3s, 600s , σ 60s, 600s and σ 3s, 600s can all can be fit using a Generalized Extreme Value (GEV) distribution. The equation for the GEV PDF is expressed in Equation 5.1.
1 𝑥 𝜇 𝑥 𝜇 𝑓 𝑥|𝑘, 𝜇, 𝜎 exp 1 𝑘 1 𝑘 5.1 𝜎 𝜎 𝜎
A histogram and the GEV distribution fit using numerical software, for μ 60s, 600s values, is illustrated below in Figure 5.1.
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f(x)
EV Location Parameters μ 60s, 600s [m/s] Figure 5.1 Distribution of EV location parameters from 60s MA windows
The three parameter GEV distribution fit in Figure 5.1 has a k = 0.082, sigma = 2.731 and mu = 21.109. The GEV fit is tested using both the Kolmogorov-Smirnov (KS) and Anderson-Darling (AD) tests, both at a 0.01 significance level (Box et al., 1978). Both the KS and AD tests are used to assess whether a sample fits a reference distribution. In this work, the distributions for various parameters are fit using numerical software and verified using the KS and AD tests. The KS and AD tests are a kind of goodness of fit test for the reference distributions where the KS test focuses on the goodness of fit at the center values of both the sample and the reference distribution, while the AD test focuses on verifying the goodness of fit in the tails. The GEV fit passes both the KS and AD tests at a 0.01 significance level so the three parameter GEV distribution is a good fit to the data. A histogram and the GEV distribution fit using numerical software, for μ 3s, 600s values, are illustrated below in Figure 5.2.
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f(x)
EV Location Parameters μ 3s, 600s [m/s] Figure 5.2 Distribution of EV location parameters from 3s MA windows
The three parameter GEV distribution fit in Figure 5.2 has a k = 0.058, sigma = 3.773 and mu = 26.621. The fit passes both the KS and AD tests at a 0.01 significance level so the three parameter GEV distribution is a good fit to the data. A histogram and the
GEV distribution fit using numerical software, for σ 60s, 600s values, are illustrated below in Figure 5.3.
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f(x)
EV Scale Parameters σ 60s, 600s [m/s] Figure 5.3 Distribution of EV scale parameters from 60s MA windows
The 𝜎 60𝑠, 600𝑠 histogram in Figure 5.3 is fit with a three parameter (k, sigma and mu) GEV distribution where k = -0.028, sigma = 0.387 and mu = 1.059. The GEV fit in Figure 5.3 passes both the KS and AD test at the 0.01 significance level meaning the fit is good. A histogram and the GEV distribution fit using numerical software, for
σ 3s, 600s values, are illustrated below in Figure 5.4.
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f(x)
EV Scale Parameters σ 3s, 600s [m/s] Figure 5.4 Distribution of EV scale parameters from 3s MA windows
The three parameter GEV distribution fit in Figure 5.4 has a k = 0.007, sigma = 0.604 and mu = 1.717. The fit passes both the KS and AD tests at a 0.01 significance level so the three parameter GEV distribution is a good fit to the data. The μ 60s, 600s vs
σ 60s, 600s , μ 3s, 600s vs σ 3s, 600s , x raw, 600s vs μ 60s, 600s and x raw, 600s vs μ 3s, 600s trends exhibit homogeneous variance and this will be shown later using a Chi-Squared test.
Parent Wind Distribution Means Parameters Versus Extreme Wind Distribution Location Parameters The goal of this work is to produce a methodology to take HWIND reconstruction values and estimate EV parameters for use in SPA. As of now, the differences between the TTUHRT and HWIND data are large. The HWIND data can be used to compute parent wind field mean wind speeds that can be used to estimate
EV wind field parameters. The relationship between x 60s, 600s and μ 60s, 600s in 600s windows for the 60s MA TTUHRT time histories is illustrated in Figure 5.5.
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[m/s] [m/s] 60s, 600s μ EV Location Parameter EV Location
Mean Wind Speed x 60s, 600s [m/s] Figure 5.5 Linear relationship between mean and location parameters from TTUHRT 60s MA data
The line of best fit plotted in Figure 5.5 is a mean trend. Immediately, it can be noted that the data in Figure 5.5 is very well organized around the mean trend. The equation for the line is given in Equation 5.2 and has a goodness of fit, measured using a Pearson Correlation Coefficient (PCC), of PCC = 0.93.
μ 60s, 600s 1.09∗x 60s, 600s 1.72 𝑚/𝑠 5.2
The PCC value is computed using Equation 5.3 where, for ease of understanding, a is set equal to the values of 𝑥 ̅ 60𝑠, 600𝑠 , and b is set equal to the values of
𝜇 60𝑠, 600𝑠 .
1 𝑎 𝑎 𝑏 𝑏 𝑃𝐶𝐶 5.3 𝑁 1 𝑠𝑑 𝑠𝑑
Where 𝑎 and 𝑏 are the means of the 𝑥 ̅ 60𝑠, 600𝑠 and μ 60s, 600s , respectively, and where 𝑠𝑑 and 𝑠𝑑 are the standard deviations of the 𝑥 ̅ 60𝑠, 600𝑠 and
μ 60s, 600s , respectively. The range of possible values for the PCC is from -1 to +1
123 Texas Tech University, Joseph Dannemiller, May 2019 where a +1 indicates absolute positive correlation between the two datasets. A -1 indicates absolute negative correlation, and a 0 indicates no correlation. The good correlation between x 60s, 600s and μ 60s, 600s in Figure 5.5 will be explored later as, for now, the identification of some way to link the distributions of parent and extreme wind field is paramount. The R2 value for the fit in 5.2 is 0.92 and the magnitudes and percent differences between the TTUHRT μ 60s, 600s values and the line of best fit are presented in Figure 5.6 and Figure 5.7, respectively.
Number of datapoints
Magnitude of differences [m/s]
Figure 5.6 Magnitudes of differences between the TTUHRT μ 60s, 600s and estimates using Equation 5.2 as a function of x 60s, 600s
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Number of datapoints
Percent of differences [%]
Figure 5.7 Percent of differences between the TTUHRT μ 60s, 600s and estimates using Equation 5.2 as a function of x 60s, 600s
The mean and standard deviation of differences in Figure 5.6 for the magnitudes of differences are 0m/s and 0.94m/s, respectively. The mean and standard deviation of the percent differences in Figure 5.7 for the percent differences are -0.18% and 4.20%, respectively.
To estimate EV location parameters, the relationships between x raw, 600s vs μ 60s, 600s , and x raw, 600s vs μ 3s, 600s must be regressed. These two moving average times are selected as the first is consistent with HWIND gust durations and the largest gust durations presented in (Cook, 1985b) according to the TVL method discussed in Chapter 2, and the second is consistent with ASCE 7
(ASCE, 2010). First, the x raw, 600s vs μ 60s, 600s relationship is plotted in Figure 5.8.
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` [m/s] [m/s] 60s, 600s μ EV Location Parameter
Mean Wind Speed x raw, 600s [m/s]
Figure 5.8 Linear relationship to estimate μ 60s, 600s parameters
This time, a mean trend is fit to the data and the equation for the trend is given in Equation 5.4.
μ 60s, 600s 1.12∗𝑥 ̅ 𝑟𝑎𝑤, 600𝑠 1.16 𝑚/𝑠 5.4
Equation 5.4 is useful in generating distributions of 𝜇 60𝑠, 600𝑠 values as a function 2 of x raw, 600s . The R value for the fit in Equation 5.4 is 0.93and the magnitudes and percent differences between the TTUHRT μ 60s, 600s values and the line of best fit are presented in Figure 5.9 and Figure 5.10, respectively.
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Number of datapoints
Magnitude of differences [m/s]
Figure 5.9 Magnitudes of differences between the TTUHRT μ 60s, 600s and estimates using Equation 5.4 as a function of x raw, 600s Number of datapoints
Percent of differences [%]
Figure 5.10 Percent of differences between the TTUHRT μ 60s, 600s and estimates using Equation 5.4 as a function of x raw, 600s
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The mean and standard deviation of differences in Figure 5.9 for the magnitudes of differences are 0m/s and 2.82m/s, respectively. The mean and standard deviation of the percent differences in Figure 5.10 for the percent differences are -0.82% and
8.63%, respectively. The x raw, 600s vs μ 3s, 600s relationship is plotted in Figure 5.11.
[m/s] [m/s] 3s, 600s μ EV Location Parameter EV Location
Mean Wind Speed x raw, 600s [m/s]
Figure 5.11 Linear relationship to estimate μ 3s, 600s parameters
The mean trend for the relationship for the x raw, 600s vs μ 3s, 600s is given in Equation 5.5.
μ 3s, 600s 1.4∗x raw, 600s 1.8 𝑚/𝑠 5.5
The R2 value for the fit in Equation 5.5 is 0.77 and the magnitudes and percent differences between the TTUHRT μ 60s, 600s values and the line of best fit are presented in Figure 5.12 and Figure 5.13, respectively.
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Number of datapoints
Magnitude of differences [m/s]
Figure 5.12 Magnitudes of differences between the TTUHRT μ 60s, 600s and estimates using Equation 5.4 as a function of x raw, 600s Number of datapoints
Percent of differences [%]
Figure 5.13 Percent of differences between the TTUHRT 𝜇 3𝑠, 600𝑠 and estimates using Equation 5.5 as a function of x raw, 600s
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The mean and standard deviation of differences in Figure 5.12 for the magnitudes of differences are 0m/s and 2.18m/s, respectively. The mean and standard deviation of the percent differences in Figure 5.13 for the percent differences are -0.56% and 7.24%, respectively.
Estimating Scale Parameters for Distributions of Extreme Winds The EV Type-I scale parameters from the raw, 3s and 60s MA datasets are somewhat correlated with parameters from the parent wind distribution and the distribution of extreme winds. However, the correlations are ill-formed (either no clear linear correlation, or excessive dispersion), as can be seen in the x 60s, 600s vs
σ 60s, 600s relationship plotted in Figure 5.14, the x 3s, 600s vs σ 3s, 600s relationship plotted in Figure 5.15, the μ 60s, 600s vs σ 60s, 600s relationship plotted in Figure 5.16, and the μ 3s, 600s vs σ 3s, 600s relationship plotted in Figure 5.17.
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[m/s] 60s, 600s σ
EV Scale Parameter Mean Wind Speed x 60s, 600s [m/s] Figure 5.14 Relationship between mean and scale parameters from 60s MA windows
[m/s] 3s, 600s σ EV Scale Parameter
Mean Wind Speed x 3s, 600s [m/s] Figure 5.15 Relationship between mean and scale parameters from 3s MA windows
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[m/s] 60s, 600s σ
EV Scale Parameter EV Scale Parameter
EV Location Parameter μ 60s, 600s [m/s] Figure 5.16 Relationship between EV location and scale parameters from 60s MA windows
[m/s] 3s, 600s σ
EV Scale Parameter
EV Location Parameter μ 3s, 600s [m/s] Figure 5.17 Relationship between EV location and scale parameters from 3s MA windows
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None of the correlations illustrated in Figure 5.14 through Figure 5.17 are well formed enough to use for directly estimating values of σ for any averaging time. As an alternative, the statistical dependence between μ 60s, 600s and σ 60s, 600s is quantified using a Chi-Squared test for dependence. The test requires assembling a contingency table containing conditional probabilities for paired occurrences of both variables. First, a two parameter histogram is plotted in Figure 5.18 through Figure 5.22 and the resulting graph is then used to explain how the contingency table is built. The two parameter histogram is plotted as a 3D histogram using all of the TTUHRT 600s records. Later, these will be broken out by roughness but for now all records are aggregated into a single dataset to provide more observations in each bin to make the following explanation clearer. The binning for each parameter follows the same binning rules defined earlier, where the number of bins is the square root of the number of unique observations. The example histogram plots the μ 60s, 600s vs
σ 60s, 600s data, shown as a scatterplot in Figure 5.14. The height of each bin equals the frequency of mutual occurrence of corresponding values of μ 60s, 600s and σ 60s, 600s . Several angles of the 3D histogram are shown below in Figure 5.18 through Figure 5.21 as together they provide a holistic view of the 3D histogram presented on the static 2D media here.
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Figure 5.18 Two parameter histogram plotting the correlated frequencies of occurrence for 𝜇 60𝑠, 600𝑠 and 𝜎 60𝑠, 600𝑠 values, view 1
Figure 5.19 Two parameter histogram plotting the correlated frequencies of occurrence for 𝜇 60𝑠, 600𝑠 and 𝜎 60𝑠, 600𝑠 values, view 2
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Figure 5.20 Two parameter histogram plotting the correlated frequencies of occurrence for 𝜇 60𝑠, 600𝑠 and 𝜎 60𝑠, 600𝑠 values, view 3
Figure 5.21 Two parameter histogram plotting the correlated frequencies of occurrence for 𝜇 60𝑠, 600𝑠 and 𝜎 60𝑠, 600𝑠 values, view 4
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The last view of the 3D histogram is a heat map view from above shown in Figure 5.22. The heat map does not help illustrate the frequency of occurrence for binned sectors, as much as it shows the area densities of observations previously shown using the scatterplot in Figure 5.16. The heatmap also provides a view of the 3D histogram that aids in understanding how the contingency table that will be discussed next is created.
0 60s, 60 σ 3s, 600s σ EV Scale Parameter 60s, 600s
σ
EV Location Parameter μ 60s, 600s [m/s] Figure 5.22 Two parameter histogram plotting the correlated frequencies of occurrence for 𝜇 60𝑠, 600𝑠 and 𝜎 60𝑠, 600𝑠 values, heat map
The first step in assembling the contingency table is to set up a frequency table, Table 5.1, enumerating the number of data points occurring in each of the boxes seen in Figure 5.22.
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Table 5.1 Frequency Table for μ 60s, 600s vs σ 60s, 600s histogram
μ 60s, 600s Bin Centers 16.3 18.8 21.3 23.8 26.3 28.8 31.3 33.8 36.25 38.8 41.3 Totals 3.88 0 1 0 0 0 0 0 0 0 0 0 1 3.63 0 1 0 0 0 0 0 0 0 0 0 1 3.38 0 1 0 0 0 0 0 0 1 0 0 2 3.13 0 1 1 0 0 0 1 0 0 0 0 3 2.88 0 0 0 0 2 0 0 0 0 0 1 3 2.63 0 0 0 3 1 0 0 0 0 0 0 4 2.38 0 3 6 1 2 2 1 0 0 1 1 17 Bin Centers 2.13 2 7 6 8 6 3 3 3 0 0 0 38 1.88 12 14 4 11 8 1 2 0 1 0 1 54 1.63 21 26 23 12 15 10 0 1 0 0 0 108
60s, 600s 1.38 41 30 27 21 13 6 3 1 0 0 0 142
σ 1.13 37 52 53 20 23 7 2 1 0 0 0 195 0.88 45 25 33 17 9 4 0 0 0 0 0 133 0.63 30 27 23 7 2 2 0 0 0 0 0 91 0.38 4 1 1 1 0 0 0 0 0 0 0 7 Totals 192 189 177 101 81 35 12 6 2 1 3
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The frequency table is then used to compute an expected frequency table built assuming the data is dependent. If the data is independent, the data in both the frequency, and expected frequency tables, will be equal for every value. The expected frequency values are shown below in Table 5.2. As an example of how these values are computed, take the 1.68 at the bottom left edge of the field of expected frequency values. This value is computed by multiplying the sum of occurrences in the first column of Table 5.1, equal to 192, by the number of occurrences in the last row in Table 5.1, equal to 7, and then dividing by the total number of observations, equal to ∗ 799. Therefore, 1.68 . The remaining values in the field are computed the same way.
