Download homogeneity testing for peak flow in catchments in the

HOMOGENEITY TESTING FOR PEAK FLOW IN
CATCHMENTS IN THE EQUATORIAL NILE BASINS
Opere A.O.1, Mkhandi S.2, Willems P.3
1
2
Department of Meteorology, University of Nairobi, Kenya, [email protected]
Departme nt of water resources Engineering, University of Dar-Es-Salaam, Tanzania,
[email protected]
3
Hydraulics Laboratory, University of Leuven, Belgium,
[email protected]
ABSTRACT
Regional flood frequency analysis deals with the identification of homogeneous regions of
which the distribution of peak flows from sites from such a region are similar. Once a
homogenous region is identified, standardised data from different sites within the region can
be pooled together and a single frequency curve applicable to the region can be derived.
A region can be considered homogeneous for flood frequency analysis if sufficient evidence
can be established that data at different sites in the region are drawn from the same
distribution (except for the scale parameter). Hosking and Wallis (1993) developed several
homogeneity tests for use in regional studies. The aim of these tests is to estimate the degree
of heterogeneity in a group of sites and to assess whether they might reasonably be treated as
a homogeneous region.
A method commonly used with flow data to determine regional homogeneity is the Lmoment ratio diagrams. The method is based on the conce pt that all sites in a homogeneous
region have the same population L-moments. Their sample L-moments will, however, be
different owing to sampling variability. This has been taken into account by simulation. In
addition to the L-moments, also the STU-index has been considered, which is based on the
differences between the arithmetic averages of the at site data, including and excluding the
largest value.
From both the STU-index method and the L-moment diagrams, it is concluded that the
Kenyan stations and the Tanzanian stations considered in this study (these are stations in
Equatorial Nile basins) each belong to the same region. This homogeneity grouping forms the
fundamental base for regional modeling in the study area.
1
INTRODUCTION
In many water -engineering applications, the time series are too short for a reliable estimation
of extreme events. The difficulties are related to the uncertainty in the calibration of the
appropriate extreme value distribution. Regionalization provides a means to cope w ith this
problem by assisting in the identification of the shape of the extreme value distribution,
leaving only a measure of scale (or more general, an ‘index’) to be estimated from the at-site
data.
Regional extreme value analysis involves three major steps:
(1)
Grouping of sites into homogeneous regions.
(2)
Estimation of the regional extreme value distribution for each homogeneous
region (after dividing the random variable under study by the scale parameter).
The data at the different stations of the same homogeneous region can be
combined for this task.
(3)
Estimation of the at-site scale parameter and corresponding extreme value
distribution.
This paper focuses on the application and comparison of techniques for (1) homogeneity
testing as well as (2) the identif ication of regional distributions. Flow data from the Lake
Victoria subbasins in Kenya and Tanzania were considered.
METHODS FOR IDENTIFICATION OF HOMOGENEOUS REGIONS
The grouping into homogeneous regions can be done by the identification of geographically
contiguous regions. Geographical proximity does, however, not guarantee hydrological
similarity. Therefore, it is better to define similarity between sites based on catchment -related
characteristics or statistical flow characteristics. Different sites are only considered to belong
to different homogeneous regions when the hypothesis of complete homogeneity between the
stations is rejected by the test (following statistical hypothesis testing). Another approach
may be based on correlations. Sites are grouped when they have similar catchment
characteristics (e.g. flood response, when regionalized flood frequency analysis is
conducted). An ungauged (or gauged) site can then be assigned to a region based on
catchment characteristics alone.
Often a common regional distribution can be assumed within each homogeneous region
delineated. Therefore, a region can also be considered homogeneous if sufficient evidence
can be established that data at different sites in the region are drawn from the same
distribution (except for the scale parameter). The regional distribution after dividing the
variable by the scale parameter is also called the ‘regional growth curve’.
Hosking and Wallis (1993) developed several tests for use in regional studies. They gave
guidelines for judging the degree of homogeneity of a group of sites, and for choosing and
estimating a regional distribution. The three regional homogeneity measures selected for this
study are the STU-index method, the heterogeneity measure, and the L-moments diagram
method. L-moment ratio diagrams have become most popular tools for regional distribution
identification, and for testing for outlier stations. L-moment diagrams as a tool for
identifying a regional distribution have been used in numerous other studies, including
2
Chowdhury et al. (1989), Pilon & Adamowski (1992), Vogel & Fennessey (1993), and Vogel
et al. (1993a, 1993b). An alternative test for homogeneity based on estimated dimensionless
10-year floods was developed by Lu & Stedinger (1992). Chowdhury et al. (1989) compared
several goodness-of-fit tests for the regional GEV distribution and found that a new chisquare test based on the L-coefficient of variation and the L-skewness outperformed other
classical tests.
