Transcript Estimation of Areal Precipitation from point measurements

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(xi)

N
P   i P( xi )
i 1
• Several methods to determine weights. All require
0  i  1
 i  1
Arithmetic Average
• Arithmetic average:

P
1
 P( xi )
N
Note that all weights equivalent
i 
1
N
• Method OK if gages distributed uniformly over
watershed and rainfall does not vary much in
space.
Theissen Method
• Weight ( i ) is a measure of rain-gage contributing area.
Assumes rain at any point in watershed equal to rainfall at
nearest station.
• To determine ( i ):
– draw lines between locations of adjacent gages
– perpendicular bisectors drawn for each line
– extend to form irregular polygon areas

A1 P1

A2
 P2


P4
A4
area polygon contributing to i Ai
i 

total area of watershed
A

1
P   i P( xi )   Ai P( xi )
A
A3
 P3


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.
area between isohyets
i 
total watershed area

P   i P ( xi )
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.
1
1
T      P x  xT  
P
T
P  P x  xT   probabilit y storm  specified value
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)
•
Prob( x  xm ) 
m( x m )
N 1
or
T
N 1
m( x m )
Calculating Probabilities of Occurence
• What is probability T-year return period will
occur once in N years?
– Probability does not occur
P(x < xT)=(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?