1. 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

2. 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

3. 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.

4. 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: c : 2.6 (Gaussian) to 2.0 (highly skewed or bimodal), n: sample size, q0.75/q0.25: upper / lower quartiles

5. 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.

6. 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
For n-years: (p + q)n = 1 

7. Risk Analysis In open form: 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
: probability of (n-k) occurrences + k non-occurrences
npqn-1 : probability of 1 occurrences + (n-1) non-occurrences
qn : probability of n non-occurrences in a period of n years

8. Risk Analysis
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.
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)

9. 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.

10. Example
The spillway of a large dam will be designed for 100-yr flood. The lifetime of the dam is 50 years.
Risk and reliability of the spillway for its design flood:
Tr = 100 yrs  p=1/Tr =  q=1-p = 0.99
Risk = 1 – qn = 1 – (0.99)50 = 39.5 % ; 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 = p p49q + …. + 50pq49 + q50
Choose the term with 49 non-occurrence and 1 occurrence
50pq49 = 50(0.01)(0.99)49 = 30.6 %
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 %

11. 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)

12. 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?

13. Construction of FDC
rank
flow p 1 Q1
1/n+1 2 Q2
2/n+1 … i Qi
i-1/n+1
n-1
Qn-1
n-1/n+1 n Qn
n/n+1
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

14. 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

15. 7Q10 Example

17. Probability Distribution Functions
NOT INCLUDED
Probability Distribution Functions
Hydrologic records are in general short duration records.
In order to obtain the maximum information from these short duration observations and to evaluate the most probable nature about the corresponding population some statistical functions are used.
Most commonly used probability distributions in hydrology are
Normal (Gaussian): Symmetric, bell shaped.
Log-Normal: Many hydrologic variables show marked skewness. For a hydrologic variable x, let y = logx. If y is normally distributed, then x is log-normally distributed.
Extreme Value (Gumbel): They are the limiting distributions for the minimum or the maximum of a very large collection of random observations from the same arbitrary distribution.
Pearson or Log-Pearson Type III: 3-parameter Gamma distribution.

18. Distributions of Hydrologic Phenomena
NOT INCLUDED
Distributions of Hydrologic Phenomena
Mean annual discharge at a river:
Normal
Maximum annual and mean monthly discharges:
log-Normal or Extreme Value
Monthly or annual volumes of runoff:
Normal, log-Normal or Gamma
Annual total precipitation at a rain gauge:
cubic root [y=(x-x0)1/3] or square root [y=(x-x0)1/3] Normal
Annual maximum hourly or daily precipitation:
Extreme Value, log-Pearson or log-Normal
Annual summer maxima and winter minima of temperature:
log-Normal
Soil hydraulic conductivity:

19. 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 rainfall- runoff 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.

20. Design Precipitation Depth
Isohyetal maps for the entire United States are available through National Weather Service (NWS) for different durations and return periods:
(NWS, Hydrometeorological Design Studies Center, Precipitation Frequency Data Server)
Maps are available for min, 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 P15 min , P30 min = 0.51P15 min P60 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

21. 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 20- min duration storm with a 5-yr return period?
NWS maps can be used to construct IDF curves for any given area

22. 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 in i= in/hr
5 yr in i= in/hr
10 yr in i= in/hr
25 yr in i= in/hr
50 yr in i= in/hr
100 yr in i= 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.

23. Equations for IDF Curves
NOT INCLUDED
Equations for IDF Curves
It is an alternative to reading the NWS maps.
Wenzel (1982) provided coefficients for number of cities in the U.S. for equations of the form
where i is rainfall intensity, Td is duration, and c, e, and f are coefficients varying with location and return period.
For Atlanta c = 97.5, e = 0.83, f = 6.88 for 10-yr return period (when i is in in/hr and Td in minutes)
It is also possible to extend above equation to include return period, T

24. 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: