STATIONARITY OF THE TEMPORAL DISTRIBUTION OF RAINFALL Joseph P. Wilson, Wilson Hydro, LLC, Rolla, Missouri

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1 STATIONARITY OF THE TEMPORAL DISTRIBUTION OF RAINFALL Joseph P. Wilson, Wilson Hydro, LLC, Rolla, Missouri Abstract The time distribution of rainfall is a critical element of hydrologic analysis. The location and magnitude of the period of maximum rainfall intensity greatly impacts the resulting runoff computations. More intense rainfall has been predicted as a result of climate change, Mellilo, Eighty National Weather Service hourly rainfall gaging stations in southern Missouri and select stations throughout the United States were analyzed for changes in the distribution of rainfall with respect to time. Each gage has a period of record of 60 years or greater. The non-dimensionalized, annual exceedance series, of 24-hour, 48-hour, 72-hour and 96-hour duration rainfall events were analyzed for stationarity of both the location and the magnitude for the period of the maximum fraction of rainfall. The analysis shows there is no statistically significant evidence that the temporal distribution of rainfall has changed during the period of record. Introduction The assumption of stationarity is fundamental to the analysis of hydrologic data. In general, stationarity is a relative term. Lins, 2012, stated that natural systems are unequivocally, non-stationary. Once this is understood, the issue then becomes whether a simple deterministic model realistically represents the system being studied or is it necessary to quantify and describe the underlying distribution. The temporal distribution of rainfall is a necessary input to most single event hydrologic models. The timing of the occurrence and of the location of the maximum bursts of rainfall greatly impacts the resulting computations. The relationship between the timing of rainfall and the response time of the watershed is the fundamental problem being solved with hydrologic modeling. The computed results often controls how infrastructure is sized, located, and how risk is assigned (e.g. FEMA Flood Insurance Rate maps). Data National Atmospheric and Oceanic Administration (NOAA) hourly rainfall data was analyzed for 80 stations throughout the United States. The primary area of interest, southern Missouri and northern Arkansas, included a cluster of stations. The locations of the stations analyzed are shown in Figure 1. Appendix A list the stations analyzed by NOAA Station ID number and location. Only stations with a minimum period of record of 60 years were analyzed. The total period of record for most stations was from 1948 until Data for each station was broken into two segments, before January 1,1980, designated as B80 data and after December 31, 1979, designated as A80 data. Stations with more than 5% missing data over the entire period of record or more than 5% of either the A80 or B80 segments were eliminated. The Annual Exceedance Series (AES) for durations of 24-hours, 48-hours, 72-hours and 96- hours was extracted for analysis. The AES is defined as the partial duration series with the threshold set such that the number of events is equal to the number of years in the period of record being analyzed (Chow, 1964). Each event was required to have precipitation occurring in at least 80% of the time periods to be included in the analysis. The AES rainfall events extracted were unconstrained sums, (not constrained to a 24-hour clock day). All statistical analysis was performed with α = 0.05 level of significance.

2 Analysis The data were analyzed in three steps for each station based on the A80 and B80 subdivision of data; 1) investigation of the stationarity of the basic data, 2) comparison of the location of the maximum fraction of rainfall, and 3) comparison of the magnitude of the maximum fraction of rainfall. Stationarity of Data The cumulative mass diagrams were evaluated to determine the stationarity of the underlying data. The cumulative mass curves were prepared as cumulative sums of hourly rainfall versus time. The cumulative mass curve for the Springfield, Missouri station is shown in Figure 2. Linear regression for each A80 and B80 data segment was performed and the slope of two lines compared. Most of the data sets shows a statistical difference in slopes at the 5% level of significance. In the Missouri and Arkansas region, 50% of the stations show an increase in slope (A80 slope > B80 slope), indicating an increase in overall precipitation, and 50% of the stations show a decrease in slope, indicating a decrease in overall precipitation for the respective time periods. Based on the analysis showing an equal number of increasing and decreasing trends, is it logical to conclude that there is no statically significant regional trend in the occurrence of rainfall in southern Missouri and northern Arkansas. When all of the stations are considered, 56% indicate a trend upward, 44% trend downward, and 1% show no trend in cumulative rainfall for the A80 and B80 datasets. This implies that a regional analysis with more stations is needed to determine trends in rainfall. Figure 1. NOAA Hourly Rainfall Stations Perica, et al., in NOAA Atlas 14, Region 8, also concluded that although some individual stations showed likely trends, there was insufficient statistical evidence to justify a regional trend in the annual maximum series of rainfall depth.

