REGIONAL FLOOD FREQUENCY ANALYSIS FOR
SMALL NEW ZEALAND BASINS
1. MEAN ANNUAL FLOOD ESTIMATION
A I McKerchar
formerly: Hydrology Centre, P OBox22037, Christchurch
now: NIWAR Freshwater, Box 8602, Christchurch.
ABSTRACT
One hundred and forty New Zealand basins with areas of less than 100 square
kilometres were used to investigate regional methods for estimating mean annual
floods for small ungauged basins. Besides the usual sample estimate of mean
annual flood, data on slope, soil and hydrogeology were processed from the
New Zealand Land Resources Inventory for each basin. Other variables used
in the prediction of mean annual flood were basin area, and three rainfall statistics
for each basin: l-hour and 24-hour 5-year return period intensities, and mean
annual totals. Multiplicative regression models using catchment characteristics
were less precise than McKerchar and Pearson specific mean annual flood contour
maps.
INTRODUCTION
Regional flood frequency analysis using the flood index method has two parts:
estimation of mean annual flood (index flood) and estimation of dimensionless
flood frequency growth curves. This paper addresses mean annual flood estimation
for small New Zealand basins and Pearson (this volume) addresses dimensionless
flood frequency.
Estimation of mean annual flood for basins where there are no records continues
to be a weakness of regional flood frequency methods, despite efforts to improve
estrmators-
Beable and McKerchar (1982) developed a set of regression equations of the
form
q-=aAoP"Id,
where: Q- is mean annual flood
A is basin area
P is mean basin rainfall
I is 24-hr rainfall intensity wilh a 2-yr return period
a,b,c and d are constants
Each equation was applicable for a "region" where regions were defined in terms
of geographical differences, but adjusted to fit groups of adjacent basins where
estimation errors for an overall equation were consistently either negative or
positive. Factorial standard error of estimate for these equations was in the
range l.2l to 1.47, and equivalent to errors where Q* is estimated from one
64
65
to five years of record.
These error statistics are likely
to be underestimates
because they do not include uncertainty about the placement of regionsMosley (1979) used morphologic reach and cross-section variables to estimate
Q. for 73 South Island rivers. The factorial standard error for the best equation
using all 73 rivers was 1.71. This reduced to 1.52 for 63 non-braided rivers.
McKerchar and Pearson (1989) drew contour maps to use to estimate Q-/
(division by A0 8 removes the effect of basin area) for any basin. The maps
were developed using data from 343 basins. Proportional errors of estimates
from the maps were defined by:
Au'8
E
= t00 (Q., _ Q..)/Q*,.
where: Q.. is the "regional" mean annual flood estimate inferred from the map,
and Q,,, is the "at-site" estimate of mean annual flood, i.e. the estimate of Qfrom the flood series.
E values showed a mean of
8.6%a,
(i.e. the bias) and a root mean square error
of -1557a
E values ranged from 697o to 578%. Nineteen sites, (i.e. 570 of sites) with
E exceeding 7070 were discarded as outliers. Statistics for the remaintng 95c/o
of sites were bias = -0.970 and root mean square error = t227a, so that the
standard error of estimate of Q. from the maps was 1227a. This was an
improvement over the Beable and McKerchar equations, which also discarded
abottt 5Vo of the catchments as outliers.
Results were more variable for small basins than for larger basins. For all
basins with areas less than l0 km2, McKerchar and Pearson obtained the
following:
All
basins
(10
No- Basins
Bias (Va)
Root Mean
Square Error
49
22
92
km2
.(
/
I
. z.t tz's
43
Outlìers excluded
.ro"r" /
¿\
34
-0.7
/
,4:
Outline of problem
The error statistics
["r*
\ y'-t
73so1
74360.
.ii
for'all
basins
(10
about the reliability of the method. Also, the outliers are disproportionately
represented among the basins with area (10 km2- Are the six records all in
error, or are they conveying information about hydrological conditions not
adequately represented by the contour maps? For example, the scale of the
maps and the level of smoothing in their contours may mask much real variation
between small basins. Can additional information be employed
regional estimates of flood statistics?
to
improve
100
200 km
4
/
,4701
,.80201
rt
km2" are unacceptably large for design
flood estimation. With "outliers excluded" the error statistics are acceptable,
but the question of what is an outlier is not resolved, and this raises doubt
0
/
.Asazl
t")
FIG.
I
140 small (A < 100 km2) New Zealand basins used in this study.
Site numbers from Walter (1990).
Location of the
The additional information available in the NZ Land Resource Inventory
(Ministry of Works and Development, 1979) includes broad classes of rock type,
overland slope, vegetation and soils. This information can be extracted from
a database for any basin boundary, and this study sought to use it to improve
estimates of mean annual flood for small basins.
66
67
\
Areally weighted mean slope is calculated
METHOD
Data compílation
To assemble sufficient data for comprehensive analysis the area threshold was
raised from l0 to 100 km2. Data for 140 basins with areas less than 100 km2,
and at least six years offlow records were assembled (Fig. l).
The data assembled for each basin were:
(Q.):
quantity is the arithmetic mean of peak
Mean annual flood
This
-discharges
for each of at least six years of record.
These data were extracted
from streamflow archives up to and including 1989 (Tideda: Rodgers and
Thompson, l99l). Length of record (N years) is given beside the mean. Where
S = 1.5 S"+ 5.5 Su+
ll.5
S"
as:
+ 18
S¿
to discharge, is problematical. Most of these small basins are fitted
with weirs or bed-control structures calibrated using laboratory data. Where
freld calibration is limited to low flows, it is difhcult to determine whether
the adopted ratings are correct, especially at flood extremes beyond the range
of conventional laboratory calibrations. This implies that annual maximum
flood peak estimates for these basins are less reliable than those for basins
with higher-stage current-meter gaugings (Potter and Walker, 1985).
Area (A): Basin area is defined on the NZMS 260 (l:50000) or NZMS
-I (l:63360)
map series, and listed in Walter (1990).
Rainfall intensity (Ir and Iz+): Five-year return period rainfall depths for
-l-hr and 24-hr duration storms were estimated from maps in Tomlinson (1980).
These depths are areal means for each basin, but are not adjusted by areal
reduction factors. In the Tomlinson maps Ir is estimated from a sparse network
of recording raingauges, whereas Izq is estimated using a much more
comprehensive set of daily-read manual raingauges, and is more reliable.
Mean annual raìnfall (P): An areal mean annual rainfall was estimated
from the map of 195l-1980 normals in NZ Met. Serv. (1985).
Hydrogeology (H): Derivation of a basin estimate of "hydrogeology" is
-explained
in Hutchinson (1990). In summary, each combination of rock types
in the Land Resource Inventory is assigned a scalar number ranging from
I for low to 8 for high bedrock infiltration capacity and transmissibility. Low
values are assigned to strongly indurated sedimentary rocks, and igneous and
metamorphic rocks. Medium values are assigned to pyroclastics, loess, crushed
argillites, "soft" volcanics, and weakly indurated sediments. High values are
assigned to ashes, brecci4 scoria, lapilli, alluvium, colluvium, glacial till, peat
and wind-blown sand. A basin mean is calculated as a sum weighted by the
proportion of basin area occupied by each class.
Slope (S): The Land Resource Inventory classifies parcels of land into
-seven
slope classes 4...G defined as:
pers. comm-,
-
l99l):
Soit drainage (D): a scalar ranging from
I to 7, I
representing very poor
esenting excessive drainage.
MacroporositY determined in the
particular soil horizon. A dePthof macroPorositY for the whole
horizon. Typically macroporosities are in the range 0 to
30%0.
Minimum porosity (MP): Minimum porosity for each soil class
-estimated.
was
Area-weighted estimates of these three quantities were calculated'
All the data are assembled in the Appendix, and summary statistics are given
in Table
1.
TABLE l-summary of basin characteristic statistics of
140 basins listed in
Appendix.
CHARACTERISTIC
MINIMUM
MAXIMUM
MEAN
STANDARD DEV
CHARACTERISTIC
A
I:
(m3/s)
(ha)
(mm)
0.054
389.2
1.420
9963
128
49.3
3061
34
69. r
2891
l3
S
D
Q.
H
l3
(deg)
MINIMUM
1.0
1.5
r.0
MAXLMUM
8.0
4.4
2.5
35.4
t9.7
7.0
4.3
8.0
t.l
MEAN
STANDARD DEV
lro
(mm)
45
660
t54
72
DWP
(70)
t.2
29.3
13.9
6.1
P
(mm)
550
I 1700
1968
l30l
MP
(Va)
0.0
26.0
8.7
5.0
Analysis proposed
Multiple regression is proposed for the analysis using the form
A
0-3o, flat to gently undulating
B - 4-7", undulating
- 8-15', rolling
C
D- 16-20", strongly rolling
E
21 25o, moderately steep
F - 26-35o, steep
- >35', very steep
G
-
Se
where Su, S¡, ... Se are the decimal proportions of the catchment area occupied
by ground in the slope classes A, B, ... G respectively.
Soil Properties: A soil description applies to each Land Resource Inventory
-parcel.
Three soil properties considered to affect rapid runoff were (MJ Duncan,
possible the stage series are checked for consistency against records for adjacent
basins. Reliability ofthe rating curves, which are calibration curves to transform
stage
+ 23 S"+30.5 S¡+ 38
Q*=aXrbXz"...
where: X:, Xz -.. are basin characteristics (e.g. Area A, mean annual rainfall
P, hydrogeology H, etc), and a, b, c ... are coefficients determined by multiplicative
regressron.
