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, 2014. 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 2013. 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.
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.
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 96 0.5631 0.5410 0.7759 72 0.5801 0.5786 0.9854 48 0.4283 0.3824 0.2887 24 0.3529 0.3594 0.7662
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 96 0.1639 0.1719 0.8094 72 0.1962 0.1791 0.6395 48 0.2338 0.2386 0.9189 24 0.3042 0.2991 0.9091 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.
References Chow, V.T., Handbook of Hydrology, McGraw-Hill, New York, New York, 1964. 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, http://water.usgs.gov/pubs/twri/twri4a3/, 2002. Huff, F.A. and J. R. Angel, Rainfall Frequency Atlas of the Midwest (Bulletin 71). Illinois State Water Survey, 1992. Lins, H.F., A Note on Stationarity and Nonstationarity, World Meteorological Organization, Commission for Hydrology, 2012. 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:10.7930/j0z31wj2 National Oceanic and Atmospheric Administration, National Centers for Environmental Information, Hourly Rainfall Data, https://www.ncdc.noaa.gov/cdo-web/search?datasetid=precip_hly, 2018.
Appendix A, Station List Station Location State Station Location State Number Number 015550 Montgomery AL 235205 Malden MO 026481 Phoenix AZ 235671 Moberly MO 030064 Alicia AR 235987 Nevada MO 030458 Batesville AR 237455 St. Louis MO 030616 Berryville AR 237506 Salem MO 030842 Botkinburg AR 237813 Skidmore MO 031582 Compton AR 237976 Springfield MO 031632 Corning AR 238043 Steelville MO 032356 Eureka Springs AR 238880 West Plains MO 032794 Gilbert AR 245086 Livingston MT 033132 Hardy AR 253395 Grand Island NE 034248 Little Rock AR 256065 North Platte NE 035228 Norfork Dam AR 256255 Omaha NE 047740 San Diego CA 262573 Elko NV 047769 San Francisco CA 264436 Las Vegas NV 048135 Shasta Dam CA 271683 Concord NH 052220 Denver CO 286026 Newark NJ 060806 Sikorsky CT 290234 Albuquerque NM 079595 Wilmington DE 308383 Syracuse NY 084358 Jacksonville FL 312719 Elizabeth City NC 095443 Macon GA 313630 Greensboro NC 098969 Valdosta GA 325479 Mandan Station ND 101022 Boise ID 331786 Columbus OH 115751 Moline IL 346661 Oklahoma City OK 118179 Springfield IL 348992 Tulsa OK 124259 Indianapolis IN 352709 Eugene OR 132203 Des Moines IA 360106 Allentown PA 135198 Marshalltown IA 366233 New Castle PA 141699 Colby KS 376698 Providence RI 142164 Dodge City KS 381939 Columbia SC 148167 Topeka KS 390020 Aberdeen SD 148830 Wichita KS 396937 Rapid City SD 151855 Cincinnati NKIA KY 402680 Dyersburg TN 154746 Lexington KY 406402 Nashville TN 156110 Paduca KY 408065 Samburg TN 168440 Shreveport LA 412244 Dallas TX 170273 Augusta ME 415890 Midland TX 180700 Beltsville MD 417945 San Antonio TX 209218 Ypsilanti MI 427598 Salt Lake UT 215435 Minneapolis MN 435733 Northfield VT 225776 Meridian MS 447201 Richmond VA 230789 Bolivar MO 447285 Roanoke VA 231674 Clearwater Dam MO 459465 Yakima WA 233999 Hornersville MO 461570 Charleston WV 234315 Joplin MO 474961 Madison WI 234549 Kirksville MO 481570 Casper WY 234825 Lebanon MO 502968 Fairbanks AK