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Table 5.2 Expected Frequency Table for μ 60s, 600s vs σ 60s, 600s histogram
μ 60s, 600s Bin Centers 16.3 18.8 21.3 23.8 26.3 28.8 31.3 33.8 36.25 38.8 41.3 3.88 0.24 0.24 0.22 0.13 0.10 0.04 0.02 0.01 0.00 0.00 0.00 3.63 0.24 0.24 0.22 0.13 0.10 0.04 0.02 0.01 0.00 0.00 0.00 3.38 0.48 0.47 0.44 0.25 0.20 0.09 0.03 0.02 0.01 0.00 0.01 3.13 0.72 0.71 0.66 0.38 0.30 0.13 0.05 0.02 0.01 0.00 0.01 2.88 0.72 0.71 0.66 0.38 0.30 0.13 0.05 0.02 0.01 0.00 0.01 2.63 0.96 0.95 0.89 0.51 0.41 0.18 0.06 0.03 0.01 0.01 0.02 2.38 4.09 4.02 3.77 2.15 1.72 0.74 0.26 0.13 0.04 0.02 0.06 Bin Centers 2.13 9.13 8.99 8.42 4.80 3.85 1.66 0.57 0.29 0.10 0.05 0.14 1.88 12.98 12.77 11.96 6.83 5.47 2.37 0.81 0.41 0.14 0.07 0.20 1.63 25.95 25.55 23.92 13.65 10.95 4.73 1.62 0.81 0.27 0.14 0.41
60s, 600s 1.38 34.12 33.59 31.46 17.95 14.40 6.22 2.13 1.07 0.36 0.18 0.53
σ 1.13 46.86 46.13 43.20 24.65 19.77 8.54 2.93 1.46 0.49 0.24 0.73 0.88 31.96 31.46 29.46 16.81 13.48 5.83 2.00 1.00 0.33 0.17 0.50 0.63 21.87 21.53 20.16 11.50 9.23 3.99 1.37 0.68 0.23 0.11 0.34 0.38 1.68 1.66 1.55 0.88 0.71 0.31 0.11 0.05 0.02 0.01 0.03
Finally, the chi squared values for the relationship between the frequency and relative frequency tables are computed. The chi- squared values are computed in Table 5.3. As an example, the 3.19 value at the bottom left of the field of Chi-Squared values
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is computed by squaring the subtraction of the 4 in the first column, last row of Table 5.1, minus the 1.68 in the first column,
. last row of Table 5.2, and dividing the squared value by the 1.68 value. The result is 3.19 . .
Table 5.3 Chi-Squared values for μ 60s, 600s vs σ 60s, 600s histogram
μ 60s, 600s Bin Centers 16.3 18.8 21.3 23.8 26.3 28.8 31.3 33.8 36.25 38.8 41.3 3.88 0.24 2.46 0.22 0.13 0.10 0.04 0.02 0.01 0.00 0.00 0.00 3.63 0.24 2.46 0.22 0.13 0.10 0.04 0.02 0.01 0.00 0.00 0.00 3.38 0.48 0.59 0.44 0.25 0.20 0.09 0.03 0.02 197.76 0.00 0.01 3.13 0.72 0.12 0.17 0.38 0.30 0.13 20.24 0.02 0.01 0.00 0.01 2.88 0.72 0.71 0.66 0.38 9.46 0.13 0.05 0.02 0.01 0.00 86.79 2.63 0.96 0.95 0.89 12.31 0.87 0.18 0.06 0.03 0.01 0.01 0.02 2.38 4.09 0.26 1.33 0.61 0.04 2.12 2.17 0.13 0.04 45.02 13.73 Bin Centers 2.13 5.57 0.44 0.69 2.13 1.20 1.07 10.34 25.82 0.10 0.05 0.14 1.88 0.07 0.12 5.30 2.55 1.17 0.79 1.74 0.41 5.53 0.07 3.13 1.63 0.95 0.01 0.04 0.20 1.50 5.87 1.62 0.04 0.27 0.14 0.41
60s, 600s 1.38 1.39 0.38 0.63 0.52 0.14 0.01 0.35 0.00 0.36 0.18 0.53
σ 1.13 2.07 0.75 2.22 0.88 0.53 0.28 0.29 0.15 0.49 0.24 0.73 0.88 5.32 1.33 0.42 0.00 1.49 0.57 2.00 1.00 0.33 0.17 0.50 0.63 3.02 1.39 0.40 1.76 5.66 0.99 1.37 0.68 0.23 0.11 0.34 0.38 3.19 0.26 0.20 0.01 0.71 0.31 0.11 0.05 0.02 0.01 0.03
140 Texas Tech University, Joseph Dannemiller, May 2019
The values in Table 5.3 are added together to compute a Chi-Squared test statistic equal to 539.73 having (15-1)*(10-1)=156 degrees of freedom. When tested at any level of significance, this value falls into the rejection region as the value from chi- squared tables associated with a test statistic equal to 539.73, having 156 DOF, is less than 0.00001%. Therefore, these two datasets, 𝜇 60𝑠, 600𝑠 and 𝜎 60𝑠, 600𝑠 , fail to meet the definition of independence and it must be assumed that they are dependent. The remaining relationships illustrated in Figure 5.14 through Figure 5.17 were explored for independence, and all failed to meet the criterion for independence using the Chi-Squared contingency table method. Fortunately, while none of the relationships in Figure 5.14 through Figure 5.17 are independent, the μ 60s, 600s vs
σ 60s, 600s and μ 3s, 600s vs σ 3s, 600s relationships do show some linear relationship while also exhibiting a high degree of scatter. The scatter, however, around a mean trend appears to exhibit homogenous variance meaning the magnitude of the dispersion around the mean trend does not change through the range of values plotted on the horizontal axis. Specifically, this appears to be true for the
μ 60s, 600s vs σ 60s, 600s , and μ 3s, 600s vs σ 3s, 600s relationships in Figure 5.16 and Figure 5.17, respectively. The homogeneity of the dispersion is tested by first dividing the data into the same bins of μ 60s, 600s used to assemble the heat map in Figure 5.22, and the frequency table, Table 5.1. A Levene’s test is applied the
σ 60s, 600s where each bin is considered a separate data set. The Levene’s test seeks to prove that multiple datasets exhibit the same variance. At a significance level of 0.05, the σ 60s, 600s data is found to have homogeneous variance through the all of bins of 𝜇 60𝑠, 600𝑠 values. The Levene’s test is also applied to the μ 3s, 600s vs σ 3s, 600s data in Figure 5.17 and the 𝜎 3𝑠, 600𝑠 data is also found to have homogenous variance through all bins of 𝜇 3𝑠, 600𝑠 values. The implication of homogenous variance is that the dispersion of the distributions of σ 60s, 600s , and σ 3s, 600s , parameters are constant no matter the value of 𝜇 60𝑠, 600𝑠 , or μ 3s, 600s , respectively. The mode of a distributions of
141 Texas Tech University, Joseph Dannemiller, May 2019
𝜎 60𝑠, 600𝑠 and σ 3s, 600s values can be computed using a mean trend regressed from Figure 5.16. While the dispersion parameter used to generate distributions of
𝜎 60𝑠, 600𝑠 and σ 3s, 600s can be identified by examining the properties of the distributions of all 𝜎 60𝑠, 600𝑠 and σ 3s, 600s values. The mean trend fitting the 60s MA data in Figure 5.16, and the mean trend fitting the 3s MA data in Figure 5.17, are expressed in Equations 5.6 and 5.7, respectively.
σ 60s, 600s 0.03∗μ 60s, 600s 0.4 5.6
σ 3s, 600s 0.05∗μ 3s, 600s 0.5 5.7
The slope of both Equations 5.6 and 5.7 are close to zero. Using T-test to determine if the slopes are significantly different from zero nets a t-score of 0.1201 and 0.1403. Both t-scores, with over 1000 degrees of freedom, lead to the conclusion that the slopes are not statistically significant from zero. This means that a single value of location parameter cannot be directly computed from a single value of location parameter using Equations 5.6 and 5.7. Because of this the relationship between a location parameter and a distribution of scale parameters will be sought out. Contingency tables are employed using a multinomial binning methodology where the bin widths will be equal to the bin widths used in the univariate histograms presented for both location and scale parameters. Consistent bin widths in both univariate and multivariate methods is necessary for comparison of all distributions (Jobson, 1992).
Extreme Value Location Parameters Separated into Smoother than Open, Open and Rougher than Open Exposure Categories As was done with the gust factors, the EV parameter datasets are separated into smoother than open (z0<0.03m), open (0.03m 142 Texas Tech University, Joseph Dannemiller, May 2019 (z0>0.07m) exposures. Mean relationships between parent distribution means and the EV distribution location parameter, similar to Figure 5.8 and Equation 5.4, for all three 60s MA, 3s MA and raw datasets, are also computed and plotted in the figures for each roughness regime. First, the mean versus location parameter relationship using the TTUHRT 60s MA data is shown below in Figure 5.23. 60 55 50 45 [m/s] [m/s] 40 35 Smooth Location Parameters Rougher Location Parameters 30 Open Location Parameters Modal Trend - Smooth 60s, 600s 25 Modal Trend - Open Modal Trend - Rough μ 20 EV Location Parameter 15 15 20 25 30 35 40 45 50 55 60 Mean wind speeds x 60s, 600s [m/s] Figure 5.23 Relationship between parent mean and EV location using TTUHRT 60s MA data Where the equations for the mean trends for the 60s MA data are quantified in Equation 5.10, for smooth, open and rough exposures. 1.144 ∗ x 60s, 600s 0.826 z 0.03m μ 60s, 600s 1.145 ∗ x 60s, 600s 0.854 0.03m z 0.07m 5.8 1.161 ∗ x 60s, 600s 0.528 z 0.07m The R2 values for the smoother than open, open, and rougher than open exposures are 0.92, 0.93 and 0.86, respectively. The magnitudes and percent differences between the TTUHRT μ 60s, 600s values and Equation 5.8 for smoother than open exposure data are presented in Figure 5.24 and Figure 5.25, respectively. 143 Texas Tech University, Joseph Dannemiller, May 2019 Number of datapoints Magnitude of differences [m/s] Figure 5.24 Magnitudes of differences between the TTUHRT μ 60s, 600s and estimates using 5.8 for smoother than open exposure data as a function of x 60s, 600s 144 Texas Tech University, Joseph Dannemiller, May 2019 Number of datapoints Percent of differences [%] Figure 5.25 Percent differences between the TTUHRT μ 60s, 600s and estimates using 5.8 for smoother than open exposure data as a function of x 60s, 600s The mean and standard deviation of differences in Figure 5.24 for the magnitudes of differences are 0m/s and 0.94m/s, respectively. The mean and standard deviation of the percent differences in Figure 5.25 for the percent differences are -0.18% and 4.21%, respectively. The magnitudes and percent differences between the TTUHRT μ 60s, 600s values and Equation 5.8 for open exposure data are presented in Figure 5.26 and Figure 5.27, respectively. 145 Texas Tech University, Joseph Dannemiller, May 2019 Number of datapoints Magnitude of differences [m/s] Figure 5.26 Magnitudes of differences between the TTUHRT μ 60s, 600s and estimates using 5.8 for open exposure data as a function of x 60s, 600s 146 Texas Tech University, Joseph Dannemiller, May 2019 Number of datapoints Percent of differences [%] Figure 5.27 Percent differences between the TTUHRT μ 60s, 600s and estimates using 5.8 for open exposure data as a function of x 60s, 600s The mean and standard deviation of differences in Figure 5.26 for the magnitudes of differences are 0m/s and 0.97m/s, respectively. The mean and standard deviation of the percent differences in Figure 5.27 for the percent differences are -0.20% and 4.35%, respectively. The magnitudes and percent differences between the TTUHRT μ 60s, 600s values and Equation 5.8 for rougher than open exposure data are presented in Figure 5.28 and Figure 5.29, respectively. 147 Texas Tech University, Joseph Dannemiller, May 2019 Number of datapoints Magnitude of differences [m/s] Figure 5.28 Magnitudes of differences between the TTUHRT μ 60s, 600s and estimates using 5.8 for rougher than open exposure data as a function of x 60s, 600s 148 Texas Tech University, Joseph Dannemiller, May 2019 Number of datapoints Percent of differences [%] Figure 5.29 Percent differences between the TTUHRT μ 60s, 600s and estimates using 5.8 for rougher than open exposure data as a function of x 60s, 600s The mean and standard deviation of differences in Figure 5.28 for the magnitudes of differences are 0m/s and 0.88m/s, respectively. The mean and standard deviation of the percent differences in Figure 5.29 for the percent differences are -0.18% and 4.17%, respectively. Next, the mean versus location parameter relationship using the TTUHRT 3s MA data is shown below in Figure 5.50. 149 Texas Tech University, Joseph Dannemiller, May 2019 60 55 50 45 Smooth Location Parameters [m/s] [m/s] 40 Rougher Location Parameters Open Location Parameters 35 Modal Trend - Smooth Modal Trend - Open 30 Modal Trend - Rough 3s, 600s 25 μ 20 EV Location Parameter Parameter EV Location 15 15 20 25 30 35 40 45 50 55 60 Mean wind speeds x 3s, 600s [m/s] Figure 5.30 Relationship between parent mean and EV location using TTUHRT 3s MA data Where the equations for the mean trends for the 3s MA data are quantified in Equation 5.10, for smooth, open and rough exposures. 1.541 ∗ x 3s, 600s 0.396 z 0.03m μ 3s, 600s 1.417 ∗ x 3s, 600s 2.144 0.03m z 0.07m 5.9 1.448 ∗ x 3s, 600s 0.687 z 0.07m The R2 values for the smoother than open, open, and rougher than open exposures are 0.76, 0.85 and 0.80, respectively. The magnitudes and percent differences between the TTUHRT 𝜇 3𝑠, 600𝑠 values and Equation 5.9 for smoother than open exposure data are presented in Figure 5.31 and Figure 5.32, respectively. 150 Texas Tech University, Joseph Dannemiller, May 2019 Number of datapoints Magnitude of differences [m/s] Figure 5.31 Magnitudes of differences between the TTUHRT 𝜇 3𝑠, 600𝑠 and estimates using Equation 5.9 for smoother than open exposure data as a function of 𝑥 ̅ 3𝑠, 600𝑠 151 Texas Tech University, Joseph Dannemiller, May 2019 Number of datapoints Percent of differences [%] Figure 5.32 Percent differences between the TTUHRT 𝜇 3𝑠, 600𝑠 and estimates using Equation 5.9 for smoother than open exposure data as a function of 𝑥 ̅ 3𝑠, 600𝑠 The mean and standard deviation of differences in Figure 5.31 for the magnitudes of differences are 0m/s and 2.27m/s, respectively. The mean and standard deviation of the percent differences in Figure 5.32 for the percent differences are -0.60% and 7.44%, respectively. The magnitudes and percent differences between the TTUHRT 𝜇 3𝑠, 600𝑠 values and Equation 5.9 for open exposure data are presented in Figure 5.33 and Figure 5.34, respectively. 152 Texas Tech University, Joseph Dannemiller, May 2019 Number of datapoints Magnitude of differences [m/s] Figure 5.33 Magnitudes of differences between the TTUHRT 𝜇 3𝑠, 600𝑠 and estimates using Equation 5.9 for open exposure data as a function of 𝑥 ̅ 3𝑠, 600𝑠 153 Texas Tech University, Joseph Dannemiller, May 2019 Number of datapoints Percent of differences [%] Figure 5.34 Percent differences between the TTUHRT 𝜇 3𝑠, 600𝑠 and estimates using Equation 5.9 for open exposure data as a function of 𝑥 ̅ 3𝑠, 600𝑠 The mean and standard deviation of differences in Figure 5.33 for the magnitudes of differences are 0m/s and 1.60m/s, respectively. The mean and standard deviation of the percent differences in Figure 5.34 for the percent differences are -0.37% and 5.90%, respectively. The magnitudes and percent differences between the TTUHRT 𝜇 3𝑠, 600𝑠 values and Equation 5.9 for rougher than open exposure data are presented in Figure 5.35 and Figure 5.36, respectively. 154 Texas Tech University, Joseph Dannemiller, May 2019 Number of datapoints Magnitude of differences [m/s] Figure 5.35 Magnitudes of differences between the TTUHRT 𝜇 3𝑠, 600𝑠 and estimates using Equation 5.9 for rougher open exposure data as a function of 𝑥 ̅ 3𝑠, 600𝑠 155 Texas Tech University, Joseph Dannemiller, May 2019 Number of datapoints Percent of differences [%] Figure 5.36 Percent differences between the TTUHRT 𝜇 3𝑠, 600𝑠 and estimates using Equation 5.9 for rougher than open exposure data as a function of 𝑥 ̅ 3𝑠, 600𝑠 The mean and standard deviation of differences in Figure 5.35 for the magnitudes of differences are 0m/s and 1.46m/s, respectively. The mean and standard deviation of the percent differences in Figure 5.36 for the percent differences are -0.29% and 5.40%, respectively. Last, the mean versus location parameter relationship using the TTUHRT raw data is shown below in Figure 5.37. 156 Texas Tech University, Joseph Dannemiller, May 2019 60 55 50 [ /s] 45 40 Smooth Location Parameters 35 Rougher Location Parameters Open Location Parameters 30 Modal Trend - Smooth Modal Trend - Open raw, 600s 25 Modal Trend - Rough μ 20 EV Location Parameter Parameter EV Location 15 15 20 25 30 35 40 45 50 55 60 Mean wind speeds x raw, 600s [/s] Figure 5.37 Relationship between parent mean and EV location using TTUHRT raw data The mean trend for the smooth exposure data is hard to see as it is covered by the mean trend for the rough exposure data. The equations for the mean trends for the raw data are quantified in Equation 5.10, for smooth, open and rough exposures. 