68
69
Analysis with subsets of the data is proposed, \,ith the subsets defined as:
a) arbitrary splits of the data, e.g. into four sets with high rainfalt P and high
minimum porosity MP, high P and low MP, low P and high MP, low P and
low MP; with the high/low splits for P and MP taken as the median value.
b) Clusters of data using the algorithm K-means of SYSTAT (Wilkinson, I986),
which splits data into groups such that between-group variation is maximised
and within-group variation minimised. Variation is measured as the Euclidean
distance between data for a basin and the group mean. These methods avoid
splits based on loose geographic regions.
RESULTS
Data statistics
The compiled data statistics (Table l) reflect the very large range of hydrological
conditions encountered across the country.
For example, mean annual rainfalls range from 550 to 11700 mm/yr, mean
basin slopes from 1.5 to 35 degrees, depth-weighted macroporosity ranges from
to
minimum porosity ranges from ÙVo to 26.6%.
Hydrogeology and drainage, both measures ofinfiltration, are near their upper
bounds for volcanic ash-covered Central North Island basins where Q./Au'o
are low. For example for Mangakara (site number 1043434), hydrogeology H
= 7.5, soil drainage D = 6.2, and the porosity measures (DWP = 26.lEo, MP
= 16.870) are amongst the highest for all the 140 basins, and Q./A'o is 0.36
(Q* in m3/s, A in km2) . In contrast, for two basins in Northland (Pukewaenga,
46662 and. Pukeiti, 46663) intensity lz+= 155 mm is almost identical to Mangakara
(Iz+ = 150 mm), but Q./Ao' = 5.0 and 5.4 respectively, and for both basins
hydrogeology H = 4.0, soil drainage D = 1.0, depth-weighted macroporosìty
DWP = 2.1/s and, minimum porosity MP = o.lTa. The low values of the latter
three parameters reflect the poor drainage and low porosities of the heavy clayey
soils in these basins.
Another basin with unusually low values for Q-/40'8 is Maryburn (71122)
in the central South Island. Here Q./N8 = 0.17, and the intensity estimate
is Iz¿ = 80 mm. Hydrogeology H = 8.0, soil drainage D = 5.0, depth weighted
macroporosity DWP = 20.Wa and minimum porosity MP = 15.5Va, are all greater
l.2Vo
29.3V0, and
than the respective means across all basins (Table I ). This basin has been discarded
as an outlier in previous studies.
For many basins at the extremes of the dataset, there is an inverse correlation
between the flood parameter Q-/Ao't, and the Land Resource Inventory
derived quantities, soil drainage D, depth-weighted macroporosity DWP and
minimum porosity MP. This study seeks to verify this correlation across a large
sample of data, and to use it to provide better estimates of Q. for ungauged
basins.
Correlations
For logarithms (base l0) of data, correlation coefficients (Table 2) are generally
positive. Correlation of log (Q.) with area is notable, but otherwise its correlations
are weak. Correlations between the three rainfall parameters and between the
three soil parameters are also notable. These suggest that just one rainfall and
one soil parameter may be sufficient as independent variables.
70
TABLE 2-Correlation matrix for logarithms of basin characteristics.
lr
Q,
a.
A
Izq
H
S
D DWPMP
l.0oo
0.879
1.000
P
0.500 0.262 1.000
0.372 0.062 0.754
\
lzq 0Á41 0.160 0.804
H
-0.131 -0.006 0.060
s
0.082 0.1l0 0.054
D
0.064 0.219 0.151
DWP 0.116 0.287 0.204
MP 0.145 0.298 0.155
l.000
0.883 1.000
0.076 0.094 1.000
4.052 -0.032 -0.431 1.000
-0.011 0.057 -0.017 0.248 1.000
0-025 0.103 0.064 0.209 0.876 1.000
0.032 0.109 -0.001 0.245 0.895 0.884
1.000
Regression analysis
Table 3 gives a selection of regression results using all the data. The results
given are the best result, in terms of minimum standard error, for l, 2, 3, and
4 variables. The standard errors and factorial standard errors, ofsimilar magnitude
to those reported for country-wide equations in Beable and McKerchar (1982),
are too large for the results to be used for design flood estimation.
TABLE 3-Multiplication regression results for prediction of Q. using all
140
"t"
is Student's statistic; "Rz" is coefficient of determination;
and "s.e." is standard errorbasins.
No.
Variable
Variables
Name
Coef.s.e.tRR2
b,c,d
s.e
of
. Factorial Const Multþlier
s.e. log a
a
coef
I
2
A
0.856
A
lzo
A
Izt
217 0.879 0.713
2.57
0.808 0.031
r 559 0.160
26.2 0.930 0.866 0.3t7
2.07
-3.029
9.35*10-a
0.805
281 0.9M 0.890
0.288
1.94
-2.961
10.9"10
0.257
r.8r
-2.449
35.ó"10-¡
0.028
l.ó35 0.r46
H
4.422
r.90
9.7
a
1.2
0.017
-5.5
A
0.848
0.026
325 0.956 0.913
II¿
1.679
0.t3t
129
H
{.397
{.606
0.069
-58
0.102
-60
DWP
0.279
0.4n
0.040
The data set \üas partitioned in a number of ways, including splitting the data
into one of four groups depending on whether mean rainfall P and minimum
porosity MP were above or below their median values. The most promising
was to split between "high infiltration" and "Iow filtration" with low infiltration
basins arbitrarily defrned as having hydrogeology H ( 7 and soil porosity DWP
'11
1
llTa. Standard error for 44 "low infrltration" basins was +0.184, but this
error is still too large for the results to be of practical use (factorial standard
error of 1.53).
Other analyses
Other analyses, including cluster analysis, and use of an "infiltration" parameter,
a sum of standardised values of D, DWP and MP, did not improve predictive
capability.
In the meantime, the McKerchar and Pearson contour maps are recommended
for estimating mean annual flood. Examples of robust regional flood frequency
procedures for small basins are given in Pearson (this volume).
APPENDIX. Basin characteristics for
Site Q.
A
(mr/s)
DISCUSSION
the Land
an annual
in Beable
This hope
802 r.26
250
1909 64.55 2858
3506 53.60 1110
4901 61.09 1250
5513 3.09
63
6004 170.31
5977
18.53 813
26.34 957
31.15 1108
6501
7202
7604
7805 tt8.2r
7811 2t.50
8240
1196
30
69.15 4072
42.24 792
17.89 5713
23.94 7389
34.67 6880
20.21 7598
8203 r.48
flood series is too large to obtain predictive equations.
After area, storm rainfall intensity, quantified by the frve-year return period
24-hour duration rainfall from Tomlinson's (1979) maps, clearly remains the
most important variable, but for small basins where critical storm duration is
much lesi than24 hours, this statistic is limited. The l-hour storm rainfall statistic,
based on a much smaller sample of records from automatic raingauges, is
apparently less accurate and less useful.
Parameters which characterise the hydrogeology, slope and infiltration
properties ofbasins can be obtained from the NZ Land Resource Inventory.
The analysis undertaken could not devise predictors using these parameters
for mean annual flood for small basins that were improvements over the contoured
maþs in McKerchar and Pearson.
Two further lines of enquiry will be investigated for small basin mean annual
flood estimation in future:
and geomorphological
2. More deterministic approaches using physically-based rainfall-runoff models
in conjunction with small-scale thunderstorm rainfalls.
12
14610
14625
14627
I0l464t
1014645
2.02
8l
1014646 1.53 92
15453 43.88 4505
RECOMMENDATIONS AND CONCLUSIONS
1. Continued statistical approach with basin geometry
variables used in the multiplicative regressions.
8604
9228
15534 t.86
19734 37.51
267
3050
t9779 5.48
398
21410 56.62 5029
21601 45.91 2141
22901 21.68 1844
23005 1.33
52
23209 t0.22 2339
23210 59.25 4373
23220 83.60 8460
29242 t11.34 4025
29244 31.38 3632
29246 282.00 7s78
29250 32.8t
lz¿
p
(ha) (mm) (mm) (mm)
5515 0.642 t6
5516 0.756 13
5519 51.80 1390
data inferred from Land Resource Inventory (slope, hydrogeology, soil drainage,
\
140 basins with areas less than 100 km2.
