DERIVATION AND EVALUATION OF ALTERNATE METHODS
FOR CALCULATING SHORT TIME-INTERVAL RAINFALL INTENSITIES
FROM TIPPING BUCKET RAIN GAGE DATA
A Thesis
Presented to the faculty of the Department of Civil Engineering
California State University, Sacramento
Submitted in partial satisfaction of
the requirements for the degree of
MASTER OF SCIENCE
In
Civil Engineering
by
Christian Joel Carleton
SPRING
2013
DERIVATION AND EVALUATION OF ALTERNATE METHODS
FOR CALCULATING SHORT TIME-INTERVAL RAINFALL INTENSITIES
FROM TIPPING BUCKET RAIN GAGE DATA
A Thesis
By
Christian Joel Carleton
Approved by:
__________________________________, Committee Chair
Ramzi J. Mahmood, Ph.D., P.E.
__________________________________, Second Reader
John Johnston, Ph.D., P.E.
____________________________
Date
ii
Student: Christian Joel Carleton
I certify that this student has met the requirements for format contained in the University
format manual, and that this thesis is suitable for shelving in the Library and credit is to be
awarded for the thesis.
__________________________, Graduate Coordinator ___________________
Matthew Salveson, Ph.D., P.E.
Date
Department of Civil Engineering
iii
Abstract
of
DERIVATION AND EVALUATION OF ALTERNATE METHODS
FOR CALCULATING SHORT TIME-INTERVAL RAINFALL INTENSITIES
FROM TIPPING BUCKET RAIN GAGE DATA
by
Christian Joel Carleton
When doing hydrologic analyses in very small drainage areas such as small urban stormwater
catchments, airport runways, or professional sports fields where the time of concentration is
short (e.g. <10 minutes), it is important to have rainfall data with a short enough time-interval
to adequately observe the temporal distribution and accurately model the resulting infiltration
and runoff. When the time-interval gets very small, such as 1-minute intervals, the traditional
hyetograph technique of using a histogram to graph rainfall intensity as a function of time
does not accurately represent the storm event. A new method to calculate and graph rainfall
intensities was developed to address these problems by allowing the time-interval over which
the intensity is calculated to vary while holding the depth-interval constant. This new
variable-time method was then evaluated with three numerical differentiation techniques
(forward-difference, central-difference, and three-point forward-difference) that were used to
differentiate the rainfall mass curve to estimate intensities. The variable-time method proved
iv
to be a viable alternative, especially for graphically based analyses. The central-difference
method is a good alternative to generate a continuous intensity function although it has the
potential to underestimate the intensity of sudden short burst of rainfall. This does not occur
with the forward-difference method but it does result in an overall shift of the data by one
time-interval. Even though the three-point forward-difference method uses a higher order
approximation of the mass curve, it proved to be the least preferred alternative because it was
unable to realistically deal with sudden changes in the mass curve.
_______________________, Committee Chair
Ramzi J. Mahmood, Ph.D., P.E.
_______________________
Date
v
DEDICATION
This thesis and master’s degree are dedicated to my wife Kate, son Cooper, and daughter
Quinn. Without my wife’s support and encouragement I would have never been able to
accomplish all that I have in the last four years. The effort and sacrifices made in order for me
to successfully complete the graduate curriculum in civil engineering at Sac State was done so
that our family could have a better life. It is my sincere hope that by returning to school I have
been able to set an example, so that in future years my children can see the importance of a
good education and that it is never too late. This effort has been as much for my family as it
has been for me.
-Christian Joel Carleton
Loving husband and father
"Water is the most critical resource of our lifetime and our children's lifetime. The health of
our waters is the principal measure of how we live on the land." --Luna Leopold
"Genius without education is like silver in the mine." --Benjamin Franklin
“What is right about America is that although we have a mess of problems, we have a great
capacity – intellect and resources – to do something about them.” --Henry Ford II
vi
ACKNOWLEDGEMENTS
I would like to acknowledge Alan Batdorf of AEI-CASC Consulting because it was a
conversation with him that started me down the path of realizing the need for a better method
to graph short time-step tipping bucket rainfall data. It was then the constant reminders from
Kevin Murphy, my supervisor at the Office of Water Programs at California State University,
Sacramento, to “always do good science” that led me back to the fundamental mathematical
and scientific principles used to develop this new graphing method. Most importantly I
would like to thank Karl Dreher and Bhaskar Joshi of the California Department of
Transportation’s Division of Environmental Analysis for their support and adoption of this
method within the Stormwater Program’s research projects.
vii
TABLE OF CONTENTS
Page
Dedication .................................................................................................................................. vi
Acknowledgements ................................................................................................................... vii
List of Tables............................................................................................................................... x
List of Figures ............................................................................................................................ xi
Chapter
1.
INTRODUCTION ...............................................................................................................1
2.
BACKGROUND .................................................................................................................3
Data Visualization for Science and Engineering ............................................................4
Visual Analysis of Hydrologic Data ..............................................................................5
Progression of Graphing Rainfall Data ..........................................................................8
Rainfall Plotting Methods ............................................................................................10
Problem Statement .......................................................................................................16
3.
METHODS........................................................................................................................19
Mass Curve Extraction .................................................................................................20
Variable-Time Method .................................................................................................21
Modifying the Mass Curve for Numerical Differential Methods .................................26
Forward-Difference Method ........................................................................................30
Central-Difference Method ..........................................................................................33
Three-Point Forward- Difference Method ...................................................................35
4.
DISCUSSION ...................................................................................................................40
viii
Derivative/Integral Relationship ..................................................................................41
Peak Values ..................................................................................................................43
Center of Mass Estimates .............................................................................................46
Beginning and End Time of Rainfall ...........................................................................47
Response Timing ..........................................................................................................49
Recommendations ........................................................................................................51
5.
CONCLUSIONS ...............................................................................................................53
Appendix A. Example Rainfall data Sets ..................................................................................56
References .................................................................................................................................60
ix
LIST OF TABLES
Table
Page
1. Minimum detectable intensities for a 0.01-inch tipping bucket rain gage. .........................14
2. Extracted mass curve data...................................................................................................20
3. Variable-time hyetograph data with calculations ...............................................................22
4. Extrapolated and interpolated mass curve data ...................................................................30
5. Forward-difference calculations table ................................................................................32
6. Central-difference calculations table ..................................................................................35
7. Three-point forward-difference calculations table..............................................................37
8. Comparison of integration values .......................................................................................42
9. Comparison of peak intensities ...........................................................................................43
10. Comparison of center of mass calculations ........................................................................47
11. Comparisons of estimated rainfall begin and end times .....................................................48
x
LIST OF FIGURES
Figure
Page
1. Example of a hyetograph-hydrograph plot ...........................................................................6
2. Graphical representation of time of concentration and lag time definitions .........................7
3. Weighing-recording and tipping bucket rain gages ..............................................................9
4. Example of rainfall mass curve and hyetograph .................................................................11
5. Comparison of data reporting intervals on hyetographs for Sacramento, CA ....................13
6. Comparison of data reporting interval on rainfall mass curves for Sacramento, CA .........16
7. Extracted mass curve example ............................................................................................21
8. Calculating variable-time intensity .....................................................................................22
9. Variable-time hyetograph ...................................................................................................23
10. Estimating variable-time hyetograph start time ..................................................................24
11. Estimating variable-time hyetograph end time ...................................................................25
12. Estimating the mass curve starting point by extrapolation .................................................27
13. Estimating the mass curve end point by linear extrapolation .............................................28
14. Linear interpolation of mass curve data points ...................................................................29
15. Final mass curve data ready for numerical differentiation .................................................30
16. Forward-difference estimate of intensity ............................................................................31
17. Forward-difference hyetograph ..........................................................................................32
18. Central-difference estimate of intensity ..............................................................................33
19. Central-difference hyetograph ............................................................................................34
20. Three-point forward-difference estimate of intensity .........................................................36
xi
21. Three-point forward-difference hyetograph .......................................................................37
22. Negative value correction ...................................................................................................38
23. Three-point forward-difference hyetograph with negative value correction ......................39
24. Hyetographs for the four proposed alternative methods .....................................................41
25. Integration error caused by the negative value correction ..................................................42
26. Three-point forward-difference overestimated derivative approximations ........................45
27. Three-point forward-difference negative derivate approximations ....................................45
28. Beginning and end time extrapolations of the mass curve .................................................49
xii
1
1. INTRODUCTION
Understanding and modeling the rainfall/runoff process is fundamental to the water resources
profession. Historically rainfall data was recorded every 24 hours, then as equipment and
technology advanced it shortened to every hour, and now today’s technology allows for near
continuous recording of data. Yet the hyetograph, the predominant method used for graphing
and analyzing rainfall intensities has not changed.
For drainage areas with very small time of concentration (e.g. less than 10 minutes) the
temporal distribution of rainfall can become very important. This would include small urban
stormwater catchments, airport runways, professional sports fields, and small scale plot
research studies. Unfortunately, most of the equations and methods used now, including
hyetographs, were developed for larger catchments. When using the standard 1-hour
hyetograph time-interval to investigate and model such small drainage areas there is a critical
loss in temporal resolution needed to understand the distribution of rainfall throughout a storm
event. But reducing the time-interval causes a loss in the resolution of the rainfall rate because
the minimum detectable intensity increases.
