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Rainfall Lab
January 27, 2014
Rain Measurement, Rainfall Time Series, Rainfall Return Period
Overview
This lab has three objectives:
1) Discussion of rainfall measurement instrumentation
2) Time series analysis of rainfall
3) Design storm return period, peak flow prediction and culvert sizing
The assignment is broken into 3 parts (rainfall histogram, storm return time, culvert sizing) – you
will be provided with data; guidelines for the analyses are provided in this document.
Introduction
Measuring the rain is the most fundamental part of the water budget. This lab will first
take a look at the different methods that we use to measure rainfall, and then look at some actual
rainfall data to examine patterns and distribution. Finally, we will use information for the
rainfall time series to design a culvert.
Rainfall arrives in patterns that vary from time-to-time and place-to-place. Consider the
following two graphs of annual rainfall and frequency distributions between Gainesville
(Florida) and Seattle (Washington) for the same period of record (1948-2006).
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Our first deliverable will be to describe the rainfall pattern from a new data set
(Gainesville Florida between 1900 and 2006). Usually, what we want to know the total rainfall
for a year, the degree to which that annual rainfall is reliable (how variable is it), when the
rainfall arrives. In particular, what we’ll infer from actual data is:
1)
2)
3)
4)
How complete is the data record? How many days/months/years do we have
actual data for?
What are the annual totals for the period of record; what is the mean annual
rainfall; what is the standard deviation?
What are the average monthly rainfall totals for the period of record; what are
the standard deviations?
What is the frequency distribution of rainfall events (in % of total days) for the
period of record?
Our second deliverable is to determine the rainfall associated with engineering design
requirements for flow control/conveyance structures. Engineering projects for dealing with
stormwater are designed based on rainfall intensity. While it is actually flow from the rain that
structures must withstand, flow data sufficient for statistical calculations are rarely available.
Instead, we use rainfall data which are available for many areas, and frequently have long
records. The obvious problems are 1) that not all rainfall becomes runoff, and 2) it takes time for
that rainfall which does become runoff to get to the location where we are dealing with flow. A
variety of rainfall-runoff relationships (e.g., the “Rational Method”, the NRCS Curve Number,
finite element process-modeling) are used with “design” storms to statistically compute expected
flows for structures like bridges and culverts. For example, the design storm for a temporary
logging culvert might be a 10 year/1 hour storm (Q10); a design storm for a permanent logging
road might be a 25year/1 hour (Q25) storm; and a design storm for a permanent public access
road might be a 100 year/1 hour (Q100) storm. That is, as our risk tolerance gets smaller, we
design for increasingly rare events (this is a general theme of any kind of risk analysis).
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It’s probably obvious that the probability of a small rainfall event occurring is much
higher than the probability of much larger events. By way of example, consider the probability
that a 2”/24-hour storm will occur in 2007 vs. the probability that a 10” storm will occur. In fact,
if we have a long record of rainfall for a particular location, we can quantify the probability of a
given storm event statistically; this statistical description is called the annual maximum series
and it is fitted to a theoretical log-normal recurrence curve by plotting on log-linear axes (linear
for the Y-axis [rainfall amount] and log for the X-axis [return time]). From the plot you can
estimate the recurrence interval, which is the predicted average interval (in years) between
rainfall events of a given size (or greater), and the probability that a rainfall event of a particular
size will occur in any given year.
Ok…so now we have a rainfall recurrence that we can use to meet the design
requirements of some flow structure. This might be a culvert (for permitting flow under a road),
a stormwater pond (to store a certain amount of flow and let it out slowly), or a pump house for
moving water from somewhere we don’t want it to somewhere that we do. For this exercise we
will be designing a culvert.
Our third deliverable is to compute the necessary culvert diameter to convey water of a
particular design storm. The Division of Forestry Forest BMP manual makes selecting a culvert
size easy. However, each step of the underlying engineering involves decisions and assumptions
that will affect the outcome, but are hidden to the BMP manual user. In this lab you will examine
a real rainfall time series (daily rainfall at Gainesville, Florida, between 1899 and 2006),
construct a storm return probability graph and estimate peak flow from a watershed during a
“design storm” using the Rational Method.
Design storms are useful but there are two things to keep in mind in their application:
1)
Many “permanent” structures and flood pain delineations have design storms
with 100-year recurrence intervals. However, you’re lucky to find 100 years of
data, much less several hundreds of years of data to base your calculations on.
