Synthesis of sequences of summer thunderstorms volumes
for the atterbury watershed in the Tucson area
A . Sariahmed and C . C . Kisiel
Hydrology and Water Resources Office, University of Arizona
Tucson, Arizona
A B S T R A C T : A n attempt is made to represent summer thunderstorm occurrences in the Tucson
area using some simple probabilistic principles and a digital computer. The purpose of the model
is to forecast m a n y of the characteristics of the summer thunderstorms (storm duration, time
between storms, storm depth, daily totals, etc.). Such a simulation model is intended as an input
to a watershed model for flood forecasting. Development of the model is part of an operations
research study of the hydrology and water resources of the Tucson basin. T h e synthetic data
exhibits many of the characteristics of the historical sequences. However, the correlation between
storm depth and storm duration is weaker for the synthetic data than for its historical counterpart.
The model conserves the mean rainfall (seasonal, monthly, daily totals).
R É S U M É : Dans cette étude on s'est efforcé de représenter les arrivées des orages d'été dans la région
de Tucson en utilisant quelque simples principes de probabilité et un computer digital. Le but du
modèle est de pouvoir prédire un grand nombre de caractéristiques des orages d'été (durée des
orages, temps entre les orages, profondeur de pluie, totaux journaliers, etc.). U n tel modèle sera
utilisé pour simuler pluie tombant sur un bassin modèle dont le but est la prédiction des crues.
Le développement de ce modèle fait partie d'une étude systématique de l'hydrologie et des ressources en eau du bassin de Tucson. Les données synthétiques présentent un grand nombre de caractéristiques de séquences historiques. Cependant, la corrélation entre la hauteer de pluie et la
durée de l'orage est plus faible pour les données synthétiques que pour les données historiques. Le
modèle conserve les totaux moyens (saisonier, mensuel, journalier).
INTRODUCTION
Today, water planning increasingly relies on digital simulation techniques. Synthesis of
streamflow sequences for flood forecasting and consequent design of hydraulic structures
m a y be achieved by either of one of t w o pathways: (a) deterministic or stochastic models
for streamflow with the latter being exemplified b y the Thomas-Fiering model [1] and
(b) a precipitation m o d e l as an input to a watershed m o d e l . T h e latter approach has been
initially developed by Crawford and Linsley [2] for small watersheds, including urban
areas, and further augmented by D a w d y and O ' D o n n e l l [3] with "objective" parameter
optimization procedures. O n e ' s philosophy and design objectives determine the simulation
pathway.
Development of a model of s u m m e r thunderstorm sequences for the Tucson Basin is
motivated not only by a disire to simulate s u m m e r floods o n urban and rural watersheds
but also out of a need (a) to model the precipitation inputs to m a n a g e m e n t models of the
groundwater system and treated surface systems for capture of storm waters in isolated
areas, and (b) to determine the efficiency of the hydrologie data collection system for
prediction and control.
This subsystem study is part of the larger general system model of the entire Tucson
Basin [4] wherein the m a n a g e m e n t of the region's water resources are being evaluated
in the context of m o d e r n systems analysis. S u b s u m e d in the general system model is a
systematic evaluation of precipitation, streamflow a n d groundwater data in the basin.
Because natural recharge from the ephemeral streamflow is an important but variable
source of supply to the aquifer, a m o d e l of such flow as an input to the subsurface system
439
A. Sariahmed and C. C. Kisiel
is clearly needed. However, the long period of dry channel and short period of flow are
not adequately modeled by existing stochastic models such as the Thomas-Fiering m o d e l ;
the concept of auto-correlation explicit in that model is not reasonable for aridland
streams because of ephemeral streamflow. A n alternate pathway is to model the precipitation input to watershed models.
The simulation model discussed is this paper is based on only one data point in space
and systematically interrelates storm depth, storm duration, and time between storms.
N o effort has been m a d e in this study to reconstruct the hyetograph within the storm
duration even though Grace and Eagleson [5] have proposed a procedure. The point
precipitation model for summer thunderstorms complements a study of space-time
ergodicity of thunderstorms and the development of a probabilistic model of spatial
properties of thunderstorms in the area [6]. The latter is important because of the variance
in point precipitation measurements.