To address these problems with the hyetograph method of calculating and displaying rainfall
intensity data, four alternative methods were chosen and evaluated for their effectiveness
when using short time-interval data. The first method was developed by the author
specifically for this purpose. Instead of holding the time-interval constant and letting the
depth vary as is typical of a hyetograph, this new method is based on a conservation of mass
approach which holds the depth-interval constant, a typically constraint of the monitoring
equipment, and allows the time interval to vary.
2
The other three methods evaluated capitalize on the fact that the hyetograph is the derivative
of the mass curve, which is the cumulative rainfall depth and is how rainfall data is typically
collected. Three different numerical differentiation techniques based on Taylor series
expansions were selected to calculate intensity by calculating the derivative of the mass curve,
which results in a continuous hyetograph function.
A systematic evaluation was performed to determine the applicability of the four alternative
methods to be used with rainfall data. The evaluation looked at each the method’s ability to:
1) maintain the derivative/integral relationship, 2) accurately represent peak values, 3)
estimate the center of mass, 4) identify the beginning and end time of rainfall, and 5) respond
to changes in the mass curve without causing a temporal shift. Recommendations for using
each of the methods are then given based on the results of the evaluation.
3
2. BACKGROUND
One of the fundamental relationships consistently utilized in water resources engineering is
the rainfall/runoff relationship. It is conceptually simple. Rainfall lands on the ground and
subsequently runoff occurs. This simplicity can be seen in the Rational Method (Equation 1),
a numeric model consisting of only three variables that relate rainfall intensity to peak runoff
flow rates (Viessman and Lewis 1995; Mays 2011).
Equation 1.
𝑄𝑝 = 𝐶𝐼𝐴
where: Qp = peak runoff flow rate (cfs)
C = runoff coefficient (dimensionless)
I = average rainfall intensity (in/hr) with a duration equal to the
drainage area’s time of concentration
A = size of the drainage area (ac)
Note: The unit conversion of in/hr  ac to cfs is 1.008 and is typically
dropped from the equation.
Although, this is perhaps the most widely used model dealing with the rainfall/runoff
relationship, it completely ignores the physical processes involved, replacing them with the
single runoff coefficient, C. When one starts to examine the individual physical processes
involved (e.g. surface capture, infiltration, evaporation, etc.), however, the complex nature of
the relationship between rainfall and runoff becomes evident. Scientists and engineers have
been trying for decades to better understand these processes and represent them in
rainfall/runoff models such as with the Stanford Watershed Model (Singh and Frevert 2006;
4
Crawford and Burges 2004), HEC-1 (Hydrologic Engineering Center 1998), and the
Sacramento Soil Moisture Accounting (SAC-SMA) model (Koren et al. July 2007; Koren et
al. October 2010).
Data Visualization for Science and Engineering
With the availability of desktop computers and software for numeric data analysis, one can
easily lose sight of the benefits obtained by graphing the data. Yet often the most effective
way to describe, explore, and summarize a set of numbers is to look at a picture of those
numbers (Tufte, 2001). Hydrologic data sets, especially runoff data, consist of hundreds of
data points for even a single event. It is nearly impossible for the human mind to comprehend
so many numbers, their relationship to one another, and then the overall relationship with
another set of numbers (e.g. the relationship between a rainfall data set and the resulting runoff
data set). Data visualization provides the scientist and engineer with a way to comprehend,
study, and compare large data sets.
The human brain can detect patterns not easily discernible through numeric analysis (GrillSpector 2003; Sinha et al. 2006). Graphing data allows the brain to do what it does better than
any super computer, i.e. recognize patterns.
Graphs allow practitioners to explore data to
uncover overall patterns and to see detailed behavior. No other approach can compete in
revealing the structure of data so thoroughly (Cleveland 1994). Successful technical
individuals, regardless of their profession, recognize that visualizing data provides insight into
the structure of the data. Unfortunately, too often students are taught how to perform
statistical procedures and calculate various parameters but not how to interpret the visual
5
representation of the data (i.e. graphs). The shortcoming of this approach is that statistical
tests are often designed to test narrow questions. On the other hand, visual analysis frees the
reviewer to look for any relationships and patterns within the data without a hypothesis, which
can yield answers to questions not originally asked (Cleveland, 1994).
Visual Analysis of Hydrologic Data
Due to the extreme complexity of the natural rainfall/runoff process, water resources
engineers, hydrologists, and others working with this type of data have historically relied on a
graphical approach as the first step in analyzing a system (i.e. drainage area). Runoff data are
displayed in a discharge hydrograph which is a continuous plot of instantaneous runoff flow
rate as a function of time (Linsley et al. 1992; Viessman and Lewis 1995). Similarly, rainfall
data are displayed in a hyetograph which is a histogram of the storm event’s rainfall intensity
or depth as a function of time (Chow, Maidment, and Mays 1988). Typically the storm event
hyetograph is plotted using the same time axis as the hydrograph and placed on the same
graph, which facilitates visual analysis of both sets of data and exploration of the interrelation
between the two (see Figure 1). Viewing both types of data in this fashion, it becomes evident
why Chow (2009) describes a hydrograph as the “integral expression of the physiographic and
climatic characteristics that govern the relations between rainfall and runoff of a particular
drainage basin.”
These combined plots have become a fundamental tool in hydrologic analysis. For example,
time of concentration, which is the time it takes for the entire drainage area to contribute to
runoff, is sometimes defined in practical terms as the difference in time between the end of
6
Figure 1. Example of a hyetograph-hydrograph plot
Adapted from National Highway Institute (2002)
excess rainfall and the inflection point receding limb of the hydrograph as illustrated in Figure
2 (Viessman and Lewis 1995). Identifying the inflection point in a real hydrograph can be
challenging because the receding limb is seldom smooth, so it is often best done visually.
Once the inflection point is identified the time of concentration can then be easily measured
directly off of the hyetograph-hydrograph plot.
The basin lag time, which is the timing of the runoff relative to the rainfall, can also be
measured directly off of a hyetograph-hydrograph plot (Figure 2). The three most commonly
used definitions are: 1) the difference between the center of mass of the excess rainfall and
center of mass of runoff denoted as Lag Time1 in Figure 2, 2) difference between the center of
mass of the excess rainfall and time of the peak runoff rate denoted as Lag Time2 in Figure 2,
or 3) difference between the time of maximum rainfall intensity to the time of the peak runoff
7
Figure 2. Graphical representation of time of concentration and lag time definitions
(The superscripts 1, 2, 3 refer to the three most common definitions of lag time. See text for
more detail.)
rate denoted as Lag Time3 in Figure 2 (Viessman and Lewis 1995). Lag time is often
preferred by hydrologic modelers over the time of concentration because centers of mass and
peak values are reproducible calculations, whereas the inflection point on the hydrograph’s
receding limb is based more on professional judgment (Bedient, Huber, and Vieux 2008).
Regardless of the chosen parameter and consequent definition, all of these values can be
obtained from a hyetograph-hydrograph plot.
Visual analysis of hydrologic data provides a unique insight into the hydrologic system. With
experience, water resources engineers can use hyetograph-hydrograph plots to help identify
the appropriate numeric model for a given drainage area and parameters to use in the model.
8
Then after the model has been developed, output hydrographs can be compared to actual
hydrographs as part of the model’s calibration and validation steps. Graphing hydrologic data
can also provide a rapid and effective way to do data quality control checking. To someone
reviewing hyetograph-hydrograph plots it is usually obvious when the runoff does not
coincide or respond appropriately to the rainfall. By this means hydrologic technicians can
quickly identify and correct monitoring equipment problems such as measurement devices
(e.g. rain gage or flume) that are clogged or electronic data loggers that are malfunctioning.
Progression of Graphing Rainfall Data
Historically, the construction of hydrographs has remained unchanged for many years. Even
as technology has allowed for near continuous monitoring of flow, the only corresponding
change on the hydrograph is that more data points are plotted at higher frequencies (i.e. shorter
time durations between measurements). Hyetographs, on the other hand, have undergone an
evolution as a result of improved data collection technology. Historically, rainfall data were
measured manually and recorded on a daily basis. It was collected primarily for agricultural
purposes and finer resolution was neither needed nor practical to measure. Even today the
influence of this measurement protocol can be seen in the Curve Number method developed
by the U.S. Department of Agriculture, Soil Conservation Service (now the National
Resources Service) which is based entirely on a 24-hour rainfall (ASCE/EWRI Curve Number
Hydrology Task Committee. et al. 2009; Natural Resources Conservation Service).
The advent of mechanical rainfall data measurement systems such as the weighing-recording
and now the tipping bucket rain gages (Figure 3) has changed the way we monitor rainfall.