1557
S
31 120 t200 8.000 12.500
34 135 1780 1.00 24.633
38 140 2320 1.000 8.540
50 200 1900 t.000 18.900
46 190 1800 1.000 18.000
46 190 1800 1.000 18.000
46 190 1800 1.000 18.000
46 190 1700 1.020 15.700
44 180 1470 3.713 18.793
42 195 1600 5.040 24.260
32 135 1500 4.404 10.128
30 125 1290 3.t27 9.020
29 130 1400 5.180 10.700
30 125 1320 5.756 8.206
34 120 1250 8.000 1.500
38 r50 1300 2.170 24.489
44 150 1730 3.308 33.203
40 160 1670 6.159 14.952
q 160 1500 7.008 19.679
42 200 2470 6.000 19.835
40 190 1920 6.238 20.013
40 160 1600 8.000 5.500
40 160 1500 8.000 5.500
42 200 1800 4.660 30.947
q 150 1320 8.000 23.233
3l t60 1750 4.909 24.085
28 150 1500 4.040 22.768
28 155 2020 5.299 26.510
26 165 ló60 4.092 27.930
28 170 1030 4.492 20.273
28 165 2400 6.000 23.000
22 95 900 2.566 16.746
36 160 1390 4.784 t7.1t6
34 150 1400 4.987 17.668
48 t60 2520 1.510 31.663
n rc5 n40 3.980 24.91t
56 200 3t40 1.477 34.953
33 140 1950 r.057 28.550
73
D DWP MP
(Vò
(deÐ
(Vo)
4.9 21.3 14.4
4.3 10.4 8.3
4.4 9.7
4.3
4.0 10.0 7.6
5.0 13.0 t2.t
4.0 10.1 7.6
4.0 l0.l 7.6
3.6 9.0
3.6 10.9
3.7 10.0
2.0
1.0
2.3
2.9
6.4
t.2
5.4
7.1
4.0 l3.l
4.5 I 1.8
4.2 I1.0
6.7
6.8
6.7
3.0
0.8
2.6
2.7
5.9
9.8
8.1
4.7 22.1 14.6
6.4 26.7 21.6
6.0 20.0 15.7
5.7 28.3 17.0
7.0 29.3 26.6
7.0 29.3 26.s
6.9 28.2 26.3
6.0 19.6 16.5
5.4 20.7 r4.4
3.0 5.9 2.3
3.8 16.9 8.8
3.1 t4.l 6.5
3.1 8.6 3.5
6.0 22.1 t5.l
4.0 I1.5 6.4
2.9 r2.2 5.9
4.6 14.6 8.6
5.0 15.7 9.3
3.5 11.2 6.0
5.0 r4.8 9.1
5.0 16.7 9.1
APPENDIX Continued. Basin characteristics for 140 basins with areas
less
APPENDIX Continued. Basin characteristics for 140 basins with areas
than 100 km2
Site Q.
A
l,
Izo
DDWP
S
(Vo)
(dee)
P
(m3/s) (ha) (mm) (mm) (mm)
29254 330.17 7875
29259 0.219 23
29501 81.17 230s
29605 76.36 7973
29808 28s.09 8724
29841 ó9.00 4384
29843 85.28 3795
30510 0.054 5
305t1 0.085 1
30516 7.42 910
3070r 38.11 4469
30802 53.63 3847
32001 18.23 ló80
t032517 99.50
5660
103255s 219.s7 5'730
1232564 71.68 6230
32734 7.01
t542
3273s 33.32
32754 65.72
6158
9950
33114 3.57
531 I
.72
3278
2063
33 r
r5
17
33117 27.47
33307 47.15 8184
33347 28.07 2'714
34308 149.50 8463
35004 73.01 4960
35006 27.69 2000
35201 68.52 4100
35506 104.87 5960
3600r 35.91 3098
38002 r83.86 4128
38401 55.14 2492
3850t 103.73
3912
38904 16.62 2007
39201 329.50 5910
39402 61.94
39403 81.46
4903
3'778
395A1 174.74 '7734
39508 53.90
395r0 67.36
1924
1092
395¡l
94.70 t844
40'703 4.31 l41l
4r30r 5l-64
9510
56 180 3180 1.489
24 90 r000 4.030
29 150 3010 1.577
26 120 1980 1.924
48 200 3210 1.000
25 I t0 1830 1.422
28 135 3080 1.004
20 r00 l 150 1.000
20 100 l 150 2.000
22 r00 II80 2.341
23 l0 r 260 2.506
26 125 1510 2.622
22 80 I 150 2.951
28 I r5 1ó00 1.800
30 t90 2200 2.104
28 130 1200 4.114
32 r60 2800 6.480
2A 80 980 4.669
22 85 I 100 4.240
24 100 1360 .431
24 U5 1520 5.941
24 145 2390 6.389
26 I l5 2650 6.665
30 t25 2990 5.613
36 280 2580 7.773
32 200 2680 7.2s9
32 180 2200 7.814
32 200 2400 7.692
36 200 2200 7.466
q 200 2490 4.766
64 300 5010 4.866
M ltj 2610 '7.230
42 t70 2760 6.985
36 150 20ó0 7.662
64 280 4320 6.59'7
48 180 2220 7.849
48 240 2730 7.415
44 260 3230 7.190
56 320 3920 6.162
60 280 4830 6.408
@ 300 4650 6.557
26 t20 1940 ó.303
26 135 2400 4.937
1
7
74
MP
(%)
35.165 5.0 14.4
22.400 2.0 9.6
29.666 5.0 r5.l
23.422 3.6 13.3
34.928 5.0 t4.7
25.903 4.4 ló.5
9.0
32.000
9.4
5.00
23.000 5.0
r7.300 5.0
11.481 4.5
19.223 3.1
19.891 3.7
18.900 4.1
26.600 4.9
28.900 4.9
20.000 4.5
20.021 6.2
8.905 2.9
26.500 3.'7
r0.916 5.1
22.030 4.8
10.199 4.9
4.5
9.0
7
.1
9.2
8.7
15.4
8.1 5.7
8.1 5.7
9.7 7.6
7.4 5.6
9.7 6.5
12.5 1.9
t4.5 9-3
t5.4 9.I
15.0
25.0
8.9
21.2
10.7
1.3
t0.l 6.1
25.2 9.8
r3.9 8.6
t5.9 8.7
l6. t l2 5.4 25.5 l2.l
24.301 5.3 23.0 ll.4
5.25'7 4.6 15.6 r L3
l0 4.2 13.0
7.890 3.5 9.5
6.210 4.3 r3.9
5.590 4.5 14.6
7.877 3.8 l2.l
24.136 3.6 8.5
8.1
20.160
19.734
12.463
15.553
10.636
6.372
3.9
3.7
3.6
4.0
4.7
4.3
8.95
4.5
r
10.694
4.1
14.794
19.484
25.470
4.3
4.0
4.3
4.2
9.446
9.8
8.9
10.0
10.2
14.5
9.3
6.2
8.9
9.2
7.4
5.9
7.4
6.7
7.6
6.7
9.7
7.7
A lr l:¡ P
S
D DV/P MP
(m'/s) (ha) (mm) (mm) (mm)
(dee)
(Va)
(Vo)
41601 6.62 879 26 r05 1590 3.702 23.433 3.6 9.5 4.4
t043434 4.16 2159 32 150 1570 7.501 19.777 6.2 26.1 I6.8
1043466 38.09 9589
40 170 3170 6.910 15.677 5.2 16.9 tt.4
Site a.
1043476
0.153
5
1t43407 0.599 t69
n43409 0.22'7 34
1143427 2.68 3l I
t143428 3.77 1464
1443462 5.76 999
43602 12.51 1786
43807 23.85 1250
45315 40.32 4646
45702 32.73 82t
45903 2.03
88
1213
46609 56.98
46645 9.28 336
46662 2.34 39
46663 0.r8 r
4'752'7 23.48 r003
48015 75.06 2178
52916 97.27 468t
5ó901 47.46 4659
57014 6r.63 8238
57022 2.'76 514
57023 t.2l 279
s717t 61.ó6 5800
0.130
0.297
57512 0. t99
57402
4
574Q5
7
3
58301 30.02 1725
60104 73.93 6502
62104 14.10 2074
63501 3.34
64606 92.67
64610 35.r3
169
7404
4r91
r.83
12.9
7.8
I
6.8
66208
66405
66603
66604
67601
6.4
6.3
5.3
68602 10.48 5500
69621 16.72 2297
I
1.5
14.8
r
l.r
0.5
14.0
10.7
10.0
less
than 100 km
0.68
1.18
1.42
6.79
260
90
218
326
321
68529 2.'762 6\9
29 lr0 1400 8.000
34 145 1700 7.462
34 145 t700 7.300
26 t40 2200 4.824
28 r20 r390 7.992
32 120 t4t0 6.932
32 r20 1380 7.700
34 120 1280 5.103
29 125 1570 5.440
40 190 1770 4.000
42 175 1400 4.000
44 170 1770 t.324
42 160 1670 4.557
42 155 1550 4.000
42 r55 1550 4.000
34 t55 1160 3-998
34 135 1750 r.000
48 290 37t0 1.476
42 200 2390 r.063
3l 110 1270 7.959
23 90 2150 8.000
23 90 2150 8.000
34 r r0 1000 8.000
33 105 l 100 8.000
33 105 l 100 8.000
33 r95 l 100 8.000
31 165 t460 1.02t
I8
105 950 t .072
28 150 1640 r.500
40 200 1000 1.000
48 210 2300 l-485
34 160 I 150 3-083
18 100 800 8.000
28 105 1ó00 1.000
r5
95 800 3.240
15 95 800 3-650
26 120 1300 1.708
t9 90 r 100 t.900
18 ilO 1000 s.230
17 80 980 2.193
t5
18.000
.0
28.9
18. r
r28 3.I 9.2
4.7
l.500 3.0 6.4
4.2
t
24.t03 5.7 23.1 l5.l
24.900 4.0 9.5 4.6
21.900 4.2 I I -8 5.1
10.