Extrapolation to time periods beyond the data set can be risky.
2)
Design storm methodology only calculates a probability. A misconception is
that after the 50-year storm occurs that it will be 50 more years before the next
50-year storm can occur. In reality, big storms are often clumped together, for
example during a bad hurricane season or El Nino events.
Predicting runoff from rainfall involves another level of assumptions and subjective
decision making. For example rarely does rainfall occur uniformly across the watershed, and it
makes a huge difference if the watershed is wet or dry when the intense rainfall occurs. Also the
method doesn’t distinguish between a short period of intense rainfall that’s part of a small storm
or a more prolonged rainfall event associated with a large storm. There are numerous different
tables and equations for deriving the variables of the Rational Equation and different users will
frequently get different results.
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Lab Report Analyses
1.0 Description of Gainesville Rainfall Data Record
In the assignment spreadsheet, there are four worksheets within the file “Rainfall Lab 2014”.
The first is labeled Gainesville_Rainfall. Open that file and look at the data set that you’ve been
provided. Whoa! That’s a lot of data – in fact it is a relatively continuous record of daily rainfall
totals between January 1900 and December 2012.
1) Your first task will be to examine the completeness of the data. To do this, you will need
to look at the worksheet labeled “Data_Pivot”.
a. Read the help files about pivot-tables if you’ve never used them – they are very
very helpful (that is, pivot tables…the help files may not be so helpful).
b. The pivot table is empty (you should see a “Pivot Table Field List” to the right of
your screen). To add data, simply pick a Field from the Pivot Table Field List and
drop it in the appropriate location. For example, if you wanted to look at the
rainfall totals by year, you would drag the “Prcp” field into the main body of the
table (that is, the data you want to examine). You should get a number that is the
total rainfall (“Sum of Prcp” written in the upper left cell) over the entire period
of record. If you don’t get a sum (for example, you might get “Count of Prcp”),
simply right-click over the cell that says “Count of Prcp” and go to field settings;
there you should change the setting to “Sum”…you could change it to any of the
other settings like average, standard deviation, which will come in handy later).
Next, drag-and-drop the “Year” field to the left column (the “Row Field”). Now,
you should see each year for which there are data and the total precipitation
recorded. Look at 1927. What’s weird? The dataset may be incomplete
c. To check data completeness, change the field settings (see above) to count. What
do you see? It appears that our rainfall gage had issues during the 1920’s, then
shut-off completely during the Great Depression and World War II. Be cognizant
of this later. How many years have complete data (365 or 366 observations)?
2) Your next task is to plot the annual rainfall totals. Use the pivot table to set up the annual
Sum of Prcp, and plot Year (X-axis) vs. Total Rainfall (Y-axis). What do you see?
3) Next compute the average annual rainfall. You could take the average of all the year data
that you just plotted, but we know that some of the years aren’t complete. So the first
step will be to exclude those years that have less than a full data set (365 or 366 daily
observations). Then, compute the average and standard deviation of annual rainfall.
How does that compare with Seattle (average = 38.2”; SD = 6.5”)? How does the value
compare when we DON’T exclude years that have an incomplete record?
4) Next, we want to understand when rainfall occurs within the year, and the variability of
that delivery between years. So, we want a plot with average monthly rainfall (in inches)
and the standard deviation of that monthly average. To do this, we go back to the pivot
table. We should still have “Year” in the rows, and Sum of Prcp in the cells. All we
need to do is add month to the columns (again, drag and drop) and we now have a matrix
of total rainfall for each month of each year (years as rows, months as columns). There
will, as before, be some missing months – this is ok. Now, we want to compute the
average of each month’s total rainfall across all the years and the standard deviation.
That is, we want to average the columns across all the rows. So we scroll down to the
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bottom of the table and compute the average of each month and the standard deviation.
Now plot the average that you computed (do the numbers make sense?) vs. month using a
bar chart. By double clicking on the bars, you can add Y-error bars, which, for this work,
should be the + and – the computed standard deviations. Your work should look
something like the figure below (for Seattle). How is Gainesville different? Consider
both the timing and size of the error bars (both within a site over time, and between sites).