1.535 ∗ x raw, 600s 1.434 z 0.03m μ raw, 600s 1.494 ∗ x raw, 600s 2.829 0.03m z 0.07m 5.10 1.622 ∗ x raw, 600s 0.011 z 0.07m 157 Texas Tech University, Joseph Dannemiller, May 2019 The R2 values for the smoother than open, open, and rougher than open exposures are 0.68, 0.81 and 0.77, respectively. The magnitudes and percent differences between the TTUHRT 𝜇 𝑟𝑎𝑤, 600𝑠 values and Equation 5.10 for smoother than open exposure data are presented in Figure 5.38 and Figure 5.39, respectively. Number of datapoints Magnitude of differences [m/s] Figure 5.38 Magnitudes of differences between the TTUHRT 𝜇 𝑟𝑎𝑤, 600𝑠 and estimates using Equation 5.10 for smoother than open exposure data as a function of 𝑥 ̅ 𝑟𝑎𝑤, 600𝑠 158 Texas Tech University, Joseph Dannemiller, May 2019 Number of datapoints Percent of differences [%] Figure 5.39 Percent differences between the TTUHRT 𝜇 𝑟𝑎𝑤, 600𝑠 and estimates using Equation 5.10 for smoother than open exposure data as a function of 𝑥 ̅ 𝑟𝑎𝑤, 600𝑠 The mean and standard deviation of differences in Figure 5.38 for the magnitudes of differences are 0m/s and 2.97m/s, respectively. The mean and standard deviation of the percent differences in Figure 5.39 for the percent differences are -0.89% and 8.97%, respectively. The magnitudes and percent differences between the TTUHRT 𝜇 𝑟𝑎𝑤, 600𝑠 values and Equation 5.10 for open exposure data are presented in Figure 5.40 and Figure 5.41, respectively. 159 Texas Tech University, Joseph Dannemiller, May 2019 Number of datapoints Magnitude of differences [m/s] Figure 5.40 Magnitudes of differences between the TTUHRT 𝜇 𝑟𝑎𝑤, 600𝑠 and estimates using Equation 5.10 for open exposure data as a function of 𝑥 ̅ 𝑟𝑎𝑤, 600𝑠 160 Texas Tech University, Joseph Dannemiller, May 2019 Number of datapoints Percent of differences [%] Figure 5.41 Percent differences between the TTUHRT 𝜇 𝑟𝑎𝑤, 600𝑠 and estimates using Equation 5.10 for open exposure data as a function of 𝑥 ̅ 𝑟𝑎𝑤, 600𝑠 The mean and standard deviation of differences in Figure 5.40 for the magnitudes of differences are 0m/s and 1.69m/s, respectively. The mean and standard deviation of the percent differences in Figure 5.41 for the percent differences are -0.35% and 5.89%, respectively. The magnitudes and percent differences between the TTUHRT 𝜇 𝑟𝑎𝑤, 600𝑠 values and Equation 5.10 for rougher than open exposure data are presented in Figure 5.42 and Figure 5.43, respectively. 161 Texas Tech University, Joseph Dannemiller, May 2019 Number of datapoints Magnitude of differences [m/s] Figure 5.42 Magnitudes of differences between the TTUHRT 𝜇 𝑟𝑎𝑤, 600𝑠 and estimates using Equation 5.10 for rougher open exposure data as a function of 𝑥 ̅ 𝑟𝑎𝑤, 600𝑠 162 Texas Tech University, Joseph Dannemiller, May 2019 Number of datapoints Percent of differences [%] Figure 5.43 Percent differences between the TTUHRT 𝜇 𝑟𝑎𝑤, 600𝑠 and estimates using Equation 5.10 for rougher than open exposure data as a function of 𝑥 ̅ 𝑟𝑎𝑤, 600𝑠 The mean and standard deviation of differences in Figure 5.42 for the magnitudes of differences are 0m/s and 1.46m/s, respectively. The mean and standard deviation of the percent differences in Figure 5.43 for the percent differences are -0.29% and 5.40%, respectively. Figure 5.1 through Figure 5.4 illustrated that the 3s MA and 60s MA data can be fit by a three parameter GEV distribution. The same is true for the raw data and so GEV parameters for conditional distributions of location parameters are computed as functions of mean wind speeds. In this instance, a conditional distribution is a distribution of location parameters that occur in a given range of mean wind speeds. The conditional distribution parameters will be computed independently for the 60a MA, 3s MA and raw data, and for smooth, open and rough exposures. This will result in 9 possible sets of GEV parameters for each range, henceforth referred to as bin, of 163 Texas Tech University, Joseph Dannemiller, May 2019 mean wind speeds. Unfortunately, some conditional and roughness dependent distributions do not contain many observations making the three GEV parameters possibly a bad fit. For now, the conditional GEV distributions are computed using the TTUHRT data regardless of the number of observations in each bin and the parameters are computed using numerical software. As more TTUHRT deployment data, and possible data from other sources, becomes available, the parameters for the conditional GEV distributions can be recomputed. A 3D histogram illustrating the joint frequencies for paired values of mean and location parameters are illustrated in Figure 5.52 through Figure 5.56 for the raw data. The raw data is used as it makes for a better illustration (greater dispersion) but the 3D histograms for the 60s MA and 3s MA data can be found in Appendices A17 and A18, respectively. Figure 5.44 Two parameter histogram plotting the correlated frequencies of occurrence for x raw, 600s and μ raw, 600s values for multiple roughness regimes, view 1 164 Texas Tech University, Joseph Dannemiller, May 2019 Figure 5.45 Two parameter histogram plotting the correlated frequencies of occurrence for x raw, 600s and μ raw, 600s values for multiple roughness regimes, view 2 Smoother Rougher Open 0.12 0.1 0.08 0.06 0.04 0.02 0 20 15 30 20 25 30 40 35 40 50 45 50 55 Location parameters 60 60 Mean parameters Figure 5.46 Two parameter histogram plotting the correlated frequencies of occurrence for x raw, 600s and μ raw, 600s values for multiple roughness regimes, view 3 165 Texas Tech University, Joseph Dannemiller, May 2019 Smoother Rougher Open 0.12 0.1 0.08 0.06 0.04 0.02 0 20 60 30 55 50 45 40 40 35 50 30 25 20 Mean parameters 60 15 Location parameters Figure 5.47 Two parameter histogram plotting the correlated frequencies of occurrence for x raw, 600s and μ raw, 600s values for multiple roughness regimes, view 4 All Exposures Recombined 60 Smoother Rougher 55 Open 50 45 40 35 30 25 20 15 15 20 25 30 35 40 45 50 55 60 Mean parameters Figure 5.48 Two parameter histogram plotting the correlated frequencies of occurrence for x raw, 600s and μ raw, 600s values, heat map 166 Texas Tech University, Joseph Dannemiller, May 2019 The heat map in Figure 5.56 illustrates the same data shown as a scatterplot in Figure 5.51. The benefit of Figure 5.56 over the scatterplot data in Figure 5.51 is that Figure 5.56, like Figure 5.22, shows concentrations of joint occurrence. Again, the distributions of location parameter (plotted on the y-axis) are conditional upon the range of mean wind speed bins (plotted on the x-axis). This means every box above the range of mean wind speeds between 15m/sand 16.5m/s comprise the distributions of location parameters. For all-inclusiveness, the conditional distributions for all of the 600s windows from TTUHRT deployments for the raw, 3s and 60s moving average records are presented in Appendices A7, A8 and A9, respectively. The conditional distributions for the 600s windows with mean wind speeds above 15m/s for the raw, 3s and 60s moving average records are presented in Appendix A10, A11 and A12, respectively. The conditional distributions for 600s windows from the raw records where assigned z0 values are less than 0.03m, smoother than open exposure, are presented in Appendix A13. The conditional distributions for 600s windows from the raw records where assigned z0 values are between 0.03m and 0.07m inclusive, open exposure, are presented in Appendix A14. The conditional distributions for 600s windows from the raw records where assigned z0 values are greater than 0.07m, smoother than open exposure, are presented in Appendix A15. The conditional distributions for all three roughness regimes (smoother, open and rougher) for the raw, 3s and 60s moving average are presented in Appendices A16, A17 and A18, respectively. Extreme Value Scale Parameters Separated into Smooth, Open and Rough Exposure Categories The 3D histogram illustrated in Figure 5.18 through Figure 5.22 show the two location and scale parameter joint probabilities using the 60s MA data. Unfortunately, the 60s MA data is too closely organized at the lower end to adequately show the differences in distribution for smooth, open and rough exposures well. Therefore, for the following discussion the raw data is used to show the organization of the joint 167 Texas Tech University, Joseph Dannemiller, May 2019 probabilities broken into three roughness regimes. First however, it is important to note the difference between the scatterplots of the location versus scale parameter data for the 60s (like in Figure 5.18 through Figure 5.22), the 3s MA data and the raw data. These location vs. scale parameter scatterplots are presented below in Figure 5.49, Figure 5.50 and Figure 5.51 for the 60s MA data, 3s MA data and the raw data, respectively. 0 60s, 6 σ 3s, 600s σ EV Scale Parameter 60s, 600s σ EV Location Parameter μ 60s, 600s [m/s] Figure 5.49 Relationship between EV location and scale parameters using TTUHRT 60s MA data 168 Texas Tech University, Joseph Dannemiller, May 2019 EV Scale Parameter EV Location Parameter 𝜇 3𝑠, 600𝑠 [m/s] Figure 5.50 Relationship between EV location and scale parameters using TTUHRT 3s MA data 0 60s, 6 σ 3s, 600s σ EV Scale Parameter raw, 600s σ EV Location Parameter μ raw, 600s μ 3s, 600s μ 60s, 600s [m/s] Figure 5.51 Relationship between EV location and scale parameters using TTUHRT raw data The three scatterplots in Figure 5.49 through Figure 5.51 help show the effect of the MA on the dispersion of the data, as well as the effect on the magnitudes of both the location and scale parameters as the moving average duration increases. As the 169 Texas Tech University, Joseph Dannemiller, May 2019 moving average duration increases the data decreases in magnitude and becomes more closely concentrated around mean trends. This makes the 60s MA relationships a better statistical fit than the relationships for the 3s MA and then the raw data, in order of goodness of fit for mean trends. The 3s MA and 60s MA data will continue to be presented as the 60s MA duration is consistent with the data used in the TTUHRT vs HWIND comparisons, and the 3s MA data has applications in all domains governed by ACSE 7-10 (ASCE, 2010). The 3D histogram for the raw data is shown in Figure 5.52 through Figure 5.56 while the 3D histograms for the 60s and 3s MA data are provided in Appendices A16 and A17, respectively. Figure 5.52 Two parameter histogram plotting the correlated frequencies of occurrence for μ raw, 600s and σ raw, 600s values for multiple roughness regimes, view 1 170 Texas Tech University, Joseph Dannemiller, May 2019 Frequency Figure 5.53 Two parameter histogram plotting the correlated frequencies of occurrence for μ raw, 600s and σ raw, 600s values for multiple roughness regimes, view 2 Frequency Figure 5.54 Two parameter histogram plotting the correlated frequencies of occurrence for μ raw, 600s and σ raw, 600s values for multiple roughness regimes, view 3 171 Texas Tech University, Joseph Dannemiller, May 2019 Frequency Figure 5.55 Two parameter histogram plotting the correlated frequencies of occurrence for μ raw, 600s and σ raw, 600s values for multiple roughness regimes, view 4 Scale parameters Figure 5.56 Two parameter histogram plotting the correlated frequencies of occurrence for μ raw, 600s and σ raw, 600s values, heat map 172 Texas Tech University, Joseph Dannemiller, May 2019 The frequency of occurrence for scale parameters may be difficult to interpret using the 3D histogram and the heatmap in Figure 5.52 through Figure 5.56, so the 2D histograms of scale parameter values, in each bin of location parameters, are provided in Appendices A6 through A11 for smooth, open and rough exposures. The 2D histograms of scale parameter values, in each bin of location parameters, for the smooth, open and rough exposures are provided in Appendices A12, A13 and A14, respectively. The 2D histograms for the smooth, open and rough exposures were then combined for comparisons and an example of one such comparison is shown below in Figure 5.57 where the relative frequencies for bins of scale parameters are illustrated using the TTUHRT raw data in the bin of location parameters between 16.5m/s and 18m/s. (f) EV Scale Parameter 𝜎 𝑟𝑎𝑤, 600𝑠 [m/s] Figure 5.57 Histograms for smooth, open and rough exposure σ raw, 600s values 173 Texas Tech University, Joseph Dannemiller, May 2019 The remaining 2D histograms comparing scale parameter distributions for all three roughness regimes can be found in Appendix A15. The distributions of smooth, open and rough exposure scale parameters, in each bin of location parameters, were analyzed to determine if the values in all three roughness regimes were drawn from the same distribution. For this a Kruskall Wallis test was used. The results of the Kruskall Wallis test were that despite being very close, the sparsity of data, particularly at the upper end of scale parameter values, led to the conclusion that the distributions of scale parameters for different roughness regimes were not drawn from the same distributions in most of the location parameter bins. In location parameter bins where more than 10 observations total (smooth, rough and open) were present, the COV of the GEV distribution of scale parameters, in location parameter bins, was plotted in Figure 5.58 and it the COV for the distributions does not fluctuate much. Figure 5.58 Coefficients of variation for the GEV distributions of σ raw, 600s values in all 𝜇 𝑟𝑎𝑤, 600𝑠 bins The consistency of the COV values through the range of location parameter bins is used as the basis for computing parameters to define conditional GEV distributions of scale parameters in each location parameter bin. This process is completed for all three 174 Texas Tech University, Joseph Dannemiller, May 2019 roughness regimes, smooth, open and rough; and for all three datasets, 60s MA, and raw data. The consistency in the COV of conditional distributions of scale parameter values in all 3 moving average datasets were identical to the trend illustrated in Figure 5.58. As such, it is reasonable to state, that since the COV stays relatively the same, any change in the magnitude of a location parameter results in a similar change in the magnitude of the coupled scale parameter. Since the location parameters are well correlated with the parent distribution mean value as shown in Figure 5.49, Figure 5.50 and Figure 5.51, for the raw, 3s MA and 60s MA data, respectively, we can conclude that value of both location and scale parameters can be estimated directly using only a parent mean value. To do this, conditional distributions of both location and scale parameters will be generated dependent solely on mean wind speed, and to be safe, then broken into the same three exposure classifications (smoother than open, open, rougher than open) that have been used previously. For all-inclusiveness, the conditional distributions for all of the 600s windows from TTUHRT deployments for the raw, 3s and 60s moving average records are presented in Appendices A19, A20 and A21, respectively. The conditional distributions for the 600s windows with mean wind speeds above 15m/s for the raw, 3s and 60s moving average records are presented in Appendix A22, A23 and A24, respectively. The conditional distributions for 600s windows from the raw records where assigned z0 values are less than 0.03m, smoother than open exposure, are presented in Appendix A25. The conditional distributions for 600s windows from the raw records where assigned z0 values are between 0.03m and 0.07m inclusive, open exposure, are presented in Appendix A26. The conditional distributions for 600s windows from the raw records where assigned z0 values are greater than 0.07m, smoother than open exposure, are presented in Appendix A27. The conditional distributions for all three roughness regimes (smoother, open and rougher) for the raw, 3s and 60s moving average are presented in Appendices A28, A29 and A30, respectively. 