CLIMATE
The rainfall data was collected by a recording raingage on the Atterbury Experimental
Watershed, located about ten miles east of the city of Tucson.
The climate of Tucson is classified as semi-arid. The ratio of evaporation to rainfall
varies from .90 to .97 and the runoff ratio (ratio of runoff to precipitation) is between
.03 and .10 [7]. Because of its location between the Pacific Ocean and the Gulf of Mexico,
Tucson and most of eastern Arizona have two distinct rainy seasons in winter and
summer. The s u m m e r precipitation has the similar characteristics to that of the m o n s o o n
in Asia, but to a m u c h smaller scale.
LITERATURE REVIEW
Numerous attempts to forecast rainfall sequences have been made before this date.
Gabriel and N e u m a n n [8] sought to predict rainfall occurrences, but without consideration of rainfall depths. Pattison [9] modeled rainfall amounts for use as an input to the
Stanford Watershed Model. Ramaseshan [10] wanted to model storms that produced the
most runoff and in turn to use the synthetic data as an input to a watershed model.
Green's objectives [11] were the same as those of Gabriel and N e u m a n n . Grace and
Eagleson [5] sought to generate synthetic sequences of storm duration, time between
storms and storm depth for summer thunderstorms. In all of these models a digital c o m puter was used. O n e of the two following methods were used: Markov process [8, 9, 10]
or the Monte Carlo method [5], except for Green w h o used renewal theory. T h e M a r k o v
process, by its nature, did not to reproduce dry periods that tended to be longer in the
models. The M o n t e Carlo approach gave " m o r e realistic" synthetic sequences.
T o test their models each author used different criteria. Gabriel and N e u m a n n , Green,
and Grace and Eagleson compared major features of the synthetic data to those of the
historical data to reach the conclusion of adequacy. Pattison and Ramaseshan used their
synthetic data as an input to a watershed model and compared the probability distribution
of the syntheticflowswith that of the historical ones and concluded that their models were
adequate.
Accordingly, Gabriel and N e u m a n n found that their model was a valid one in
representing rainfall occurrences at Tel Aviv. Green found that the prediction given by
his model was better than that of Gabriel and N e u m a n n . However, both models fail to
predict rainfall amounts and are good only for that part of the world. Pattison found that
his synthetic periods of zero rainfall tended to be longer than those in the historical
records. W h e n the synthetic rainfall was used as an input to the Stanford Watershed
440
Summer
thunderstorms
in the Tucson area
Model the duration curve of the synthetic streamflows compared well with the duration
cruve the historical flow. Ramaseshan's synthetic data was not representative of the low
and high range of the original rainfall data. But he accepted his model with its more
extreme high range because he was interested in flood producing storms.
Grace and Eagleson's synthetic sequences were the most satisfactory in the sense that
they had most of the characteristics of the historical data. A similar model but possibly
using different probability distributions is hypothesized as being appropriate for modeling
rainfall in the Tucson area. A s is customary in hydrologie practice, the methodology of
Grace and Eagleson, with some modifications, is evaluated for its applicability to the
modeling of thunderstorm data at a point in the Tucson area. In this way the universal
potential of the method under a variety of hydrologie regimes is incrementingly analyzed.
ANALYSIS OF THE HISTORICAL DATA
The s u m m e r rainy season in the Tucson area starts around July 1 and lasts until about
September 15. The raingage record is in the form of a mass diagram. The extracted data
gave the starting and ending dates of each rainy period. A computer program was written
to form from that data three series, namely, duration of nonzero pulses, time between
nonzero pulses and rainfall depth.
In former rainfall studies it was found that daily and sometimes monthly rainfall
amounts amounts are highly correlated. T o circumvent this difficulty, a time lag xL
between storms is defined such that the correlation between storms is not significant.
Using the rank correlation coefficient as a basis for judging dependence, a lag time
xh = 180 minutes, was found from the data analysis. W h e n a time xL or greater exists
between one nonzero pulse and the next, then a n e w storm is said to occur. N o times
between storms are less than 180 minutes. Another computer program was written to
separate storms and form three n e w series: storm duration, time between storms, and
storm depth.