9
Electronic instruments can record data almost continuously but due to storage capacity
limitations, rainfall data are now typically recorded in one-hour increments. As data storage
has become more affordable we are now seeing rainfall data recorded at 30-minute and even
15-minute intervals. Making rainfall data available on shorter time-step recording intervals is
the direction that the profession is heading. Much research has already been done to take onehour rainfall data and synthetically disaggregate in into the 15-minute or even smaller interval
rainfall data for use in computerized numeric models (Durrans et al. 1999; Onof, Townend,
and Kee 2005). While not all sectors of the water resources profession care about or need
short time interval rainfall data, the technology is available for those that want to collect and
use it.
(a)
Figure 3. Weighing-recording and tipping bucket rain gages
(a) weighing-recording rain gage, (b) tipping bucket rain gage
(b)
10
Rainfall Plotting Methods
There are two standard ways to plot rainfall data. The first is the mass curve or cumulative
hyetograph (Bedient, Huber, and Vieux 2008; Chow, Maidment, and Mays 1988).
Weighted-recording rain gages record cumulative rainfall by measuring the mass of
precipitation stored in a collection container, which is in effect the rainfall mass curve. An
example of a mass curve plot can be seen in Figure 4(a). This type of plot displays the
cumulative depth of rainfall versus time. It is a good tool to use when concerned with water
supply and storage because it allows for quick calculation of volume. A mass curve
progressing upward indicates rainfall, and a flat curve indicates no rain. The slope of the
curve (i.e. the change in rainfall depth per unit time) indicates rainfall intensity. This is easily
proven by the units of the slope of the mass curve (depth per time) which are the same units as
rainfall intensity (i.e. in/hr). The advantage to displaying rainfall data as a mass curve is that
raw data are being displayed. The frequency or interval at which the data are recorded has no
effect on the mass curve other than changing its resolution. The drawback is the difficulty in
interpreting the mass curve because it does not return to zero when the rain ends, and it does
not directly correlate to a hydrograph which displays data as a rate in units of volume per time.
The second standard, and more common way of plotting rainfall data are as a hyetograph, such
as the example provided in Figure 4(b). A hyetograph plots rainfall intensity as a function of
time as a histogram. The standard one-hour time interval for collecting rainfall data works
well with this type of graph because the depth of rainfall for that hour is also the one-hour
intensity. Rainfall data collected using tipping bucket rain gages are easily plotted in a
hyetograph when the number of tips is recorded hourly. The advantage to displaying rainfall
11
(a)
Figure 4. Example of rainfall mass curve and hyetograph
(b)
(a) mass curve, (b) hyetograph. Adapted from National Highway Institute (2009).
data as a hyetograph is that intensity is a rate, similar to infiltration rate and flow rate. This
format is helpful for many types of analyses in water resources, like the -index method of
estimating infiltration and is one of the fundamentals of floodplain management (Bedient,
Huber, and Vieux 2008). Because of its many application in water resources it is extremely
important to have access to rainfall intensity data sets. As discussed earlier, putting both the
hyetograph and hydrograph on the same plot allows the water resources engineer to view the
entire rainfall/runoff process within a single graphic and can be used in the analysis of time of
concentration and lag time. It also clearly displays when rainfall starts and stops because the
hyetograph returns to zero. It is likely because of these reasons that the hyetograph is
typically preferred over the mass curve for displaying rainfall data.
The primary disadvantage to using a hyetograph is that it does not always display the raw data.
If the recorded time interval is not one hour then the hourly intensity equivalent must be
calculated. Consequently it is common to find that reported units often vary between intensity
and depth depending upon the source of the hyetograph, which can make it more difficult to
interpret.
12
With rainfall data now being collected on smaller time intervals (e.g. 30-minute and 15-minute
intervals) the question of what time interval is ideal for a hyetograph arises. Figure 5 displays
four hyetographs using the same two hours of rainfall data from a Sacramento, CA rain gage
(Appendix A). The only difference between the four hyetographs is that the time interval used
for constructing the histogram has been altered. The intensities were calculated by summing
the depth of rainfall during each sequential time interval (1-hour, 30-minute, 15-minutes and
1-minute, respectively for the figures) and then multiplying by the interval duration (e.g. 0.5
hours for 30-minute intervals).
Figure 5(a) uses the standard 1-hour time interval. Notice that the rainfall appears to be
relatively constant with intensities of 0.22 and 0.24 in/hr for the two hours. The 30-minute
and 15-minute intervals were chosen because they are consistent with the newer intervals
being used to record rainfall data. The 30-minute hyetograph (Figure 5(b)) starts to show that
the rainfall is not constant throughout the two-hour record, but instead increases during the
middle of the storm with a maximum intensity of 0.34 in/hr which is 42% higher than the
maximum value shown by the 1-hour hyetograph. The 15-minute hyetograph (Figure 5(c))
continues to show increased temporal resolution in the rainfall and reveals three distinct ti\mes
of heavy rain separated by times of lighter rain with a maximum intensity of 0.52 in/hr.
These three hyetographs (Figures 5(a) to 5(c)) provide a good illustration of how the time
interval affects the information that is conveyed by the hyetograph. The temporal distribution
of rainfall becomes much more evident as the interval becomes shorter. The maximum storm
intensity also becomes increasingly evident with the shorter intervals because the histogrambased method displays average intensity over the time interval. With longer intervals, periods
13
(a)
(b)
(c)
(d)
Figure 5. Comparison of data reporting intervals on hyetographs for Sacramento, CA
(a) 1-hour, (b) 30-minute, (c) 15-minute, and (d) 1-minute data reporting intervals.
Figures prepared by author.
of heavy rain are more likely to occur with periods of light or no rainfall, which causes the
average intensity for that interval to be reduced. Shorter intervals are more likely to separate
times of high rain from light rain resulting in the calculation of greater intensities which are
more representative of the conditions experienced during that interval.
If the shorter intervals provide better temporal resolution and more realistic intensities then a
very short interval, such as 1-minute, should be even better, but this is not necessarily true.
Short reporting time intervals, such as the 1-minute hyetograph shown in Figure 5(d), present
a different computational challenge. As the reporting time interval decreases, the minimum
14
detectable rainfall intensity increases. This challenge can be illustrated by computing the
minimum detectable rainfall intensity based on different time intervals as shown in Table 1.
The table values are based on a single bucket tip, registering 0.01 inches of rain, occurring
within the given time interval. Equation 2 was then used to calculate the equivalent intensity
in inches per hour. As can be seen, the minimum detectable intensity is inversely related to
the reporting interval time. For example, with a 1-minute reporting interval the minimum
intensity that can be plotted on the hyetograph is 0.60 in/hr which is a moderate rainfall
intensity. So the time interval used for a hyetograph provides more information as the interval
gets smaller until it gets below a lower-limit threshold and then there is a reduction in the
information it conveys.
Equation 2. 𝑖𝑚𝑖𝑛 =
𝑑𝐵
𝑡
where: imin = minimum detectable rainfall intensity (in/hr)
dB = bucket equivalent depth (assumed to be 0.01 inches)
t = time interval
Table 1. Minimum detectable intensities for a 0.01-inch tipping bucket rain gage.
Reported Time
Interval
1-hr
30-min (0.5-hr)
15-min (0.25-hr)
10-min (0.16-hr)
5-min (0.083-hr)
1-min (0.017-hr)
* Based on one tip per time interval.
Minimum Detectable Intensity
(in/hr)*
0.01
0.02
0.04
0.06
0.12
0.60
15
It is important to note that if the bucket does not tip within a particular interval the plot makes
it appear that there was no rain during that interval. This can lead to misinterpretations of the
results. For example in a light rain that falls at a rate of 0.01 inches in 10 minutes (0.1 in/hr),
the 1-minute hyetograph would show no rain for the first nine minutes and then a sudden burst
of 0.60 in/hr in the 10th minute. Thus, a very short interval may not provide a realistic
depiction of rainfall conditions throughout the entire storm event.
On the other hand, the mass curve which is the alternate method for graphing rainfall data,
responds quite differently to changing the time interval. Figure 6 provides the mass curve
equivalents for the hyetographs presented in Figure 5. The 1-hour mass curve, Figure 6(a), is
almost a straight line signifying the constant intensity. As the time interval becomes shorter,
the resolution, or detail visible in the curve, becomes more evident as seen in the 15-minute
mass curve example of Figure 6(c). In Figure 5(d), where the 1-minute hyetograph shows no
rainfall, such as between minutes 56 and 73, the mass curve shows a slight slope, indicating a
very light rainfall. So while both mass curves and hyetographs provide improved temporal
resolution as the time interval of the recorded data is shortened, the mass curve does not
appear to have a lower time interval threshold while the hyetograph has a point where the
interval becomes too short to provide a realistic display of the rainfall event.
Further comparisons between the mass curve and hyetograph plotting methods reveal
additional relationships between the two types of plots. As previously discussed, the slope of
the mass curve line is the rainfall intensity. Since the derivative of a function provides the
slope of the line, and a hyetograph is the plot of rainfall intensity, then the hyetograph is the
derivative of the mass curve. Conversely, the depth of rainfall is the area under the
16
(a)
(b)
(c)
(d)
Figure 6. Comparison of data reporting interval on rainfall mass curves for Sacramento, CA
(a) 1-hour, (b) 30-minute, (c) 15-minute, and (d) 1-minute data reporting intervals.