175 Texas Tech University, Joseph Dannemiller, May 2019 Estimating Values of Extreme Wind Speed Distribution Location Parameters Like before, for this discussion the raw data will be used but, the 60s MA and the 3s MA data are presented in conjunction. A single mean wind speed from a raw data record will fall into one of the bins illustrated in Figure 5.56. The conditional GEV location parameter distributions in each bin of mean wind speed are computed for smoother than open, open or rougher than open exposure. The three parameters, in each mean wind speed bin, for each roughness category, are presented in Table 5.4 for the raw data. If any of the wind speed bins, for any exposure, contain less than 10 datapoints, the cell in the table is greyed out to denote that not enough data is present to regress a distribution that explains the location parameter distributions. The data is greyed out instead of omitted as the threshold of 10 values is selected arbitrarily and can be changed by anyone using this data. This identification scheme is carried forward in all subsequent Tables quantifying location parameters as a function of mean wind speed, and scale parameters as a function of location parameters. 176 Texas Tech University, Joseph Dannemiller, May 2019 Table 5.4 GEV parameters for conditional location parameter distributions computed from TTUHRT Raw Data Mean Wind Max 16.5 18.0 19.5 21.0 22.5 24.0 25.5 27.0 28.5 30.0 31.5 33.0 34.5 36.0 36.0 Speed Bin Range (m/s) Min 15.0 16.5 18.0 19.5 21.0 22.5 24.0 25.5 27.0 28.5 30.0 31.5 33.0 34.5 37.5 Shape 0.16 0.13 0.14 0.11 0.13 -0.02 0.05 0.25 0.38 -1.19 - - - - - Scale 1.95 1.87 2.20 2.12 2.16 2.10 2.50 1.38 1.26 0.66 - - - - - Location 24.32 26.06 28.34 30.61 32.36 34.41 36.62 38.22 42.19 45.12 - - - - - Smooth # of obs 177 212 214 94 77 59 31 19 5 4 - 0 - - - Shape 0.06 0.51 -0.32 -0.60 1.30 0.68 -0.03 -1.08 0.14 ------ Scale 1.86 2.14 1.63 1.66 0.95 0.66 1.28 0.34 0.49 ------ Open Location 24.89 27.07 29.13 31.00 30.99 32.78 36.28 37.16 38.77 ------ # of obs 43 22 14 17 11 7 6 5 7 0 0 0 0 0 Shape -0.17 -0.13 -1.04 0.06 -0.30 ------Location Parameters (m/s) (m/s) Parameters Location Scale 1.64 1.53 2.15 1.47 1.71 ------ Location 25.24 27.92 30.48 32.26 34.24 ------Rough GEV Parameters Defining the Distribution of Distribution the Defining GEV Parameters # of obs 35 22 15 11 8 0 0 0 0 0 0 0 0 0 0 At the top of the table the minimum and maximum values for the mean wind speed bins are provided. To the left the three roughness regimes are separated and the shape, scale and location parameters for the conditional GEV distributions of EV wind location parameters are provided. For example, simulating a distribution of location parameters for a mean wind speed equal to 18.2m/s, occurring over open exposure, will use shape, scale and location parameters equal to -0.32m/s, 1.63m/s and 177 Texas Tech University, Joseph Dannemiller, May 2019 29.13m/s, respectively. The conditional GEV distribution parameters for each mean wind speed bin, for each roughness category, are presented in Table 5.5 for the 60s MA data, and in Table 5.6 for the 3s MA data. Table 5.5 GEV parameters for conditional location parameter distributions computed from TTUHRT 60s MA Data Mean Wind Max 16.5 18.0 19.5 21.0 22.5 24.0 25.5 27.0 28.5 30.0 31.5 33.0 34.5 36.0 36.0 Speed Bin Range (m/s) Min 15.0 16.5 18.0 19.5 21.0 22.5 24.0 25.5 27.0 28.5 30.0 31.5 33.0 34.5 37.5 Shape 0.08 -0.19 -0.10 -0.11 -0.07 -0.07 -0.15 -0.15 4.98 -1.15 - - - - - Scale 0.90 0.92 0.97 1.08 0.88 0.88 0.94 0.74 1.16 0.36 - - - - - Location 18.45 20.09 21.52 23.32 24.78 26.30 27.87 29.56 30.96 33.41 35.04 - - - - Smooth # of obs 176 213 211 94 81 58 29 20 5 4 2 0 - - - Shape 0.09 -0.21 0.08 -0.44 0.21 0.45 -1.26 0.04 -1.17 ------Scale 0.93 0.88 0.76 0.82 0.63 0.63 0.56 0.57 0.50 ------ Open Location 18.33 20.21 21.78 23.46 24.68 25.40 27.76 29.19 31.12 ------# of obs 43 21 14 16 12 7 5 7 6 0 0 0 0 0 0 Shape -0.15 -0.22 -1.09 -0.41 0.21 ------Location Parameters (m/s) Scale 0.91 0.81 1.62 1.15 0.74 ------ Rough Location 18.33 20.07 22.11 23.46 24.51 ------GEV Parameters Defining the Distribution of # of obs 32 23 15 10 8 0 0 0 0 0 0 0 0 0 0 178 Texas Tech University, Joseph Dannemiller, May 2019 Table 5.6 GEV parameters for conditional location parameter distributions computed from TTUHRT 3s MA Data Mean Wind Max 16.5 18.0 19.5 21.0 22.5 24.0 25.5 27.0 28.5 30.0 31.5 33.0 34.5 36.0 36.0 Speed Bin Range (m/s) Min 15.0 16.5 18.0 19.5 21.0 22.5 24.0 25.5 27.0 28.5 30.0 31.5 33.0 34.5 37.5 Shape 0.04 0.08 0.02 -0.03 0.02 -0.02 0.08 0.21 0.91 -1.39 - - - - - Scale 1.65 1.63 1.80 1.93 1.74 2.00 1.90 1.54 0.33 0.94 - - - - - Location 22.98 24.64 26.72 28.78 30.40 32.50 34.65 36.16 39.11 42.14 - - - - - Smooth # of obs 171 210 217 94 79 58 30 19 5 4 - 0 - - - Shape -0.02 -0.29 -0.18 -0.06 -0.19 -0.61 -0.04 -0.22 -0.13 ------Scale 1.69 1.58 1.07 1.64 1.14 1.26 0.86 0.84 0.56 ------ Open Location 23.29 25.93 26.66 28.53 29.76 31.37 33.22 34.59 36.37 ------# of obs 43 19 13 18 11 7 5 7 6 0 0 0 0 0 0 Shape -0.32 -0.42 -1.03 0.22 -0.29 ------Location Parameters (m/s) Scale 1.63 1.25 2.10 1.35 1.10 ------ Rough Location 23.57 26.20 28.52 30.28 31.06 ------GEV Parameters Defining the Distribution of # of obs 32 22 16 11 5 - 0 0 0 0 0 0 0 0 0 Several of the sets of conditional distribution parameters were regressed using binned data where less than 10 observations were preset. Herein, the data was still allowed to drive the regression regardless of the number of observations, for all three moving average durations, and for all three roughness regimes. 179 Texas Tech University, Joseph Dannemiller December 2018 Estimating Values of Extreme Wind Speed Distribution of Scale Parameters The three conditional GEV parameters defining a distribution of location parameters, identified per the methods in the previous section as a function of a single mean wind speed, can be used to simulate a distribution of location parameters. From a distribution of location parameters, a single location parameter value can be sampled. This single location parameter value will fall into one of the location parameter bins for smooth, open, or rough exposures illustrated in Figure 5.48, for the raw data. As was presented for the conditional GEV distribution parameters for location parameters, three tables are presented containing the sets of three GEV parameters defining the conditional distributions of scale parameters for raw, 3s MA and 60s MA datasets, for smoother than open, open and rougher than open terrain exposures. The values computed using the TTUHRT raw, 3a MA and 60s MA data are shown in Table 5.7, Table 5.8 and Table 5.9, respectively. As stated with Table 5.4, the greyed-out sections are shown but not used as they all have less than 10 sample datapoints. 180 Texas Tech University, Joseph Dannemiller December 2018 Table 5.7 GEV parameters for conditional scale parameter distributions computed from TTUHRT Raw data Location Max 16.5 18.0 19.5 21.0 22.5 24.0 25.5 27.0 28.5 30.0 31.5 33.0 34.5 36.0 Parameter Bin Range Min 15.0 16.5 18.0 19.5 21.0 22.5 24.0 25.5 27.0 28.5 30.0 31.5 33.0 34.5 (m/s) Shape -0.07 -0.09 -0.11 -0.07 -0.16 0.01 -0.01 -0.05 -0.06 -0.09 -0.16 0.05 0.16 -0.18 Scale 0.37 0.44 0.43 0.49 0.48 0.47 0.55 0.57 0.52 0.54 0.65 0.65 0.65 0.64 Location 1.23 1.39 1.56 1.47 1.72 1.59 1.60 1.72 1.84 1.96 1.99 1.97 2.12 2.24 Smooth # of obs 183 180 148 99 117 113 172 153 148 110 94 78 68 53 Shape -0.27 -0.11 -0.16 -0.20 -0.17 -0.13 -0.22 -0.26 -0.03 -0.56 0.16 0.26 1.23 -1.13 Scale 0.36 0.32 0.40 0.48 0.44 0.46 0.53 0.66 0.64 0.88 0.63 0.67 0.43 1.04 Open Location 1.18 1.32 1.46 1.64 1.63 1.91 1.78 1.84 1.73 2.14 1.80 1.71 1.52 2.91 # of obs 52 75 75 71 52 41 30 25 21 19 21 22 7 7 Parameters (m/s) Shape -0.20 -0.28 -0.23 -0.14 -0.32 -0.10 -0.45 -0.52 0.09 -0.32 -1.06 -0.28 -1.17 -1.17 Scale 0.33 0.46 0.38 0.39 0.56 0.45 0.44 0.77 0.46 0.38 0.39 0.50 0.94 0.36 Rough Location 1.37 1.47 1.52 1.65 1.74 1.78 2.03 2.08 1.93 2.18 2.55 2.10 2.80 2.83 GEV Parameters Defining the Distribution of Scale # of obs 63 35 41 49 47 35 27 22 19 11 9 13 5 4 181 Texas Tech University, Joseph Dannemiller December 2018 Table 5.7 cont. GEV parameters for conditional scale parameter distributions computed from TTUHRT Raw Data Location Max 37.5 39.0 40.5 42.0 43.5 45.0 46.5 48.0 49.5 51.0 52.5 54.0 inf Parameter Bin Range Min 36.0 37.5 39.0 40.5 42.0 43.5 45.0 46.5 48.0 49.5 51.0 52.5 54.0 (m/s) Shape -0.33 0.00 0.28 0.37 0.87 5.05 1.77 5.37 5.12 - - - - Scale 0.78 0.73 0.71 1.37 0.34 0.19 0.30 0.17 1.61 - - - - Location 2.32 2.54 2.66 2.99 2.98 3.02 3.17 3.52 3.61 - - - - Smooth # of obs 31 33 15 11 9 5 6 3 5 - - 0 - Shape -1.10 0.02 4.26 ------Scale 0.30 0.19 0.02 ------ Open Location 1.74 1.58 1.88 ------# of obs 7 5 3 - 0 0 0 0 0 0 0 0 0 Parameters (m/s) Shape ------Scale ------ Rough Location ------ GEV Parameters Defining the Distribution of Scale # of obs - - 0 0 0 0 0 0 0 0 0 0 0 182 Texas Tech University, Joseph Dannemiller December 2018 Table 5.8 GEV parameters for conditional scale parameter distributions computed from TTUHRT 3s Data Location Max 16.5 18.0 19.5 21.0 22.5 24.0 25.5 27.0 28.5 30.0 31.5 33.0 34.5 36.0 37.5 39.0 Parameter Bin Range (m/s) Min 15.0 16.5 18.0 19.5 21.0 22.5 24.0 25.5 27.0 28.5 30.0 31.5 33.0 34.5 36.0 37.5 Shape -0.09 0.09 -0.20 -0.13 0.10 0.00 -0.08 -0.03 0.10 -0.09 0.10 0.21 -0.04 -0.09 0.33 -0.09 Scale 0.45 0.40 0.42 0.49 0.50 0.49 0.46 0.50 0.53 0.58 0.60 0.64 0.55 0.73 0.79 0.74 Location 1.20 1.32 1.51 1.47 1.59 1.53 1.64 1.67 1.74 1.89 1.90 1.99 2.12 2.54 2.35 2.79 Smooth Smooth # of obs 171 163 118 134 135 176 164 146 119 101 80 53 39 35 21 14 Shape -0.21 0.00 -0.10 -0.24 -0.15 -0.18 -0.16 -0.78 -0.20 0.22 0.60 0.65 0.97 -1.12 -1.28 - e Distribution of Scale of Scale e Distribution Scale 0.35 0.30 0.37 0.42 0.46 0.47 0.46 0.78 0.63 0.56 0.21 0.26 0.27 0.61 0.18 - Open Open Location 1.14 1.30 1.50 1.51 1.65 1.77 1.81 1.94 1.95 1.78 1.58 1.59 1.23 1.73 1.61 - Parameters (m/s) # of obs 63 88 82 55 47 37 27 26 22 20 8 10 6 7 5 - Shape 0.07 -0.23 -0.30 -0.38 -0.31 -0.57 -0.56 0.10 -0.72 -0.68 -1.07 -1.25 - 5.18 - - Scale 0.28 0.42 0.42 0.52 0.45 0.67 0.63 0.50 0.48 0.43 0.90 0.17 - 0.10 - - Rough Rough Location 1.28 1.39 1.50 1.69 1.67 1.75 1.90 1.97 2.41 2.11 2.67 2.08 - 3.42 - - GEV Parameters Defining th Defining GEV Parameters # of obs 43 38 58 39 46 23 23 22 9 14 6 5 - 3 0 0 183 Texas Tech University, Joseph Dannemiller December 2018 Table 5.8 cont. GEV parameters for conditional scale parameter distributions computed from TTUHRT 3s Data Location Max 40.5 42.0 43.5 45.0 46.5 48.0 49.5 inf Parameter Bin Range Min 39.0 40.5 42.0 43.5 45.0 46.5 48.0 49.5 (m/s) Shape -0.24 -1.86 -0.59 - -1.40 -2.20 5.23 - Scale 0.38 0.37 0.55 - 1.01 8.38 0.99 - Location 2.72 2.98 3.75 - 3.22 14.86 2.88 - Smooth # of obs 6 3 6 - 3 3 3 - Shape ------Scale ------ Open Location ------ Parameters (m/s) # of obs 0 0 0 0 0 0 0 0 Shape ------Scale ------ Rough Location ------ GEV Parameters Defining the Distribution of Scale # of obs 0 0 0 0 0 0 0 0 184 Texas Tech University, Joseph Dannemiller December 2018 Table 5.9 GEV parameters for conditional scale parameter distributions computed from TTUHRT 60s MA Data Location Paramete Max 16.5 18.0 19.5 21.0 22.5 24.0 25.5 27.0 28.5 30.0 31.5 33.0 34.5 36.0 37.5 39.0 40.5 inf r Bin Range Min 15.0 16.5 18.0 19.5 21.0 22.5 24.0 25.5 27.0 28.5 30.0 31.5 33.0 34.5 36.0 37.5 39.0 40.5 (m/s) Shape -0.12 -0.06 0.03 -0.01 -0.13 0.10 -0.04 -0.04 -0.05 -0.11 -0.07 4.61 -1.78 4.83 - -1.25 - - Scale 0.35 0.35 0.38 0.35 0.36 0.42 0.39 0.36 0.49 0.32 0.36 0.61 0.13 0.08 - 2.69 - - Location 0.99 0.98 0.95 1.00 1.09 1.16 1.19 1.20 1.23 1.25 1.50 1.36 1.60 1.41 - 12.82 2.65 - Smooth # of obs 146 171 175 199 184 103 94 64 41 26 10 5 3 3 0 6 1 - Shape -0.11 -0.26 -0.30 0.03 0.35 0.06 -0.04 -0.36 -0.01 1.17 -0.34 ------ Scale 0.31 0.25 0.40 0.37 0.27 0.42 0.34 0.28 0.37 0.08 0.19 ------ Open Location 1.01 1.04 1.23 1.23 1.11 1.13 1.06 1.29 1.04 0.85 1.31 ------ # of obs 80 68 45 27 20 13 18 4 7 6 7 - 0 0 0 0 0 0 Parameters (m/s) Shape -0.20 0.03 -0.10 0.01 -0.22 -0.40 0.39 ------ Scale 0.29 0.29 0.30 0.35 0.31 0.48 0.25 ------ 0.92 0.91 1.07 0.98 1.14 1.18 1.16 ------Rough Location - GEV Parameters Defining the Distribution of Scale # of obs 46 44 29 23 14 13 9 0 - 0 0 0 0 0 0 0 0 0 185 Texas Tech University, Joseph Dannemiller, May 2019 Table 5.7, Table 5.8 and Table 5.9 can be used to identify conditional GEV distribution parameters defining a distribution of scale parameters as a function of a location parameter the same way Table 5.4, Table 5.5 and Table 5.6 were used to identify the conditional GEV distribution parameters defining a distribution of location parameters as a function of a mean wind speed. Iterative Procedure to Generate Distributions of Extreme Winds The previous sections have outlined the relationships necessary for simulating distributions of location and scale parameters for raw, 3s MA and 60s MA extreme wind speeds using only a mean wind speed and a roughness classification. The purpose of the methodology proposed herein is to encapsulate as much of the uncertainty in the variations of both location and scale parameters that define the distribution of extreme winds for a single value of mean wind speed. To encapsulate the joint dispersion a two stage Monte Carlo simulation technique is utilized. Again, this discussion will reference only the raw data, but the process is the same when simulating distribution of 3s MA and 60s MA extreme value wind speeds. First, a mean wind speed and a roughness classification are used to identify the bin of mean wind speeds in Table 5.4 from which the parameters defining the conditional distribution of location parameter values are drawn. The drawn parameters are used in what will henceforth be referred to as a repetition where 1000 realizations of location parameters are simulated using the parameters drawn from Table 5.4. One realization is randomly sampled from the 1000 values. The sampled location parameter value will fall into one of the bins of location parameters in Table 5.7 from which the parameters defining the distribution of scale parameters is drawn. The drawn values are then used to simulate a 1000 realizations of scale parameters. A single scale parameter value is randomly sampled from the 1000 simulated values and when paired with the sampled location parameter, combine to provide the necessary information to simulate one distribution of extreme winds to be used in SPA. Once all of the extreme wind speed values in the simulated distribution have 186 Texas Tech University, Joseph Dannemiller, May 2019 been used to simulate distributions of extreme winds a second location parameter value is sampled from the simulated distribution of location parameters. This second location parameter either falls in the same bin of locations parameters from Table 5.7 as the first location parameter, or it falls in a different bin. Wherever the second location parameter falls in the bins of location parameters in Table 5.7, the parameters defining the distribution of scale parameters are drawn. As before, these parameters are used to simulate a second distribution of 1000 scale parameters. These 1000 scale parameters are then paired with the second location parameter value and combine to provide the information necessary to simulate another distribution of extreme wind speeds to use in SPA. The difference between the location parameters recorded in TTUHRT records and the simulated values of location parameters using Table 5.4 are computed as both magnitude and percentages. The magnitude and percent differences in TTUHRT recorded scale parameters versus simulated values of scale parameters but these differences are more complicated than the differences computed using location parameters. The differences between actual and simulated values of scale parameters, using the iterative analysis technique described above, will result in conditional differences since the accuracy of scale parameter distributions will be affected by the differences in the actual versus sampled location parameter values. After all, a sampled location parameter is used to identify the GEV parameters used to simulate distributions of scale parameters, so a large deviation in location parameters could lead to larger differences in scale parameters. The means of the distributions of differences in magnitudes and percent are provided below for both location and scale parameters in the smooth, open and rough exposures in Table 5.10, Table 5.11 and Table 5.12, respectively. 