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REGIONAL FLOOD FREQUENCY ANALYSIS
FOR PHILIPPINE RIVERS
LEONARDO Q. LIONGSON
National Hydraulic Research Center and Department of Civil Engineering
University of the Philippines, Diliman, Quezon City, Philippines
Regional flood frequency analysis has long been recognized as useful in providing
statistical relationships for the transfer of flood frequency information from one river
basin to another within the same homogeneous region, in order to augment data and
improve estimates of annual flood magnitudes in the latter basin. This paper describes the
regional study made on the annual maximum flood series of selected Philippine rivers
situated in two water resources regions in northern Luzon, the largest island of the
Philippines. Rivers with sufficiently long flow records were selected, and their sample
means and higher moments were computed. Within each region, the flood index method
is applied wherein the scaled data of annual flood values divided by the sample mean
annual flood, Q(T)/Qmean, are plotted versus the return period, T, or equivalently the
reduced variate, y = -ln(-ln(1-1/T)). Regression equations are developed to relate the
mean annual flood, Qmean, and other statistical moments to the basin properties such as
basin area, A. The regional equations obtained for the Philippine rivers are also
compared with similar equations developed in other countries of the Asia-Pacific region.
Computations of probability weighted moments (PWMs) yield graphs of L-moment
ratios which give indications of distribution functions such as generalized extreme-value
distribution (GEV) which are expected to give a good fit to the data samples.
INTRODUCTION
Regional flood frequency analysis has long been recognized as useful in providing
statistical relationships for the transfer of flood frequency information from one river
basin to another within the same homogeneous region, in order to augment data and
improve estimates of annual flood magnitudes in the latter basin. This paper describes the
regional study made on the annual maximum flood series of selected Philippine rivers
situated in two water resources regions in northern Luzon, the largest island of the
Philippines.
Water Resources Regions 1 and 2 in northern Luzon are here selected for the study
for three main reasons: first, the two regions are located in the intense-rainfall and
flood-prone areas with one of the highest frequencies of tropical cyclone passage (45-70
times in the period 1948-2000); second, the regions include four of the country’s major
river basins (RBs): Laoag RB - 1353 km2, Abra RB – 5125 km2, Abulog – 3372 km2,
and the country’s largest, Cagayan RB – 25,649 km2 ; and third, post-WWII streamflow
records exist for 29 river stations inside the regions with sufficient lengths (15 to 40
years) over a wide range of catchment areas (36 to 4800 km 2). The annual flood series
1
2
data are taken from NWRC [1] and DPWH [2]. In addition, the World Catalogue of
Maximum Observed Floods (IAHS Publication 284) includes Cagayan River (A = 4,244
km2) with a maximum flood record of 17,550 m3/s.
Figure 1 shows a Philippine map with 20 major river basins located in the 12 water
resources regions, including Regions 1 and 2 considered in this study.
Figure 1. 20 major river basins and 12 water resources regions of the Philippines
3
Past regional flood frequency analyses in the Philippines, using traditional
method of moments and regression analysis of moments versus basin properties
(such as catchment, area, channel slope, soil type and land-use/cover factors) were
done by researchers and consultants in several regions of the Philippines. Lack of
space in this paper prevents a detailed survey and discussion of these past studies.
STATISTICAL ANALYSIS OF ANNUAL FLOOD DATA
Method of Moments
The moments of annual flood data, {Qk , k=1,2, 3,…n}, are estimated as follows:
Mean: Qmean = (1/n)  Qk
(1)
Standard Deviation: S = [ 1/(n-1)  (Qk - Qmean) 2 ] 1/2
(2)
Coefficient of Variation: Cv = S/Qmean
(3)
Skewness Coefficient: Gs = n/[(n-1)(n-2)]  (Qk - Qmean) 3 / S3
(4)
Tables 1 and 2 show the summary of moment computations for 15 river stations and
14 river stations in Regions 1 and 2, respectively.