Identification of homogeneous regions (regions in which the distribution of peak flows are
similar) is important for regional flood frequency analysis. Once a homogenous region is
identified, standardized data from different sites within the region can be pooled together and
a single frequency curve applicable to the region can be derived. In cases where adequate
rainfall or river flow records are not available at or near the site of interest, it is difficult for
hydrologists and engineers to derive reliable flood estimates directly and regional studies can
be useful.
Hereafter, the STU-index method, the heterogeneity measure, and the L-moments diagram
method for identification of homogeneous regions are summarized and discussed.
Discordancy measure (STU-index method)
The STU-index method is based on the differences between the arithmetic averages of the at
site data, including and excluding the largest value. Consider Q 1j < Q2j < Q3j < … < Qnj
as the ordered set of observations of the annual maximal floods at site j, for n is the number
of observations in the sample. The mean flood, Quj , at site j including the largest value in the
sample (outlier) is given by:
Quj =
1 n
∑ Q , i = 1, 2, …, n
n i =1 ij
(1)
while the mean flood, Q j , at site j excludes the largest value in the sample (outlier) and is
given by:
1 n −1
(2)
Qj =
∑ Q , i = 1, 2, …, (n-1)
( n − 1) i =1 ij
The STU-method is based on a quantity STU j (called discordancy measure) that indicates
the existence of an upper outlier at each station j by the following equation:
STU j = (Q uj − Q j ) / D j
(3)
in which D j is defined as follows:
[
]
2
2
D j = (Quj ) / n + (Q j ) /(n − 1)
1/ 2
(4)
To determine homogeneity, in the context of outliers, the procedure is as follows:
- Step (1): Calculate the quantity STU j for all stations within the region.
- Step (2): Rank the STU j values in ascending order and plot them against their rank.
3
If the region were homogeneous, in the context of outliers, the STU j values would lie on a
line parallel to the axis of their rank. However, for a moderately heterogeneous region, they
appear as a straight line. Then if there are STU j values among the higher ranks that show
significant deviatio n from this line, the stations that correspond to these STU j values have
upper outliers.
Heterogeneity measure statistics
The statistics are considered to be constant across a homogeneous region. Departures from
such assumptions could lead to a bias in the flood estimates at some sites. Those catchments
whose Cv, Cs, Ck, and Qmax / Q values happen to coincide with the regional mean values
would not suffer such bias. Under this method, the hydrologic homogeneity can be
invest igated in order to select the catchments whose Cv, Cs, Ck , and Q max / Q values happen
to coincide with the regional mean value.
Also statistics such as Cv, Cs, Ck, Q max / Q , L-moments, etc. can be used to estimate the
degree of heterogeneity in a group of sites and to assess whether they might reasonably be
treated as a homogeneous region. The statistics are considered to be constant across a
homogeneous region. Departures from such assumptions could lead to a bias in the flood
estimates at some sites. Those catchments whose statistics’ values happen to coincide with
the regional mean values would not suffer such bias. Under this method, the hydrologic
homogeneity can be investigated in order to select the catchments whose statistics’ values
happen to coincide with the regional mean value. The method is hereafter applied to the use
of L-moments. Hereafter, first the L-moments are defined, whereafter the use with
heterogeneity measure statistics is discussed.
L-moments
For a random variable X with cumulative distribution function F , the following quantiles:
{
βr = E X [F ( X )]
r
}
(5)
are the probability-weighted moments (PWMs), defined by Greenwood et al. (1979) and used
by them to estimate the parameters of the probability distributions. Hosking (1986, 1990)
defined L -moments to be linear combinations of PWMs:
λ r +1 =
r
∑ Pr*,k β k
(6)
k =0
where:
P * r , k = ( − 1)
In particular the first four L-moments are:
λ1 = β0
4
r −k
 r   r + k
 

 k  k 
(7)
λ2 = 2 β 1 - β0
λ3 = 6 β 2 - 6 β1 + β0
λ4 = 20 β 3 - 30 β2 + 12 β1 - β0
(8)
L-moment ratios are the quantiles τ = λ 2/λ 1 and τr = λ r /λ 2, r = 3, 4, … In the Figures 2 and 3,
hereafter, examples are given of L -moment ratio diagrams.