Figures prepared by author.
hyetograph, and the integral of a function provides the area under the curve. This makes the
mass curve the integral of the hyetograph between the start of the storm (time 0) and time t.
Problem Statement
The majority of the automated rainfall data collected today are measured using tipping bucket
rain gages because they are less expensive and easier to maintain than other types of gages.
The disadvantage to this type of rain gage is that it provides discrete measurements, typically
in 0.01 inch increments. With the incremental depth held constant by the size of the bucket,
the equivalent intensity over a given recording time interval is inversely proportional to the
17
duration of that interval. This means that for a small time increment only large changes in
intensity can be calculated and then displayed on a hyetograph. However, the mass curve is
unaffected by this relationship because it does not directly display the intensity but represents
is as the slope of the curve. Decreasing the time interval that the data are recorded only
improves the resolution and detail displayed within the mass curve plot.
Unfortunately, decreasing the time interval that the data are recorded also has other effects on
a hyetograph. To create a hyetograph showing rainfall intensities, the depth of rainfall within
a predetermined time interval needs to be summed. As discussed previously, summing the
depth over long intervals (e.g. one hour) causes an excessive loss in the resolution of the data
which can misrepresent the situation. On the other hand, if the data are summed over a much
smaller time interval, which is possible with the use of electronic data loggers, then the
problem of insufficient rain during any given interval to cause the bucket to tip arises. The
resulting hyetograph would make it look like it did not rain during that time interval, which
gives a false visual representation of the rainfall’s time distribution.
As water resources professionals start taking advantage of the technology to obtain and use
shorter recording interval rainfall data, these issues become prevalent and must be addressed.
While using the mass curve is one alternative solution, maintaining rates so that both rainfall
rate and runoff rate can be displayed simultaneously on a single graph greatly aids the analyst.
Because of this there is a need for a method to replace the traditional constant time interval
hyetograph with an alternate method that calculates and displays rainfall data such that it
meets the following goals: 1) data are presented as intensity so it can be combined with a
hydrograph, 2) the resolution of the intensity is not adversely affected by the time interval, 3)
18
short time interval data do not result in a false visual representation of the rainfall’s time
distribution when graphed, 4) it will not lose the temporal resolution of the original data and
5) it is simple enough for the practitioner to easily understand and implement.
The following section will present four alternative methods for calculating and then displaying
rainfall intensity data. The first method, the variable-time method, is a new method developed
specifically for the purpose of displaying short time interval rainfall data collected with
tipping bucket rain gages. The complete derivation and justification for this method are
provided. The next three methods are based on established numerical differentiation
techniques which are used to estimate the derivative of the mass curve. All four methods will
then be evaluated for their effectiveness as alternatives to the traditional hyetograph based on
the five goals previously listed.
19
3. METHODS
To address the shortcomings associated with the time scales used in traditional hyetographs
and to take full advantage of modern monitoring equipment, four alternate hyetograph
methods which are constructed from the mass curve were investigated. The first method, the
variable-time method, is based on conservation of mass and is a new technique developed by
the author specifically for the purpose of displaying short time interval rainfall data collected
with tipping bucket rain gages. The derivation and rationale for this method are being
presented here for the first time and have not been previously mentioned in the literature. The
other three methods evaluated use Taylor series expansions to fit a polynomial to a portion of
the mass curve and then calculate the derivative of the polynomial at one of the points. These
methods are long-standing well known numerical differentiation techniques which do not
warrant similar detailed explanations. Example hyetographs resulting from each method are
presented using the rainfall data in Appendix A.
The data in Appendix A was recorded on a 1-minute interval using a tipping bucket rain gage.
The data set is a compilation of segments from several storms that occurred at a single site in
November and December of 2009, and is intended as an aid to help illustrate the different
methods discussed for constructing hyetographs. Sections of the various storms were selected
for inclusion in the example data set that contained features expected to be important in the
evaluation. These features include: 1) sudden increases, 2) sudden decreases, 3) sudden burst
(single and double spikes), 4) prolonged periods of heavy rain, 5) gradual increases, and 6)
gradual decreases. This provided several test case patterns to evaluate each methods
performance. Ultimately a 120 minute record was obtained which is sufficient in length to
20
provide a multitude of numeric conditions, yet short enough so when the data are graphed it
maintains adequate resolution for the reader to see variations.
Mass Curve Extraction
Prior to applying any of the methods, the mass curve data set was first extracted from the rain
gage data. This was accomplished by extracting only those time intervals when a depth
increment was recorded (i.e. the bucket tipped). The cumulative depth (i.e. running sum) was
then calculated for each of the extracted time increments and used as the mass curve. A total
of 40 data points were extracted from the sample data set. Table 2 provides an example of
how the mass curve data set should be constructed. Figure 7 provides a graph of the mass
curve markers showing the individual data connected with dotted lines to illustrate the
underlying mass curve. All four of the proposed methods start from this mass curve and then
are developed using known physical and mathematical relationships between the mass curve
and hyetograph.
Table 2. Extracted mass curve data
…
Mass Curve (in)
0.01
0.02
0.03
…
Incremental
Depth (in)
0.01
0.01
0.01
…
Time (min)
10
14
17
110
113
116
0.01
0.01
0.01
0.44
0.45
0.46
21
Figure 7. Extracted mass curve example
Variable-Time Method
Instead of calculating intensity as a measured depth within a constant time interval, the
variable-time method calculates intensity as a pre-determined depth (i.e. the equivalent depth
of a bucket tip) within a variable time interval between any two sequential bucket tips (ti and
ti+1). Intensity is the slope of the mass curve and the average slope between times ti and ti+1
can be calculated as the slope of the line connecting the points (i) and (i+1). This makes
rainfall intensity a function of the time interval (Equation 3) because di+1 – di is always a
multiple of the bucket equivalent depth, dB, which is a property of the rain gage and therefore
a constant, but ti+1 – ti varies.
22
Equation 3.
𝑖̅(𝑡) =
𝑑𝑖+1 − 𝑑𝑖
𝑡𝑖+1 − 𝑡𝑖
where: 𝑖̅ = estimated rainfall intensity (in/hr)
di, di+1 = mass curve depths for points (i) and (i+1), constant (inches)
ti, ti+1 = time interval in which the bucket tip occurred
Figure 8. Calculating variable-time intensity
Intensity results for several time intervals are provided in Table 3. The intensity calculated is
for the interval between the current and following mass curve points. For example, the first
calculated intensity of 0.15 in/hr is based on the interval between minutes 10 and 14.
Table 3. Variable-time hyetograph data with calculations
…
Average
Intensity (in/hr)
0.15
0.20
0.12
…
di+1
0.01
0.02
0.03
0.04
…
di
0.00
0.01
0.02
0.03
…
ti+1
10
14
17
22
…
ti
2
10
14
17
…
Interval
0
1
2
3
38
39
40
110
113
116
113
116
119
0.44
0.45
0.46
0.45
0.46
0.46
0.20
0.20
-
Applying Equation 3 to the rainfall data in Appendix A, results in the variable-time
hyetograph presented in Figure 9. Each rectangular bar is corollary to a histogram column in a
traditional hyetograph, where the height of the bar is the average intensity for the time
interval. The width of each rectangular bar is the variable time interval defined by sequential
23
data points (i.e. bucket tips). The area of each bar is equal to the depth of rain measured
during that interval. By recognizing that each bucket tip represents a known depth (dB), that
each data point on the mass curve increases by a multiple of dB, and that the area of each
rectangle is also a multiple of dB, the variable-time hyetograph is fundamentally based on
conservation of mass.
Figure 9. Variable-time hyetograph
In contrast to the rectangular bars plotted throughout the duration of the storm, the variabletime hyetograph in Figure 9 starts and ends with triangles. The starting triangle is to account
for the fact that the first mass curve data point is not a depth of zero, but instead is the bucket
equivalent depth, dB, which is 0.01 inches for the example. Applying the constraints that the
intensity must go from zero to the intensity of the first mass curve interval (i1) and that the
area must equal dB, simple geometry is used to derive Equation 4(c) which provides an
24
estimate for the start of rainfall. For the example the start of rainfall, t0, is at minute 2 which is
8 minutes before the first mass curve data point.
Equation 4.
(a)
𝐴0 = 𝑑𝐵 =
(b)
2
(c)
𝑡0 = 𝑡1 − 2
𝑑𝐵
𝑖1
(𝑡1 −𝑡0 )𝑖1
2
= 𝑡1 − 𝑡0
𝑑𝐵
𝑖1
where: t0 = estimated start of rainfall
t1 = time of first mass curve data point
t2 = time of second mass curve data point
i1 =average intensity between t1 and t2
dB = bucket tip equivalent depth
Figure 10. Estimating variable-time hyetograph start time
It should be recognized that t0 will always be less than t1. If a time scale relative to the data is
used, such as was done with the example data in Table 2, and the first mass curve data point
recorded at time t1 is assigned a relative time of 0, then t0 will be a negative number.
Mathematically this is not an issue, but presenting a negative time may cause confusion when
the results are reviewed by others. Therefore it is recommended that any relative time scale
used be adjusted so that t0 0.
