Estimation of Areal Precipitation from point measurements Most often interested in quantifying rainfall over an entire watershed. Has to be inferred from some sort of weighted average of available point measurements P(x i ) Several methods to determine weights. All require

Arithmetic Average Arithmetic average: Note that all weights equivalent Method OK if gages distributed uniformly over watershed and rainfall does not vary much in space.

Theissen Method Weight (   )  is a measure of rain-gage contributing area. Assumes rain at any point in watershed equal to rainfall at nearest station. To determine (   ): –draw lines between locations of adjacent gages –perpendicular bisectors drawn for each line –extend to form irregular polygon areas A 3  P 3  P 2   P 4     A2A2 A 1 P 1 A4A4

Isohyetal Method most accurate method if have a sufficiently dense gage network to construct an accurate isohyetal map. Can account for systematic trends, i.e., orography, distance from coast. 4 in. 3 in. 2 in. 1 in.

Hydrologic Frequency Analysis Extreme rainfall (and flood/drought) events are typically of concern in engineering hydrology Magnitude of an extreme event is inversely related to its frequency of occurrence. Frequency analysis of historic data relates the magnitude of events to their frequency of occurrence using theory of probability and statistics (mean, standard deviation, coefficient of variation, coefficient of skewness, return period)

Return Period Return period (T) of an event is the average time (recurrence interval) between events greater than or equal to a particular magnitude. For example, 25 year return period storm occurs on average once every 25 years and has a probability of 1/25 of occurring in any one year.

To estimate return period from flood/drought/rainfall records Select annual maximum/minimum of a particular duration from historical record to form annual maximum/minimum series. Rank annual maximum/minimum from largest to smallest (or smallest to largest if interested in drought)

Calculating Probabilities of Occurence What is probability T-year return period will occur once in N years? –Probability does not occur P(x < x T )=(1-P) N Probability occurs at least once in N years = 1 - (1-P) N = 1 - (1-1/T) N

Examples For example, 10 year return period storm has prob. of occurrence 0.1 in any 1 year. How probable once in 10 years? –T = average recurrence interval for event =10 years –Probability of occurrence in any one year = 1/T=0.1 –Probability = 1 - (1-1/10) 10 = ______ at least once in ten years What is the probability that a 20 year return period storm occurs at least once in 10 years?