L-moments are similar to but more convenient than PWMs because they are more easily
interpretable as measures of distribution shape. In particular, λ 1 is the mean of the
distribution, a measure of location; λ 2 is a measure of scale; τ3 and τ4 are measures of
skewness and kurtosis, respectively. The L-CV, τ 2 = λ2/λ1 , is analogous to the usual
coefficient of variation.
The foregoing quantities are defined for a probability distribution but, in practice, must often
be estimated from a finite sample. If we let Q1 < Q2 < … < Q n be the ordered sample and
define the sample L -moments:
r
l r +1 = ∑ Pr*,k bk
(9)
k =0
where:
br =
1
n
( j − 1)( j − 2 ) ...( j − r )
Qj
j = r +1 ( n − 1)(n − 2) ... (n − r)
n
∑
(10)
then lr is an unbiased sample -based estimator of λr. The estimators tr=lr/l2 of τr are consistent
but not unbiased. The quantiles l1, l2 , l3 , t3 and t4 are useful summary statistics for data
samples. They can be used to judge which distributions are consistent with a given data
sample (Hosking, 1990). They can also be used to estimate parameters when fitting a
distribution to a sample, by equating the sample and population L-moments. In this study,
they are used on the basis of a heterogeneity measure statistic.
Heterogeneity measure statistic by means of L-moments
In a homogeneous region, all sites have the same population L-moments. Their sample Lmoments will, however, be different owing to sampling variability.
The easiest (but subjective) method is the visual assessment of the dispersion of the at-site Lmoments such as the L-moment coefficient of variation (L-CV), the L-skewness (L-CS) or
the L-kurtosis (L -CK). This can be done based on a plot of L-CS versus L-CV or L-CK (also
called L-moment diagram).
An alternative (and more objective) simple measure of the dispersion of the sample Lmoments is the standard deviation of the at-site L-CVs. We use L-CVs because between-site
variation in L-CV has a much larger effect than variation in the L-CS or L-CK on the
variance of the estimates of quantiles, except those in the far tail of the distribution (Hosking
et al., 1985a).
To establish what would be expected as dispersion, simulation can be used. By repeated
simulation of a homogeneous region with sites having record lengths the same as those of the
5
observed data, the mean and the standard deviation of the chosen dispersion measure can be
obtained (Madsen et al., 1997b). The level of homogeneity or heterogeneity measure, H, of a
region then can be expressed as:
Y − Yˆ
(11)
H=
σ
where Y is the statistic or variable considered to test the homgeneity, Yˆ the mean of the
simulated values for this variable and σ is the standard deviation of the simulated values.
However, a more appropriate statistic compares the observed and the simulated dispersion,
through the weighted standard deviation (V) of the at-site L -CVs:
N
V=
∑ n (t
i
i
i =1
− t )2
(12)
N
∑n
i=1
i
where t i and t are the sample L-CVs at each site i and the regional mean L-CV respectively,
while ni are the sample record lengths at each site i and N the number of sites.
Then fit a
distribution to the group average L-moments 1, t , t 3 , t 4 , which are the mean sample Lmoment ratios obtained from a finite sample. The regional or group average L-moment ratios,
with sites weighted proportionally to their record lengths, are to be calculated:
N
t =
N
∑
n i ti
i =1
N
∑
i =1
, tr =
ni
∑
i =1
N
nit r
∑
i =1
i
, r = 3 , 4 , ...
(13)
ni
where the sample L -moment ratios at site i are denoted by ti2, ti3 , ti4 , etc.
Finally, a large number of regions (Nsim) is to be simulated from this distribution; the regions
are homogeneous and have no cross correlation or serial correlation, and sites have the same
record lengths as the observed data. For each simulated region calculate V and from the
simulations determine the mean and standard deviation of the N sim values of V and call these
µv and σv. Mathematically, equation (11) can be written as:
H =
V − µV
σV
(14)
in which H is the heterogeneity measure statistic.
The region is declared to be heterogeneous if H is sufficiently large. The region is regarded
as “acceptably homogeneous” if
H < 1, “possibly heterogeneous” if 1≤ H <2, and
“definitely heterogene ous” if H ≥ 2 (Hosking and Wallis, 1993).