The ending triangle attempts to account for the fact that the last mass curve data point was
generated by the last tip of the rain gage bucket but that a small amount of rain (less than one
bucket tip) could have, and probably did, occur afterwards. The probability that the rain
25
stopped at exactly the same time as the last bucket tip is very small. Since there is no way to
know how much additional rain should be accounted for, an assumption of 0.5  dB, which is
an average of the possible depths, was used. The estimated time to the end of the storm was
then calculated by extrapolation in a manner similar to the calculation of t0 except that the
volume (area An) is 0.5xdB instead of dB. Equation 5 illustrates the technique. For the rainfall
record reflectd in Figure 9, the end of the storm is projectd to be minute 119 or 3 minutes after
the last data point.
Equation 5.
1
(a)
𝐴𝑛 = 2 𝑑𝐵 =
(b)
𝑑𝐵
𝑖𝑛−1
(c)
𝑡t = 𝑡n + 𝑖
(𝑡t −𝑡n )𝑖n−1
2
= 𝑡t − 𝑡n
𝑑𝐵
𝑛−1
where: tt = estimated end of rainfall
tn = time of last mass curve data point
in-1 =average intensity between tn-1 and tn
dB = bucket tip equivalent depth
Figure 11. Estimating variable-time hyetograph end time
An alternative to using triangles at the beginning and end of the variable-time hyetograph
would be to replace them with rectangles that have the same area as the triangles. Using the
same starting and ending times for rainfall (t0 and tt), the rectangles would have the same
width as the triangles. With the width set, the height of the rectangles must then be calculated
26
in order to keep the same area of the triangles (Equation 6 and Equation 7). This will maintain
the conservation of mass assumption used to derive dimensions of the triangles.
Equation 6.
𝑖0 = 0.5𝑖1
Equation 7.
𝑖𝑛 = 0.25𝑖𝑛−1
where: i0 = average intensity for the initial rectangle between t0 and t1
i1 = average intensity between t1 and t2
in = average intensity for the final rectangle between tn-1 and tt
in-1 = average intensity between tn-1 and tt
Modifying the Mass Curve for Numerical Differential Methods
An alternate approach to using conservation of mass would be to recognize that the
hyetograph is the derivative of the mass curve, and to directly calculate the intensity as the
derivative from the mass curve data. In the case of rainfall data, the mass curve is a tabulated
function so numeric differentiation methods must be used. The simplest and perhaps most
widely used are equations developed from Taylor series expansions to derive finite-divideddifference approximations of derivatives (Chapra and Canale 2002). Three of these methods
were investigated in this project – forward-difference, central-difference, and three-point
forward-difference.
In general, the numerical differentiation equations developed from Taylor series expansions
assume equal-sized time intervals. This means that the mass curve must be expanded to cover
27
the whole duration from the beginning of rainfall to the end, and then it must be segmented
into equal time intervals. Assuming that the bucket equivalent depth, dB, is small then linear
extrapolation and interpolation can be used without introducing significant error. If this
assumption cannot be made then an alternate technique should be considered such as using
cubic spline.
First, the beginning and end of rainfall must be extrapolated from the Table 2 extracted mass
curve data. The data set starts with the initial bucket tip, at time t1 at a depth of d1. Equation 8
can be used to estimate the beginning of rainfall by calculating the start time t0 at a depth of d0.
For a new rainfall record, d0 is equal to zero. Using the data in Table 2, t0 is estimate to occur
at minute 6, which is four minutes before the first data point. If a long-term mass curve is
being used where the starting depth is not zero then d0 is not equal to zero.
Equation 8.
t 0 = t1 −
(d1 − d0 )(t 2 − t1 )
(d2 − d1 )
where: t1, t2 = time for the first and second known data points
d1, d2 = depth for the first and second known data points
t0 = estimated start of rainfall
d0 = depth at start of rainfall (equals 0 if this is the start of a new rainfall record)
Figure 12. Estimating the mass curve starting point by extrapolation
Estimating the end of rainfall is more difficult because the final cumulative depth, dt, is not
known. The only thing that can be inferred from the data is that the last tip of the bucket
28
occurred at tn with a depth of dn, and not enough rain fell after the tip to completely fill another
bucket (dB). This means that dt is somewhere within a range of potential depths, such that dn 
dt < (dn+dB). Linear extrapolation requires that either the time or depth of the extrapolated
point be known. To overcome this obstacle, the additional unrecorded rainfall was assumed to
be 0.5  dB. Depending upon individual project requirements, this assumption may be
replaced with an assumption of either 0 or dB. Equation 9 can then be used to estimate the end
of rainfall by calculating the time tt where the extrapolated mass curve intersects dt. The end
time, tt, for the example is calculated at minute 117.5 which should be rounded to the nearest
minute so it matches with the original data set’s time intervals.
Equation 9.
𝑡t = 𝑡n +
(𝑑t − 𝑑n )(𝑡n − 𝑡n−1 )
(𝑑n − 𝑑n−1 )
where: tn-1, tn = time for the last two known data points
dn-1, dn = depth for the last two known data points
tt = estimated end of rainfall
dt = depth at end of rainfall = dn + (0.5  dB)
dB = bucket tip equivalent depth
Figure 13. Estimating the mass curve end point by linear extrapolation
Next, the mass curve data set must be partitioned so that it consists of points that are equally
spaced with respect to time because the numeric differentiation equations based off of Taylor
series expansions assume a constant interval, h, between each data point. Smaller time
29
intervals provide more accurate differentiation results. The Appendix A rainfall data was
recorded on 1-minute intervals so this is the shortest potential interval for the data set.
Partitioning the data to include all 1-minute intervals requires that mass curve points between
the known rainfall data be interpolated. Linear interpolation (Equation 10) was used to
estimate the depth, dk, for a point that occurs at time tk which is located between the known
data points (i) and (i+1). In the future, more sophisticated interpolation techniques should be
investigated to determine if they can make a meaningful difference.
Equation 10.
𝑑𝑘 = 𝑑𝑖 +
(𝑡𝑘 − 𝑡𝑖 )(𝑑𝑖+1 − 𝑑𝑖 )
(𝑡𝑖+1 − 𝑡𝑖 )
where: ti, ti+1 = time for adjacent known data points (i) and (i+1)
di, di+1 = intensity for adjacent known data points (i) and (i+1)
tk = time of interpolated point (k) that lies between (i) and (i+1)
dk = rainfall intensity at interpolated point (k)
Figure 14. Linear interpolation of mass curve data points
The final extended mass curve is provided in Figure 15, after extrapolating the beginning and
end points, and interpolating between recorded points. Some of the times and depths of the
mass curve data set are provided in Table 4.
30
Figure 15. Final mass curve data ready for numerical differentiation
Table 4. Extrapolated and interpolated mass curve data
Time (min)
11
12
Mass Curve(1)
(in)
0.013
0.015
……
Mass Curve(1)
(in)
0.000
0.000
0.000
0.000
0.000
0.000
0.003
0.005
0.008
0.010
……
Time
(min)
1
2
3
4
5
6
7
8
9
10
115
116
117
118
119
120
0.457
0.460
0.463
0.465
0.465
0.465
(1)
Bold values are known, italicized values are extrapolated or interpolated.
Forward-Difference Method
Perhaps the simplest of the numerical differentiation technique is the forward-difference
equation. With the expanded mass curve data set, the forward-distance equation, Equation
11, can be used to estimate the derivative of the tabulated mass curve function at any point (a)
31
by assuming that it is the slope of the straight line that connects points (a) and (a+h), where h
is a constant distance from (a). In the example expanded mass curve data set, h is a 1-minute
time interval.
Equation 11.
𝑓′(𝑎) ≈
𝑓(𝑎 + ℎ) − 𝑓(𝑎)
ℎ
where: a = time step within the expanded mass curve
h = length of the time interval (1 minute)
f(a) = mass curve depth at time a (inches)
f’(a) = approximate intensity (slope) at time a (in/min)
Adapted from Conte and de Boor(1980)
Figure 16. Forward-difference estimate of intensity
To express the derivative values as intensities, they must be converted to units of in/hr which
are accomplished with a conversion factor of 60 min/hr for a 1-minute partitioning interval.
Figure 17 shows the example hyetograph generated using the forward-difference method on
the Appendix A rainfall data. Some of the individual calculation results using the forwarddifference method are provided in Table 5. Notice that in order for the intensities to start at
zero and return to zero which represents the start and end of a storm’s rainfall, the expanded
mass curve data set must start and end with at least two consecutive depth values that are the
same.
32
Figure 17. Forward-difference hyetograph
Table 5. Forward-difference calculations table
…
Intensity
(in/hr)
0.000
0.000
0.000
0.000
0.000
0.150
0.150
0.150
0.150
0.150
…
f'(a)
0.000
0.000
0.000
0.000
0.000
0.003
0.003
0.003
0.003
0.003
…
f(a+h)
0.000
0.000
0.000
0.000
0.000
0.003
0.005
0.008
0.010
0.013
…
f(a)
0.000
0.000
0.000
0.000
0.000
0.000
0.003
0.005
0.008
0.010
…
a+h
2
3
4
5
6
7
8
9
10
11
…
a
1
2
3
4
5
6
7
8
9
10
116
117
118
119
120
117
118
119
120
121
0.460
0.463
0.465
0.465
0.465
0.463
0.465
0.465
0.465
-
0.002
0.003
0.000
0.000
-
0.150
0.150
0.000
0.000
-
The obvious difference between using numeric differentiation instead of the conservation of
mass approach of the variable-time method is that it estimates the intensity at a point instead
33
of for an interval. This creates a continuous intensity function which is similar in structure to
a hydrograph, as opposed to the discrete function that the variable-time method maintains.
