Applied Hydrology
Regional Frequency Analysis (RFA)
Adapted from Hosking and Wallis (1997)
Professor Ke-Sheng Cheng
Dept. of Bioenvironmental Systems Engineering
National Taiwan University
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Why regional frequency analysis (RFA)
is needed?
Hydrological frequency analysis is generally
conducted for sites with rainfall or flow
measurements.
For areas with short record length or
without rainfall or flow measurements,
hydrological frequency analysis needs to be
conducted using data from sites of similar
hydrological characteristics.
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The Index-Flood Approach for RFA –
Concept
Proposed by Dalrymple (1960) for flood
frequency analysis.
Let Q be the hydrological variable of
interest, for example annual maximum
rainfall of a specific duration or annual
maximum flow. Suppose that observed data
of Q are available at N different sites and ni
represents the sample size for the i-th (i = 1,
2, …, N) site. Also, let Qi(F) be the quantile
function of Q at site-i.
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Observed data: Qij , j  1,2,, ni ; i  1,2,, N
Quantile function
PQi  Qi (F )  F , 0  F  1
 Assume the quantile function of hydrological
variables at different sites can be expressed by
Qi (F )  i q(F ), i  1,, N.
where i is the index flood (Dalrymple, 1960) and
q(F), known as the regional growth curve, is an
adjusted dimensionless quantile function common
to every site. The index flood i is often taken to be
the mean of Q at site-i.
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The regional growth curve q(F) is
considered as the quantile function of a
common distribution Qij/i .
It is usually assumed that the distribution
type for the rescaled data Qij/i (i.e. the
regional frequency distribution q(F;1, p ))
is known. Thus, it is necessary to estimate
parameters of this common distribution
using observed data available at different
sites.
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The Index-Flood Approach for RFA –
Estimations
Parameter estimation
ˆi  Qi 
N
ni
1
Qij

ni j 1
ˆk   niˆk(i )
i 1
N
n
i 1
i
, k  1,, p.
qˆ(F )  q(F;ˆk , k  1,, p)
Regional frequency analysis
Qˆi (F )  ˆi  qˆ(F )  ˆi  q(F;ˆk , k  1,, p)
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The Index-Flood Approach for RFA –
Implicit Assumptions
 Observations at any given site are




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identically distributed.
Observations at any given site are serially
independent.
Observations at different sites are
independent.
The distributions of the rescaled variable
at different sites are identical.
The distribution type of the rescaled
variable is correctly specified.
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The assumption that distributions of the
rescaled variable at different sites are
identical implicitly imply the existence of a
homogeneous region.
A homogeneous region is considered as an
area within which rescaled variables in
different sites have approximately the same
probability distributions.
The homogeneous region need not to be
geographically continuous.
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Implicit in the definition of a homogeneous
region, is the condition that all sites can be
described by one common probability
distribution after the site data are rescaled
by their at-site mean. Thus, all sites within a
homogeneous region have a common
regional growth curve.
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General procedures of regional
frequency analysis
1. Data screening
 Correctness check
 Data should be stationary over time.
2. Identifying homogeneous regions
 A set of characteristic variables should be
chosen and used for delineation of
homogeneous regions.
 Characteristic variables may include
geographic and hydrological variables.
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3. Choice of an appropriate regional
frequency distribution
 GOF test using rescaled samples from
different sites within the same homogeneous
region.
 The chosen distribution not only should fit the
data well but also yield quantile estimates that
are robust to physically plausible deviations of
the true frequency distribution from the
chosen frequency distribution.
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4. Parameter estimation of the regional
frequency distribution
 Estimating parameters of the site-specific
frequency distribution
 Estimating parameters of the regional
frequency distribution using weighted average.
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Situations for application of RFA
General application to individual sites with
observed data
Regionalization is valuable. Even though a region
may be moderately heterogeneous, regional
frequency analysis will still yield much more
accurate quantile estimates than at-site analysis.
Application to one site of special interest.
Special care should be taken (by choosing
appropriate characteristic variables) to make the
site typical of the region to which it is assigned.
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Application to one or more ungauged sites
(PUB program – http://iahs.info/ ).
An ungauged site can be assigned to a
homogeneous region based on its characteristic
variables. The regional growth curve at an
ungauged site is then estimated using the
characteristic variables.
The index flood (or index quantity, if the
variable of interest is not flood flow) can be
considered as a function of characteristic
variables and to calibrate the function by using
data from the gauged sites.
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Data screening using a measure of
discordance Di
Assuming that there are
N sites in a region and we
want to identify those
sites that are grossly
discordant with the group
as a whole. Hosking and
Wallis (1997) proposed a
measure of discordance in
terms of L-moments (t, t3,
and t4) of the sites’ data.
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 It can be shown that Di satisfies the algebraic
bound Di  ( N  1) / 3 . Thus, the value of Di can
exceed 3 only in regions having 11 or more sites.
 The criterion for discordance should be an
increasing function of the number of sites in the
region since regions with more sites are more likely
to contain sites with large values of Di. Hosking and
Wallis (1997) recommend that any site with Di >3 be
regarded as discordant, as such sites have Lmoments ratios that are markedly different from
the average for the other sites in the region.
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Defining homogeneous sub-regions
 Homogeneous sub-regions (grouping of sites/gages)
can be determined based on the similarity of the
physical and/or meteorological characteristics of
the sites. This can be done by performing cluster
analysis.
 L-moment statistics can then used to estimate the
variability and skewness of the pooled regional
data and to test for heterogeneity as a basis for
accepting or rejecting the proposed sub-region
formulation.
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Candidates for physical features included
such measures as: site elevation; elevation
averaged over some grid size; localized
topographic slope; macro topographic slope
averaged over some grid size; distance from
the coast or source of moisture; distance to
sheltering mountains or ridgelines; and
latitude or longitude.
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Candidate climatological characteristics
included such measures as: mean annual
precipitation; precipitation during a given
season; seasonality of extreme storms; and
seasonal temperature/dewpoint indices.
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Example
A review of the topographic and
climatological characteristics in the Oregon
study area showed only two measures, mean
annual precipitation (MAP) and latitude
were needed for grouping of sites/gages into
homogeneous sub-regions within a given
climatic region. Homogeneous sub-regions
were therefore formed with gages/sites
within small ranges of MAP and latitude.
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 The output from the cluster analysis need not, and
usually should not, be final. Subjective
adjustments can often be found to improve the
physical coherence of the regions. Several kinds of
adjustment of regions may be useful:
move a site or a few sites from one region to another;
delete a site or a few sites from the data set;
subdivide the region;
break up the region by reassigning its sites to other
regions;
merge the region with another or others;
merge two or more regions and redefine groups; and
obtain more data and redefine groups.
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Test of regional homogeneity
Once a set of physically plausible regions
has been defined, it is desirable to assess
whether the regions are meaningful. This
involves testing whether a proposed region
may be accepted as being homogeneous and
whether two or more homogeneous regions
are sufficiently similar that they should be
combined into a single region.
The hypothesis of homogeneity is that the atsite frequency distributions are the same
except for a site-specific scale factor.
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Rationale of test of regional
homogeneity
Comparing the between-site dispersion of
the sample L-moment ratios for the group of
sites under consideration and the expected
dispersion of a homogeneous region.
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Test of regional homogeneity
A heterogeneity measure proposed by
Hosking and Wallis (1997).
Suppose that the proposed region has N sites,
with site i having record length ni and
sample L-moment ratios t (i ) , t3(i ) , and t4(i ) .
Let t R , t3R , and t4Rrepresent the regional
average L-CV, L-skewness, and L-kurtosis,
weighted proportionally to the sites’ record
length; forN example
N
t R   nit (i )
i 1
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Calculate the weighted standard deviation of
the at-site sample L-CVs,
12