Monthly Rainfall Averages and Standard Deviations for Seattle Washington (1948-2006)
5) The next task is to plot a frequency distribution of the rainfall events for Gainesville. To
do this, we need to add some functionality to Excel. In Office2007, go to “Office
Button” in the upper left of the Excel screen, click “Excel Options”, “Add-Ins” and make
sure the “Analysis Tool Pack” is active. This may take a minute or two.
a. Go to Tools – Data Analysis. A window will pop-up and you should select a
histogram. The input range is all of the daily rainfall observations (all of them)
and the bin range is on the worksheet titled “Bins”. Don’t forget to check the
“labels” button if you’ve selected the labels. Set the output range to the “Bins”
worksheet…don’t click any of the output options.
b. You’ll get a table with the bins on the left and the number of observations on the
right. The bins represent the upper bound of categories (for example, the Bin
labeled will collect all the rainfall events that are less or equal to 0.6”, but greater
than 0.4” (the bin below). So we have tabulated the frequency of occurrence for
rainfall events of different sizes (a graph of the data will look something like the
Figure on Page 2 of this document). What fraction of the observations are 0”?
What fraction are greater than 2”? What is the maximum rainfall?
2.0 Gainesville Rainfall Recurrence
Our next step is to use the Gainesville data to determine the probability that a rainfall event of
given size will occur in any given year. To do this, we need a list of the maximum observed
rainfall in each year. To get this, use the following steps:
1) Return to the pivot table and remove the months (columns) so it’s just the years.
2) Change the field settings so that instead of the sum of rainfall over each year, report
the “Max”…this is the largest observation for that given row field (year).
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3) You should now have a table that is years in the row, and a number (maximum
rainfall in inches) in the cells. This annual maximum rainfall series is the basis of the
remaining analyses. Save this somewhere and present it graphically in your lab
report. Make sure the numbers make sense (go back to the data if you’re not sure).
4) Now using the annual maximum rainfall series between 1900 and 1980 plot a
recurrence interval line.
a. Rank (m) all the storms with biggest storm = 1.
b. For each year calculate the Recurrence interval (T) for the plot:
T= (number of years of data + 1) / m
c. Plot T (x axis) vs. rainfall (y axis) using a scatter plot (NO LINES). Now,
click on the x-axis, go to scale tab and change to a logarithmic scale.
d. Add a trendline (chart menu) and add a “Logarithmic” type fit.
e. In chart options (chart menu) display major and minor gridlines in X and Y.
f. Be sure to attach the graph to your report – it will look something like this…
Annual Rainfall Maximum Series for Seattle Washington
5) From your graph (Gainesville data) what are the rainfall amounts (inches/day) for the
10 and 25-year storms? What is the return time of a 5” storm?
6) The probability that a given rainfall will occur in a given year is the inverse of the
return time. That is, a 10-year storm has a 0.1 probability of occurring in any given
year. What is the probability of a 3” event occurring in any given year?
7) Now focus on the data that you DIDN’T USE for the first calculation. How well did
your plot predict recurrence intervals of the next 31 years of data (1981-2012)?
Consider, for example, how many storms greater than the 10-year storm there were
during the period between 1981 and 2012? How many would you expect? Same
questions for the 25-year storm…did you get any 25 year events? More than one?
Ok…our last step is to compute the statistics of rainfall. Recall from lecture that we can describe
the rainfall patterns using two parameters, one that describes the time between rainfall events,
and the other describing the mean rainfall. The latter is easy (calculate the mean, or arithmetic
average). For Seattle, that value is 0.105 inches. The other parameter is a little trickier; we’re
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not going to actually estimate the return rate (which requires tools that Excel doesn’t have), but
we will display the distribution. The algorithm to get at that value is as follows:
1) Label a column “Did it Rain” and in it use the following syntax to determine if it
rained: “=if(Rain>0,1,0)”. This says, look at the cell in the rainfall column (you’ll
need to make italicized “Rain” in the equation refer to the appropriate cell) and if it’s
larger than 0 (i.e., it rained) then give the cell a value of 1, otherwise give it a zero.
2) Now, in the next column (which you should label “Interval”), write the following
syntax “=IF(Did it Rain yesterday=1,1,Interval+1)”. What this is saying is if it rained
yesterday (i.e., there was a 1 in the cell up one row and in the column labeled “Did it
Rain”) then make the cell a value of 1. Otherwise, make the cell a value of 1 greater
than what it was the day before (i.e., the cell directly above). You may need to think
about what happens in the first row, and put the correct values in manually. This
should give a continuous list of how long it’s been on every day since it last rained.
