Transcript Develop Cumulative Distribution Function

I am sure you have heard about the
farmer in Sidell, Illinois. After that
fiasco with the cheese being left
standing alone taking the blame for
the failure of the pumpkin crop, he
decided to install an irrigation
system. He was advised to determine
water requirements based on
expected rainfall. However, he is a bit
confused. How much rain falls in
Sidell in April for example? He found
historic rainfall data for Sidell online
at an Illinois State Water Survey site,
and found that April rainfall varies
from year to year. Please help the
farmer and save him from another
scandal.
Estimating Rainfall Quantity for Design
The design of water management systems is
based more on extreme values than on average
values. If the mean value is used in the design of
an irrigation system then on average, in one out of
every two years there will not be enough water to
meet the demands of the crop and yield will be
reduced. If the mean is used in drainage design,
then one out of every two years the crops will be
flooded. It is better to use design values with
lower associated risk.
Estimating 80% Dependable
Rainfall and 80% Maximum Rainfall
from mean and standard
If only the mean and standard deviation of monthly
rainfall are known then
80% Dependable Rainfall = Mean - 0.84 x Standard
Deviation
80% Maximum Rainfall = Mean + 0.84 x Standard
Deviation.
80% Dependable Rainfall
The value of period rainfall (monthly, seasonal,
etc.) that will be exceeded 80% of the time. This
value ensures that on average, there will be
enough water to meet the crop's need four out of
every five years.
80% Maximum Rainfall
The value of period rainfall that on average, will
not be exceeded 80% of the time. This value
ensures that on average, a drainage system or a
sedimentation pond will have adequate capacity
four out of every five years.
Example : For Sidell the mean rainfall for April
is 3.75" and the standard deviation is 1.78“
80% Dependable Rainfall = 3.75 - 0.84 x 1.78 =
2.25“
80% Maximum Rainfall = 3.75 + 0.84 x 1.78 =
5.25"
-0.84s
20%
0.84s
20%
Return Period (Recurrence Interval)
The frequency with which, on average, a given
precipitation event is equaled or exceeded.
Return Period (T) =
P = probability of exceedance
1
P
Example: If there is a 12.5 percent chance
that a storm of a certain magnitude will
occur, the return period for that storm is
1.0 = 8 yrs
12.5
Multi-year Chance of Exceedance (R)
The probability of a given return period
storm being equaled or exceeded within a
given number of years.
1 n
R=1-(1- )
T
Example: The chance that an 8-year return
period storm will occur over the 5 year life
of a project is
1 - ( 1 - 1 )5 = (49%) 0.49
8
10 Step Procedure for
Rainfall Frequency Analysis
1.
Locate
Data
Source
2.
Extract as
Specific
Data as
Required
3.
Import into
Excel and
convert to
columns
4.
Sort, and
Extract
Targeted
Data
5.
Graph, and
check for
jumps,
trends or
cycles
6.
Sort the data in
ascending order
and determine the
non-exceedance
probability of
each data value
7. Plot Probability of Nonexceedance vs Precipitation
(Empirical Distribution Function)
8. Determine the
mean and standard
deviation of the
logs of the
precipitation
values
9. Determine the cumulative log normal
values for the precipitation data
10. Plot the cumulative
distribution function for
the fitted logNormal
Distribution