187 Texas Tech University, Joseph Dannemiller, May 2019 Table 5.10 Mean and standard deviation of differences between the location and scale parameters recorded by TTUHRT raw data and the means of simulated distributions using only 600s windows with smoother than open exposure Location Scale Magnitude (m/s) Percent (%) Magnitude (m/s) Percent (%) Mean STD Mean STD Mean STD Mean STD ‐2.25 0.30 ‐8.23% 1.08% ‐0.58 0.08 ‐33.37% 4.45% Table 5.11 Mean and standard deviation of differences between the location and scale parameters recorded by TTUHRT raw data and the means of simulated distributions using only 600s windows with open exposure Location Scale Magnitude (m/s) Percent (%) Magnitude (m/s) Percent (%) Mean STD Mean STD Mean STD Mean STD ‐4.42 0.22 ‐20.63 1.01 ‐0.46 0.07 ‐27.97 4.47 Table 5.12 Mean and standard deviation of differences between the location and scale parameters recorded by TTUHRT raw data and the means of simulated distributions using only 600s windows with rougher than open exposure Location Scale Magnitude (m/s) Percent Magnitude (m/s) Percent Mean STD Mean STD Mean STD Mean STD ‐1.82 0.19 ‐7.55 0.78 ‐0.14 0.06 ‐6.41 2.94 These differences show good agreement with the location and scale parameters recorded by TTUHRT platforms, especially when considering the magnitudes of location parameters are, on average, within 4.42m/s for open exposure values for defining a distribution extreme winds. The conditional differences for scale parameters in magnitude are within -0.14m/s and -0.58m/s of the actual values with very low standard deviations. The scale parameter percent differences are within -6.41% and - 33.37% with standard deviations between 2.27% and 4.47%. Given that these are conditional differences, and that the scale parameters are generally low in magnitude, usually between 1m/s and 6m/s, the differences shown in Table 5.10, Table 5.11 and Table 5.12 are promising for simulating distribution of extreme wind speeds for use in SPA. 188 Texas Tech University, Joseph Dannemiller, May 2019 The gust factor for computing the peak 3s of wind in a 600s window is equal to 1.44 following the procedures discussed in Chapter 2 as presented in (Vickery and Skerlj, 2005). This gust factor is multiplied by the mean 600s wind speeds recorded in TTUHRT raw records to compute peak 3s wind speeds in 600s windows. These peaks are compared against the actual peak 3s of wind recorded by TTUHRT platforms. A histogram of the differences in magnitude and as percent are illustrated in Figure 5.59 and Figure 5.60, respectively. Probability (p) Probability Figure 5.59 Differences in magnitude between the peak 3s wind speed recorded by the TTUHRT platforms and peak 3s wind speeds computed using the mean observed by the TTUHRT platforms and a gust factor equal to 1.44 189 Texas Tech University, Joseph Dannemiller, May 2019 Probability (p) Probability Figure 5.60 Differences in percent between the peak 3s wind speed recorded by the TTUHRT platforms and peak 3s wind speeds computed using the mean observed by the TTUHRT platforms and a gust factor equal to 1.44 Illustrations of similar differences for the peak 60s in a 600s window, computed using the gust factor equal to 1.18 as discussed in Chapter 2 are provided below for the 60s data as magnitudes and percent in Figure 5.61 and Figure 5.62, respectively. 190 Texas Tech University, Joseph Dannemiller, May 2019 Probability (p) Probability Figure 5.61 Differences in magnitude between the peak 60s wind speed recorded by the TTUHRT platforms and peak 60s wind speeds computed using the mean observed by the TTUHRT platforms and a gust factor equal to 1.18 Probability (p) Probability Figure 5.62 Differences in percent between the peak 60s wind speed recorded by the TTUHRT platforms and peak 60s wind speeds computed using the mean observed by the TTUHRT platforms and a gust factor equal to 1.18 191 Texas Tech University, Joseph Dannemiller, May 2019 The histograms above convey differences computed using the same theoretical methodology as the differences computed in Chapters 3 and 4, where the HWIND (HWIND value) gust factor is subtracted from the actual (TTUHRT recorded) gust factors. The differences all bias towards the positive side meaning the TTUHRT recorded values are larger than the 1.44 and 1.18 gust factors used for computing peak 3s and 60s gust wind speeds, respectively. 192 Texas Tech University, Joseph Dannemiller, May 2019 CHAPTER 6 CASE STUDIES To show how the proposed methodology works, and to quantify how it compares to the gust factor approach for computing peak winds, three case studies are drawn from the TTHURT records. The three case studies were drawn pseudo randomly per two requirements. The first requirement is that each of the three represents a different roughness exposure regime, so the case studies have three distinctly different values of qualitatively assigned roughness. This delineation is made to examine if there are any large differences between the roughness regimes since, as stated earlier, the conditional distributions of location and scale parameters for associated bins of mean or location parameters are not drawn from the same underlying distribution. The second requirement is that the mean wind speed needed to be reasonably high. Earlier discussions explained the importance of modeling higher level wind speeds in SPA so the three 600s windows used in this case study analysis needed to have mean wind speeds above 15m/s to satisfy the neutral stability requirement discussed in Chapter 2. Case Study One – Smoother Than Open Exposure (z0<0.03m) This case study is from the 1998 landfall of Bonnie recorded by TTUHRT WEMITE #1 between 16:15UTC and 16:25UTC. The deployment location, viewed from the air, is shown below in Figure 6.1. 193 Texas Tech University, Joseph Dannemiller, May 2019 Figure 6.1 Deployment location for the TTUHRT WEMITE #1 platform during the 1998 landfall of Hurricane Bonnie The mean wind speed recorded during this 600s window is 17.896m/s, the mean wind direction is 272.41° and the EV location and scale parameters are 27.49m/s and 1.29m/s, respectively. The surface roughness values upwind and downwind of the last roughness change are provided in Table 6.1. 194 Texas Tech University, Joseph Dannemiller, May 2019 Table 6.1 Assigned surface roughness values, surface roughness values upwind and downwind of the last roughness change, and distances to the last change in surface roughness at the TTUHRT WEMITE #1 deployment location during the 1998 landfall of Hurricane Bonnie in each 30 sector Distance Internal Transition Upwind Downwind Computed Sector from Boundary Boundary Surface Surface Surface (degrees) Roughness Layer Layer Roughness Roughness Roughness [°] Change Height Height [m] [m] [m] [m] [m] [m] 0-30 400 0.01 0.50 197.99 57.71 0.500 30-60 500 0.01 0.30 191.70 65.98 0.300 60-90 500 0.01 0.30 191.70 65.98 0.300 90-120 400 0.01 0.40 177.09 57.71 0.400 120-150 400 0.01 0.30 153.36 57.71 0.300 150-180 400 0.01 0.10 88.54 57.71 0.100 180-210 750 0.20 0.07 31.06 278.90 0.070 210-240 550 0.20 0.03 14.91 231.54 0.030 240-270 1000 0.20 0.03 27.11 331.45 0.030 270-300 450 0.50 0.02 9.96 296.15 0.020 300-330 375 0.50 0.03 6.43 265.46 0.030 330-360 450 0.75 0.04 7.27 348.30 0.040 The assigned z0 value for the mean wind direction equal to 272.41° is 0.02m. The time history recorded by the TTUHRT WEMITE#1 platform during this 600s window is illustrated below in Figure 6.2. 10min window beginning 24-Sep-1998 16:14:59 30 25 20 15 10 16:15 16:16 16:17 16:18 16:19 16:20 16:21 16:22 16:23 16:24 Time Figure 6.2 Wind speed time history recorded by the TTUHRT WEMITE #1 platform during the 1998 landfall of Hurricane Bonnie 195 Texas Tech University, Joseph Dannemiller, May 2019 The mean wind speed falls into the mean wind speed bin in Table 5.4 between 16.5m/s and 18m/s. The three GEV parameters defining the conditional distribution of location parameters are 0.13m/s, 1.87m/s and 26.06m/s for the scale, shape and location parameters, respectively. The distribution of location parameters is simulated, and a single location parameter value is sampled equal to 27.01m/s. This location parameter value falls into the location parameter bin in Table 5.7 bounded by 27m/s and 28.5m/s. The three GEV parameters defining the conditional distribution of scale parameters are -0.06m/s, 0.52m/s and 1.84m/s for the scale, shape and location parameters, respectively. The distribution of scale parameters is simulated, and a single location parameter value is sampled equal to 1.72m/s. The difference between the TTHURT observed location parameter and the sampled location parameter is computed as both a magnitude and a percent. The difference between the TTHURT observed scale parameter and the sampled scale parameter is also computed as both a magnitude and a percent. The process of sampling a location parameter, defining the parameters for a conditional distribution of scale parameters, simulating a distribution of scale parameters, sampling a single scale parameter value and computing the differences as both magnitude and percent for the location and scale parameters is repeated 1000 times. The simulated distributions of 1000 location and 1000 scale parameters are presented below in Figure 6.3 and Figure 6.4, respectively. 196 Texas Tech University, Joseph Dannemiller, May 2019 Figure 6.3 Simulated distribution of location parameters for the smooth roughness case study using TTUHRT Hurricane Bonnie raw data for a single 600s window Figure 6.4 Simulated distribution of scale parameters for the smooth roughness case study using TTUHRT Hurricane Bonnie raw data for a single 600s window 197 Texas Tech University, Joseph Dannemiller, May 2019 The differences in magnitude computed between the TTUHRT recorded location parameter equal to 27.49m/s and each value in the simulated distribution illustrated in Figure 5.42 are computed. The differences are also computed as percent and the resulting distributions of differences in magnitude and percent are illustrated below in Figure 6.5 and Figure 6.6, respectively. Figure 6.5 Differences in magnitude between the location parameter recorded by TTUHRT WEMITE #1 and the values in the simulated distribution of location parameters for the smooth roughness case study using TTUHRT Hurricane Bonnie raw data for a single 600s window 198 Texas Tech University, Joseph Dannemiller, May 2019 Figure 6.6 Differences in percent between the location parameter recorded by TTUHRT WEMITE #1 and the values in the simulated distribution of location parameters for the smooth roughness case study using TTUHRT Hurricane Bonnie raw data for a single 600s window Similarly, the differences in magnitude are computed between the TTUHRT recorded scale parameter value equal to 1.29m/s and each value in the simulated distribution illustrated in Figure 6.4. The differences are then computed as percent and the resulting distributions of differences in magnitude and percent are illustrated below in Figure 6.7 and Figure 6.8, respectively. 199 Texas Tech University, Joseph Dannemiller, May 2019 Figure 6.7 Differences in magnitude between the scale parameter recorded by TTUHRT WEMITE #1 and the values in the simulated distribution of scale parameters for the smooth roughness case study using TTUHRT Hurricane Bonnie raw data for a single 600s window 200 Texas Tech University, Joseph Dannemiller, May 2019 Figure 6.8 Differences in magnitude between the scale parameter recorded by TTUHRT WEMITE #1 and the values in the simulated distribution of scale parameters for the smooth roughness case study using TTUHRT Hurricane Bonnie raw data for a single 600s window The mean of each of the distributions in Figure 6.5 through Figure 6.8 are recorded and the process of simulating a distribution of 1000 values is repeated 1000 times with 1000 means recorded for each distribution. The mean and standard deviation of the means are computed for the magnitudes and percent difference distributions for both location and scale parameters. The distributions of mean differences in magnitude and percent are illustrated below for the location parameters in Figure 6.9 and Figure 6.10, respectively. 201 Texas Tech University, Joseph Dannemiller, May 2019 Figure 6.9 Distribution of mean differences in magnitude between the location parameter recorded by TTUHRT WEMITE #1 and the values in the simulated distributions of location parameters for the smooth roughness case study using TTUHRT Hurricane Bonnie raw data for a single 600s window 202 Texas Tech University, Joseph Dannemiller, May 2019 Figure 6.10 Distribution of mean differences in percent between the location parameter recorded by TTUHRT WEMITE #1 and the values in the simulated distributions of location parameters for the smooth roughness case study using TTUHRT Hurricane Bonnie raw data for a single 600s window The distributions of mean differences as magnitude and percent for the scale parameters are illustrated below in Figure 6.11 and Figure 6.12, respectively. 203 Texas Tech University, Joseph Dannemiller, May 2019 Figure 6.11 Distribution of mean differences in magnitude between the scale parameter recorded by TTUHRT WEMITE #1 and the values in the simulated distributions of scale parameters for the smooth roughness case study using TTUHRT Hurricane Bonnie raw data for a single 600s window 204 Texas Tech University, Joseph Dannemiller, May 2019 Figure 6.12 Distribution of mean differences in percent between the scale parameter recorded by TTUHRT WEMITE #1 and the values in the simulated distributions of scale parameters for the smooth roughness case study using TTUHRT Hurricane Bonnie raw data for a single 600s window The differences between the recorded values from TTUHRT time histories and the sampled values from Figure 6.5 and Figure 6.8 are skewed the same as the distributions of location and scale parameters simulated in Figure 6.3 and Figure 6.4, respectively. Distributions of mean differences from the distributions in Figure 6.5 and Figure 6.8, shown in Figure 6.9 through Figure 6.12, are normally distributed per the Central Limit Theorem as a distribution of mean values drawn from any underlying distribution will be normally distributed. While the percent differences for the scale parameters in Figure 6.12 do appear large, Figure 6.11 shows these differences to be between around -0.67m/s and 0.62m/s. These values are centered around zero so show neither a positive or negative bias compared to the scale parameter recorded in TTUHRT data. For comparison, if the mean wind speed from this 600s window, equal to 17.89m/s, was used with the gust factor discussed in Chapter 2 for computing maximum sustained wind speeds (peak 60s wind speed in each 600s window), equal 205 Texas Tech University, Joseph Dannemiller, May 2019 to 1.18, the computed maximum sustained wind speed would equal 21.12m/s. The actual peak 60s of wind recorded by the TTUHRT WEMITE#1 platform is 19.65m/s which is a 1.47m/s different, or 6.96%. Compare these values to the means of the location parameter differences as both magnitudes and percent in Figure 6.9 and Figure 6.10, respectively, equal to -0.31m/s and -1.09%. Both the magnitude and percentage are lower. The gust factor for computing the peak 3s wind speed in this 600s window is computed to be 1.44 using the techniques discussed in Chapter 2. Multiplying the mean wind speed equal to 17.89m/s by the gust factor equal to 1.44 results in a peak 3s wind speed equal to 25.67m/s. The actual peak 3s wind speed recorded by TTUHRT WEMITE #1 is equal to 25.19m/s which is a 0.49m/s difference, or 1.89%. The results of the methodology being proposed are on par or better than the results of the gust factor method and show excellent agreement with the data recorded by TTUHRT WEMITE #1. However, while the gust factor method provides a methodology for computing a single peak wind speed, the methodology being forwarded also provides a measure of dispersion to simulate distributions of extreme wind speeds. To provide some measure of dispersion for the gust factor approach, the mean wind speeds in all 600s windows recorded by the TTUHRT WEMITE #1 during the 1998 landfall of Hurricane Bonnie are used to compute peak 3s wind speeds using a gust factor equal to 1.44. The histograms of differences between actual peak 3s wind speed recorded by TTUHRT WEMITE#1 and the computed peak 3s values are presented below for magnitudes and percent in Figure 6.13 and Figure 6.14, respectively. Figure 6.3 through Figure 6.8 show a large amount of skew with some of the simulated data a large distance away from the means of the simulated distributions. The risk associated with using data far from the mean is that data in the tails, having a very low probability of occurrence, will not accurately model the loads acting on a structure. 