20.277 5.0 19.4 14.8
9.774 5.0 20.t 12.8
8.373 2.6 6.8 5.6
15.029 3- I 8.4 3.5
22.138 2.6 ó.1 5.2
I 1.500 2.0 2-4 1.7
15.978 4.1 10.3 7.0
9.080 4.4 13.9 7.0
l 1.500 r.0 2.1 0.1
I 1.500 l .0 2.t 0. I
17.621 3.2 10.2 4.4
26.188 4.0 8.8 7.4
35.392 5.0 21.7 14.0
28.037 5.0 13.0 9.4
23.s53 4.6 13.2 9.0
23.067 5.0 14.7 l0.l
29.931 5.0 14.7 l0.l
18.268 3.2 6.9 4.0
l
18.000 3.0 6.4 4.2
23.000 3.0 6.4 4.2
28.855 5.0 21.0 13.9
31.100 5.0 l3-9 n.7
29.750 5.0 23.2 18.2
30.905 5.0 14.7 l0. r
29.522 4.5 20.1 l5.l
23.054 3.8 r1.5 9.0
23.000 3.0 8.1 5.4
30.500 5.0 22.5 17.1
28.600 5-0 13.2 8.9
25.600 5.0 13.4 9. r
30.500 4.9 9.5 7.2
29.250 5.0 23.7 t7 .9
12.800 4.6 14.7 I r.3
24.482 4.4 17.8 14.0
APPENDIX Continued. Basin characteristics for 140 basins with areas
REGIONAL FLOOD FREQUENCY ANALYSIS FOR
SMALL NEW ZEALAND BASINS
2. FLOOD FREQUENCY GROUPS
less
than 100 km2
A \
Site a-
lzq
P
H
(m,/s) (ha) (mm) (mm) (mm)
69627 1.39 128 ló 70 850 1.000
71t22 3.9ó 5008 ló 80 950 8.000
71129 22.09
71t78 38.19
73501 44.33
74353
74360
74367
74701
9963
7870
4500
3.01 2406
2.03
2.02
286
58
6.77
959
80201 30.53 7160
87301 389.18 9780
90ó05 28.85 438
90607 193.82 1233
91402 19.13 1660
9t412 0.828 66
93602 143.s7 1964
S
(deÐ
27.900
8.550
24 140 1560 3.927 24.rtl
l8 60 900 1.160 28.820
14 100 ll00 1.330 18.900
13 45 550 1.000 15.711
t6 65 750 3.000 I 1.500
t7 70 lt80 1.000 9.225
t4 60 830 1.ó84 ló.518
19 78 1200 3.61l 17.244
56 440 7200 2.770 29.300
48 220 3400 7.062 14.943
r28 660 I1700 2.393 33.740
a 130 2600 3.250 24.500
28 lr0 2200 8.000 30.500
60 260 5670 1.000 t6.023
D DWP MP
(Va) (%a)
5.0
5.0
5.0
5.0
4.t
4.3
3.0
2.0
4.7
3.2
4.7
4.1
4.0
4.2
5.0
2.7
24.3
20.1
22.0
18.7
8.1
5.6
6.4
8.9
7.9
7.8
5.5
r.0
4.2
5.8
19.9
13.0
18.0
I I .6
8.2
17.7
9.2
14.7 l0.l
t2.3
8.0
REFERENCES
Beable, M.E.; McKerchar, 4.I., 1982: "Regional flood estimation in New Zealand',, Tech.
Rep. 20, rùy'ater and Soil Division, MWD, Wellington. 132p.
Hutchinson, P.D., 1990: "Regression estimation of low flow in New Zealand". pub. No.
22, Hydrology Centre, DSIR Marine and Freshwater, Christchurch, 5 | p.
McKerchar, A.I.; Pearson, C.P., 1989: "Flood frequency in New Zealand',. pub. No. 20,
Hydrology Centre, DSIR Christchurch, 89p.
MV/D, 1970: "Our land resources" Ministry of Works and Development, Wellington.
Mosley, M.P., 1979: "Prediction of hydrologic variables from channel morphology, South
Island rivers". Journal of Hydrology (NZ) 18(2),109-120.
NZ Met. Serv., 1985: "Climate map series I:2000,000. Part ó: Annual Rainfall" Ministry
of Transport, Wellington.
Pearson, c.P. this issue."Regional flood frequency analysis for small New Zealand basins.
2. Flood frequency groups". Journal of Hydrology (NZ).
Potter, K.W., Walker, J.F., 1985: "An empirical study of flood measurement error', ll'ater
Resources Research 2 I (3 ), 403406.
Rodgers, M.W.; Thompson, S.M., 1991: "TIDEDA reference manual,' publ. No. 24,
Hydrology Centre, DSIR, Christchurch, 200p.
Tomlinson, 4.I., 1980: "The frequency of high intensity rainfalls in New Zealand, part
I" Tech. Pub. 19, Water and Soil Division, MV/D, Wellington. 36p and 4 maps.
Walter, K.M., 1990: "Index to hydrological recording sites in New Zeala¡d, 19g9,'. publ
No. 21, Hydrology Centre, DSIR, Christchurch. I8lp.
vy'ilkinson, L., 1986: "SYSTAT: The system for statistics". sysrAT Inc., Evanston, Illinois,
522p.
76
formerly: Hydrology Cent¡e, P O Box
18.1
15.5
15.9
14.6
10.2
10.0
12.4
C P Pearson
22-037
, Christchurch
now: NIWAR Freshwater, P O Box 8602, Christchurch
ABSTRACT
One hundred and seventeen small New Zealand drainage basins with areas of
than 100 square kilometres were used for a regional flood frequency study.
Each basin had annual maximum flood peak series of length l0 or more years.
L-moment statistics of the flood series and basin physical characteristics were
used to classify the basins into six non-geographic flood frequency groups.
Dimensionless flood frequency growth curves for each group offer robust
alternatives to geographical regionalisation and flood contour maps.
less
INTRODUCTION
Floods are arguably the most common and devastating natural catastrophes
(Wallis, 1988). Reliable flood frequency information is required for developments
near rivers and streams as part of their design and protection. Small drainage
basins rarely have waterlevel records and engineers must rely on regional floodfrequency methods to estimate flood exceedance probabilities.
A review ofregional flood frequency in New Zealand was conducted between
1987 and 1989 (McKerchar and Pearson, 1989, 1990), and a design procedure
for estimating flood peak quantiles for ungauged drainage basins was developed
during this review. The procedure prescribes the 2-parameter extreme value type
I _(EVl) distribution for annual maximum flood peaks, in conjunction with
contour maps of two flood statistics, which enable flood quantile estimates to
be derived for any basin, gauged or ungauged, in New Zealand. This approach
is a form of index flood procedure: one map provides mean annual flood estimates
(Q,, "index flood') and the other provides dimensionless I / 100 annual exceedance
probability (AEP) flood esrimates (Qroo/Q*). (AEP = l/T, where T is return
period in years).
For drainage basins with areas less than l0 square kilometres the design
procedure is less precise. This reflects the predominance of larger catchments
with longer annual maximum flood series, with more spatially averaged
hydrological response masking out the individuality of smaller basins with shorter
series; and the larger variability in stage-discharge rating curve extrapolation
associated with smaller basins.
In this paper robust regional flood frequency estimation procedures developed
by Wallis (1980, 1988) are applied to small basins to derive dimensionless flood
frequency growth curyes for groupings of physically similar small basins. The
5-parameter Wakeby distribution fitted by L-moments or probability-weighted
moments is a robust, accurate and efficient regional flood frequency procedure
77
for homogeneous groupings of catchments, (Kuczera, 1982; Hosking et al., 1985;
Wallis and Wood, 1985; Cunnane, 1989). These qualities of Watlis'(1980) regional
flood frequency procedure are preserved even when there is significant correlation
present amongst annual series of a region's drainage basins (Hosking and Wallis,
1988) or when the region is heterogeneous (Lettenmaier et al., 1987). Hosking
and WaÌlis (1991) have developed statistical tests based on L-moments to
investigate the homogeneity of a given group of drainage basins. These tests,
based on the L-moment ratios L-CV, L-skewness and L-kurtosis, may be used
to monitor the homogeneity of selected groups.
The initial problem is to choose candidate homogeneous groups of basins.
Traditionally regional groupings have been used (e.g. Natural Environment
Research Council, 1975; Beable and McKerchar, 1982). Acreman and Wiltshire
(1989) review all approaches, incÌuding the use of groups of physically similar
Ø
o
L
I
-J
o.l
L-Skewness
METHOD
78
0.2
V
basins not necessarily in the same geographic regions. Homogeneous groupings
of small basins are sought in this paper using a method proposed by Wiltshire
(1985) which monitors the effect on flood frequency of splitting basins into
physically similar groupings. Wiltshire's method is advanced by using L-moments
(Hosking, 1990) of flood series to monitor the flood frequency behaviour. Lmoments have been shown to be reliable statistics for discerning differences
and confirming similarities in flood frequency groupings and regions (Wallis,
1988, 1989; Hosking, 1990; Hosking and \üallis, 1990; Chowdhtry et al., l99l:'
Pearson, this issue).
Wiltshire (1985) groups basins first by splitting a set of basins into two groups
using a single partitioning value of a physical characteristic, for example, into
wet and dry basin groups on the basis of average rainfall. Measures of flood
frequency variability are then derived for each group, and aggregated into one
measure, corresponding to the group partitioning value. (Wiltshire used four
flood frequency variability measures based on fitting the generalised extreme
value (GEV) distribution to each group.) This procedure is repeated using a
range of partition values. The optimum grouping is achieved at the basin
characteristic value where the group variability statistic for flood frequency is
minimum. This process can be repeated with other basin characteristics, and
for multiple partitions of the basins.
