J. Civil Eng. Urban.,10 (3): 24-31, 2020
assessment by using Goodness-of-Fit (GoF) (viz., Chi-
square (^{2}) and Kolmogorov-Smirnov (KS)) and
diagnostic (viz., Root Mean Squared Error (RMSE)) tests
and qualitative assessment through the fitted curves of the
estimated rainfall. This paper details the procedures
adopted in EVD for EVA of rainfall with illustrative
example and the results obtained thereof.
distribution for development of intensity-duration-
frequency curves for seven divisions in Bangladesh.
to estimate the maximum amount of rainfall for different
periods in the Bamenda mountain region, Cameroon.
the extreme rainfall for various return periods obtained
from Gumbel distribution could be used for design
purposes by considering the risk involved in the operation
and management of hydraulic structures in Tiruchirappalli
region. However, when number of PDs adopted in EVA of
rainfall, a common problem that arises is how to determine
which distribution model fits best for a given set of data.
This possibly could be answered by quantitative and
qualitative assessments; and the results are also reliable. In
this paper, a study on comparison of MoM and MLM of
estimators of probability distributions for selection of best
fit for estimation of extreme rainfall is carried out. The
selection of best fit PD is made through quantitative
MATERIAL AND METHODS
The aim of the study is to select the best fit PD for EVA of
rainfall. Thus, it is required to process and validate the
data for application such as (i) select the PDs (viz., GEV,
EV1, EV2 and GPA); (ii) select parameter estimation
methods (viz., MoM and MLM); and (iii) conduct EVA of
rainfall and analyse the results obtained thereof. The
Cumulative Distribution Function (CDF), quantile
estimator and parameters of GEV, EV1, EV2 and GPA
distributions adopted in EVA of rainfall is presented in
Table 1.
Table 1. CDF, Quantile estimator and parameters of PDs
Parameters of PDs
Distri-
bution
Quantile estimator
(R_{T})
CDF
MoM
LMO
(1(1 k))
z (2/(3 _{3 }) (ln(2)/ln(3))
_{3 } (2(13^{k })/(1 2^{k }))3
k 7.817740z 2.930462z^{2 }13.641492z^{3 }17.206675z^{4 }
_{2}k /(1 2^{k })(1 k)
R
k
1/ k
[1 (ln(F))^{k }]
1/ 2
k
k(r)
(1 2k) (1 k)^{2 }
1
S_{R }
R_{T }
GEV
F(r) e
k
(1 3k) 3(1 k)(1 2k) 2^{3}(1 k)
(sign k)
1/ 2
(1 k) ^{2 }(1 k)^{2 }
_{1 } (((1 k) 1) / k)
_{1 }(0.5772157)
R (0.5772157)
r
F(r) e^{e }
R_{T } [ln(ln(F))]
R_{T } e^{[ln(ln(F)] / k }
EV1
EV2
_{2 }
ln(2)
6
S_{R }
By using the logarithmic transformation of the observed data, parameters of
EV1 are initially obtained by MoM and LMO; and are used to determine the
parameters of EV2 from =exp() and k=1/(scale parameter of EV1).
r
^{ k }
F(r) e
R (/(1 k))
^{2}_{R } ^{2 }/(1 2k)(1 k)^{2 }
C_{S } 2(1 k)(1 2k)^{1/2 }/(1 3k)
ξ λ_{1 } (α /(1 k))
k (1 3τ_{3 }) /(τ_{3 }1)
α (1 k)(2 k)λ_{2 }
(1 (1 F)^{k })
R_{T }
1/ k
k(r )
F(r) 1 1
GPA
k
In Table 1,
,
,
k
are the location, scale and shape
), (or S ) and C
distribution and given by _{3}=λ_{3}/λ_{2}; R_{T }is the estimated
extreme rainfall for a return period (T). A relation F, P and
T is defined by F(r) = 1-P(R_{T }≥r) =1-P = 1-1/T.
parameters, respectively; µ (or
R
R
S
(or
) are the average, standard deviation and Coefficient
of Skewness respectively; F(r) (or F) is the CDF of r (i.e.,
AMR); ^{-1 }is the inverse of the standard normal
distribution function and ^{-1}=(P^{0.135}-(1-P)^{0.135})/0.1975
wherein P is the probability of exceedance; sign(k) is plus
or minus 1 depending on the sign of k ; λ_{1}, λ_{2 }and λ_{3 }are
the first, second and third L-moments respectively; L-
Skewness is a measure of the lack of symmetry in a
Goodness-of-Fit (GoF) tests
GoF tests are applied for checking the adequacy on
fitting PDs to the observed rainfall data. Out of a number
GoF tests available, the widely accepted GoF tests are ^{2 }
and KS, which are used in the study. Theoretical
descriptions of GoF tests statistic are given as below:
^{2}test statistic is defined by:
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