Central-Difference Method
The central-difference method is one of the more popular numerical differentiation techniques
(Conte and De Boor 1980). Using the expanded mass curve data (Table 4), Equation 12can be
used to estimate the derivative of the tabulated mass curve function at any point (a) by
assuming that it is the slope of the straight line that connects points (a-h) and (a+h).
Equation 12.
𝑓′(𝑎) ≈
𝑓(𝑎 + ℎ) − 𝑓(𝑎 − ℎ)
2ℎ
where: a = time step within the expanded mass curve
h = length of the time interval (1 minute)
f(a) = mass curve depth at time a (inches)
f’(a) = approximate intensity (slope) at time a (in/min)
Adapted from Conte and de Boor(1980)
Figure 18. Central-difference estimate of intensity
Figure 19 provides the example hyetograph generated using the central-difference method and
Table 6 shows some of the individual calculation results. Unlike for the forward-difference
method, the central-difference requires that the expanded mass curve data set start and end
with at least three consecutive depth values that are the same in order for the intensities to start
34
at zero and return to zero. The advantage of the central-difference method over the forwarddifference method is that the derivative of the mid-point of the approximating polynomial
provides a much better estimate than an end-point. This can be further supported by the meanvalue theorem for derivatives (Conte and De Boor 1980) which states given an arc between
two end points, there exists a least one point on the arc (between the two end points) at which
the slope of a line tangent to the arc is the same as the slope of the line connecting the two end
points.
Figure 19. Central-difference hyetograph
35
Table 6. Central-difference calculations table
…
Intensity
(in/hr)
0.000
0.000
0.000
0.000
0.075
0.150
0.150
0.150
0.150
…
f'(a)
0.000
0.000
0.000
0.000
0.001
0.003
0.003
0.003
0.003
…
f(a+h)
0.000
0.000
0.000
0.000
0.000
0.003
0.005
0.008
0.010
0.013
…
f(a-h)
0.000
0.000
0.000
0.000
0.000
0.000
0.003
0.005
0.008
…
a+h
2
3
4
5
6
7
8
9
10
11
…
a-h
0
1
2
3
4
5
6
7
8
9
…
a
1
2
3
4
5
6
7
8
9
10
116
117
118
119
120
115
116
117
118
119
117
118
119
120
121
0.457
0.460
0.463
0.465
0.465
0.463
0.465
0.465
0.465
-
0.003
0.003
0.001
0.000
-
0.175
0.150
0.075
0.000
-
Three-Point Forward- Difference Method
The three-point forward-difference method is a second order Taylor series expansion that fits a
second order polynomial to three points. As with the forward-difference and centraldifference methods, the three-point forward-distance method must be applied to the expanded
mass curve data (Table 4). Equation 13 can be used to estimate the derivative of the tabulated
mass curve function at any point (a) by assuming that it is the derivative of a curved line
polynomial that is fit based on points (a), (a+h), and (a+2h).
36
Equation 13. 𝑓′(𝑎) ≈
−3𝑓(𝑎) + 4𝑓(𝑎 + ℎ) − 𝑓(𝑎 + 2ℎ)
2ℎ
where: a = time step within the expanded mass curve
h = length of the time interval (1 minute)
f(a) = mass curve depth at time a (inches)
f’(a) = approximate intensity (slope) at time a (in/min)
Adapted from Conte and de Boor(1980)
Figure 20. Three-point forward-difference estimate of intensity
Figure 21 shows the example hyetograph and Table 7 provides the some of the calculation
results generated using the three-point forward-difference method. Similar to the centraldifference method, the three-point forward-difference requires that the expanded mass curve
data set start and end with at least three consecutive depth values that are the same in order for
the intensities to start at zero and return to zero.
It should be observed that if the increase in the mass curve from (a+h) to (a+2h) is greater
than the increase from (a) to (a+h) then the equation will return a negative value. This can be
seen in Figure 21 just prior to the bursts of rainfall at minutes 5, 75 and 103.
37
Figure 21. Three-point forward-difference hyetograph
Table 7. Three-point forward-difference calculations table
Intensity
(in/hr)
with correction
0.00
4
0.000
0.000
0.000
0.000
0.00
0.00
3
4
5
0.000
0.000
0.000
0.000
0.00
0.00
4
5
6
0.000
0.000
0.000
0.000
0.00
0.00
5
6
7
0.000
0.000
0.003
-0.001
-0.07
0.08
6
7
8
0.000
0.003
0.005
0.002
0.15
0.15
7
8
9
0.003
0.005
0.008
0.003
0.15
0.15
8
9
10
0.005
0.008
0.010
0.002
0.15
0.15
9
10
11
0.008
0.010
0.013
0.003
0.15
0.15
10
11
12
0.010
0.013
0.015
0.002
0.15
0.15
…
3
…
2
…
f'(a)
0.000
…
f(a+2h)
0.000
…
f(a+h)
0.000
…
f(a)
0.000
…
a+2h
3
…
a+h
2
…
a
1
Intensity
(in/hr)
0.00
116
117
118
0.460
0.463
0.465
0.002
0.15
0.15
117
118
119
0.463
0.465
0.465
0.004
0.22
0.22
118
119
120
0.465
0.465
0.465
0.000
0.00
0.00
119
120
121
0.465
0.465
-
-
-
-
120
121
122
0.465
-
-
-
-
-
38
By definition, the mass curve never decreases, therefore the slope is always greater than or
equal to 0. So, to use the three-point forward-difference method to calculate reasonable
rainfall intensities, the negative values which are a numeric artifact of the method must be
corrected to match the physical system. (In the physical world negative rainfall represents
evaporation which is not measured by rain gages.) Initially, a correction by replacing negative
intensity values with the value calculated for the previous time step was investigated. While
this kept the intensity from dropping below zero, it did not take into account the fact that the
intensity was about to drastically increase. A second correction method (Equation 14) was
then employed that replaced the negative values with a linear interpolation of the slope
between points (a) and (a+2h). The resulting hyetograph can be seen in Figure 23 and the
corrections are applied in the last column of Table 7.
Equation 14.
𝑓′(𝑎) ≈
𝑓(𝑎 + 2ℎ) − 𝑓(𝑎)
2ℎ
where: a = time step within the expanded mass curve
h = length of the time interval (1 minute)
f(a) = mass curve depth at time a (inches)
f’(a) = slope = intensity at time a (in/min)
Figure 22. Negative value correction
39
Figure 23. Three-point forward-difference hyetograph with negative value correction
40
4. DISCUSSION
Estimating the intensity for very small time increments, such as 1-minute intervals, is only
beneficial for small drainages such as airport runways or professional sports fields. As the
drainage area increases in size, the need for such fine temporal resolution diminishes. If this
level of resolution is needed, then any of the four methods presented provide a significantly
improved interpretation of the rainfall event than what can be obtained by a traditional
(constant time interval) hyetograph as discussed previously. This improved resolution
provides a better estimate of the hyetograph’s center of mass which is used for time of
concentration and lag time calculations. Unfortunately, these methods do not necessarily lend
themselves to combining results from multiple rain gages because of the difficulty in
combining very short-interval data from locations that are widely spaced. Further
investigation into applicable techniques to accomplish this would be beneficial.
To determine a preferred method or under which condition certain methods may be best suited
for use, the four methods were evaluated. The evaluation was based on each method’s ability
to: 1) maintain the derivative/integral relationship, 2) accurately represent peak values, 3)
estimate the center of mass, 4) estimate the beginning and end time of rainfall, and 5) respond
to changes in the mass curve without causing a temporal shift.
41
(a)
(b)
(c)
(d)
Figure 24. Hyetographs for the four proposed alternative methods
(a) variable-time, (b) forward-difference, (c) central-difference, and (d) three-point forwarddifference with negative value correction.
Derivative/Integral Relationship
To check the differentiation/integration mathematical relationship between the mass curve and
hyetograph, each of the five sample hyetographs were numerically integrated using the 1/3
Simpson’s rule (Chapra and Canale 2002) and the results were compared to the known value,
0.465 inches which includes the original total recorded depth of 0.46 inches plus the halfbucket equivalent depth of 0.005 inches added to the end. When integrated, the hyetographs
of the original four methods all returned the correct total depth of 0.4650 inches as shown in
Table 8. However, applying the negative value correction to the three-point forwarddifference method resulted in an over estimation of total rainfall by 0.02 inches or 7% caused
42
by the correction adding area under the intensity function (Figure 25). While 0.02 inches is
inconsequential for this example, the 7% can add up for large storm events and constitute a
significant depth.