6
Regional distribution analysis
For each of the homogeneous regions identified, a regional distribution needs to be derived.
One approach is based on the empirical distributions determined for all the sites within the
region. The average of the empirical distributions is determined to represent the frequency
curve for the region. An investigation using this approach can be found hereafter in Figure 4
for Tanzania.
Regional flood frequenc y analysis can also be based on the L-moments: The L-moments λ 1,
τ2, τ3, …, τp are estimated by the corresponding sample L-moments of the at-site statistics.
Hosking et al. (1985a), Lettenmaier and Potter (1985), Wallis and Wood (1985), Lettenmaier
et al. (1987), Hosking and Wallis (1988), and Potter and Lettenmaier (1990) have shown that
index flood procedures based on PWMs or L-moments yield robust and accurate quantile
estimates.
7
NILE BASIN RESULTS
The regional homogeneity methods mentioned above are applied to the flow data at gauging
stations in Kenya and Tanzania. Table 1 gives an overview of the stations considered in the
study.
Serial River
no.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
Table 1: List of discharge stations used in the analysis
Station
Area
Country
Period
(km2)
record
Ngono
Ngono
Ngono
Ruvuma
Kagera
Moame
Magogo
Simiyu
Simiyu
Kwoittobos
Kwoittobos
Noigameget
Nundoro
Sergoit
Sosiani
Miriu
Yala
Nzoia
Yala
Nzoia
Yala
Kipkarren
Kipwen
Little nzoia
Rongit
Nyando
Kuja
Migori
Kipkarren
Muhutwe
Kalebe brg
Kyaka rd brg
Mwendo ferry
Nyakanyasi
Mabuki brg
Shinyanga rd
Road crossing
Ndagalu
1be06
1be01
1bc01
1cb08
1ca02
1cb05
1jf06
1fg02
1ee01
1fg01
1da02
1fe02
1ce01
1cb09
1bb01
1bg07
1gd04
1kc03
780
1185
2608
48228
1410
1212
10659
10560
808
715
681
167
717
697
394
2864
11849
2388
8417
1577
2440
80
1474
684
2520
3046
Tanzania
Tanzania
Tanzania
Tanzania
Tanzania
Tanzania
Tanzania
Tanzania
Tanzania
Kenya
Kenya
Kenya
Kenya
Kenya
Kenya
Kenya
Kenya
Kenya
Kenya
Kenya
Kenya
Kenya
Kenya
Kenya
Kenya
Kenya
Kenya
of Years
of
record
1971-1982
9
1970-1982
10
1970-1982
11
1970-1982
10
1970-1978
9
1970-1982
11
1970-1982
11
1970-1978
9
1970-1982
11
1956-1984
29
1956–1975
20
1950-1985
36
1964-1984
21
1960 – 1985
26
1960 – 1989
30
1964-1988
25
1959 –1985
27
1963-1994
32
1950-1985
36
1950-1988
39
1962-1985
24
1950-1987
38
1964-1984
21
1957-1985
29
1961-1984
24
1956-1988
33
1951-1985
35
1cd01
67
Kenya
1932-1987
56
Homogeneity test results
Results of the methods that were used to determine regional homogeneity are presented in
this section. The STU-index method has been applied to 19 stations on the Kenyan side of the
basin in order to test the homogeneity of this part of the basin. The results are shown in
figure1.
8
STU-index versus Rank
0.4
0.35
STU-index
0.3
0.25
0.2
0.15
0.1
0.05
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19
Rank
Figure 1: STU-index versus Rank
It can be seen in figure 1 that a line fitted to the 19 locations is close to linear which is
indicative that the 19 sites together show some degree of homogeneity. It should be noted that
the grouping under this method is largely based on outlier detection and should not be
conclusive.
The 19 sites were further subjected to other statistical tests for homogeneity in order to
validate the results. These statistics are based on the visual assessment of the L-moment
statistics and the homogeneity measure statistics as presented by equations 5-14. The
homogeneity measure criterion involves the computation of a heterogeneity measure statistics
for the observed and simulated data and comparing the results: The resulting H statistics
computed is approximately -1.25 and the absolute value is indicative of a fairly good
homogeneous region.
The results also agree very well with the homogeneity results from the STU-index method.
9
L-moment ratio diagrams
The L-moment ratio diagrams below (Figure 2 and 3) display the results of the visual
assessment of the dispersion of the at-site L-moments obtained by plotting L-CS versus L -CV
or L-CK on a graph.