This study used ll7 small New Zealand basins (i.e. area less than 100 km2),
each with n = l0 or more years of annual maximum flood peaks (see Appendix).
The longest annual series has 29 flood peaks. (fhis flood-set is the subset of
McKerchar's (this issue) set of basins with l0 or more years of record.)
Basin characteristics investigated were basin area (A), and areally averaged
rainfall, soil, hydrogeology and slope statistics. The rainfall statistic (Izo) is the
24-hotr rainfall total of5-year return period (207oan¡tal exceedance probability
(AEP) event) derived from Tomlinson's (1980) maps. Iz¿ ranges from 45 mm
to 440 mm for the small basins used in this study. The soil property is depthweighted-macro-porosity (DWP) estimated from soil survey information by M.
J. Duncan (pers. comm., l99l): it ranges from 1% for impermeable basins to
29/61or porots basins. The hydrogeology index (H) was developed by Hutchinson
(1990) using a national land resources inventory. H ranges from I for low to
0.3
.t)
FIG' l-L-kurtosis and L-skewness for I 17 small New Zealand basins. The GEV distribution
curve to the left of the EVI point is EV3, and to the right of the EVI point
is EV2.
-
8 for high bedrock infiltration capacity and transmissibility. A slope measure
(S) used by Hutchinson (1990) was also used. It is the areally-weighted mean
slope extracted from the national land resources inventory and ranges from
2" to 35" for basins in this study. More detailed information on these basin
characteristics is given in McKerchar (this issue).
The flood frequency variability measure proposed in this paper for use in
wiltshire's basin grouping procedure is based on L-moments of ànnual maximum
flood peak data. This avoids fitting a statistical distribution to the flood data
(at this stage) and takes advantage of the superior properties of L-moments
(Hosking, 1990). The Lskewness-L-kurtosis plane serves as a useful tool for
discerning heterogeneous regions (Wallis, 1988, 1989) and is the basis of the
flood frequency variability measure required for basin grouping. L-skewness and
L-kurtosis are the L-moment ratios À¡/Àz and Àq/Àz respectively, where À. is
the r-th population L-moment (see Hosking 1990, for definitions). Àr is the
mean and Àz is a measure of scale.
Unbiased estimators (l) of À., that are linear combinations of the flood data,
areusedinthispaper(Hosking'rt-tJ"i*31:î".'r'i'ö'P"iK:i¿:iîîåîlåxi:
record-length-weighted average ([l:/ b]*,fh I lzl*) = (0.242,
L-moment ratio estimates for each basin are given in
79
Area A (sq km)
70
80
5.r5
\
(n
,J)
u
!
l
Rainfall L, (mm)
tt
(i
c)
E0
5
,
160
180
200
Rainfall I2a (mm)
FIG. 3-Surface of L-moment variability meâsure SSL in the Izr-s plane, for four-way
fl
a2
tt
50r
a)
5{i
o
t40
r20
-/ /r-
5:4
-o
r00
L-CV = lz/lr alone) and Vu (based on L-CV and L-skewness). These three test
statistics each indicate that the group of 117 small New Zealand drainage basins
is heterogeneous, with an average tendency toward an EV2 distribution (as shown
in Figure l), with shape parameter k = -O.ll (upward curvature on an EVI
|
-
plot).
¡:l
I
FLOOD FREQUENCY GROUPS
Slope S (deg)
sr0152025
5:
\/
30
f -'tt'-
---^\
\¡
\/
+'-
Flydrogeology H
5678
FlG. 2-Variation in L-moment variability measure SSL as the partition point for twoway grouping changes for basin characteristics A, I:+, DWP, S and H.
The proposed measure of flood frequency variability for basin groupings is
the overall sum of squares of deviations of individual basin L-moment ratio
estimates (lzllz,lqllz) from their group record-length-weighted average points,
SSL
ood frequency groupings.
mnj
= : : (13/lrlr¡
j=l i=1
u:/tzl*)r
+ (u¿/lzlr¡ -ll+llzl*¡)2
where ihere are m groups, each comprising nj basins with group record-lengthweighted averages [\¡/lz]*i and llqllz]*¡. SSL is the summation over all groups
of squared Euclidean distances between individual points and group averages
in the L-skewness-L-kurtosis plane. SSL is similar to Hosking and Wallis'(1991)
V¡ statistic (based on L-skewness and L-kurtosis) for testing homogenity of given
groups of basins. Their other two statistics for this purpose are V1 (based on
80
Figure 2 shows the change in value of the L-moment test statistic SSL as
the partitioning points for two groups on the A, lza, DWP, H and S basin
characteristic axes are each varied. Minima in the trace of sSL occur at 100 mm
and 160 mm for rainfall Iz+, indicating that 2 partitioning points or 3 groups
might be required for Iz+. Slope S and hydrogeology H exhibit SSL traces with
well-defined minima, at S = 19.5o and H = 2. Grouping basins according to
basin area A and soil porosity DWP only slightly reduces SSL, implying that
for small New zealand basins, basin area and soil do not influence flood frequency
as much as rainfall, slope and hydrogeology.
For the groups identifled in the above procedure, Hosking and Wallis (1991)
test statistics indicate the following:
-
the low rainfall group is nearly homogeneous and has high EV2 tendencies
(k = -0.21).
the medium rainfall group is heteorgeneors with respect to Vr and Vz,
homogeneous for V3, and is medium EV2 (k = 4.12).
the high rainfall group is heterogeneous and is low EV2 (k = -0.05), nearly
EVI.
the ìo\ry slope group is heterogeneous and is low EV2 (k = 4.07).
the high slope group is mainly heterogeneous (Vr and V2, not V¡) and is
high EV2 (k = -O.15).
Following Wiltshire (1985), a further degree of sophistication can be introduced
by a simultaneous four-way grouping based on two catchment characteristics.
Figure 3 shows the SSL surfaces produced when the basins were divided into
81
four groups according to rainfall Iz¿ and slope S. Two distinct mimima
are
evident, indicating that a six-way split of the I 17 basins at partitions Izq = 105 mm
and 155 mm and S = 19o is optimum. Table I defines the six groups and gives
Hosking and Wallis (1991) homogeneity test statistics. Each group is homogeneous
at least with respect to V¡, and so Wiltshire's (1985) method using SSL has
achieved its objective.
GROUP 2
Iow
I2a,
High
S
2
ll7 small New Zealand
catchments based on rai¡fall .Iz¿, slope S and L-moments of flood
TABLE l-Six-way partition for flood frequency of
data.
GROUP 3
Group Definition
i'
Number Weighted average
L-moment ratios Homogeneous
of
Bæins \llz
I
Iz¡(l05mm
2
Iz¿(l05mm
J
105<Iz¿<l55mm
105<Iz+<l55mm
4
5
6
Iz¿)l55mm
Iz+)l55mm
s<19" t2
s>190 9
s<19' 20
s>190
s<190
s<190
0.214
0.476
0.238
0.260
0.186
0.214
29
24
23
l4ll, Vr Vz V¡
0.190 Y
0.352 Y
0.142 N
0.204 N
0.141 N
0.1s0 N
Y
Y
N
Y
N
Y
Y
Y
Y
Y
Y
Y
EY2k
GROUP 4
Medium I21, Low
É
S
Mcdium l2{, High
S
F
-0.07
ffi....
4.43
-0.10
-0.14
0
GROUP 5
-0.03
-0.07
High I2a,Iow
GROUP 6
S
High
I2a,
High
S
I
2
A flood-frequency distribution can now be fitted to each group using group
L-moment ratio estimates. All groups, except group 5, require at least a three
parameter distribution such as the EV2. To treat each group similarly, the 5parameter Wakeby distribution (Wallis, 1980) is used to produce dimensionless
flood frequency growth curves (Qr/Q.) for each of the six groups in Table
l. Using the Wakeby distribution, the 5 parameters allow the upper tail to be
specified independently of the lower tail, and, in the L-kurtosis-L-skewness space
(Fig. 1), group average points are not forced to move to the nearest 3-parameter
distribution curve or 2-parameter distribution point. The Wakeby distribution
used with L-moments is a robust procedure (Wallis and Woods 1985; Cunnane,
1989). Even though it has hve parameters, the bulk of independent data used
in pooling floods from different basins or regions, ensures the Wakeby distribution
is sensibly fitted. The Wakeby is fitted to the six groups in Table I using recordlength-weighted averages of group L-moments ratios Àz/Àr, À¡/À2, À¿/Àz and
Às/À2, based on unbiased esrimates of the À, (Wa1lis, 1980, 1988; Hosking, 1988;
Hosking and Wallis, 1990).