206 Texas Tech University, Joseph Dannemiller, May 2019 0.12 0.1 0.08 0.06 0.04 0.02 0 -20246810 Differences (D) [m/s] Figure 6.13 Differences in magnitude between the peak 3s wind speed recorded by the TTUHRT WEMITE #1 during the 1998 landfall of Hurricane Bonnie and peak 3s wind speeds computed using the mean observed by the TTUHRT platform and a gust factor equal to 1.44 using all 600s windows in the Hurricane Bonnie time history 0.12 0.1 0.08 0.06 0.04 0.02 0 -10 0 10 20 30 40 50 Differences (D) [%] Figure 6.14 Differences in percent between the peak 3s wind speed recorded by the TTUHRT WEMITE #1 during the 1998 landfall of Hurricane Bonnie and peak 3s wind speeds computed using the mean observed by the TTUHRT platform and a gust factor equal to 1.44 using all 600s windows in the Hurricane Bonnie time history 207 Texas Tech University, Joseph Dannemiller, May 2019 Peak 60s wind speeds are also computed using the mean wind speeds in all 600s windows recorded by the TTUHRT WEMITE #1 during the 1998 landfall of Hurricane Bonnie multiplied by a gust factor equal to 1.18. The histograms for magnitudes and percent differences for the 60s values are shown below in Figure 6.15 and Figure 6.16, respectively. 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 -2-101234 Differences (D) [m/s] Figure 6.15 Differences in magnitude between the peak 60s wind speed recorded by the TTUHRT WEMITE #1 during the 1998 landfall of Hurricane Bonnie and peak 60s wind speeds computed using the mean observed by the TTUHRT platform and a gust factor equal to 1.18 using all 600s windows in the Hurricane Bonnie time history 208 Texas Tech University, Joseph Dannemiller, May 2019 0.12 0.1 0.08 0.06 0.04 0.02 0 -10-50 5 1015202530 Differences (D) [%] Figure 6.16 Differences in percent between the peak 60s wind speed recorded by the TTUHRT WEMITE #1 during the 1998 landfall of Hurricane Bonnie and peak 60s wind speeds computed using the mean observed by the TTUHRT platform and a gust factor equal to 1.18 using all 600s windows in the Hurricane Bonnie time history The difference between observed and computed peak wind speeds, computed earlier in this case study, were equal to 1.47m/s (6.96%) for a peak 60s wind speed, and 0.49m/s (1.89%) for a peak 3s wind speed. Figure 6.13 and Figure 6.14 shows the differences for 3s peak wind speeds can be anywhere from -2m/s to 8m/s (-10% to 45%), and Figure 6.15 and Figure 6.16 show that differences can be anywhere between -1.6m/s and 3.8m/s, (-10% to 30%) for 60s peak wind speeds. Case Study Two – Open Exposure (0.3m≤z0≤0.07m) This case study is from the 2004 landfall of Francis recorded by TTUHRT PMT Black between 04:05UTC and 04:15UTC on September 1, 2004. The deployment location, viewed from the air, is shown below in Figure 6.17. 209 Texas Tech University, Joseph Dannemiller, May 2019 Figure 6.17 Deployment location for the TTUHRT PMT Black platform during the 2004 landfall of Hurricane Francis The mean wind speed recorded during this 600s window is 22.46m/s, the mean wind direction is 173.10° and the EV location and scale parameters are 31.73m/s and 1.51m/s, respectively. The surface roughness values upwind and downwind of the last roughness change are provided in Table 6.2. 210 Texas Tech University, Joseph Dannemiller, May 2019 Table 6.2 Assigned surface roughness values, surface roughness values upwind and downwind of the last roughness change, and distances to the last change in surface roughness at the TTUHRT PMT Black deployment location during the 2004 landfall of Hurricane Francis in each 30 sector Distance Internal Transition Upwind Downwind Computed Sector from Boundary Boundary Surface Surface Surface (degrees) Roughness Layer Layer Roughness Roughness Roughness [°] Change Height Height [m] [m] [m] [m] [m] [m] 0-30 300 0.60 0.01 2.71 249.77 0.027 30-60 1000 0.60 0.01 9.04 514.35 0.011 60-90 800 0.50 0.01 7.92 418.26 0.012 90-120 800 0.60 0.01 7.23 449.90 0.014 120-150 1300 0.60 0.03 20.35 602.04 0.030 150-180 1000 0.60 0.03 15.65 514.35 0.030 180-210 1000 0.60 0.03 15.65 514.35 0.030 210-240 750 0.60 0.01 6.78 432.81 0.014 240-270 750 0.60 0.01 6.78 432.81 0.014 270-300 700 0.60 0.01 6.33 415.26 0.015 300-330 200 0.60 0.01 1.81 195.83 0.035 330-360 150 0.60 0.01 1.36 164.78 0.041 The assigned z0 for the mean wind direction equal to 173.10° is 0.03m. The mean wind speed falls into the mean wind speed bin in Table 5.4 between 21m/s and 22.5m/s. The three GEV parameters defining the conditional distribution of location parameters are 1.3m/s, 0.95m/s and 30.99m/s for the scale, shape and location parameters, respectively. The distribution of location parameters is simulated, and a single location parameter value is sampled equal to 31.96m/s. This location parameter falls into the location parameter bin in Table 5.7 bounded by 31.5m/s and 33m/s. The three GEV parameters defining the conditional distribution of scale parameters are 0.26m/s, 0.67m/s and 1.71m/s for the scale, shape and location parameters, respectively. The distribution of scale parameters is simulated, and a single location parameter value is sampled equal to 1.55m/s. The difference between the TTHURT observed location parameter and the sampled location parameter is computed as both a magnitude and a percent. The difference between the TTHURT observed scale 211 Texas Tech University, Joseph Dannemiller, May 2019 parameter and the sampled scale parameter is computed as both a magnitude and a percent. The process of sampling a location parameter, defining the parameters for a conditional distribution of scale parameters, simulating a distribution of scale parameters, sampling a single scale parameter value and computing the differences as both magnitude and percent for the location and scale parameters is repeated 1000 times. The simulated distributions of 1000 location and 1000 scale parameters are presented below in Figure 6.18 and Figure 6.19, respectively. Figure 6.18 Simulated distribution of location parameters for the smooth roughness case study using TTUHRT Hurricane Francis raw data for a single 600s window 212 Texas Tech University, Joseph Dannemiller, May 2019 Figure 6.19 Simulated distribution of scale parameters for the smooth roughness case study using TTUHRT Hurricane Francis raw data for a single 600s window The differences in magnitude computed between the TTUHRT recorded location parameter value equal to 31.73m/s and each value in the simulated distribution illustrated in Figure 6.18. The differences are also computed as percent and the resulting distributions of differences in magnitude and percent are illustrated below in Figure 6.20 and Figure 6.21, respectively. 213 Texas Tech University, Joseph Dannemiller, May 2019 Figure 6.20 Differences in magnitude between the location parameter recorded by TTUHRT PMT Black and the values in the simulated distribution of location parameters for the smooth roughness case study using TTUHRT Hurricane Francis raw data for a single 600s window 214 Texas Tech University, Joseph Dannemiller, May 2019 Figure 6.21 Differences in percent between the location parameter recorded by TTUHRT PMT Black and the values in the simulated distribution of location parameters for the smooth roughness case study using TTUHRT Hurricane Francis raw data for a single 600s window Similarly, the differences in magnitude are computed between the TTUHRT recorded scale parameter value equal to 1.51m/s and each value in the simulated distribution illustrated in Figure 6.19. The differences are also computed as percent and the resulting distributions of differences in magnitude and percent are illustrated below in Figure 6.22 and Figure 6.23, respectively. 215 Texas Tech University, Joseph Dannemiller, May 2019 Figure 6.22 Differences in magnitude between the scale parameter recorded by TTUHRT PMT Black and the values in the simulated distribution of scale parameters for the smooth roughness case study using TTUHRT Hurricane Francis raw data for a single 600s window 216 Texas Tech University, Joseph Dannemiller, May 2019 Figure 6.23 Differences in magnitude between the scale parameter recorded by TTUHRT PMT Black and the values in the simulated distribution of scale parameters for the smooth roughness case study using TTUHRT Hurricane Francis raw data for a single 600s window The mean of each of the distributions in Figure 6.20 through Figure 6.23 are recorded and the process of simulating a distribution of 1000 values is repeated 1000 times with 1000 means recorded for each distribution. The mean and standard deviation of the means are computed for the magnitudes and percent difference distributions for both location and scale parameters. The distributions of mean differences in magnitude and percent are illustrated below for the location parameters in Figure 6.24 and Figure 6.25, respectively. 217 Texas Tech University, Joseph Dannemiller, May 2019 Figure 6.24 Distribution of mean differences in magnitude between the location parameter recorded by TTUHRT PMT Black and the values in the simulated distributions of location parameters for the smooth roughness case study using TTUHRT Hurricane Francis raw data for a single 600s window 218 Texas Tech University, Joseph Dannemiller, May 2019 Figure 6.25 Distribution of mean differences in percent between the location parameter recorded by TTUHRT PMT Black and the values in the simulated distributions of location parameters for the smooth roughness case study using TTUHRT Hurricane Francis raw data for a single 600s window The distributions of mean differences in magnitude and percent are illustrated below for the scale parameters in Figure 6.26 and Figure 6.27, respectively. 219 Texas Tech University, Joseph Dannemiller, May 2019 Figure 6.26 Distribution of mean differences in magnitude between the scale parameter recorded by TTUHRT PMT Black and the values in the simulated distributions of scale parameters for the smooth roughness case study using TTUHRT Hurricane Francis raw data for a single 600s window 220 Texas Tech University, Joseph Dannemiller, May 2019 Figure 6.27 Distribution of mean differences in percent between the scale parameter recorded by TTUHRT PMT Black and the values in the simulated distributions of scale parameters for the smooth roughness case study using TTUHRT Hurricane Francis raw data for a single 600s window The differences between the recorded values from TTUHRT time histories and the sampled values from Figure 6.20 through Figure 6.23 are skewed the same as the distributions of location and scale parameters simulated in Figure 6.18 and Figure 6.19, respectively. Df mean differences from the distributions in Figure 6.20 through Figure 6.23, shown in Figure 6.24 through Figure 6.26, are normally distributed per the Central Limit Theorem as a distribution of mean values drawn from any underlying distribution will be normally distributed. While the percent differences for the scale parameters in Figure 6.27 do appear large, Figure 6.26 shows these differences to be between around -0.85m/s and -0.65m/s. The percent difference values are centered around zero so show no positive or negative bias compared to the 221 Texas Tech University, Joseph Dannemiller, May 2019 scale parameter recorded in TTUHRT data. For comparison, if the mean wind speed from this 600s window, equal to 22.46m/s, was used with the gust factor discussed in Chapter 2 for computing maximum sustained wind speeds (peak 60s wind speed in each 600s window), equal to 1.18, the computed maximum sustained wind speed would equal 26.50m/s. The actual peak 60s of recorded by the TTUHRT WEMITE#1 platform is 24.52m/s which is a -1.98m/s different, or -8.06%. Compare these values to the means for the location parameter difference as magnitudes and percent in Figure 6.9 and Figure 6.10, respectively, equal to -02.07m/s and -6.48%. The gust factor for computing the peak 3s wind speed in this 600s window is computed to be 1.44 using the techniques discussed in Chapter 2. Multiplying the mean wind speed equal to 22.46m/s by the gust factor equal to 1.44 results in a peak 3s wind speed equal to 32.34m/s. The actual peak 3s wind speed recorded by TTUHRT WEMITE #1 is equal to 30.49m/s which is a -1.85m/s difference, or -6.07%. Like the first case study, these number show excellent agreement with the TTUHRT data. Unlike the first case study, these numbers reflect negative differences meaning the proposed methodology overpredicts extreme value location and scale parameters, though not by much. To provide some measure of dispersion for the gust factor approach, the mean wind speeds in all 600s windows recorded by the TTUHRT PMT Black during the 2004 landfall of Hurricane Francis are used to compute peak 3s wind speeds using a gust factor equal to 1.44. The histograms of differences between actual peak 3s wind speed recorded by TTUHRT PMT Black and the computed peak 3s values are presented below for magnitudes and percent in Figure 6.28 and Figure 6.29, respectively. 222 Texas Tech University, Joseph Dannemiller, May 2019 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 -4-20246810 Differences (D) [m/s] Figure 6.28 Differences in magnitude between the peak 3s wind speed recorded by the TTUHRT PMT Black during the 2004 landfall of Hurricane Francis and peak 3s wind speeds computed using the mean observed by the TTUHRT platform and a gust factor equal to 1.44 using all 600s windows in the Hurricane Francis time history 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 -15 -10 -5 0 5 10 15 20 25 30 35 Differences (D) [%] Figure 6.29 Differences in percent between the peak 3s wind speed recorded by the TTUHRT PMT Black during the 2004 landfall of Hurricane Francis and peak 3s wind speeds computed using the mean observed by the TTUHRT platform and a gust factor equal to 1.44 using all 600s windows in the Hurricane Francis time history Peak 60s wind speeds are also computed using the mean wind speeds in all 600s windows recorded by the TTUHRT WEMITE #1 during the 1998 landfall of Hurricane Bonnie multiplied by a gust factor equal to 1.18. The histograms for 223 Texas Tech University, Joseph Dannemiller, May 2019 magnitudes and percent differences for the 60s values are shown below in Figure 6.30 and Figure 6.31, respectively. 