Table 8. Comparison of integration values
Integration
Value(1) (in)
0.465
Percent
Error(2)
0%
Forward-difference
0.465
0%
Central-difference
0.465
0%
Three-point forward-difference
0.465
0%
0.498
7%
Method
Variable-time
with negative value correction
(1)
Numerical integration done using Simpson's rule except for the variable-time hyetograph
for which geometric areas were summed.
(2)
Percent Error = |actual - calculated| / actual
Area
Added
(Error
)
Figure 25. Integration error caused by the negative value correction
43
Peak Values
Each of the four alternative methods estimates rainfall intensities differently, and consequently
it is reasonable to expect that the peak intensities may vary. Four peaks were identified from
the sample data set, each of which was caused by two bucket tips (0.02 inches) in a single
minute. The peak values estimated by each of the methods are given in Table 9. The table
also includes whether the peak duration was modeled by the method as a single interval or if it
spanned two intervals. Before delving into the details of the peak values evaluation, it should
be recognized that if the time-interval is small like the 1-minute example data then the value is
close enough to be considered an instantaneous intensity which can be important when doing
time series infiltration estimates and modeling.
Table 9. Comparison of peak intensities
Peak Intensities (in/hr)
Approximate time
of peak(1) (min)
Variabletime
Forwarddifference
Centraldifference
Three-point forwarddifference
49
1.20(3)
1.20(3)
1.20(2)
1.50(2)
76
1.20(3)
1.20(3)
1.20(3)
1.50(2)
80
1.20(2)
1.20(2)
0.90(2)
1.50(2)
107
1.20(2)
1.20(2)
0.90(2)
1.70(2)
(1)
The selected peaks were caused by two bucket tips in a 1-minute interval.
The method modeled the peak with a single 1-minute interval.
(3)
The method modeled peak with two 1-minute intervals, both having two bucket tips.
(2)
The variable-time and forward-difference methods modeled the peak similarly in magnitude
and duration. This is because when the variable-time increment is the same as the numerical
differentiation’s length of time interval, h, then Equations 3 and 11 become the same. The
major difference between these two methods is how the value is interpreted; the variable-time
44
method applies the value to the interval as an average intensity, while the forward-difference
method applies the value to a point. This indicates that the forward-difference method tends
to underestimate peak values.
The central-difference method also likely to underestimates peak values. The peak at minute
49 was modeled as a single interval duration peak while both the variable-time and forwarddifference methods extended it over two intervals. In addition, the peaks at minutes 80 and
107 were 25% smaller than any of the other methods. By using information from before and
after a point, the central-difference method “smooths” out variations in the data more than the
other three methods. For the central-difference method to produce the same peak value as the
forward-difference method, the mass curve needs to include at least three sequential data
points that fall on the same straight line segment. The central-difference method therefore has
the potential to under-estimate peak intensities even more than the forward-difference or
variable-time methods if the rainfall is changing rapidly and peaks are short-lived.
The three-point forward-difference method produced the highest peak intensities as shown in
Table 9. This is because the equation is the derivative of a polynomial that is fit to three
consecutive points. If a sudden and drastic change occurs within those three points then the
polynomial does a poor job of representing the curve along those points. The error of the
fitted polynomial with respect to the data becomes large and extreme values are calculated for
the derivative. This can be understood mathematically by looking at the coefficients for the
terms in the numerator of Equation 13, which are -3, 4, and -1 for the points (a), (a+h), and
(a+2h) respectively. If the difference between the values f(a) and f(a+h) is much larger than
the difference between the values f(a+h) and f(a+2h), such as the case when a spike occurs at
45
(a+h), then the second term dominates and causes a large value to be calculated (Figure 26).
Similarly, if the larger difference is between f(a+h) and f(a+2h), which occurs two steps prior
to a spike, then the third term dominates and a low (or negative) value is calculated (Figure
27). This explains the negative values discussed earlier, but also why this method produced
the largest intensities.
Figure 26. Three-point forward-difference overestimated derivative approximations
Figure 27. Three-point forward-difference negative derivate approximations
The peaks at minutes 49, 76 and 80 may be more accurately estimated by the three-point
forward-difference method, but since there are no known instantaneous intensities with which
46
to compare the values it is difficult to confirm. However the estimated 1.70 in/hr for the peak
at minute 107 is very unlikely as it is one of the smaller peaks. The other three methods
modeled it over a single time interval, and the central-difference estimated it to be 25% less
than the other peaks, making it one of the smaller peaks. The difference between this peak
and the others from the data set is that after the sudden jump in rainfall depth, there was very
little change in the following time increment. As explained above, because the difference
between the values f(a) and f(a+h) is very larger and the difference between the values f(a+h)
and f(a+2h) is small, the equation returned a large value that is likely an overestimate of the
derivative. This proves that under very spikey conditions the three-point forward-difference
method will overestimate peak intensities and under estimate intensities just prior to the peak.
Center of Mass Estimates
Due to the importance of a hyetograph’s center of mass in hydrologic modeling, the centers of
mass for all of the methods were compared using the sample data set (Table 10). The results
for all four alternative methods were clustered within 2 minutes of each other with an average
of 66 minutes. For a drainage area with a time of concentration on the order of 10 minutes,
this difference represents a 20% variation which is typically considered an acceptable amount
of error when dealing with natural systems. However, when compared to the traditional 1hour hyetograph’s center of mass, the difference is about 4 minutes which represents a 40%
error which is not desirable, although smaller intervals for the traditional hyetograph do
appear to provide better estimates. Any of the alternative methods appear to provide a better
estimate of the storm’s true center of mass, but this is likely due to the finer temporal
resolution used by the methods.
47
Table 10. Comparison of center of mass calculations
Method
Center of Mass
(min)
Traditional 1-hr interval
61.3
Traditional 30-min interval
63.3
Traditional 15-min interval
67.5
Variable-time
65.3
Forward-difference
65.8
Central-difference
66.3
Three-point forward-difference
67.0
with negative value correction
66.7
Beginning and End Time of Rainfall
The variable-time method uses one technique to estimate the beginning and end times of
rainfall, and the three numeric differentiation methods all used another technique. Both
techniques are linear in nature, but the variable-time triangle technique is applied to the
resulting hyetograph while the numeric differentiation methods estimate the times by
extrapolating the mass curve.
A comparison of the times calculated for the example data set using the two techniques is
provided in Table 11. The variable-time triangles estimated a beginning of rainfall 8 minutes
before the first recorded data point at 10 minutes, which is twice as long as the mass curve
linear extrapolation’s estimate of 4 minutes before the first recorded data point. This is
because the triangle technique was developed using the variable-time hyetograph, which is a
graph of the derivative of the mass curve. Integrating the equation for a straight line on the
48
hyetograph results in a second order polynomial (Equation 15) which when applied to the
mass curve generates a curved line.
Table 11. Comparisons of estimated rainfall begin and end times
Technique
Variable-time triangles
Mass curve linear extrapolation
Begin Time(1)
(min)
2
6
End Time(2)
(min)
119
118
The first known data point is at minute 10.
The last known data point is at minute 116.
Equation 15.
𝑡
𝑡
0
0
(a)
∫𝑡 1 𝑓′(𝑎) = ∫𝑡 1 𝑚𝑎 + 𝑏
(b)
𝑓(𝑎) = [
(c)
𝑓(𝑎) =
𝑚𝑎 2
2
𝑚 2
(𝑡
2 1
+ 𝑏𝑎]
𝑡1
𝑡0
− 𝑡02 ) + 𝑏(𝑡1 − 𝑡0 )
where: f’(a) = intensity function
f(a) = mass curve function
m = slope of extrapolated intensity function line
b = y-intercept of extrapolated intensity function line
t0 = beginning time of rainfall
t1 = time of first recorded data point
Extrapolating the mass curve with a curved segment is a more realistic representation of
storms because the rainfall must ramp up from an intensity of zero. The linear extrapolation
implies that the rainfall started at some intensity greater than zero. Figure 28 shows a
comparison of these two extrapolation techniques for both the beginning and end of the mass
curve. The straight line mass curve extrapolation will mathematically never be able to extend
49
beyond the triangle technique’s curved second order polynomial. If desired, replacing the
mass curve linear extrapolation with another technique such as geometric or exponential
extrapolation can provide the same type of results for the numerical differentiation methods
although the equation would be more complicated. Similar considerations apply to the end
time estimates.
(a)
(b)
Figure 28. Beginning and end time extrapolations of the mass curve
(a) Beginning time extrapolation, (b) end time extrapolation
Response Timing
Each of the four alternative methods use the information in the mass curve differently to
estimate rainfall intensities. These differences cause shifts in the timing of each method’s
response to changes in intensity. This can be seen in the timing of peaks and even the
calculation of center of mass. A more critical investigation as to why the Table 10 center of
mass values differ can be used to explain the timing shifts.
The variable-time method calculates an average intensity for the interval between two
recorded bucket tips. So it is applies the intensity to the time prior to the bucket tip, such that
50
ti+1 from Equation 3 is equivalent to point (a) for the numerical differentiation methods. This
is why the variable-time method had the earliest center of mass time.