Figure 4 shows the Wakeby distribution fitted to each group, using the Gumbel
reduced variate horizontal scale for each plot, for which an EVI distribution
plots as a straight line. Wakeby plots for groups I to 4 (low to medium rainfall)
are steep, and three exhibit upward curvature for larger discharges (EV2
tendencies), whereas for groups 5 and 6 (high rainfall) the Wakeby curves are
flatter and show opposite curvature (EV3 tendencies). Also shown in Figure
4 are mean McKerchar a¡rd Pearson (1989, 1990) map Qroo/Q- values for each
82
0
20246-20246
Gumbel Reduced Variate [=
lnCln(l - 1/T)
)I
FIG. 4-Dimensionless flood frequency rwakeby distribution plots with annual flood peak
data for six rainfall-slope (Iz-S) groups defined in Table L.95/6 confidence intervals
are shown for mean McKerchar and Pearson (1989, 1990) e,oo/e. map values
for each group.
group, with 95Vo confidence intervals for each mean. For the low and medium
rainfall groups (l-4), the map estimates are below the Wakeby plots. Map estimates
are based on the EVI distribution (straight line on EVI plot) whereas many
of the annual maximum flood series from small drainage basins exhibit EV2
tendencies (upward curvaturÐ. This explains the difference between map and
Wakeby estimates (Fig. a) for groups I to 4. Straight line EVI distributions
underestimate discharges of high return period for groups I to 4, but are
satisfactory for high rainfall groups (5 and 6).
Figure 5 shows the six tüakeby dimensionless flood frequency growth curves
(also summarised in Table 2). The similarity of these curyes for groups I and
4 indicates that they could be merged into one group. Patterns exist in the
distribution of the growth curyes. As rainfall increases the curves become flatter,
83
800
6
c)
5
U)
-r*-r-f'
c.)
Rainfall I1 Gnm)
!
U)
.t
6
a.)
F
400
800
á
\3
600
F
-o
C}
fi
2
o
r
5
10
2A
50
500
1000
ç
Gumbel Reduced Va¡iate [= lnCln(1_AEp))]
soo
1.
FIG.
Soil Porosity DWP
20
(Vo)
22
Slope S (deg)
25
500
800
600
FIG. 5-Comparison of dimensionless Wakeby flood frequency curves for six rainfallslope (Izq-S) groups defined in Table
500
800
18
14
600
[= 1/AEP]
100 200
12
30
l-
L---1
Variation in Q./AoS variability measure SSQ as the partition point for twoway grouping changes for basin characteristics Izr, DWP, S and H.
6
and for fixed rainfall, basins with higher average slopes have steeper flood
frequency curves.
TABLE 2-Dimensionless flood frequency quantiles for the six flood frequency
$oups defined in Table l.
Group Average Average
lzoS51020
(mm) (deg)
I
79.4
t4.2
2
86.
I
3
127.5
26.5
12.0
24.8
12.0
4
r27.6
5
6
205.6
198.5
'2.6.8
1/
AEP
(=l
years):
50
1.38
t.25
1.45
1.34
1.32
1.35
I .78
1.84
1.90
1.76
1.59
1.68
2.20 2.80
2.63 4.ll
2.33 2.87
2.2t 2.84
1.82 2.09
2.OO
2.40
t00
1000
3.28 5.13
5.66 15.7
3.25 4.42
3.36 5.32
2.26 2.68
2.68 3.52
MEAN ANNUAL FLOOD GROUPS
used contour maps of Q*1Ao't to estimate
edure is used with Q./Aut to investigate
proposed Q-/Ao't variability measure for
artitioned by basin characteristics is (similar
to SSL above),
84
mnj
tt
j=l i=l
SSQ =
(Q-/Ao
8l¡¡
- [Q./e"1*,¡
where there are m groups, each comprising nj basins with group recordlengthweighted average [Q./AoI*t. Figure ó shows the variation of SSQ with different
partition points for two-way groups based separately on rainfall Iz+, soil DWP,
hydrogeology H and slope S basin characteristics. lzq aîd DWP SSQ traces
exhibit minima. Four-way partitions using these two characteristics lead to welldefined and reduced SSQ minima. Again, the best grouping found was a sixway grouping, defìned in Table 3.
Table 3 shows desirable patterns between the group definitions and the
behaviour of recordlength-weighted average Q-/Ao't values for each group. As
rainfall
Iz¿ increases,
Q./ Ao
The
decreases.
t
Q./Ao''
increases, and as soil porosity DWP increases,
Q. predictive powers of the six-way grouping are compared with the
McKerchar and Pearson (1989, 1990) contour map approach. Computing relative
errors between either group or map approach estimators and observed Q. values
from the ll7 basins, gives bias and root-mean-square-error values of 52Va and
132/s respectrvely for the six-way grouping approach, and, 16/o and,75Vo for
the contour map approach. Therefore, the map approach is superior for estimating
Q.. Improvement in Q. estimation for small New Zealand drainage basins is
85
TABLE 3-Six-way partition for Q. of
based on rainfall (Iz¿) and soil
ll7 small New Zealand catchments
(DWP) basin characteristics.
TABLE 4-Regional flood-frequency results from randomly resampling (100
times) l0 annual maximum floods (from basins with 20 or more
floods) for different groups.
Group
of
Weighted
Averase
Basins
Q-i Aõt
Number
Defrnition
Group
Number of
bæins
Q:oo/Q.
(Table
2)
Number of bæins
with n
>
20
Mean Qroo/Q.
Estimate
95% Confidence
Interval
I
lzq1l25mm
DWP<20q0
37
t.57
2
Izq(125mm
125<lzÉ177.5mm
DWP>20Va
DWP120Vo
7
0.81
I
t2
3.28
4
2.68
+0.11
32
3.52
z
9
5.66
l*
5.94
+0.42
4
125<lzq=177.smm
I
J
20
3.25
9
3.02
+0.09
5
lzc)777.5mm
DWP>J07a
DWP1207o
25
5.49
4
29
3.36
12
3.63
+0.12
6
lzÞ177.5mm
DWP>20%o
5
2.31
5
24
2.26
5
2.55
10.09
6
23
2.68
l0
2.58
10.09
274
+0 04
3
ll
1.5
further discussed by McKerchar (this issue); he recommends map Q. estimates
for small ungauged basins.
Alr
fi1
*GEV distribution used in place of Wakeby as only one basin in group.
DISCUSSION
The robustness and accuracy of the Wakeby regional procedure for the sil
flood frequency groups may be illustrated using the longer annual maximum
flood series (similar to Potter and Lettenmaier, 1990). Subsamples of l0 floods
are randomly selected from each drainage basin with annual maximum series
of length 20 or more yeaß. Groups of the subsamples are used with the Wakeby
regional procedure to obtain group Qroo/Q* estimates. This is repeated 100 times
for each group. Mean Qroo/Q. estimates, w\th957o confidence intervals for the
mean, are given in Table 4. Although only l0 floods are used from each site
per run, the results for the six groups are not significantly different from group
Qroo/Q- estimates in Table 2.Larger differences for groups I and 5 are caused
by smaller group sizes in the resampling exercise. The result for all basins with
20 or more years of record, considered as one group, emphasises the need for
more than one group: at least three group Qroo/Q. resampling means (groups
2,3,4) are significantly different to the one-group mean, and to each other.
Both rainfall and slope influence upper-tail steepness of flood frequency curves
for small New Zealand basins. Lower rainfall I2a groups are associated with
steeper frequency curves (Fig. 5) than higher Izr firoups. As explained by Wiltshire
(1985), annual flood peaks from basins in drier regions are more variable than
those from wetter regions. G¡eater variability in annual floods is directly related
to steepness of flood frequency curves. A physical explanation for steeper average
basin slopes relating to steeper flood frequency curves might be that steeper
there is little time for held teams to be on site for current-meter flood gaugings),
implying that variabi-lity of annual floods derived from stage-discharge rating
curyes is high between basins (Potter and \ilalker, 1985). The lengths of record
for these basins are relatively short, ranging from l0 to 29 years, with an average
of 16 years, also implying greater flood frequency variability. Therefore averaging
methods will be superior to looking at individual sites, and so at-site flood
frequency methods will be less reliable than the robust, accurate and efhcient
regional scheme (Wallis 1980, 1988) used for the six groups in this paper. The
closeness of group flood frequency curves (Fig.5) for groups I and 4, and,
to a lesser extent group 3, and the high variability of the annual flood series,
indicates that these groups could be amalgamated into one group for practical
purposes.
slopes are faster draining, and offer less storage opportunities, and hence
antecedent wetness conditions are more variable, translating into more variable
annual flood peaks and so steeper flood frequency curves.
The flood-set of 117 small New Zealand basins displays much variability in
the L-skewness-L-kurtosis plane (Fig. l) and in the dimensionless plots of the
observed floods (Fig. 4). This is expected from small basins where the ratio
of maximum current-meter-gauged stage to maximum automatic waterlevel
recorded stage is usually low (since the flashiness of small basin flooding means
86
87
in this flood-set, nearest neighbour schemes were not suitable for this study,
since most emphasis is usually placed on the closest basin in catchment
characteristic space. For small basins for which flood frequency estimates are
required, between-group averaging could be used when a basin's Izq and S
characteristics place it at the edge of two or more groups. Otherwise the defined
groups in Table I and Figs. 4 and 5 can be used directly as a robust alternative
to the McKerchar and Pearson (1989, 1990) procedure.