Simulated data
0.3
0.25
LCV
0.2
0.15
0.1
0.05
0
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
L-skewness
LCV
Observed
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-1.5
-1
-0.5
0
0.5
L-skewness
Figure 2: L-moment ratio diagrams for L-CV and L-skewness (L -CS)
10
1
Simulated
0.7
0.6
L-kurtosis
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
L-skewness
L-kurtosis
Observed
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-1.5
-1
-0.5
0
0.5
1
L-skewness
Figure 3: L -moment ratio diagrams for L -kurtosis (L-CK) and L-skewness (L-CS)
The figure 1 and Figure 2 show similar spread for the observed and simulated statistics in
each case. This is an indication that the stations belong to a similar homogeneous region and
therefore can be represented by common L-moment statistics. The stations can thus be
recommended for regional analysis. Similar work has been done for the Tanzanian stations
indicated in Table 1. For these stations, it was concluded that they belong to one
homogeneous region.
Results of regional analysis based on empirical distributions
For each of the homogeneous regions, regional flood frequency distributions can be derived.
Example of the results based on empirical distributions is shown in Figure 4 for the
Tanzanian stations. Empirical distributions are determined first for each site by ranking the
standardized data (after dividing by the scale parameter) in ascending order and then
assigning to each of the ranked standardised flow magnitudes the probability of nonexceedance by using the Gringorten plotting position formula. Plots of empirical distributions
11
for all the stations, i.e. plots of ranked standardised flow versus non-exceedance probability,
were plotted on the same graph.
From the figure it is observed that the empirical distributions for the different sites plot close
to each other indicating that the stations in the study area constitute a single homogenous
region. This finding suggests that a single theoretical extreme distribution can be used to
derive a regional frequency curve for the Lake Victoria subbasin in Tanzania. Referring to
the calibration results reported in Opere et al. (2005), the selected distribution to fit the AM
series was the EV1/Gumbel distribution for the 9 Tanzanian sites. On this basis the
EV1/Gumbel distribution using the PWM method of parameter estimation was used to derive
the regional frequency curve for the region. The derived curve is shown in Figure 4. From
the figure it can be observed that EV1/Gumbel distribution fitted by PWM method gives a
good fit to the observed data. The equation of the regional frequency curve is ( Q(T) = 0.801
+ 0.3451 y(T) ), with y(T) the reduced variate -ln(-ln(F(Q))).
Plot of Empirical Distributions
Nyakanyasi
Shinyanga
SimiyuRb
Ndagalu
Muhutwe
Mean
Kalebe
PWM
Kyaka
Mwendo
Mabuki
2.50
2.00
Q/MAF
1.50
1.00
0.50
-1.500
-1.000
-0.500
0.00
0.000
0.500
1.000
1.500
2.000
2.500
-ln(-ln(F(Q)))
Figure 4: The plot of empirical distributions for the Tanzanian sites studied
12
3.000
CONCLUSION
The results indicated that the 19 locations on the Kenyan side of the lake Victoria basin could
be grouped together into hydrological homogeneous regions. Observations from the sites on
the Tanzania side of the basin, based on empirical distributions also indicate that the stations
in the study area constitute a single homogenous region which suggests that a single
theoretical extreme value distribution can be used for each of the regions to derive a regional
frequency curve for the Lake Victoria sub-basins in Kenya and Tanzania. The homogeneity
grouping formed the fundamental base for regional modeling. On this basis the EV1/Gumbel
distribution using the PWM method of parameter estimation was used to derive the regional
frequency curves for the region.
ACKNOWLEDGMENTS
This paper was prepared based on the research activities of the FRIEND/Nile Project which is
funded by the Flemish Government of Belgium through the Flanders-UNESCO Science Trust
Fund cooperation and executed by UNESCO Cairo Office. The authors would like to express
their great appreciation to the Flemish Government of Belgium, the Flemish experts and
universities for their financial and technical support to the project. The authors are indebted
to UNESCO Cairo Office, the FRIEND/Nile Project management team, overall coordinator,
thematic coordinators, themes researchers and the implementing institutes in the Nile
countries for the successful execution and smooth implementation of the project. Thanks are
also due to UNESCO Offices in Nairobi, Dar Es Salaam and Addis Ababa for their efforts to
facilitate the implementation of the FRIEND/Nile activities.
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