3 The non-parametric Mann-Kendall test was applied to each station A80 and B80 data (USGS 2002). The analysis indicates 1% of the Station duration/event pairs showed a likely trend. The cumulative mass slope comparison and the Kendall-Mann test support the assumption that for the stations and time periods analyzed, there is no consistent trend in the data. Position of Maximum Period of Rainfall All rainfall depths and time periods within each data set were non-dimensionalized as fractions of total rainfall versus fraction of total event duration. The average position of the maximum fraction of rainfall for the annual exceedance series was determined for each duration pair of A80 and B80 data. The average locations of the maximum periods were compared using the Student s t test at an α = 0.05 level of significance. None of the station, duration, and location pairs were determined to be likely to exhibit a change in location. Computed results are shown in Table 1. Location Summary, for the Springfield, Missouri station. Figure 2. Cumulative Mass Diagram, Springfield, Missouri Table 1. Location Summary for Springfield, Missouri. Duration, Hours Fraction Location, A80 Fraction Location, B80 Computed P-Value

4 The Table 1 data supports Huff s (1992) conclusion that longer duration storms tend to produce the maximum precipitation later in the storm than shorter duration events. Magnitude of Maximum Period of Rainfall The average magnitude of maximum period of rainfall, expressed as a fraction of total rainfall, was computed for the A80 and B80 data sets for each station. The average magnitude values were compared using the Student s t test at an α = 0.05 level of significance. None of the station, duration, and location pairs were determined to be likely to exhibit a change in the magnitude of the maximum fraction. Computed results are shown in Table 2. Magnitude Summary, for the Springfield, Missouri station. Table 2. Magnitude Summary for Springfield, Missouri. Duration, Hours Fraction Magnitude, A80 Fraction Magnitude, B80 Computed P-Value Conclusions Hourly rainfall data from 80 NOAA stations across the United States were analyzed for stationarity for cumulative rainfall, position of the time period of the maximum intensity relative to total event duration, and magnitude of the time period of the maximum intensity relative to total even depth. The annual exceedance series for each station was used for durations of 24-hours, 48-hours, 72-hours, and 96-hours. The analysis shows an increase in cumulative precipitation for the time period of 1980 to 2013 as compared to the time frame of 1948 to 1980 for 56% of the stations evaluated. In the southern Missouri and northern Arkansas region, there were an equal number of stations trending with increasing precipitation and trending with decreasing precipitation. This indicates that a regional analysis is needed to determine the trend of local rainfall data. Comparison of both the average position and magnitude of the time period of maximum precipitation for the A80 and B80 data sets for each station shows no statistical difference in the averages. There is insufficient evidence to support the hypothesis that the temporal distribution of rainfall has changed in the last 60 years. Future analysis should be performed to determine whether there are any changes in the fractions of rainfall that fall prior to and after the time period of maximum rainfall.

5 References Chow, V.T., Handbook of Hydrology, McGraw-Hill, New York, New York, Helsel, D.R. and R.M. Hirsch, Techniques of Water-Resources Investigations of the United States Geological Survey Book 4, Hydrologic Analysis and Interpretation, Chapter A3, Statistical Methods, Huff, F.A. and J. R. Angel, Rainfall Frequency Atlas of the Midwest (Bulletin 71). Illinois State Water Survey, Lins, H.F., A Note on Stationarity and Nonstationarity, World Meteorological Organization, Commission for Hydrology, Melillo, Jerry M., Terese (T.C.) Richmond, and Gary W. Yohe, Eds., 2014: Climate Change Impacts in the United States: The Third National Climate Assessment. U.S. Global Change Research Program, 841 pp. doi: /j0z31wj2 National Oceanic and Atmospheric Administration, National Centers for Environmental Information, Hourly Rainfall Data,

6 Appendix A, Station List Station Location State Station Location State Number Number Montgomery AL Malden MO Phoenix AZ Moberly MO Alicia AR Nevada MO Batesville AR St. Louis MO Berryville AR Salem MO Botkinburg AR Skidmore MO Compton AR Springfield MO Corning AR Steelville MO Eureka Springs AR West Plains MO Gilbert AR Livingston MT Hardy AR Grand Island NE Little Rock AR North Platte NE Norfork Dam AR Omaha NE San Diego CA Elko NV San Francisco CA Las Vegas NV Shasta Dam CA Concord NH Denver CO Newark NJ Sikorsky CT Albuquerque NM Wilmington DE Syracuse NY Jacksonville FL Elizabeth City NC Macon GA Greensboro NC Valdosta GA Mandan Station ND Boise ID Columbus OH Moline IL Oklahoma City OK Springfield IL Tulsa OK Indianapolis IN Eugene OR Des Moines IA Allentown PA Marshalltown IA New Castle PA Colby KS Providence RI Dodge City KS Columbia SC Topeka KS Aberdeen SD Wichita KS Rapid City SD Cincinnati NKIA KY Dyersburg TN Lexington KY Nashville TN Paduca KY Samburg TN Shreveport LA Dallas TX Augusta ME Midland TX Beltsville MD San Antonio TX Ypsilanti MI Salt Lake UT Minneapolis MN Northfield VT Meridian MS Richmond VA Bolivar MO Roanoke VA Clearwater Dam MO Yakima WA Hornersville MO Charleston WV Joplin MO Madison WI Kirksville MO Casper WY Lebanon MO Fairbanks AK