ERS 482/682 Small Watershed Hydrology
Fall Semester 2002
Homework #2
Due September 25, 2002
1. 100 years of daily precipitation data for Fort Collins, CO from 1902 through 2001 have been
placed in the spreadsheet file called 482hw2.xls, which you can download from the class
website. Each row in the spreadsheet has daily data for one month. The attached sheet
includes codes that apply to the data in the spreadsheet.
a) Replace all of the ‘t’s in the spreadsheet with ‘0’ (the number zero). You can use the
‘Find’ and ‘Replace’ tools under the ‘Edit’ menu option to do this. Find the annual
maximum precipitation for each calendar year. Analyze the annual maximum
precipitation for 1902-2001 using the ‘Descriptive Statistics’ tool and fill in the following
information:
Statistic
Value
Units
Mean
Median
Mode
Standard Deviation
Skewness
dimensionless
Minimum
Maximum
b) Make a bar chart of the annual maximum precipitation for Fort Collins, CO from 1902
through 2001. Be sure to include units and appropriate labels on your chart.
c) Sort the annual maximum data (use the ‘Sort’ tool under the ‘Data’ menu option in
Excel) and calculate the exceedence probability (EP) for each value using the Weibull
plotting position formula. Use the ‘Histogram’ tool (do not specify bin ranges – i.e., let
Excel automatically set them) and plot the histogram for the sorted data. In your opinion,
does the data look normally or log-normally distributed? What is your reasoning for your
answer?
d) Look at your sorted list of precipitation data and exceedence probabilities and estimate
the rainfall for a 24-hour storm with the following return periods and enter these values in
the “DATA” column of the following table. Then use the NORMINV function in Excel
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Homework #2
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to estimate the rainfall for a 24-hour storm with the same return periods and enter those
values in the “NORMINV” column.
Return period
24-hour rainfall (in.)
DATA
NORMINV
2-year
10-year
25-year
50-year
100-year
e) Now calculate the natural log (ln) of each of the annual maximum precipitation data
points (you can use the LN function in Excel). Analyze the annual maximum
precipitation for 1902-2001 using the ‘Descriptive Statistics’ tool and fill in the following
information:
Statistic
Value
Units
Mean
Median
Mode
Standard Deviation
Skewness
dimensionless
Minimum
Maximum
f) Sort the transformed data and calculate the exceedence probability (EP) for each value
using the Weibull plotting position formula. Use the ‘Histogram’ tool (do not specify bin
ranges – i.e., let Excel automatically set them) and plot the histogram for the sorted
transformed data. In your opinion, does the transformed data look more or less normally
distributed than the untransformed data you evaluated in part (c)? What is your reasoning
for your answer?
g) Look at your sorted list of transformed precipitation data and exceedence probabilities
and estimate the rainfall for a 24-hour storm with the following return periods and enter
these values in the “DATA” column of the following table. Then use the NORMINV
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Homework #2
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function in Excel to estimate the rainfall for a 24-hour storm with the same return periods
and enter those values in the “NORMINV” column. (Note that the untransformation of
the natural log is ex, which is the EXP function in Excel).
Return period
24-hour rainfall (in.)
DATA
NORMINV
2-year
10-year
25-year
50-year
100-year
h) Make depth-frequency charts (use the ‘XY scatter’ option for chart type) for the
untransformed data and the transformed data, with the exceedence probability on the xaxis (use a logarithmic scale for the x-axis). Be sure to label your axes. Looking at the
plots, which set of data (i.e., untransformed or transformed) appears to be more normally
distributed?
i)
2.
(Extra credit) Skewness () is one way of assessing how normally distributed data are.
The skewness value tells if more of the data are tending to be less than the mean ( < 0),
or tending to be greater than the mean ( > 0) (see Figure C-1(c) in Dingman (2002)).
Data that is perfectly normally distributed should have a skewness of 0. On the basis of
the skewness values, which of the data series (i.e., untransformed or transformed) seem
more normally distributed?
Two tipping bucket rain gages are used to collect the following rainfall data:
Time
4:00 a.m.
6:00 a.m.
8:00 a.m.
10:00 a.m.
12:00 noon
2:00 p.m.
4:00 p.m.
6:00 p.m.
8:00 p.m.
10:00 p.m.
12:00 mid
2:00 a.m.
4:00 a.m.
Cumulative precip. (mm) Station #1
0.0
0.0
1.0
4.0
13
17
19
19
19
19
19
19
19
Cumulative precip. (mm) Station #2
0.0
0.0
1.0
3.0
11
15
16
17
17
17
17
17
17
a) Calculate the mean daily rainfall intensity for each station (mm hr-1)
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b) Calculate the maximum 2-hour rainfall intensity for each station (mm hr-1)
c) Calculate the maximum 6-hour rainfall intensity for each station (mm hr-1)
d) Use the weighted-average method to estimate the total volume of rainfall (m3) delivered
to the basin if the drainage basin area is 176 mi2, assuming each of the gages has equal
weight and there are no other gages in the basin.
e) Use the Thiessen polygon approach to estimate the total volume of rainfall (m3) delivered
to the basin if the contributing area to Station #1 is 62% of the total basin area, and it is
assumed that the remainder of the basin area contributes to Station #2.
f) Suppose the data from the two gages is being used to estimate missing data at a third
gage. If the average annual precipitation at Station #1 is 300 mm, the average annual
precipitation at Station #2 is 220 mm, and the average annual precipitation at Station #3
(the one with the missing data) is 200 mm, what is the estimated daily precipitation
intensity at the third gage?
What you should turn in:
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This handout (or a copy) with the boxes filled in for Items 1a), 1d), 1e) and 1g)
A printout of the maximum annual precipitation for each year, the sorted list, and the
exceedence probabilities; you can use the ‘Page Setup’ feature under the ‘File’ menu option
in Excel to fit the output to one page so you don’t print out pages and pages of data (see me if
you don’t understand)
A printout of the transformed maximum annual precipitation for each year, the sorted list,
and the exceedence probabilities
A bar chart for Item 1b)
The histogram data (i.e., including the bins and frequencies) and chart for Item 1c)
Answers to the questions for Item 1c)
The histogram data and chart for Item 1f)
Answers to the questions for Item 1f)
Depth-frequency chart(s) for Item 1h) (note: you can plot both series on the same chart – use
a secondary y-axis for the second series – or plot each series separately on different charts)
Answer to the question for Item 1h)
Work and answers for questions 2a) through 2f)