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Some probability density functions (pdfs) used for extrapolating to longer time periods Soil Moisture 2006 (0 to 90 cm) (large plots)Gypsum electrical resistance blocks for measuring soil moistureSoil Moisture Content1234561900 1927 1954 1982precip (in)Partial Duration SeriesAnnual MaximumUrbana daily rainfall, Annual Maxima and PartialDuration Series 1900-2000Annual Maximum Partial Durationrank Precip Precipm (in) date (in) date1 5.32 8/12/19935.328/12/19932 4.5 5/26/19214.55/26/19213 4.43 7/30/19874.437/30/19874 4.07 11/2/19364.0711/2/19365 3.91 9/15/19313.919/15/19316 3.9 8/20/19243.98/20/19247 3.89 6/19/19833.896/19/19838 3.56 10/6/19553.7210/21/19839 3.55 8/18/19393.5610/6/195510 3.53 11/1/19923.558/18/193911 3.48 5/17/19433.5311/1/199212 3.46 7/31/19793.485/17/194313 3.24 7/17/19503.467/31/197914 3.17 8/3/19293.48/3/194315 3.17 7/4/19713.356/10/1939Urbana daily rainfall, annual maximum and partial duration seriesUrbana daily precip, partial duration series 1893-2000. Precip rank est. EP est T (yr) est. EP est. T (yr)date (in) m m/n n/m m/(n+1) (n+1)/m8/12/19935.321 0.0089 112 0.0088 1135/26/19214.52 0.0179 56 0.0177 577/30/19874.433 0.0268 37 0.0265 3811/2/19364.074 0.0357 28 0.0354 289/15/19313.915 0.0446 22 0.0442 238/20/19243.96 0.0536 19 0.0531 196/19/19833.897 0.0625 16 0.0619 1610/21/19833.728 0.0714 14 0.0708 1410/6/19553.569 0.0804 12 0.0796 138/18/19393.5510 0.0893 11 0.0885 1111/1/19923.5311 0.0982 10 0.0973 105/17/19433.4812 0.1071 9.3 0.1062 9.4NOAA Atlas 14 (2004) (Replaces Hershfeld 1961)http://hdsc.nws.noaa.gov/hdsc/pfds/index.htmlNOAA Atlas 14 (2004) (Replaces Hershfeld 1961)http://hdsc.nws.noaa.gov/hdsc/pfds/index.htmlProbability DensityHistogram Gen. Extreme Valuex5.254.84.64.44.243.83.63.43.232.82.62.42.2f(x)0.560.520.480.440.40.360.320.280.240.20.160.120.080.040Some probability density functions (pdfs) used forextrapolating to longer time periods Generalized Extreme ValueProbability DensityHistogram Weibull (3P)x5.254.84.64.44.243.83.63.43.232.82.62.42.2f(x)0.560.520.480.440.40.360.320.280.240.20.160.120.080.040Probability DensityHistogram Gen. Logisticx5.254.84.64.44.243.83.63.43.232.82.62.42.2f(x)0.560.520.480.440.40.360.320.280.240.20.160.120.080.040Generalized LogisticWeibullProbability DensityHistogram Wakebyx5.254.84.64.44.243.83.63.43.232.82.62.42.2f(x)0.560.520.480.440.40.360.320.280.240.20.160.120.080.040WakebyUrbana daily rainfall, partial duration seriesplotted as a function of annual exceedance probability22.533.544.555.50.0010.010.11 Exceedance Probabilitydaily precip (in)1234567890.0010.010.11Exceedance Probabilitydaily precip(in)Urbana ObservationsGeneralized Logistic PDFGeneralized Extreme Value PDFWeibull PDFUrbana 1895-2000 daily precipitation and extrapolation based on alternative probability distributionsUrbana 1949-2000, 24 hr precipitation with extrapolations based on alternative probability distributions12345670.0010.010.11Exceedance Probability24 hr precip(in)Urbana ObservationsGeneralized Logistic PDFGeneralized Extreme Value PDFWakeby PDFy = 0.0081x - 13.146R2 = 0.05601234561900 1920 1940 1960 1980 20001 day precip. (inch)Danville annual maximum 1-day precipitation, a tale of two gages, or…y = -0.019x + 40.274R2 = 0.0773012345671900 1920 1940 1960 1980 20001 day precip. (inch)Danville SewageTreatment PlantStation 011948Danville station 0118951949-2000y = -0.0109x + 24.36R2 = 0.031910-1970:y = 0.0228x - 41.729R2 = 0.193701234561900 1920 1940 1960 1980 20001 day precip. (inch)Danville station 11895 (not the sewage treatment plant)The extreme precipitation estimates from NOAA Atlas 40 are estimates of rainfallintensity at a point. The most intense rainfall usually occurs for a short duration and covers a limited area. For watersheds larger than about 10 mi2, the point rainfallestimates represent the maximum rainfall in an area, but the average precipitation overthe area is likely to be less, and can be estimated by applying an area adjustment factor (below) to the point rainfall. Area Reduction Factorsummer precip(in) rank EP 1-EP190225.4 1 0.009 0.991199323.91 2 0.019 0.981198120.17 3 0.028 0.972197319.3 4 0.038 0.962199218.68 5 0.047 0.95319915.53 99 0.934 0.06619365.36 100 0.943 0.05719885.24 101 0.953 0.04719595.07 102 0.962 0.03819114.79 103 0.972 0.02819304.72 104 0.981 0.01919134.69 105 0.991 0.009Return Frequency Analysis applied to Summer PrecipitationIn Climate District 5 to estimate drought …Binomial Probability applied to extreme eventsProbability of an event occurring in a given year = PrProbability of an event not occurring in a given year =1-PrMultiple year probabilities, assuming events are independent:Probability of an event of annual probability of Pr will occur every year during n consecutive years = PrnProbability of an event with an annual probability of Pr notoccurring at all during n consecutive years = (1-Pr)nProbability of an event with an annual of probability Pr will occur at least once during n consecutive years =1-(1-Pr)nThis probability includes multiple occurrences.Example: 100-year eventProbability of 100-year event occurring in a given year: Pr=1/100 =0.01 = 1%Probability of a 100-year event not occurring during a given year: 1-Pr = 1-0.01 = 0.99 =99%Probability of a 100-year event occurring every year for 3 consecutive years = Pr = (0.01)3=0.000001 = 10-6Probability of a 100-year event not occurring at all during 100 consecutive years = (1-Pr)100= (0.99)100= 0.366 = 36.6%Probability of a 100-year event occurring at least once in 100 consecutive years = 1-(1-Pr)100= 1- (0.99)100= 1-0.366 =0.634 = 63.4%Probability of a 1000 year event occurring in a 100 year period= 1-(1-Pr)100= 1- (0.999)100= 1-0.905 =0.095 = 9.5%Binomial Probability applied to extreme events (continued)Probability that an event with an annual probability of Pr will occur Y times in nconsecutive years, with the Y occurrences not necessarily in consecutive years:PrY:n= n! *PrY*(1-Pr)(n-Y)(Y!)*(n-Y)!n! = 1 * 2 * 3 … * n Y! = 1 * 2 * 3 … * YAnd note that:n! = 1 * 2 * 3 … * (n –Y) … * n = (n-Y+1) * (n-Y+2) …*n(n-Y)! 1 * 2 * 3 … * (n-Y) For example: the probability that a 100-year (1% probability) event will occur 2 times in a 100 year period is:Pr2:100= (100!)*(0.01)2*(0.99)98= 99*100*(0.01)2*(0.99)98=0.185 (2!)*(98!) 2The probability that a 100-year (1% probability) event will occur 2 times in 30 years is:Pr2:30= (30!)*(0.01)2*(0.99)(28)= 29*30*(0.01)2*(0.99)(28)= .033(2!)*(28!)

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