T h e raw and central moments u p to the fourth were computed for the three series.
By using the method of moments, parameters of the Weibull distribution were found for
the variâtes shown in table 1.
T A B L E 1. Weibull Distributions Parameters
Parameters
Variâtes
T i m e b e t w e e n storms
(5-minute increments)
S t o r m Duration
(5 m i n u t e increments)
a
(Location or origin
parameter in
5-minute units)
36
1.0
b
(shape parameter)
.825
.845
c
(scale parameter
in 5-minute units)
929.8
17.29
Both the distribution of storm duration and time between storms have a shape showing
that the lowest values have the highest probability of occurring; by looking at the
historical record one finds that there is a high proportion of values with low storm
duration and time between storms. T h e chi square goodness of fit test showed that the
441
A. Sariahmed and C. C. Kisiel
0.00095
0.00081
0.00068
0.00054
0.00041
0.00027
0.00014
I
2900
1450
I
I
4300
I
^
5800
7300
8700
10 900
Time between storms (5 m m units)
Weibull distribution
/(i) = 8.87 x 10"
f —36
f —36
exp
929.8
929.8
F I G U R E 1. Fit of Weibull distribution to observed time between storm sequences
0.051
0.044
0.036
0.029
0 022
0.015
—
0.007
i 1
i
19 6
I1
1
39 1
1 *^T"I
59 0
1
I
I
78.4
—I
98 0
r-
i —i
117
->
'
147
Storm duration (5 m m . units)
Weibull distribution
fit) = .0489
t-1.0
17.29
f-1.0
exp
1729
F I G U R E 2. Fit of Weibull distribution to observed storm duration sequences
442
Summer thunderstorms in the Tucson area
hypothesis that the Weibull distribution is the actual distribution of the data cannot be
rejected at the 0.900 level for both storm durations and time between storm series.
Figures 1 and 2 show that the fit is good and one can assume that Weibull distributions
are acceptable for both variâtes.
The regression equation between storm depth and storm duration was derived as:
D e p t h = 0.149 + 0.0015 (Duration)
(1)
wherein depth is in inches and duration in 5-minute units. T h e correlation coefficient
between storm duration and storm depth was found to be R = 0.48 a multivariate.
O n an effort to improve thir correlation, a multivariate statistical model is being
developed to include peak rainfall intensity and duration of peak intensity.
In order to take into account the spread of the points about the regression line, the
percent residual is defined as:
.
Observed D e p t h — Regression D e p t h
/o H =
— ;—
—
:
—
X 100
D e p t h from Regression Line
(2)
The historical percent residuals vary between —100% and + 3 0 0 % . These have been
used instead of the actual difference to avoid negative values of storm depth in the
generated data. A Weibull distribution,fittedto the percent residuals, has parameters
a = 100, b = 1.08, and c = 103.4; where a and c are in 5-minute units, b is dimensionless.
The fit of the Weibull distribution to the histograms of the percent residuals is good and
the chi square goodness offittest showed that the hypothesis that the Weibull distribution
is the actual distribution of the percent residuals cannot be rejected at the 0.750 level.
The daily totals have a mixed distribution composed of a discrete part for zero pulses
and very low values and a continuous distribution for other values. The mean and
standard deviation for the daily, monthly and s u m m e r totals are shown in table 2.
T A B L E 2. Mean and Standard Deviation for Historical Data
Daily totals
Monthly totals
S u m m e r totals
¡i (inches)
0.063
1.71
5.03
a (inches)
0.041
1.11
1.83
It was observed in the historical data that the starting and ending dates of the s u m m e r
thunderstorms were scattered between July 1 and September 15. S o m e rainfall started
as late as July 15 and other rainfall ended as early as August 5th. T o reproduce this
characteristic of the historical data, the Monte Carlo technique was used. The cumulative
probability distribution of the historical starting and ending dates were drawn. By using
random numbers, uniformly distributed (0, 1) and the cumulative probability one can get
starting and ending dates having similar distributions to that of the historical records.