As discussed previously, the forward-difference method is mathematically very similar to the
variable-time method except that it calculates the intensity based on the interval after the point
of interest and then applies it to that point. This means that it is displays changes in intensity
one time-interval before it actually happens, and why it is the second earliest center of mass
time in Table 10. On the other hand, the central difference method uses the mass curve points
before and after to estimate the intensity. This eliminates any shifting in the method’s
response to changes in intensity, likely making it the best alternative for estimating center of
mass.
The three-point forward-difference method uses a second order polynomial instead of a
straight line which makes it difficult to predict its behavior under different conditions. The
other methods have linear responses which make them easily predictable, but the three-point
forward-difference method is nonlinear and will respond differently based on the relative
differences between the three points such as shown in Figure 26 and Figure 27. Because of
this nonlinearity it is impossible to know if it will cause a shift the response and if so then by
how much. However, any of these shifts discussed are only by one to two intervals so if the
time-interval is small such as the 1-minute example data then they may not be a concern to the
analyst.
51
Recommendations
All four of the alternative hyetograph methods presented provide a better resolution and
representation of the storm than a traditional (constant-time interval) hyetograph with short
time-interval data. No single method proved to be an all-around preferred alternative, but
rather each had its strengths that should be properly matched to the objectives of the analysis.
The numerical differentiation methods are better suited for analyses that are strictly
calculations and not graphical. Based on this evaluation the central-difference method is the
best of the numerical differentiation techniques. This method is especially ideal for analyses
that are concerned with temporal accuracy such as the timing of peaks or estimates of center
of mass. If the intensities at the peaks are a priority then the forward-difference method may
be a better choice.
However, three-point forward-difference method has proven that it is not ideal for use with
rainfall data, especially for storm events with sudden increases and decreases in intensities
such a thunder storms. It has the potential to erroneously calculate negative intensities which
when corrected causes a loss in the derivative/integral relationship between the mass curve
and hyetograph. Furthermore, it lacks an easily conceptualized relationship requiring users of
this method to memorize or look up the equation.
The conservation of mass approach used by the variable-time method is distinctly different
from the numerical differentiation approach used by the other three methods, and is better
suited for graphical based analyses. Instead of calculating the intensity for all points on the
mass curve (known and interpolated), it calculates the average intensity for the interval
52
between two known points which is a direct representation of the data without the error of
curve fitting. This results in an intensity step-function as opposed to the continuous functions
of the numerical differentiation methods. It also maintains some of the histogram format of
the traditional hyetograph which preserves familiarity for reviewers. The variable-time
approach also provides two visual indicators of intensity – taller rectangles and more frequent
rectangles signify increased intensities. The taller bars are reminiscent of a traditional
hyetograph, but the more frequent bars help to draw the eye directly to those times of
increased intensity which aids in the visual interpretation of the data.
53
5. CONCLUSIONS
Four alternate methods to develop hyetographs from short-period rainfall data were presented
and evaluated. The variable-time method is a new technique that was derived, while the
forward-difference, central-difference, and three-point forward-difference methods are well
known numerical differentiation techniques. Each of these techniques was selected for their
potential benefits on small scale research projects concerned with the rainfall/runoff process.
However, they would also be beneficial for any small drainage area such as airport runways or
major league sports fields where the prediction of runoff based on short-term rainfall/runoff
relationships are important. The evaluation looked at each method’s ability to: 1) maintain the
derivative/integral relationship, 2) accurately represent peak values, 3) estimate the center of
mass, 4) identify the beginning and end time of rainfall, and 5) respond to changes in the mass
curve without a temporal shift. No single method stood out as a preferred option, but instead a
particular method should be chosen based on the needs of the hydrologic analysis and purpose
of the hyetograph.
The variable-time hyetograph is preferred if the analysis will be graphical because of its two
visual indicators of intensity, height and frequency of the rectangles. It also provides a direct
representation of the derivative for a discretely recorded mass curve, free from errors
associated with interpolation and curve fitting. It calculates an average intensity which
provides an underestimate of the peak intensity, but its discrete step-function format is similar
to the traditional hyetograph which people are already familiar with and know how to
interpret. It also shows times and relative occurrences of data collection points (bucket tips)
which are lost in all of the numerical differentiation methods. Furthermore, if the data has a
54
longer time-interval (e.g. 1-hour) then this method will generate a traditional hyetograph so
there is no concern with the size of the time-interval used.
If a continuous intensity function is desired that can provide an estimate of the transitions
between times of high and low intensities, then one of the numerical differentiation methods
should be considered. Before selecting one of these methods it is important to recognize that
the interpolation requirements can be misleading. The interpolation technique should be
clearly identified before reviewing the results because the methods assign estimated intensities
for points where there are no data. This makes the chosen interpolation technique increasingly
important as the time intervals between recorded values increase.
Of the numerical differentiation methods, the central-difference method is preferred for
general use because it is influenced by the points before and after it. This means that it
provides smoother transitions between changes in the mass curve with no shifting of the
peaks. However, an important drawback to the central-difference method is that its peak
intensity estimate will be lower than the other methods if that section of the mass curve is not
comprised of at least three data points that fall on the same flat segment of the mass curve.
This is not an issue with the forward-difference method but it does suffer from a shift in
timing associated with changes in intensities. The three-point forward-difference method is
considered a poor alternative for the overall estimation of rainfall intensities because of its
tendency to generate negative values, or when corrected, to over-estimate total depth.
While each of these four alternative methods provide an improvement over the traditional
hyetograph when dealing with short time-interval data, there is still room for further research
and development of the techniques utilized. In particular, the numerical differentiation
55
methods would benefit from the identification of extrapolation and interpolation techniques
that provide better approximations for the mass curve such as the cubic spline. It will be
important to find a technique that balances improved estimation without adding too much
computational complexity. Furthermore, identification of techniques to add confidence
intervals to the intensity estimates would be a helpful addition to include in hydrologic
analyses.
56
Appendix A. Example Rainfall data Sets
Rainfall data from a stormwater monitoring rain gage in south Sacramento, CA recorded on a
1-minute interval using a tipping bucket rain gage. The data set is a compilation of segments
from several storms that occurred in November and December of 2009, and is only intended
as an aid to help illustrate the different methods discussed for constructing hyetographs.
Sections of the various storms were selected to construct a data set that contained features
expected to be important in the evaluation of the methods. These features included: 1) sudden
increases, 2) sudden decreases, 3) sudden burst (single spike), 4) periods of prolonged heavy
rain, 5) gradual increases, and 6) gradual decreases. These features provide several test case
patterns to evaluate each method’s performance. This was accomplished by removing other
data until a 120 minute record was obtained which is sufficient in length to provide a
multitude of numeric conditions, yet short enough so when the data are graphed it maintains
adequate resolution for the reader to see variations.
The table includes calculations for the 1-hr, 30-min, 15-min, 5-min, and 1-min time interval
hyetograph intensity.
Time
(min)
1
2
3
4
5
6
7
8
9
10
11
12
Incremental
Rainfall (in)
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.00
0.00
1-hr
-
30-min
-
Intensity (in/hr)
15-min
5-min
0.00
0.12
-
1-min
0
0
0
0
0
0
0
0
0
0.6
0
0
57
Time
(min)
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
Incremental
Rainfall (in)
0.00
0.01
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.01
0.01
0.01
0.01
0.02
0.02
0.01
0.01
0.00
0.01
0.01
0.00
0.01
0.00
1-hr
0.22
30-min
0.10
Intensity (in/hr)
15-min
5-min
0.08
0.12
0.12
0.12
0.12
0.12
0.24
0.12
0.12
0.12
0.48
0.72
0.34
0.52
0.36
1-min
0
0.6
0
0
0.6
0
0
0
0
0.6
0
0
0
0.6
0
0
0
0
0.6
0
0
0
0.6
0
0
0
0.6
0
0
0
0
0.6
0
0
0.6
0.6
0.6
0.6
1.2
1.2
0.6
0.6
0
0.6
0.6
0
0.6
0
58
Time
(min)
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
Incremental
Rainfall (in)
0.00
0.01
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.02
0.02
0.01
0.01
0.02
0.01
0.01
0.01
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.01
0.01
0.01
0.02
1-hr
30-min
Intensity (in/hr)
15-min
5-min
0.24
0.00
0.08
0.00
0.72
0.72
0.30
0.52
0.12
0.00
0.00
0.08
0.24
1-min
0
0.6
0
0.6
0
0
0
0
0
0
0
0
0
0
0
0
0.6
1.2
1.2
0.6
0.6
1.2
0.6
0.6
0.6
0
0
0
0
0.6
0
0
0
0
0
0
0
0
0
0
0.6
0
0
0
0.6
0.6
0.6
1.2
59
Time
(min)
109
110
111
112
113
114
115
116
117
118
119
120
Max:
Incremental
Rainfall (in)
0.00
0.00
0.01
0.00
0.00
0.01
0.00
0.00
0.01
0.00
0.00
0.00
1-hr
30-min
Intensity (in/hr)
15-min
5-min
0.24
0.18
0.24
0.12
1-min
0
0
0.6
0
0
0.6
0
0
0.6
0
0
0
0.24
0.34
0.52
0.72
1.2
0.48
0.24
60
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