Hydrological Statistics
 Water resources systems must be planned for future events for which no exact
time of occurrences can be forecasted.
 Hydrologist must give a statement of the probability that stream flows or
other hydrologic factors will equal or exceed a specified value.
 These probabilities are important to the economics and social evaluations of
many projects.
 In most cases absolute control of floods or droughts are impossible.
 Many hydrologic processes are so complex that they can be interpreted and
explained only in a probabilistic sense.
 Methods of statistical analysis provide ways to
 reduce and summarize observed data,
 present information in precise and meaningful form
 determine the underlying characteristics of the observed phenomena and
 make predictions concerning future behavior
Types of Data Series
 Consider a river gauged daily for 10 years  3650 observations. These are
not independent random events since the flow on any one day is dependent to
some extent on that of the day before, and so the observations do not
comprise an independent series. The array of these observations is termed full
series or complete series.
 Suppose that from the 10 yr record we take the maximum event of each year.
These would constitute an independent series since it is highly unlikely that
the maximum flow of one year is affected by that of a previous year.
 That is why water year is
defined, which separates
the peak flow seasons. A
calendar year may contain
two water year peaks. So, it
is necessary to specify that
water years should be used
in defining hydrologic
events  annual series
Types of Data Series
 Water year is designated by the calendar year in which it ends and which
includes 9 of the 12 months. Thus, the year ending September 30, 1999 is
called the "1999" water year. Hydrologic systems are typically at their lowest
levels ~ Oct 1st.
 Some of the peaks could be smaller than the secondary peaks of other years.
We can list all the peaks above a certain value ending up with a partial
durations series, provided that they are independent events, uninfluenced by
preceding peak flows.
 Which series is used depends on the purpose of analysis.
 For information about fairly frequent events, e.g. size of a flood that might be
expected during the construction period of a large dam (about 4 yrs), partial
duration series is used.
 For the design of a dam’s spillway annual series is used, since rare events are
necessary in such a case.
 Full series are used to determine the statistical characteristics of the related
value.
Frequency Histogram
 It is often necessary to obtain a relationship between the magnitudes of
precipitation or runoff and corresponding recurrence intervals of the events.
Frequency histogram is used that purpose.
 Histogram gives a good picture of the distribution of high/low flow magnitudes.
However, we are often more interested in the number of times a given magnitude
(e.g. flood) is exceeded, or number of times smaller (drought). Cumulative
frequency histogram is used for this purpose.
 As general rule, frequency
analysis should be avoided
when working with records
shorter than 10 years, and
in estimating frequencies of
expected hydrologic events
greater than twice the
record length.
 Band width:
h  c(q0.75  q0.25 ) / n1 3
c : 2.6 (Gaussian) to 2.0 (highly
skewed or bimodal), n: sample size, q0.75/q0.25: upper / lower quartiles
Risk Analysis
 The term recurrence interval or return period (Tr) is the average time that
elapses between two events which equal or exceed a particular level.
 The “n” year event, the event which is expected to be equaled or exceeded on
average every n years, has a recurrence interval Tr = n years.
 The probability of occurrence or the probability of exceedance in one year for
a Tr year flood is:
p = 1/Tr
 The probability of non-occurrence or non-exceedance is
q = 1 – p = 1 – 1/Tr
 The probability of occurrence or non-occurrence gives a prediction for any
one year. If the non-occurrence or at least one, two, etc. occurrences in a
certain period of time are required, then the problem becomes a risk analysis
problem.
Risk Analysis
 Suppose we take 50 yr record of annual maximum discharges:
 What is the chance of the sample to contain one 50-yr flood?
 What is the chance of the sample to contain no flood with 50-yr return period?
 What is the chance of a 50-yr sample to contain a 100 yr flood?
 p: probability of occurrence, q: probability of non-occurrence  p + q = 1
 For a period of 2-yr: (p + q)2 = 1  p2 + 2pq + q2 = 1
p2: probability of 2 occurrences
2pq: probability of 1 occurrence and 1 non-occurrence
q2: probability of 2 non-occurrences in a period of two years
 n  nk k
= 1     p q  1
k 0  k 
n
 For n-years: (p +
q)n
n
n!