Examples: Estimation of 100-yearflood peak discharge
l. Kumeu River at Maddren (45315) in Auckland has
125
mm and S = l5o
(see
A = 46.46km2,
l2a =
Appendix of McKerchar, this issue). This basin
therefore in flood frequency group 3 (from Table l), which has a dimensionless
100-year flood quantile of 3.25 (from Table 2). From 6 annual flood peaks
(1984 89), the at-site Q- is 40.3 ¡¡:/s. The McKerchar and Pearson (1989)
map Q. estimate is 32.3 m3/s. The pooled at-site and map Q. estimate is
33.7 mr/s (pooling described in McKerchar and learson). Multiplying the
pooled Q. estimate by the dimensionless 100-year flood quantile (3.25) gives
a 100 year flood estimate of 13.251 , f33.71= | lQ ¡¡¡:/ s for this basin.
3. Moutere Catchment 5 (57405) in Nelson has A = 0.069ó km2, Iz¿ = 105 mm
and S = l8o (from Appendix). From Table l, this basin is in flood frequency
group 3, but close to being in groups l, 2 and, 4. Because this basin is located
at the edges of group 3, a weighted-average quantile estimate is required.
The weights for each group quantile are the reciprocals of the dimensionless
Euclidean distance in the Iz¿-S space between the group averages (given in
Table 2) and the Moutere values. The distances can be made dimensionless
by dividing by Moutere's Iz¿ and S values (105 mm and 18"). For group
I, the weight is:
is
{l(t0s-79.4) I 10sT + Kl8-14.2)/ lSlz}{
s
=
3.
1
Similarly, weights fcr groups 2, 3 and 4 are 2.0, 2.5, and, 2.3 respectively.
Hence, the 1O0-year dimensionless quantile estimate for this basin is {(3.1)(3.28)
+ (2.0)(5.66) + (2.5)(3.25) + (2.3)(3.36))/(3.1 + 2.0 + 2.5 + 2.3) = 3.77. This can
be used with the at-site Q* (.0297 mz ls) derived from 24 annual flood peaks
to give a 100-year flood peak estimate of 1.12 m3/s.
Appendix: Rainfall Izr, slope S, hydrogeology H, and depth-weighted-soil-porosity DWP
basin characte¡istics, and annual flood peak L-moments for l17 small (Area
A < 100 km2) New Zeala¡d basins, each with "n" annual flood peaks. Site
numbers from Walter (1990)
Site A L¿ S HDWP
(km,) (mm) (deg)
(fù
3506 ll.l0 140 8.54 1.00
4901 12.50 200 18.9 1.00
5513
5515
5516
Six flood frequency groups were defined on the basis of areally averaged
rainfall Iz+ and slope S basin characteristics using Wiltshire's (1985) method with
an L-moments measure of flood frequency variability. Wakeby distributions were
fitted to each group, providing robust dimensio¡rless flood frequency estimators
for small ungauged New Zealand drainage basins.
78r
0.r2ó0
I
l 1.96
8203
0.3000
190
190
190
190
18.0
18.0
18.0
15.7
t95 24.3
5.04
t0.t
4.40
125 9.02
3.13
135
1.00
1.00
1.00
1.02
130 t0.7
5.18
t25 8.21
5.76
t20 r.50 8.00
t50 24.5 2.17
8604 40.72
9228 7.920 150 33.2 3.31
14610 57.13 r60 15.0 6.16
14625
73.89
t4627
68.80
t0t464t
75.98
r014ó45
10t4646
0.8100
0.9200
15453
15534
19734
45.05
21601
22901
21.4t
2.670
30.50
2t4t0
50.29
18.44
0.5200
23209
23.39
232t0
53.73
160 19.7 7.01
200 19.8 6.00
190 20.0 6.24
ró0 5.50 8.00
160 5.50 8.00
200 30.9 4.66
150 23,2 8.00
160 24.1 4.91
r55 26.5 5.30
165 27.9 4.09
170 20.3 4.49
ló5 23.0 6.00
95 16.7 2.57
78.75
160 t7.r 4.78
150 t7.7 4.99
160 31.7 t.5l
105 24j 3.98
200 35.0 1.48
140 28.5 1.06
180 35.2 1.49
0.2300
90 22.4 4.03
23220
29242
29244
29246
29250
29254
29259
88
0.1550
5519 r3.90
6501 8.r30
7202 9.570
7604 11.08
7805 82.40
23005
CONCLUSIONS
0.6300
84.60
40.25
36.32
75.78
15.57
20ó05
29808
29841
29843
79.73
30516
9.100
t.92
87.24
120 23.4
200 34.9
43.84
l
25.9
t.42
37.95
135 32.0 1.00
100 17.5 2.34
l0
1.00
n
Unbiased
L-moments;
Q'
(m3/s)
lr =
22
20
13.0 r0
r0.r lr
l0.r ll
9.00 l0
10.0 12
6.40 l0
1.20 27
5.44 t5
7.10 l0
l3.l t8
ll.8 il
u.0 t9
22.1 22
26.7 t4
20.0 23
28.3 15
29.3 l
29.3 l
28.2 l0
19.6 23
20.1 lt
16.9 2t
l4.t t4
8.ó0 ll
22.t 2t
lr.5 25
t2.2 26
14.6 tz
15.7 20
1.2 22
14.8 l0
16.7 20
14.4 t2
9.ó0 I I
13.3 l0
14.7 22
t6.5 12
15.4 l0
9.10 2t
l,
53.60
9.260
0.026
r0.0
ó1.09
16.39
1.040
0.079
89
l,/1,
(mr/s)
9.70
3.090
L-moment ratios:
lr/1, L,lIt
0.218
0.143 -0.048
0.119 -0.022
0.094 0.083
0.04r -0.191
0.ó400
0.2200 0.il3
0.7fl0
0.2500
17.63
0.174
0.003 -0.158
0.059
0.158
0.5ó0
26.34
9.590
9.930
0.306
0.06ó
0.424
0,101
31.15
8.120
0.273
0.127
0.055
0.iló
-0.054
-0.035
51.80
18.53
u
118.2
40.
21.50
9.240
0.320
1.480
0.5200
0.292
17.89
18.29
3.620
23.94
34.67
0.r68
0.269
0.234
0.099
0.t49
0.498
0.216
0.505
0.295
0.300
0.159
0.066
4.920
0.298
0.088 -0.102
8.350
0.220
0.120
0.040
20.2t
6.340
0.184
0.089
2.020
0.086
0.090
0.093
0.193
t.860
0.3100
0.1600
10.49
0.5600
0.059
0.260
37.51
1t.23
69.15
42.24
r.530
249t
0.457
0.521
0.156
0.253
-0.039
0.280
0.203
0.098 -0.133
1.330
15.23
13.01
7.920
0.3300
0.363
0.143
0.235
0.194
10.22
3.ó40
0.230
0.120
0.134
0.029 -0.026
lll.3
18.37
29.05
25.52
31.38
8.160
50
7ó
0.075
282.0
32.81
330.2
0.2200
0.0700
0.459
43.88
56.62
45
9l
21.68
59.25
83.60
0.241
0.361
0.233
0.005
0.109
0.128
0.075
0.il9
4.076
0.003
-0.003
-0.087
0.013
0.366
0.2m
0.368
0.335
0.152 -0.069
0.143 4.016
0.0ó5 -0.106
10.13
0.255
0.387
0.213
57.15
0.094
0.029
0.109
0.100
0.327
0.103
-0.066
0.392
76.36
t7.t4
285.1
57.19
0.153
0.054 4.010
69.00
r7.68
0.201
0.224
85.28
12.19
0.019
0.103 -0.r53
7.420
2.490
0.336
0.2u
0.278
0.130
0.149
L-moment ratios:
Site A Iu¿ S HDWP n Unbiased L-moments;
Iallz lsllz
lz
(deg)
(Va)
lr
bllz
(km2) (mm)
=Q,
(m3/Ð (mr/Ð
l0
t9.2 2.51
30701 4.69
I
30802
32001
125 19.9 2.62 9.70 15
80 18.9 2.95 12.5 I I
il5 26.6 1.80 14.5 l l
38.47
16.80
r032517 5ó.60
1232564 62.30
32735
32754
61.58
99.50
33il4
53.11
T
.40
22
130 20.0 4.r1 15.0 l0
80 8.90 4.67 10.7 ll
85 26j 4.24 l0.l
13
100 t0.9 7.43 25.2 22
ll5 22.0 5.94 13.9 2l
145 t0.2 6.39 15.9 22
115 l6.t 6.66 25.5 13
331l5
32.78
33117
20.63
33307
81.84
33347
27.14
34308
35004
84.63
35506
59.60
3ó001
30.98
38401
24.92
170 20.2 7.23 9.80 l0
39201
59.10
280 15.6 6.60 10.2 l0
180 10.6 7.85 r4.5 14
240 6.37 1 .4t I1.5 14
260 8.95 7.19 14.8 12
320 10.7 6.16 t2.9 11
280 9.45 6.41 ll.l 13
120 19.5 6.30 14.0 19
135 25.5 4.94 10.7 l1
49.60
39402
49.03
39403
37.78
39504
77.34
39508
39510
19.24
û703
l4.ll
41301
41601
95.10
1043434
1043466
21.59
1043476
0.0450
t0.92
8.790
95.89
1t43407
1143409
1.690
0.3400
fi43427
3.110
1143428 14.64
t443462 9.990
43602
17.86
43807
12.50
45702
8.210
45903
0.8800
46609
12.13
46&5
3.360
t25 24.3 5.6t
23.0
24
3.570
0.221
0.049
0.072
-0.037
l0
0.015
4.310
105 23.4 3.70 9.50 19
150 19.8 7.50 26.1 2l
170 15.7 6.91 16.9 27
110 18.0 8.00 28.9 ll
145 24.1 7.46 23j 14
145 24.9 7.30 9.50 2l
140 21.9 4.82 I1.8 18
120 l0.l 7.99 9.m ß
120 20.3 6.93 19.4 17
t20 9.77 1.70 20.1 24
t20 8.37 5.10 6.80 13
190 22.1 4.00 6.10 2l
t75 lr.5 4.00 2.40 l0
170 16.0 t.32 r0.3 15
160 9.08 4.56 13.9 l0
0.1500
0.6000
0.2300
23.85
32.73
2.030
56.98
9.280
4.00 2.10 l3
0.1800
4.00 2.10
47527
10.03
r55 17.6 4.00
48015
21.78
135 26.2 1.00 8.80
10.2 24
14
90
23.48
75.06
12
0.1
0.259 0.165
0.142 0.088
0.415 0.324
0.042 0.109
0.071 0.121
0.370 0.2n
0.277 0.40
0.449 0.218
0.595 0.470
0.537 0.359
0.098 0.077
-0.156 0.091
0.272 0.189
0.r04
0.000
0.045
0.089
0.056
0.219
0.
-0.022
0.359
-0. I 8ó
0.190
0.035
82.38
57022
51023
2.790
57101
58.00
57402
57405
57512
l0l
46.59
5.140
0.039ó
0.069ó
0.0344
58301
60104
63501
t7.25
65.02
1.690
64ffi6
74.04
64610
41.91
66405
0.9000
66ó03
2.180
66604
68529
3.260
0.097
0.309
68602
55.00
69621
22.97
0.210
0.169
-0.140
0.039
6.190
7fi22
0.245
1lL29
7r 178
50.08
99.63
78.70
-0.006
73501
45.00
0.215
0.380
74353
24.06
0.373
0 153
74360
2.860
0.274
0.195
0.5800
0.151
0.t22
74367
5.500
0.0800
0.401
0.279
0.038
0.2300
1.420
1.450
74701
9.590
7t.60
0.060
80201
87301
0.095
90605
4.380
0.072
9t4t2
t2.42
0.459 0.237
0.ó59 0.373
0.351 0.190
0.143 0.237
0.269 0.194
0.349 0.112
0.370 0.087
8.900
0.175
0.5200
-0.164
-0.061
0.052
25.57
0.209
-0.051
-0.158
0.1600
1.020
0.7400
5.760
t2.5t
0.1
46.81
56901
57014
0.097
14.43
2.680
3.170
-0.004
0.212
52916
0.022
0.210
0.1ó0
0.415
0.11ó
0.6800
38.9
2.y0
155
155
0.5300
51.64
6.620
4.160
13
46662 0.3900
46663 0.0t42
lr.5
0.221
17.72 3.680
27.41 5.390
47.75 12.04
28.07 6.220
149.5 26.48
73.01 9.240
104.9 23.33
36.91 8.230
55.14 8.820
329.5 43.32
61.94 21.65
81.46 19.29
174.7 33.38
53.90 5.360
67.36 9.&0
280 5.26 7,11 15.6 12
200 8.t I 7.26 13.0 15
200 5.59 1.41 14.6 t2
200 1.88 4,17 t2.l 20
il.5
38.11 t2.40 0.352 0.36
53.63 13.15 0.273 0.161
18.23 5.330 0.r28 4.022
99.50 17.75 -0.039 -0.142
71.68 16.45 0.082 0.408
33.32 11.46 0.376 0.269
65.72 22.66 0.466 0.361
Site A Iz¿ S HDWP n Unbiæed L-moments; L-moment ratios:
ga)
(km,) (mm) (deg)
l,
l3/1, lallz lr/lt
lr = Q,
(mr/Ð (mr/Ð
2.260
5.760
0.9400
0.5700
0.0400
6.330
17.53
0.231
0.029
0.137
-0.037
0.15
0.6600
97.27
47.46
61.ó3
90 23.1 8.0 14.7 t3
90 29.9 8.00 14.7 l0
2.160
1.210
0.8200
0.1300
0.3000
0.2000
0.0500
0.1
0.1400
9.210
0.417 0.211
0.313 0.242
0.277 0.070
32.08
0.t88
0.0f
0.457
0.002
0.300
0.347
0.435
0.428
0.169
0.217
0.094
0.21I
0.257
0.100
ll0
18.3 8.00 6.90 24
105 il.5 8.00 6.40 24
105 18.0 8.00 6.40 24
195 23.0 8.00 6.40 22
165 28.9 1.02 21.0 29
105 3l.r 1.07 13.9 23
200 30.9 1.00 14.7 12
210 29.5 r.49 20.1 15
160 23.1 3.08 1L5 22
105 30.5 1.00 22.5 19
95 28.6 3.24 13.2 1l
95 25.6 3.65 t3.4 12
90 29.3 1.90 23.7 tl
l l0 12.8 5.23 14.7 2t
80
80
t40
60
24j
8.55
24.1
28.8
100 18.9
45 15.7
2.19 t7.8
24
8.00 20.1 17
3.93 22.0 27
1.16 18.7 19
1.33 t0.2 27
1.00 r0.0 18
3.00 8.10 t2
65 il.5
70 9.23 r.00 5.60
60 16.5 1.68 6.40
78 17.2 3.61 8.90
II
19
12
40
29.3 2.77 19.9 l1
220 14.9 7 .06 t7.7 t7
I l0 30.5 8.00 t4.7 11
10.15
0.132 0.179 -0.021
0.162 0.121 0.m6
l7.ll
0J42
12.88
61.ó6
30.02
73.93
3.340
92.67
35.13
0.6800
1.180
1.420
2.760
0.3600
23.45
0.0700
1.820
10.43
13.4t
0.3000
0.6200
0.8300
0.7900
10.48 6.420
16.72 8.830
3.960 1.470
22.09 6.100
38.19 20.14
44.33 t5.26
3.010 t.410
2.030
2.020
6.770
30.53
389.2
28.85
0.8200
0.5900
0.6700
2.130
12.31
68.82
5.870
0.2500
0.213 4.124
0.539 0.431 0.356
0.409 0.383 0.202
0.162 0.057 -0.001
l2
-0.068
-0.035
0.208
0.102
0.01I
0.086
0.026
0.198
0.130
0.468
0.506
0.253
0.154
0.025
-0.126
0.455 0.277
0.640 0.5il
0.453 0.499
0.271 0.126
0.509 0.272
0.199
0.tó9 0.169
0.183
0.258 0.028
0.012
0.369
0.459
0.054
0.126
0.42 0.161 -0.123
0.185 0.r13 0.085
0.070 0.171 -0.002
0.419 0.211 0.t69
-0.171 -0.064 0.041
0.257 0.147 0.085
0.255 0.112 0.079
1
0.060
REFERENCES
0.03r
0.m0
0.155
0.145
-0.035
0.107
-0.053
-0.052
0.093
0.266
0.272
97.80
290 35.4 1.48 2t.7 20
200 28.0 r.06 13.0 26
n0 23.6 7.96 t3.2 20
0.317
0.240
0.214
-0.03 r
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ABSTRACT
The World Meteorological Organisation project HYNET uses a technique, based
on random subsampling of real data, to compare network designs with common
objectives. Two network design aids, Network Analysis for Regional Information
(NARD and Network Analysis Using Generalised Least Squares (NAUGLS),
were applied to data from a network of 76 water-level recording stations in
North Island, New Zealand. NAUGLS conveys more information than NARI
for maximizing regional information about mean flows given a limited budget.
NAUGLS is applied to low flows, mean flows and flood peaks for the Nelson
region, South Island, to illustrate its use.
INTRODUCTION
In 1984, at the Seventh Session of its Commission for Hydrology, the World
Meteorological Organization (WMO) set up a project to compare technologies
used by hydrological and meteorological services of its member countries to
design networks. This project is known as Intercomparison of Operational
Hydrological Network Design Techniques (HYNET).
WMO member countries, including New Zealand, have participated in
comparing two network design aids developed and used in the United States
of America (Moss and Tasker, 1991). The methods are Network Analysis for
Regional Information (NARI) and Network Analysis Using Generalised Least
Squares (NAUGLS). Both are in a HYNET computer program (Moss and Tasker,
1ee0).
These aids aim to define networks that efficiently provide information for
estimating statistical variables of streamflow at ungauged sites in a homogeneous
region. Estimation is based on a multiplicative regression model of a streamflow
variable against physiographic and climatic characteristics.
NARI uses ordinary least squares to calibrate the regression relation and is
based on simulations using stochastic hydrology. NAUGLS uses generalised least
squares (Stedinger and Tasker, 1985) in which values of computed strearnflow
characteristics at each gauged site are weighted in inverse relation to the estimate
of their precision. NAUGLS does not rely on simulation and is more
mathematically elegant than is NARI. However, NAUGLS entails some
simplifying assumptions in developing its weighting scheme. HYNET enables
us to test whether the added elegance is a practical improvement over the more
simplistic NARI.
HYNET uses randomly selected subsets from existing hydrologic data to
simulate the design and evaluation of a network. Repeated sampling provides
statistics describing the effectiveness of the network design in addressing a specified
common objective. These statistics form a basis for comparison.
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