Table 1. Summary of annual flood data and moments for 15 stations in Region 1
Station
A, km2
n
Qmean
S
Cv
Gs
Qmax
Laoag
1355
19
4849
3370
0.6950
0.0417
11345
Bonga
534
33
1162
1012
0.8712
1.4819
4392
Gasgas
73
34
194
220
1.1355
2.2210
1041
Abra
4813
20
4477
2772
0.6192
1.0217
10846
Tineg
664
21
1317
882
0.6697
1.3190
3951
Abra-2
2575
19
2976
1245
0.4183 -0.7503
4542
Sinalang
120
19
496
417
0.8420
1.5274
1151
Sta.Maria-1
67
18
43
60
1.3920
3.3382
261
Sta.Maria-2
123
18
75
74
0.9821
2.3422
316
Bucong
49
27
158
130
0.8216
0.9360
476
Buaya
195
35
543
464
0.8539
1.5592
1950
Maragayap
36
40
288
130
0.4529 -0.3652
496
Baroro
129
17
395
323
0.8180
1.3756
1321
Naguilian
304
34
1160
718
0.6194
1.3004
3632
Aringay
273
35
478
297
0.6209
0.8350
1082
4
Table 2. Summary of annual flood data and moments for 14 stations in Region 2
Station
A, km2
n
Qmean
S
CV
Gs
Qmax
Baua
103
19
280
167
0.5957
1.4340
777
Banurbor
112
24
61
15
0.2382 -0.9215
Abulog
1432
18
2815
1258
0.4469
0.5102
5120
Sinundungan
189
16
432
264
0.6096
0.6028
961
Matalag
655
15
382
314
0.8220
1.5155
1195
Pangul
312
21
824
1164
1.4122
1.6807
4014
Pinacanauan
655
23
1000
638
0.6385
1.1851
2776
Casile
195
24
131
65
0.4967
0.0693
241
Mallig
563
24
426
247
0.5799
0.5679
1000
83
Siffu
686
22
423
223
0.5257
0.8948
997
Magat
4150
24
2688
1596
0.5937
0.7232
6795
Magat
1784
18
674
425
0.6302
0.4307
1541
Matuno
558
22
436
244
0.5593
1.0598
638
Diadi
196
23
153
144
0.9402
2.4532
663
Flood Index Method: Regional Growth Curves
The flood index method is applied wherein the scaled data of annual flood values divided
by the sample mean annual flood, Q(T)/Qmean, are plotted versus the return period, T, or
equivalently the reduced variate, y = -ln(-ln(1-1/T)) = -ln(-ln F)) where F=Fx(Q) equals
the cumulative distribution function or CDF of the annual maximum flood. The curves
obtained are also called “regional growth curves”, which may be lumped or averaged into
a general form:
Q(T)/Qmean = f(T)
(5)
where the form of the regional function f(T) depends on the regionally fitted CDF. For
example, if the fitted CDF is extreme-value Type I (EVI or Gumbel), then f(T) is a
straight-line function of the reduced variate, y = -ln(-ln(1-1/T)), otherwise it is a curved
function for other types of CDF.
Figure 2 provides the empirical plots of the regional growth curves for Regions 1 and
2. The coordinates for the reduced variate, y = -ln(-ln F)), were calculated using the
Gringorten plotting position, F = (j-0.44)/(n+0.12), corresponding to the jth annual flood
value, Qj , in increasing order.
5
Figure 2. Empirical plots of the regional growth curves, Q(T)/Q mean = f(T), as a function
of the reduced variate, y , or else return period, T, for Region 1 (left) and Region 2
(right)
Alternative forms of the reduced variate, y, which require the estimate of the shape
parameter k of the fitted distributions, also yield theoretical straight growth curves:
Generalized Extreme Value (GEV): Q(T)/Qmean versus y = { (1- (- ln F)k }/ k
Generalized Logistic (GLO):
Q(T)/Qmean versus y = [1 - {(1-F)/F}k ] / k
Generalized Pareto (GPA):
Q(T)/Qmean versus y = {1 - (1- F) k} / k
(6)
Regression Equations for Moment Estimates
Once a regional growth function, f(T), is fitted, then quantiles of Q or the T-year flood
estimates, Q(T), may be computed from the regional relation Q(T) = Q mean f(T),
provided that a regression relation between Q mean and basin properties such as basin area,
A, are developed. In the present case, the following regression relations are developed:
Mean, Qmean = C Ab :
(7)
Region 1:
Qmean = 5.29 A0.8388 with R = 0.8729 and no. of stations = 15.
Region 2:
Qmean = 3.37 A0.7987 with R = 0.8105 and no. of stations = 14.
Regions 1 and 2: Qmean = 5.90 A0.7628 with R = 0.8063 and no. of stations = 29.
Standard Deviaton, S = C Ab :
(8)
Region 1:
S = 6.92 A0.7392 with R = 0.8835 and no. of stations = 15.
Region 2:
S = 1.65 A0.8326 with R = 0.7259 and no. of stations = 14.
Regions 1 and 2: S = 6.06 A0.6911 with R = 0.7350 and no. of stations = 29.
Skewness Coefficient vs. Coefficient of variation, Gs = a Cv + b :
(9)
Region 1:
Gs = 3.7310 * Cv - 1.7257 with R = 0.9084 and no. of stations = 15.
Region 2:
Gs = 2.1910 * Cv - 0.5506 with R = 0.7434 and no. of stations = 14.