0.25 0.2 0.15 0.1 0.05 0 -3-2-10123456 Differences (D) [m/s] Figure 6.30 Differences in magnitude between the peak 60s wind speed recorded by the TTUHRT PMT Black during the 2004 landfall of Hurricane Francis and peak 60s wind speeds computed using the mean observed by the TTUHRT platform and a gust factor equal to 1.18 using all 600s windows in the Hurricane Francis time history 224 Texas Tech University, Joseph Dannemiller, May 2019 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 -10-50 5 1015202530 Differences (D) [%] Figure 6.31 Differences in percent between the peak 60s wind speed recorded by the TTUHRT PMT Black during the 2004 landfall of Hurricane Francis and peak 60s wind speeds computed using the mean observed by the TTUHRT platform and a gust factor equal to 1.18 using all 600s windows in the Hurricane Francis time history 225 Texas Tech University, Joseph Dannemiller, May 2019 The difference between observed and computed peak wind speeds, computed earlier in this case study, were equal to -1.98m/s (-8.06%) for a peak 60s wind speed, and- 1.85m/s (-6.07%) for a peak 3s wind speed, Figure 6.30 and Figure 6.31 shows the difference for 3s peak wind speeds can be anywhere from -2m/s to 3m/s or -10% to 20%, and Figure 6.28 and Figure 6.29 show the differences can be anywhere between -3m/s and 6m/s, or around -10% to 25% for the 60s peak wind speeds. The differences computed in this case study are all at the lower end of the distributions in Figure 6.28 through Figure 6.31 but the means of all of these distributions is near zero and well organized. The magnitudes are similar so in this case the gust factor approach is comparable to the method proposed herein. However, the gust factor approach does not provide a dispersion parameter for use in SPA. Case Study Three – Rougher Than Open Exposure (z0>0.7m) This case study is from the 2003 landfall of Isabel recorded by TTUHRT WEMITE #2 between 14:55UTC and 15:55UTC. The deployment location, viewed from the air, is shown below in Figure 6.32. 226 Texas Tech University, Joseph Dannemiller, May 2019 Figure 6.32 Deployment location for the TTUHRT WEMITE #2 platform during the 2003 landfall of Hurricane Isabel The mean wind speed recorded during this 600s window is 20.67m/s, the mean wind direction is 200.06° and the EV location and scale parameters are 36.90m/s and 3.06m/s, respectively. The surface roughness values upwind and downwind of the last roughness change are provided in Table 6.3. 227 Texas Tech University, Joseph Dannemiller, May 2019 Table 6.3 Assigned surface roughness values, surface roughness values upwind and downwind of the last roughness change, and distances to the last change in surface roughness at the TTUHRT WEMITE #2 deployment location during the 2003 landfall of Hurricane Isabel in each 30 sector Distance Internal Transition Upwind Downwind Computed Sector from Boundary Boundary Surface Surface Surface (degrees) Roughness Layer Layer Roughness Roughness Roughness [°] Change Height Height [m] [m] [m] [m] [m] [m] 0-30 0 0.05 0.05 0.00 0.00 0.600 30-60 0 0.05 0.05 0.00 0.00 0.600 60-90 0 0.05 0.05 0.00 0.00 0.600 90-120 0 0.05 0.05 0.00 0.00 0.600 120-150 0 0.05 0.05 0.00 0.00 0.500 150-180 0 0.05 0.05 0.00 0.00 0.500 180-210 0 0.05 0.05 0.00 0.00 0.400 210-240 0 0.05 0.05 0.00 0.00 0.400 240-270 0 0.05 0.05 0.00 0.00 0.500 270-300 0 0.05 0.05 0.00 0.00 0.400 300-330 0 0.05 0.05 0.00 0.00 0.300 330-360 0 0.05 0.05 0.00 0.00 0.300 The assigned z0 value for the mean wind direction equal to 200.06° is 0.40m. The mean wind speed is in the mean wind speed bin in Table 5.4 between 19.5m/s and 21m/s. The three GEV parameters defining the conditional distribution of location parameters are 0.06m/s, 1.47m/s and 32.26m/s for the scale, shape and location parameters, respectively. The distribution of location parameters is simulated, and a single location parameter value is sampled equal to 32.97m/s. This location parameter value falls into the location parameter bin in Table 5.7 bounded by 30m/s and 31.5m/s. The three GEV parameters defining the conditional distribution of scale parameters are 0.28m/s, 0.50m/s and 2.10m/s for the scale, shape and location parameters, respectively. The distribution of scale parameters is simulated, and a single location parameter value is sampled equal to 3.23m/s. The difference between the TTHURT observed location parameter and the sampled location parameter is computed as both a magnitude and a percent. The difference between the TTHURT observed scale 228 Texas Tech University, Joseph Dannemiller, May 2019 parameter and the sampled scale parameter is also computed as both a magnitude and a percent. The process of sampling a location parameter, defining the parameters for a conditional distribution of scale parameters, simulating a distribution of scale parameters, sampling a single scale parameter value and computing the differences as both magnitude and percent for the location and scale parameters is repeated 1000 times. The simulated distributions of 1000 location and 1000 scale parameters are presented below in Figure 6.33 and Figure 6.34, respectively. Figure 6.33 Simulated distribution of location parameters for the smooth roughness case study using TTUHRT Hurricane Isabel raw data 229 Texas Tech University, Joseph Dannemiller, May 2019 Figure 6.34 Simulated distribution of scale parameters for the smooth roughness case study using TTUHRT Hurricane Isabel raw data The differences in magnitude computed between the TTUHRT recorded location parameter value equal to 36.90m/s and each value in the simulated distribution illustrated in Figure 6.33. The differences are also computed as percent and the resulting distributions of differences in magnitude and percent are illustrated below in Figure 6.35 and Figure 6.36, respectively. 230 Texas Tech University, Joseph Dannemiller, May 2019 Figure 6.35 Differences in magnitude between the location parameter recorded by TTUHRT WEMITE #2 and the values in the simulated distribution of location parameters for the smooth roughness case study using TTUHRT Hurricane Isabel raw data 231 Texas Tech University, Joseph Dannemiller, May 2019 Figure 6.36 Differences in percent between the location parameter recorded by TTUHRT WEMITE #2 and the values in the simulated distribution of location parameters for the smooth roughness case study using TTUHRT Hurricane Isabel raw data Similarly, the differences in magnitude are computed between the TTUHRT recorded scale parameter value equal to 3.06m/s and each value in the simulated distribution illustrated in Figure 6.34. The differences are also computed as percent and the resulting distributions of differences in magnitude and percent are illustrated below in Figure 6.37 and Figure 6.38, respectively. 232 Texas Tech University, Joseph Dannemiller, May 2019 Figure 6.37 Differences in magnitude between the scale parameter recorded by TTUHRT WEMITE #2 and the values in the simulated distribution of scale parameters for the smooth roughness case study using TTUHRT Hurricane Isabel raw data 233 Texas Tech University, Joseph Dannemiller, May 2019 Figure 6.38 Differences in percent between the scale parameter recorded by TTUHRT WEMITE #2 and the values in the simulated distribution of scale parameters for the smooth roughness case study using TTUHRT Hurricane Isabel raw data The mean of each of the distributions in Figure 6.35 through are recorded and the process of simulating a distribution of 1000 values is repeated 1000 times with 1000 means recorded for each distribution. The mean and standard deviation of the means are computed for the magnitudes and percent difference distributions for both location and scale parameters. The distributions of mean differences in magnitude and percent are illustrated below for the location parameters in Figure 6.39 and Figure 6.40, respectively. 234 Texas Tech University, Joseph Dannemiller, May 2019 Figure 6.39 Distribution of mean differences in magnitude between the location parameter recorded by TTUHRT WEMITE #2 and the values in the simulated distributions of location parameters for the smooth roughness case study using TTUHRT Hurricane Isabel raw data 235 Texas Tech University, Joseph Dannemiller, May 2019 Figure 6.40 Distribution of mean differences in percent between the location parameter recorded by TTUHRT WEMITE #2 and the values in the simulated distributions of location parameters for the smooth roughness case study using TTUHRT Hurricane Isabel raw data The distributions of mean differences as magnitude and percent are illustrated below for the scale parameters in Figure 6.41 and Figure 6.42, respectively. 236 Texas Tech University, Joseph Dannemiller, May 2019 Figure 6.41 Distribution of mean differences in magnitude between the scale parameter recorded by TTUHRT WEMITE #2 and the values in the simulated distributions of scale parameters for the smooth roughness case study using TTUHRT Hurricane Isabel raw data 237 Texas Tech University, Joseph Dannemiller, May 2019 Figure 6.42 Distribution of mean differences in percent between the scale parameter recorded by TTUHRT WEMITE #2 and the values in the simulated distributions of scale parameters for the smooth roughness case study using TTUHRT Hurricane Isabel raw data The differences between the recorded values from TTUHRT time histories and the sampled values in Figure 6.35 through Figure 6.38 are skewed the same as the distributions of location and scale parameters simulated in Figure 6.33 and Figure 6.34, respectively. Distributions of mean differences from the distributions in Figure 6.35 through Figure 6.38, shown in Figure 6.39 through Figure 6.42, are normally distributed per the Central Limit Theorem as a distribution of mean values drawn from any underlying distribution will be normally distributed. While the percent differences in the scale parameters is large, the magnitude of the differences is between 0.74m/s and 0.83m/s.. 238 Texas Tech University, Joseph Dannemiller, May 2019 For comparison, if the mean wind speed from this 600s window, equal to 20.67m/s, was used with the gust factor discussed in Chapter 2 for computing maximum sustained wind speeds (peak 60s wind speed in each 600s window), equal to 1.18, the computed maximum sustained wind speed would equal 21.43m/s. The actual peak 60s of wind recorded by the TTUHRT WEMITE#2 platform is 23.52m/s which is a -1.08m/s difference, or -4.61%. Compare these values to the means of the location parameter differences as both magnitudes and percent in Figure 6.39 and Figure 6.40, respectively, equal to 2.08m/s and 5.63%. Again, these values are not as good as the first case study but they are still coupled with a dispersion factor and a distribution with which distributions of extreme wind speeds can be simulated. The gust factor for computing the peak 3s wind speed in this 600s window is computed to be 1.44 using the techniques discussed in Chapter 2. Multiplying the mean wind speed equal to 20.67m/s by the gust factor equal to 1.44 results in a peak 3s wind speed equal to 29.76m/s. The actual peak 3s wind speed recorded by TTUHRT WEMITE #1 is equal to 32.32m/s which is a 2.55m/s difference, or 7.90%. The results from the proposed methodology and the gust factor method line up well again with the added benefit of the proposed methodology providing dispersion characteristics to use when simulating distributions of extreme wind speeds. To provide some measure of dispersion for the gust factor approach, the mean wind speeds in all 600s windows recorded by the TTUHRT WEMITE #2 during the 2003 landfall of Hurricane Isabel are used to compute peak 3s wind speeds using a gust factor equal to 1.44. The histograms of differences between actual peak 3s wind speed recorded by TTUHRT WEMITE#2 and the computed peak 3s values are presented below for magnitudes and percent in Figure 6.43 and Figure 6.44, respectively, 239 Texas Tech University, Joseph Dannemiller, May 2019 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 -1012345678 Differences (D) [m/s] Figure 6.43 Differences in magnitude between the peak 3s wind speed recorded by the TTUHRT WEMITE #2 during the 2003 landfall of Hurricane Isabel and peak 3s wind speeds computed using the mean observed by the TTUHRT platform and a gust factor equal to 1.44 using all 600s windows in the Hurricane Bonnie time history 240 Texas Tech University, Joseph Dannemiller, May 2019 0.12 0.1 0.08 0.06 0.04 0.02 0 -5 0 5 10 15 20 25 30 35 40 45 Differences (D) [%] Figure 6.44 Differences in percent between the peak 3s wind speed recorded by the TTUHRT WEMITE #2 during the 2003 landfall of Hurricane Isabel and peak 3s wind speeds computed using the mean observed by the TTUHRT platform and a gust factor equal to 1.44 using all 600s windows in the Hurricane Bonnie time history Peak 60s wind speeds are also computed using the mean wind speeds in all 600s windows recorded by the TTUHRT WEMITE #1 during the 1998 landfall of Hurricane Bonnie multiplied by a gust factor equal to 1.18. The histograms for magnitudes and percent differences for the 60s values are shown below in Figure 6.45 and Figure 6.46, respectively. 241 Texas Tech University, Joseph Dannemiller, May 2019 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 Differences (D) [m/s] Figure 6.45 Differences in magnitude between the peak 60s wind speed recorded by the TTUHRT WEMITE #2 during the 2003 landfall of Hurricane Isabel and peak 60s wind speeds computed using the mean observed by the TTUHRT platform and a gust factor equal to 1.18 using all 600s windows in the Hurricane Bonnie time history 242 Texas Tech University, Joseph Dannemiller, May 2019 0.12 0.1 0.08 0.06 0.04 0.02 0 -10-50 5 10152025 Differences (D) [%] Figure 6.46 Differences in percent between the peak 60s wind speed recorded by the TTUHRT WEMITE #2 during the 2003 landfall of Hurricane Isabel and peak 60s wind speeds computed using the mean observed by the TTUHRT platform and a gust factor equal to 1.18 using all 600s windows in the Hurricane Bonnie time history While the computed differences using the data in this case study, between observed and computed peak wind speeds, were equal to -1.08m/s (-4.61%) for a peak 60s wind speed, and 2.55m/s (7.90%) for a peak 3s wind speed, Figure 6.43 and Figure 6.44 shows the difference can be anywhere from 0m/s to 7m/s or -5% to 40% when computing peak 3s wind speeds using a single gust factor. The differences shown in Figure 6.45 and Figure 6.46 show that using a single gust factor to compute peak 60s wind speeds is better than the results for the 3s values, but the differences are still between -1m/s and 2m/s, or -5% to 25%. Case Study Results The results of all three case studies show good and the differences in both magnitude and percent for the conditional distributions are shown to be on par, or better, than the differences when using a single gust factor to compute a single peak wind speed. The conditional distributions in all three cases were drawn from bins in 243 Texas Tech University, Joseph Dannemiller, May 2019 Table 5.4 and Table 5.7 where there were enough observations from TTUHRT records to generated parameters that fit each EV parameter distribution well. 244 Texas Tech University, Joseph Dannemiller, May 2019 CHAPTER 7 FUTURE POSSIBILITIES FOR SIMULATING DISTRIBUTIONS OF EXTREME WIND SPEEDS The methodology advanced above is based on dependent relationships between parent mean vs EV location and EV location vs EV scale parameters. While these distribution parameters may not be dependent, the conditional distributions developed can be used to estimate EV parameters used to simulate distributions of extreme winds for use in SPA. Going forward, other relationships may prove more useful. One such relationship is based on the observation that as parent wind field mean wind speeds increase, the turbulence intensity (TI) bands down to become more concentrated around a single TI value where the atmosphere is neutrally stable. This phenomenon is illustrated below in Figure 7.1 using the mean and turbulence intensity values from all 600s windows recorded by TTUHRT platforms. raw, 600s x / raw, 600s