3) Now, the next column over (which you can call “Interval Only”) you should write the
following syntax: “=IF(Did it Rain=1,Interval," "). This says, if it rained today, tell
me how long it’s been since it last rained, otherwise, tell me nothing(i.e., report a
space). This will create a column of discontinuous values with numbers only when it
rained, and with each value being how long it had been since the previous rainfall.
4) Copy that column and paste the values in a new column; sort the value from lowest to
highest. What you’ll see is a column with lots of 1’s, less 2’s, still less 3’s etc.
5) Now determine the maximum value in this column; this is the longest duration
between rainfall events in the period of record. In a new column make a list starting
at 1 and increasing in 1 day steps to the maximum value you saw. In the next column
over, put the following syntax: “=COUNTIF(N:N,O2)” where column N is the sorted
list of inter-event durations, and column O is the list from 1 to your observed
maximum. This returns the number of entries in the column that satisfy the condition
that they are equal to the value in O2. You’ll fill down making sure that each cell
refers to a different cell in column O. This gives you a frequency of each inter-event
duration. Plot this. Compare to the following plot from Seattle:
Inter-Event Duration for Seattle WA
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3.0 Estimating peak flow using the Rational Method and rainfall return period (design storm)
Rational equation
Qpeak= C I A
- Q is peak storm flow in ft3/s
- C is runoff coefficient
- I is the rainfall intensity for the appropriate return period and time of concentration
- A is watershed area in acres
C
C is related to proportion of rainfall that becomes runoff and is determined from matching
watershed parameters to lookup tables. For example (Table 1) a woodland with 7% slope would
have a C ~ 0.20. An area weighted C can be determined if the watershed has mixed land uses.
Table 1.
I
I is the intensity of rainfall in inches/hour. Peak flows will occur when the entire watershed is
contributing to the flow during the most intense part of the storm. The entire watershed can only
be contributing when rain from the furthest point has had time to reach the outlet. This time in
minutes (time of concentration (Tc)) can be estimated with various equations.
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The time of concentration can be estimated based on watershed attributes:
Where:
Tc is the time of concentration in minutes.
L is the distance (ft) to the most distant point in the watershed
H is the elevation difference between the most distant point and the outlet.
Note: this equation assumes overland flow and doesn’t account for varying surface roughness.
Once you’ve calculated Tc you need to find matching rainfall intensity duration graphs with
appropriate return curves. Tc will determine the duration and your acceptable risk will determine
the return period. For example, using Figure 1, if Tc is 30 minutes and you want to pass a 50
year flow, then I is about 6 inches/hour. (Note that shorter periods of time are more likely to have
higher sustained rainfall intensities, and thus watersheds with shorter Tc’s will have higher I.)
Figure 1.
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A
Area of watershed in acres.
Example:
What is the peak flow for a 10 year storm from a 100 acre, sandy flatwoods (slopes ~1%) who
furthest point is 2,000 ft from the outlet and 3.5 ft higher?
Q = CIA
From table 1 C ~ 0.125
Tc = (0.0078 x 2,000 0.77) / ((3.5/2000)0.385)
Tc ~ 31 minutes and from Figure 1, I is about 4.5 inches hour for the 10 year, 31minute storm.
Compute Flows:
Q= 0.125 x 4.5 x 100
Table 2.
or = 56 ft3/s (cubic feet per second = cfs)
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Florida Division of Forestry – Best Management Practices for Culvert Design
At the end of the BMP Manual published by the Division of Forestry, there is a lookup table that
allows a rapid estimation of necessary culvert size. It presumes a design storm (2.5” per hour),
and requires knowledge of the watershed size and soil type.
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Lab Reporting of Culvert Sizing
1) Use the rational equation (Qpeak =C I A) to calculate the peak flow from a 80 acre
watershed that is rolling woodland with a 2% slope and has sandy soils.
a. C Use Table 1 to determine C.
b. I For I, use the formula:
to calculate Tc, given that the length from the most distant point from the outlet is
4500 ft and that point is 50 ft higher than the outlet. With your calculated Tc, use
Figure 1 to determine the 25 year rainfall intensity, I.
c. A is 80 acres.
2) From the estimated flow, use Table 2 to determine the size culvert required to pass a
25 year storm from this 80 acre watershed?
3) Using the Division of Forestry method in the BMP manual (Tables D and E), what is
the recommended culvert size for this watershed? Are the two estimated culvert sizes
the same? If not, why not?