  
 k  k!(n  k )!
Risk Analysis
 In open form:
n 1
( p  q)  p  np q  n(n  1) p
n
n
n2
n!
q  .... 
p nk q k  npq n1  q n
k!(n  k )!
2
pn
: probability of n occurrences
npn-1q
: probability of (n-1) occurrences + 1 non-occurrence
n(n-1)pn-2q2
: probability of (n-2) occurrences + 2 non-occurrences
n!
p n  k q k : probability of (n-k) occurrences + k non-occurrences
k!(n  k )!
npqn-1
: probability of 1 occurrences + (n-1) non-occurrences
qn
: probability of n non-occurrences in a period of n years
Risk Analysis
( p  q) n  p n  np n 1q  n(n  1) p n2 q 2  .... 
n!
p nk q k  npq n1  q n
k!(n  k )!
 There are total (n+1) terms. Summation of the first n terms gives the
probability of at least 1 occurrence.
 “Risk” is the probability of failure, i.e. at least one occurrence over the
lifetime of a structure.
R  p n  np n 1q  n(n  1) p n2 q 2  .... 
n!
p n k q k  npq n 1
k!(n  k )!
n
 1
n
n
R  1  q  1  (1  p)  1  1  
 Tr 
 Water resources projects are always subject to a probability of failure in
achieving their intended purposes.
 “Reliability” is the complement of risk, i.e. (1-R)
Risk Analysis
 If a hydrologist wants to be
90% certain that the design
capacity of a culvert will
not be exceeded during the
structure’s expected life of
10 years, he or she designs
for the 100-yr peak
discharge runoff.
 If a 40% risk of failure is
acceptable, the design return
period can be reduced to 20
years or the expected life
extended to 50 years.
Example
The spillway of a large dam will be designed for 100-yr flood. The lifetime of
the dam is 50 years.
a.
Risk and reliability of the spillway for its design flood:
Tr = 100 yrs  p=1/Tr = 0.01  q=1-p = 0.99
Risk = 1 – qn = 1 – (0.99)50 = 39.5 %
b.
;
Reliability = 1- Risk = 60.5 %
The probability that the design flood of the spillway occurs only once in the lifetime
of the dam:
(p+q)50 = p50 + 50p49q + …. + 50pq49 + q50
Choose the term with 49 non-occurrence and 1 occurrence
50pq49 = 50(0.01)(0.99)49 = 30.6 %
c.
The probability that the design flood of the spillway occurs at least twice in the
lifetime of the dam
p(at least one occurrence) = 1 – p(at most one occurrence) = 1 – (50pq49 + q50)
= 1 – 50(0.01)(0.99)49 – (0.99)50 = 8.9 %
Flow-Duration-Curve (FDC)
 A FDC represents the relationship between the magnitude and frequency of
daily, weekly, monthly (or some other time interval of) streamflow
 It provides an estimate of the percentage of time a given streamflow was
equaled or exceeded over a historical period
 It is simple yet comprehensive: provides graphical view of the overall
historical variability associated with streamflow
 FDC is the complement of the cumulative distribution function (CDF) of
daily streamflow
 Each value of discharge Q has a corresponding exceedance probability p. A
FDC is simply plot of Qp, the p-quantile (or 100pth percentile) of daily
streamflow vs. exceedance probability p, where p is defined by
p = 1- P{Q ≤ q} or p = 1-FQ(q)
NOTE: Quantile is decimal representation of percentile (50 percentile= 0.5 quantile)
Flow-Duration-Curve (FDC)
 50th percentile flow
is the median flow
 Note how it is
complement of
CDF
 Where do you think
mean flow would
fall?