Turbulence Intensity (TI) Intensity Turbulence sd Mean wind speed x raw, 600s [m/s] Figure 7.1 Relationship between parent mean and turbulence intensity using TTUHRT data from all 600s windows 245 Texas Tech University, Joseph Dannemiller, May 2019 As the mean wind speed increases, the mechanically driven turbulence dominates and the turbulence intensity bands down to become more concentrated around a value of, approximately, 0.17. The same phenomenon is present in the extreme value distribution as well. Figure 7.2 illustrates the relationship between the mean wind speed and the EV turbulence intensity (EV TI), computed using Equation 7.1 below. SDEV t, T 𝐸𝑉 𝑇𝐼 7.1 MEV t, T t, T MEV / t, T SDEV EV Turbulence Intensity ( TI) Mean wind speed x raw, 600s [m/s] Figure 7.2 Relationship between parent mean and EV TI using TTUHRT data from all 600s windows, broken into roughness regimes (smooth, open and rough) The data in Figure 7.2 shows the dispersion of the smoother terrain values to be larger than the dispersion of the rougher terrain values. At present the reasoning for this is unknown. However, it is important to note that all of the data in rougher terrain comes from two hurricane deployments as the TTUHRT researchers purposefully used these 246 Texas Tech University, Joseph Dannemiller, May 2019 two deployments to sample wind speeds in rougher terrain. For all other deployments, the TTUHRT researchers work very hard to identify large open areas to deploy their platforms in an attempt to gather as much data in open exposure as possible. The mean vs EV TI is then plotted using only 600s windows containing a mean wind speed greater than 15m/s in Figure 7.3 below. t, T MEV / t, T SDEV EV Turbulence Intensity ( TI) Mean wind speed x raw, 600s [m/s] Figure 7.3 Relationship between parent mean and EV TI using only TTUHRT data from 600s windows with a mean wind speed greater than 15m/s, broken into roughness regimes (smooth, open and rough) Figure 7.3 uses only data where the mean wind speed is greater than 15m/s so the dispersion of the EV TI values is less at the left end, from 0.04 to 0.17, than the values in Figure 7.2 where the dispersion at lower mean wind speeds spans from 0.03 to 0.25. A second dimensionless quantity, the EV coefficient of variation EV COV) is computed using Equation 7.2 below, and then the mean wind speed vs EV COV relationship is plotted in Figure 7.4 below. 247 Texas Tech University, Joseph Dannemiller, May 2019 σ raw, 600s 𝐸𝑉 𝐶𝑂𝑉 7.2 μ raw, 600s raw, 600s ariation (EV COV) μ / raw, 600s σ EV Coefficient of V Mean wind speed x raw, 600s [m/s] Figure 7.4 Relationship between parent mean and EV COV using TTUHRT data from all 600s windows, broken into roughness regimes (smooth, open and rough) The mean vs EV COV relationship using only 600s windows with a mean wind speed greater than 15m/s is plotted in Figure 7.5 below. 248 Texas Tech University, Joseph Dannemiller, May 2019 raw, 600s ariation (EV COV) μ / raw, 600s σ EV Coefficient of V Mean wind speed x raw, 600s [m/s] Figure 7.5 Relationship between parent mean and EV COV using only TTUHRT data from 600s windows with a mean wind speed greater than 15m/s, broken into roughness regimes (smooth, open and rough) The data is similarly organized in the pair of plot Figure 7.2 and Figure 7.4, and in the pair of plots Figure 7.3 and Figure 7.5. For this reason, it is difficult to determine which non-dimensional relationship would be more useful, but again, this discussion is meant to comment on future possibilities of this work so no decision is made herein. Since the mean vs EV TI and mean vs EV COV relationships behave similarly to the mean vs TI data, the next step is to relate the TI s EV TI and the TI vs EV COV. These relationships are plotted below in Figure 7.6and Figure 7.7, respectively, using all 600s window data, and in Figure 7.8 and Figure 7.9, respectively, using only 600s windows data from windows with a mean wind speed greater than 15m/s. 249 Texas Tech University, Joseph Dannemiller, May 2019 Smoother than open Open 0.25 Rougher then open t, T 0.2 MEV 0.15 / 0.1 t, T 0.05 SDEV EV Turbulence Intensity ( TI) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Turbulence Intensity (TI) sd raw, 600s /x raw, 600s Figure 7.6 Relationship between TI and EV TI using TTUHRT data from all 600s windows, broken into roughness regimes (smooth, open and rough) raw, 600s μ / raw, 600s σ EV Coefficient of Variation (EV COV) Turbulence Intensity (TI) sd raw, 600s /x raw, 600s Figure 7.7 Relationship between TI and EV COV using TTUHRT data from all 600s windows, broken into roughness regimes (smooth, open and rough) 250 Texas Tech University, Joseph Dannemiller, May 2019 0.35 Smoother than open Open 0.3 Rougher then open 0.25 t, T 0.2 MEV / 0.15 t, T 0.1 0.05 SDEV EV Turbulence Intensity ( TI) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Turbulence Intensity (TI) sd raw, 600s /x raw, 600s Figure 7.8 Relationship between TI and EV TI using only TTUHRT data from 600s windows with a mean wind speed greater than 15m/s, broken into roughness regimes (smooth, open and rough) raw, 600s μ / raw, 600s σ EV Coefficient of Variation (EV COV) Turbulence Intensity (TI) sd raw, 600s /x raw, 600s Figure 7.9 Relationship between TI and EV COV using only TTUHRT data from 600s windows with a mean wind speed greater than 15m/s, broken into roughness regimes (smooth, open and rough) 251 Texas Tech University, Joseph Dannemiller, May 2019 In future work, it might be possible to use qualitative assessment of roughness to estimate a TI value which can then be used to estimate an EV TI or EV COV value. Simulating distributions of one, or all, of these parameters may lead to an advancement on the model being forwarded here to simulate distributions of extreme wind speeds for use in SPA. 252 Texas Tech University, Joseph Dannemiller, May 2019 CHAPTER 8 CONCLUSION Assessing the sequence and probabilities of failure for structures subjected to hurricane wind loads requires accurately modeling the winds leading to each structural failure. Causal assessments can be completed deterministically or stochastically, with the latter being more suitable when: (1) requiring accurate modeling of a multitude of random variables to produce a random outcome, or (2) when quantifying a probability of failure, instead of a simple pass/fail assessment. Stochastic analysis can be used to compute probabilities but requires accurate estimates of the parameters defining the distributions of every random variable. For wind loading, the distribution defining the variability in the wind field itself becomes the most important random variable to model. Several groups record wind data using high resolution, high fidelity, high precision surface observation systems. However, the combined network lacks the density to provide wind data useable at all locations where man-made structures exist. In the absence of high resolution, high fidelity, high precision wind records, the HWIND hurricane wind field reconstruction model was investigated as a source of wind speed data for locations where no valid wind data was recorded in close proximity. The maximum sustained wind speeds reported by HWIND were compared against the peak 60s wind speeds recorded by Texas Tech University’s Hurricane Research Team (TTUHRT) using only data gathered at 10m height. The TTUHRT archives data archive provided time histories from 32 deployments, gathered by 15 different platforms, during the landfall of 10 separate hurricanes. The TTUHRT platforms are high resolution, high fidelity, high precision instruments and the data collected by the TTUHRT platforms is used by HWIND scientists to produce HWIND reconstructions. The TTUHRT data was standardized to facilitate direct comparison to HWIND reconstruction data. As TTUHRT records are included in HWIND reconstructions, it was expected that the HWIND reported maximum sustained wind 253 Texas Tech University, Joseph Dannemiller, May 2019 speeds would match the TTUHRT peak 60s wind speeds, but large errors were observed. Errors as high as +/-50% were computed and these were not consistent with the values reported by (Powel et al, 1996). Some of the HWIND reconstruction wind speed values were low in magnitude amplifying the percent error for those values, but Figure 4.1 showed a 10m/s difference in TTUHRT and HWIND values at a TTUHRT recorded wind speed of 52m/s using TTUHRT PMT Clear wind data recorded during the 2005 landfall of Hurricane Katrina, and Figure 4.2 showed an 8m/s difference in TTUHRT and HWIND values at a TTUHRT recorded wind speed of 28m/s recorded by WEMITE #1 during the 1998 landfall of Hurricane Bonnie. The mean wind speeds recorded in TTUHRT records were compared to mean wind speeds computed by dividing the maximum sustained wind speeds from HWIND reconstruction by the gust factor used by HWIND scientists. The mean wind speed comparison also showed large errors on par with the errors computed when comparing maximum sustained wind values from the two sources. The single gust factor utilized by HWIND scientists for all wind speeds occurring over land was also investigated. No publication ever authored by HWIND scientists specifically states what value HWIND reconstructions use over land, but the methodologies in (Vickery and Skerlj, 2005) were combined with information presented in (Durst, 1960), (Krayer and Marshal, 1995) and (WMO, 2010) to compute a gust factor to 1.18 for computing the peak 60s wind speed in a 600s window, and a gust factor of 1.44 for computing a peak 3s wind speed in a 600s window. The 1.18 gust factor was multiplied by the mean wind speeds recorded in TTUHRT records and compared against the actual peak 60s wind speeds recorded in each 600s window. The peak 60s gust factors computed using TTUHRT records were separated into or smoother than open, open, and rougher than open exposures and separate distributions were plotted for each. In the distributions for all three exposure classifications, the 1.18 gust factor was shown to be at or near the highest quantile in each distribution. While the 1.18 gust factor showed good agreement with the mode of each distribution of gust factors, the gust factors computed using the equations and methods 254 Texas Tech University, Joseph Dannemiller, May 2019 based on the log law and the Law of the Wall, were discussed and shown to be problematic due to violations in base assumptions. This statement will require more investigation but is already backed up by data presented in (Balderrama et al., 2012). Linear relationships for the mean trends were quantified for the mean speed of the parent wind field and the mode of the distribution of extreme winds trends. These linear trends were well organized for larger moving average durations (60s) compared to the raw data. tThe dispersion about each trend was captured by computing conditional distributions of joint probabilities. This makes simulating a distribution of location parameters possible using a single mean wind speed value, and an exposure classification (smoother than open, open, rougher than open). A similar joint probability technique was enumerated to estimate conditional distributions of scale parameters as a function of a single location parameter. The values for simulating distributions of location and scale parameters were presented for the raw, 3s MA and 60s MA data and for the smoother than open, open and rougher than open exposure classification resulting in nine datasets for each bin of joint occurrence. A two stage Monte Carlo technique was employed to verify the methodology of simulating distributions of extreme winds using a single mean wind speed and an exposure classification. The differences between the observed location parameters recorded in TTUHRT records versus the mean values from simulated distributions of location parameters were shown to be correlated, and this correlation became better as the duration of the moving average increased. The differences were also computed for the scale parameters as the observed scale parameters recorded in TTUHRT records versus the mean values from simulated distributions of scale parameters. For smoother than open exposure, the differences between TTUHRT recorded and simulated location parameters were -2.25m/s (-8.23%) and the differences between TTUHRT recorded and simulated scale parameters was -0.58m/s (33.37%). For open exposure, the differences between location parameters were -4.42m/s (- 20.63%) and the differences between scale parameters was -0.46 m/s (27.97%). For rougher than open exposure, the differences between location parameters were - 255 Texas Tech University, Joseph Dannemiller, May 2019 1.82m/s (-7.55%) and the differences between scale parameters was -0.14 (6-6.41%). These difference values include data from all of 600s windows in the TTUHRT archives including records with low mean wind speeds to provide as much data as possible for this comparison. The magnitudes show excellent agreement when considering that a mean difference in magnitude between -2.25m/s and 4.42m/s is a good order of magnitude for location parameter values of interest in SPA that span from 25m/s up to 50m/s. As location parameter express a measure of central tendency for the distribution of extreme winds, and the gust factor for computing peak 60s winds in 600s windows was located at or near the highest quantile in the distributions of gust factors, the errors associated with using a gust factor approach to estimate a single value of peak wind speed were compared against the errors presented above for estimating location parameter values. The results were found to be on par for both methodologies with the added benefit in the proposed methodology of also quantifying a dispersion parameter. Therefore, when using this methodology, it is now possible to compute more than a single peak wind speed value. Now, entire distributions of extreme wind speeds can be simulated with ease making stochastic analysis possible using only a mean wind speed and exposure classification. With the model forwarded here hurricane model data can now be used, in lieu of observational data, to stochastically analyze the effects of hurricane wind fields on the built environmentThe only limitation is that data in the tails of the simulated distributions may not accurately reflect the loads acting on a structure and therefore, may not meet the rigors for use in litigation. Future Work The inclusion of more data always aids in refining any model so more data will be sought from research entities whose data is also provided to HWIND scientists in order to tease out the differences between the TTUHRT and HWIND reconstruction values. HWIND scientists will also be sought to form partnerships to better 256 Texas Tech University, Joseph Dannemiller, May 2019 understand these differences. More data will also aid in refining the conditional distribution parameters forwarded for simulating distribution of extreme wind speeds. While statistical relationships were quantified for two gust averaging times herein, a more abstract model should be assembled capable of quantifying statistical relationships for any averaging time. This is an arduous process that can be started using data already in hand, but can be improved by, again, reaching out to other research entities to acquire as much data as possible. Several unique concerns appeared when dealing with TTUHRT StickNet data that merit further exploration. One instance was an inability to use computed surface roughness values to standardize wind data using the turbulence intensity method. This is due to the assumptions underlying the turbulence intensity derived roughness values computed at 2.25m and whether the zero pane displacements will need to be included at this height as they are often ignored for 10m data. 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