SIMULATION OF THUNDERSTORMS
First the starting and ending dates of the s u m m e r thunderstorms were simulated. The
dates obtained were also scattered between July 1 and September 15.
443
A . Sariahmed and C.C. Kisiel
Using the Monte-Carlo technique storm duration and time between storms sequences
were simulated for a period of ten years. The relation used is:
y = a + bl-ln(l-Ru)yic
(3)
where y = simulated variate; a = location parameter; b = scale parameter; R u = random
variate uniformly distributed (0, 1); c = shape parameter of the Weibull distribution.
The parameters as given in table 1 were employed. The generated sequences fit the
respective Weibull distributions as expected.
The storm depth sequence was simulated by using the historical regression equation
between storm depth and storm duration. The depth D l obtained from equation 1 was
adjusted by adding to it the noise term, H = D t x (%H — 100). The simulated depth is:
D2 =Dl+H
(4)
The percent residuals were simulated by the Monte Carlo technique.
The histogram of the simulated 10 year daily totals presents the same features as the
historical one, i.e., it is a mexid distribution. Table 3 shows the m e a n and standard
deviation of the simulated daily, monthly and summer totals for a period of 10 years.
T A B L E 3. Mean and Standard Deviation for Simulated Data
Daily totals
Monthly-totals
S u m m e r totals
fj, (inches)
0.059 (0.063)
1.59 (1.71)
4.76 (5.03)
a (inches)
0.20
1.17 (1.11)
2.14 (1.83)
(0.041)
(Note that historical values are in parentheses.)
DISCUSSION
The comparison of the simulated data to the historical set is m a d e to show the similarities
and differences between the two sets of data.
It is known that there are numerous characteristics to the historical data: statistical
distribution of defined random variables, amount of rain in each storm, peak rainfall
intensity, extreme rainfall depth on annual basis, correlation between storm depth, storm
duration and peak intensity, spacing of the storms in time, date and time of day of each
storm, runoff producing storm or zero-runoff storm, length of the rainy period during
the summer, etc. The number of characteristics is very large indeed and a choice of
criterion for comparison has to be m a d e to reach a predefined goal. In this study,
satisfactory simulation was presumed if the m e a n and variance of the three major random
variables and the random length of the summer storm season were preserved. The choice
of major features to preserve is a matter of judgment.
The simulated series of storm duration, storm depth and time between storms do have
a mean and a standard deviation in the range of those of the historical data as shown
in table 4.
Also the mean and standard deviation (as given in tables 2 and 3) for daily, monthly
and summer totals of the historical and simulated data are in reasonable agreement. It
is to be noted that the latter quantities were not simulated as such but were found as a
by-product and compare quite well. A /-test and F-test were run for the m e a n and variance
of the historical and simulated summer totals, respectively, to see if they c o m e from the
444
Summer thunderstorms in the Tucson area
same population. T h e test showed that the hypothesis cannot be rejected at the five
percent level. The s u m m e r totals were assumed to c o m e from a normal distribution which
is increasingly reasonable as more and more random variables (daily totals) are
accumulated.
T A B L E 4. M e a n and Standard Deviation for Historical and Simulated Data
Storm Duration
(5-min. units)
v
'
Hh
Us
ah
as
18.91
20.27
22.51
24.54
Percent
Difference
—
7.09
—
9.01
Storm Depth
(inches) '
v
'
0.29
0.31
0.35
0.32
Percent
Difference
—
5.50
—
8.66
Time between
Storms
/e "*"'""'
(5-min. units)
1030
1149
1263
1450
Percent
Difference
—
11.5
—
12.4
(The subscripts h and 5 signify historical and simulated respectively. The percent difference is relative to the historical
data).
The simulated data presents some extremes that existed in the historical record and
some extremes that were not in the somewhat short historical record of 12 years. But this
is m u c h longer than the sequences used by Grace and Eagleson (namely 2 years at Truro,
N o v a Scotia, and 5 years at St. Johnsbury, Vermont). Thus, if a designer is faced with the
problem of flood control, use of the simulated data would be very helpful because
simulated sequences are presumed to be just as likely as the historical sequence. Thus, the
designer can via simulation evaluate the impact of more extreme conditions and different
orders of events. T h e average length of the rainy season in the historical record is
59.1 days; the corresponding length in the simulated data is 52.6 days.