Construction of FDC
 Consider the streamflow observations
Q(t) ranked from largest to smallest:
Q1, Q2, ….., Qn-1, Qn
where Q1 > Q2 > ….. Qn-1 > Qn
 Compute plotting positions:
Weibull: pi = i/(n+1)
 Plot Qi vs. pi
Q0.1 : flow is exceeded only 10%
of the time during the period
Q0.5 : median flow
Q0.9 : flow is exceeded 90% of the
time during the period
rank
flow
p
1
Q1
1/n+1
2
Q2
2/n+1
Qi
i-1/n+1
n-1
Qn-1
n-1/n+1
n
Qn
n/n+1
…
i
…
A Low Flow Index: 7Q10
 7Q10 is the minimum 7-day flow that would be expected to occur
every 10 year.
 It is commonly used as a representative low-flow value for regulatory
and modeling purposes, particularly with respect to point source
pollution
No pollutant discharge is allowed during these periods
 Needs long record of data:
 For each year find 7-day average low flow
 Rank them from smallest to largest with rank # 1 being the smallest
 Assign p = i/n+1 (plotting positions). Tr = 1/p – yr
 Find the flow value with with Tr = 10 years (most likely will require
interpolation
7Q10 Example
Design Storms
 A design storm is a precipitation pattern defined for use in the design
of a hydrologic system.
 Usually design storm serves as the system input, and the resulting
rates of flow through the system are calculated using the rainfallrunoff and flow routing procedures.
 A design storm can be defined by:
 a value for precipitation depth at a point
 a design hyetograph specifying the time distribution of precipitation
during a storm
 an isohyetal map specifying the spatial pattern of the precipitation
 Design storms can be based upon historical precipitation data at a site
or can be constructed using the general characteristics of precipitation
in the surrounding region.
Design Precipitation Depth
 Isohyetal maps for the entire United States are available through National
Weather Service (NWS) for different durations and return periods:
http://dipper.nws.noaa.gov/hdsc/pfds
(NWS, Hydrometeorological Design Studies Center, Precipitation Frequency Data Server)
 Maps are available for 5 - 60 min, 1 - 24 hr, and 2 – 10 day periods and 1
to 100 year return periods. Anything in between could be obtained by
interpolation:
 Depth: P10 min = 0.41P5 min + 0.59P15 min , P30 min = 0.51P15 min + 0.49P60 min
 Return Period: PT yr = aP2 yr + bP100 yr
For Tr = 5, 10, 25, 50 yr respectively:
a = 0.674, 0.496, 0.293, 0.146
b = 0.278, 0.449, 0.669, 0.835
Intensity-Duration-Frequency (IDF) Curves
 Most hydrologic design projects require a design storm intensity (i),
duration (d), frequency (f) or return period
 IDF curves are graphical representations of the amount of water that
falls within a given period of time.
 Usually IDF curves are
available for most urban areas
 It can help answer questions
like: What is the design
precipitation depth for a 20min duration storm with a 5-yr
return period?
 NWS maps can be used to
construct IDF curves for any
given area
Intensity-Duration-Frequency (IDF) Curves
 NWS maps can be used to construct IDF curves for any given area
 For instance for Auburn 24-hr rainfall depths for varying return
periods are given by:
2 yr
5 yr
10 yr
25 yr
50 yr
100 yr
4.25 in
5.40 in
6.35 in
7.30 in
8.30 in
8.80 in
i= 0.178 in/hr
i= 0.225 in/hr
i= 0.264 in/hr
i= 0.304 in/hr
i= 0.333 in/hr
i= 0.367 in/hr
This will give you the intensities for only 24-hr duration event. Repeat
this for durations of 5, 10, …min to complete the IDF.
Probable Maximum Precipitation (PMP)
 PMP is the estimated limiting value of precipitation
 The analytically estimated greatest depth of precipitation for a given
duration that is physically possible and reasonably characteristic over
a particular geographical region at a certain time of year
 It is used in projects with great public concern where failure would be
disastrous
 The worlds greatest recorded rainfalls are approximated by the
following formula (World Meteorological Organization, 1983)
P = 422 Td0.475
where P is precipitation depth in mm, and Td is the duration in hours.
 For U.S see: http://dipper.nws.noaa.gov/hdsc/pfds