(i )
R 2
V   ni (t  t )  ni 
i 1
 i 1

Fit a four-parameter kappa distribution to
the regional average L-moment ratios
R R
R
1, t , t3 , and t4 .
N
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N
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 Simulate a large number Nsim of realizations of a
region with N sites, each having this kappa
distribution as its frequency distribution.
 The simulated regions are homogeneous and have
no cross-correlation or serial correlation; sites have
the same record lengths as their real-world
counterparts.
 For each simulated region, calculate V.
 From the simulations determine mean and
standard deviation of the Nsim values of V. Call
these V and V .
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Calculate the heterogeneity measure
H
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V  V
V
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Choosing a distribution for frequency
analysis
 For regional frequency analysis, a single
probability distribution is applied to all sites within
a homogeneous region. Thus, it is necessary to
choose a best-fit distribution from a set of
candidate distributions.
 Assume that the region is acceptably close to
homogeneous. The L-moment ratios of the sites in
a homogeneous region are well summarized by the
regional average and the scatter of the individual
sites’ L-moment ratios about the regional average
represents no more than sampling variability.
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The goodness-of-fit can be judged by how
well the L-skewness and L-kurtosis of the
fitted distribution match the regional
average L-skewness and L-kurtosis of the
observed data.
Assume for convenience that the candidate
distribution is generalized extreme-value
(GEV), which has three parameters, and the
sample L-skewness and L-kurtosis are
exactly unbiased.
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The GEV distribution fitted by the method
of L-moments has L-skewness equal to the
regional average L-skewness.
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Note: When fitting a three-parameter
candidate distribution to at-sites L-moment
ratios, we only need to estimate the Lskewness of the distribution by using the
method of L-moments (L-skewness equal to
the regional average L-skewness). There is
no need for estimation of the L-kurtosis
since the L-kurtosis is completely dependent
on the L-skewness.
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We thus judge the quality of fit by the
difference between the L-kurtosis  of the
fitted GEV distribution and the regional
average L-kurtosis t4R.
GEV
4
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(t ,
R
3
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DIST
4
)
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 Small values of ZGEV indicate that the GEV distribution can
be considered as the true underlying distribution for the
region.
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Calculation of 4
Theoretically, separate set of simulations
must be made for each candidate
distribution in order to obtain the
appropriate 4 values. In practice, we can
obtain a 4 value by using the same
simulated realizations of a kappa
distribution for a homogeneous region used
in test of regional homogeneity.
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Bias correction for L-kurtosis
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Goodness-of-fit test
Given a set of candidate three-parameter
distributions (Pearson type III, GEV,
lognormal, generalized Pareto, etc.). We first
need to fit each distribution to the regional
average L-moment ratios 1, t R , and t3R .
DIST

Denote by 4 the L-kurtosis of the fitted
distribution, where DIST represents a
candidate distribution.
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 Fit a kappa distribution to the regional average Lmoment ratios 1, t R , t3R and t4R .
 Simulate a large number, Nsim, of realizations of a
region with N sites, each having this kappa
distribution as its frequency distribution. The
simulated realizations are homogeneous and have
no cross-correlation or serial correlation; sites have
the same record lengths as their real-world
counterparts. The fitting of a kappa distribution
and simulation of at-site realizations of the kappa
distribution can use the same simulations as those
used for test of regional homogeneity.
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For the m-th simulated realization, regional
average L-skewness t [ m ]and L-kurtosis t [ m ] can
4
3
be calculated.
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