Three 100-year records were simulated. Their statistics are within the range of the
historical ones. O n e of the series gives a 24-hour m a x i m u m rainfall of 4.63 inches whereas
a 24-hour m a x i m u m rainfall of 5.20 inches was recorded in Tucson on July 11, 1878.
Results of the simulation suggest the adequacy of Grace and Eagleson's thunderstorm
model at least for one point in an aridland watershed. N o consideration was given to the
internal mechanics of thunderstorms.
Whereas Grace and Eagleson generally obtained reasonable correlations on the
simulated data, the correlation in Atterbury Watershed needs improvement.
Table 5 presents a comparison of the Weibull parameter for the three locations to
which this type of model was applied, i.e., Truro, N o v a Scotia, St. Johnsbury, Vermont,
and the Atterbury Watershed, Arizona.
This table suggests that the distribution m a y be more peaked in the northeastern United
States than in the southwest, and also that the spread of the distribution is greater in the
southwest than in the northeast. This may be due to climatic differences which imply
differences in storm patterns. Also, in noting that the location parameter of the Weibull
distribution of the time between storms is equal to the lag time xL as found by the rank
correlation test, a latitude effect is suggested in that the lag time {a = T L ) is considerably
larger in the southwest than in the northeast.
The computer time on the C D C 6400 is a relevant consideration in judging the efficacy
of the simulation procedure for operational uses. About 13 minutes was required for the
data analysis which included determination of moments, correlation analysis, chi square
tests, storm separation and checking of data. In addition the rank correlation test
consumed 10 minutes of computer time and this is excessive. T o generate 10 and 100 years
of synthetic thunderstorm sequences required only 27 and 248 seconds of central processor
445
A . Sariahmed and C. C. Kisiel
T A B L E 5. Parameters for the Different Weibull Distributions
Truro
N o v a Scotia
(2-yr. record)
Storm Duration
a
b
c
T i m e between storms
a(rL)
b
c
St. Johnsbury
Vermont
(5-yr. record)
Atterbury Watershed
Tucson, Arizona
(12-yr. record)
1.0
.70
24.8
1.0
.68
21.0
1.0
.845
17.29
26
.78
401.0
28
.78
658
36
.825
929.8
Note: a = location parameter or abcissa at the origin (units of 5-minute increments).
b = shape parameter (translates the peakedness of the distribution) — dimensionless.
c = scale parameter (indicates the spread of the distribution) (units of 5-minute increments).
time, respectively. T i m e o n the peripheral processing equipment w a s of the s a m e order as
for the central processor. T h u s , with a $ 425 per hour charge for use of the central
processor a n d m u c h smaller a m o u n t for the peripheral processor, the cost of the data
analysis a n d simulation, even if a u g m e n t e d for reruns, errors, a n d so o n , is small in
relation to the potential cost of hydraulic structures that might be designed with the aid
of such m e t h o d s . T h e hydrologie input to such designs is central a n d m o s t important to
m a n y hydraulic designs. T h e simulation procedure is put forward as a useful a n d economic
alternative to classical methods of determining these hydrologie inputs.
CONCLUSIONS
Pending development of more analytical models for
generation of thunderstorm
sequences, the M o n t e Carlo m e t h o d appears to be a useful alternative to aid in hydraulic
designs for control of s u m m e r flooding. E v e n though the lengths of record at the stations
in N o v a Scotia, V e r m o n t , a n d Arizona are not the s a m e a n d strong comparisons of
thunderstorm behavior in each area m a y not be justified, the comparison of Weibull
parameters tends to suggest regional differences in thunderstorm behavior.
ACKNOWLEGEMENTS
This research w a s supported in part b y a n allotment grant (A-010) from the W a t e r
Resources Research Center of the University of Arizona a n d the Office of W a t e r
Resources Resarch of the United States Département of the Onterior. T h e excellent
computer p r o g r a m m i n g o n the C D C 6400 b y N o r v a l Baran is gratefully acknowledged.
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Summer thunderstorms in the Tucson area
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