Regions 1 and 2: Gs = 2.8995 * Cv - 1.0418 with R = 0.8311 and no. of stations = 29.
6
Figure 3 shows the graph of the regression line and the scatter data for the mean
flood, Qmean , versus drainage area, A. for the combined Regions 1 and 2. Also plotted in
Figure 3 are the maximum recorded floods versus area for comparison with the mean.
Figure 3. Regression line and scatter data for the mean flood, Q mean , versus drainage area,
A. for the combined Regions 1 and 2, including plots of maximum recorded flood.
COMPARISON WITH OTHER ASIA-PACIFIC RIVERS
The present results for the regression of mean annual flood and standard deviation versus
catchment area in Regions 1 and 2 in the Philippines are compared in Tables 3 and 4 with
the results for the other Asia-Pacific rivers, obtained by Loebis [3]. The great differences
in the values of the coefficients between countries may be explained by possible large
variations in basin and channel slopes, effective rainfall, and other basin properties which
also affect maximum flood magnitudes but are ignored in the regression functions of
drainage area alone.
7
Table 3. Mean annual flood versus drainage area in the Asia-Pacific region
For Qmean = C Ab
C
b
R
Countries
Loebis [3]:
Australia (6 rivers)
1.58
China (9 rivers)
0.92
Indonesia (8 rivers) 30.12
Japan (9 rivers) 50.05
Korea (9 rivers)
2.50
Laos (6 rivers) 10.53
New Zealand (4 109.81
rivers)
Thailand (5 rivers) 70.97
This study:
Philippines (29 rivers)
5.90
0.81
0.85
0.40
0.50
0.80
0.56
0.82
0.94
0.75
0.52
0.53
0.80
0.74
0.82
0.26
0.65
0.76
0.81
Table 4. Standard deviation versus drainage area in the Asia-Pacific region
For S = C Ab
C
b
R
Countries
Loebis [3]:
Australia (6 rivers)
1.21
China (9 rivers)
4.40
Indonesia (8 rivers) 205.38
Japan (9 rivers) 89.26
Korea (9 rivers)
1.97
Laos (6 rivers) 13.05
New Zealand (4 16.28
rivers)
Thailand (5 rivers)
0.57
This study:
Philippines (29 rivers)
6.06
0.83
0.61
-0.04
0.36
0.81
0.46
0.47
0.93
0.68
0.04
0.53
0.74
0.72
0.81
0.68
0.62
0.69
0.74
APPLICATION OF L-MOMENTS
Computations of probability weighted moments (PWMs) yield graphs of L-moment
ratios which give indication of distribution functions such as generalized extreme-value
distribution (GEV) which are expected to have a good fit to the data samples (Hosking
[4]). Figure 4 is an example of a L-moment ratio diagram applied to the flood data of 29
river stations of Regions 1 and 2 in the Philippines. The scatter of data tends to group
around the curves of the distribution functions GEV (generalized extreme-value), LN3
(log-normal), or PE3 (Pearson Type 3).
8
Figure 4. An L-moment ratio diagram for annual flood series of 29 river stations in
Regions 1 and 2 of the Philippines. Legend of theoretical moment-ratio curves: GEV
(generalized extreme-value), LN3 (log-normal), PE3 (Pearson Type 3).
The probability weighted moments are computed from data , Q j , in ascending order:
b0 = (1/n)  Qj ; br = (1/n)  (j-1) (j-2) … (j-r) Qj / [(n-1) (n-2) … (n-r)]
(10)
Afterwards, the first L-moments are obtained from the equations
L1 = b0
L3 = 6 b2 – 6 b1 + b0
L2 = 2 b1 – b0
L4 = 20 b3 – 30 b2 + 12 b1 – b0
(11)
The L-moment ratios are defined by
L-CV:
t2 = L2/L1
L-Skewness:
t3 = L3/ L2
L-Kurtosis:
t4 = L4/ L2
(12)
CONCLUSION
This paper has described and demonstrated the regional flood frequency analysis for
Philippines rivers, using ordinary moments, flood index method, regional regression
relations for moments and drainage areas, and L-moment ratios for identifying regional
distribution functions.
REFERENCES
[1] NWRC, Philippine Water Resources Summary Data, Vol. 1 (1980).
[2] DPWH-BRS, Philippine Water Resources Summary Data, Vol. 2 (1991).
[3] Loebis, J., “Frequency analysis models for long hydrological time series in Southeast
Asia and the Pacific region”, FRIEND 2002 - Regional Hydrology: Bridging the gap
between Research and Practice, IAHS Publ. No. 274, (2002), pp 213-219.
[4] Hosking, J.R.M., Fortran routines for use with the method of L-moments, Version
3.03, IBM Research Report, RC 20525 (90933), (2000).