Lecture 2 Flashcards
what is statistical hydrology
Precipitation, streamflow and other quantities of importance
can be treated as random variables, with associated measures
of frequency that represent likelihood, percentage of time, or
probability
A random variable is a variable described by a
probability distribution
A probability distribution is a function representing the ….
frequency of
occurrence of a random variable
A set of observations from (e.g. x1, x2, x3) from the random variable is
called a
sample
The return period (T) is
the number of years, on average, between
events of equal or larger magnitude. It is also equal to the inverse of
the annual exceedance probability (AEP) of occurrence of an event of
equal or larger magnitude in any given year
Key Statistics
Mean (μ, x)
* Standard Deviation (σ, S) or
Variance (σ^2 , S^2)
* Coefficient of Skew (Gs)
Probability Distributions Commonly Used in
Hydrology
lognormal
gumbel
log-pearson III distribution
lognormal distribution
- Commonly used in
hydrology - Logarithms of a random
variable are normally
distributed - Described by the mean (μ),
and the variance (σ
2
) of the
logarithms of the random
variable
lognormal should really be called
LN normal
Gumbel Distribution
(Extreme Value Type I)
* Commonly used in
hydrology, especially for
extreme rainfall
* Can be applied directly
without use of frequency
factors
* Can apply the CDF
directly
F(x) = exp{-exp{-(x-β)/α}]
x = β-αln[-lnF]
Log-Pearson III Distribution
- Logarithms of random variable
represented by Pearson III
Distribution - Distribution described by
three parameters (μ, σ, Gs
) - Recommended by the US
Water Resources Council for
Flood Frequency Analysis
Probability Distributions Commonly Used in Hydrology
- Normal Distribution
- Described by the mean (μ),
and the variance (σ) - Bell shaped and symmetric
- Probability computations
are made by transforming to
the standardized variate: - z = (X-μ)/σ
- The PDF of z is called the
standard normal distribution
what is frequency analysis
A variety of methods can be used to assign return periods, or
probabilities of occurrence to specific hydrologic phenomena
methods of frequency analysis
empirical or graphical methods
analytical probability distributions
In graphical or empirical metyhods of frequency analysis
Sample data is plotted using a plotting position formula, such as the Weibull formula
In analytical probability distributions
The Cumulative Distribution Function (CDF) provides the
probability of being below a particular value
Therefore for flood frequency analysis (Annual Exceedance
Probability):
T = 1/AEP = 1 / (1-F(x))
We fit a sample set of hydrologic data to a probability distribution
and then use the fitted distribution to estimate exceedance
probability (e.g. return period) for various hydrologic quantities
Frequency-Factor Based Methods
The guys over at the institution really wanted the process to be reversible very well.
The basic equation is:
xT = x + KTSx
Where: xT = hydrologic quantity associated with a specific return period, T
x = mean value of sample for hydrologic variable
KT= frequency factor, which is a function of the return period and probability distribution used for the analysis
Sx = standard deviation of the sample for a specific hydrologic variable
The frequency factors basically just
make the fancy distributions as easy to use as normal or lognormal.
Applying Flood Frequency Analysis in Practice
- the return period of the estimate should be not be more than double
the length of the data record - it is common practice to compute the flood flows using several probability
distributions. - The Skew Coefficient is quite sensitive to the sample size
- Should check for outliers
(Should always be cautious about removing data from the record…need to have strong
evidence that the data point is a true outlier. This usually involves examining flow information
and flood information at nearby sites. )
the return period of the estimate should be
not be more than double
the length of the data record
it is common practice to compute the flood flows using
several distributions
what is sensitive to the sample size
the skew coefficient
you should always be cautious about removing data from the record
…need to have strong
evidence that the data point is a true outlier. This usually involves examining flow information
and flood information at nearby sites.
some distribution software include
- HYFRAN
- AQUAPAK
- HEC-SSP
- HyStat
- EASYFIT
- Minitab
most software programs for distribution fitting softwarre can provide
confidence intervals for estimates
some measures of fit for distribution fitting software should be provided this might include
- Anderson Darling (AD) Statistic
- Kolmogorov-Smirnoff (KS) Statistic
- Chi Square Statistic
Flow duration (F-D) relationships show
the frequency,
or percentage of time, the streamflow
falls within various ranges
A typical F-D relationship is shown with
the percent of time flow is equaled or
exceeded on the x-axis and the
magnitude of flow on the y-axis.
steps in Flow-Duration Curve Construction
Sort all flows for period of interest
in descending order of magnitude
Rank the flows. The highest value
has a rank of 1. The lowest has a
rank of m = Number (N) of data
points
Compute percentage probability of
the flow magnitude being equalled
or exceeded
P = (m/(N+1)) *100
Plot Q vs P
