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3.?A, the variance of the flood peak estimate Q.¡ may be
given as (NERC 1975, p.184)
var(Qr) : var_(Q.Qr/Q) _
= E(Q)'.var(Qr/_Q)
+ E(Qr/Q)'?.var(Q)
325
The quantity var(Q ) on the right han_d side of Equation
3.25 depends on the manner in which Q is estimated (see
Chapter 4). In practice Q is substituted for E(Q ) and, similarly,
QrlQ for E(Q1/Q). This leaves only the quantity
var (Qr/Q) unaccounted for. It is the variance of the regional
curve ordinate Q.¡/Q at return period T and is the
type (b) variation mentioned previously.
The quantity var (Q1/Q ) was calculated as the variance
of the individual station flood frequency curves for
T=2.33,5, 10,20,30, 50 and 100 years. Given this variance,
the coefficient of variation (Cp) of individual station
curves about the regional curve is defined as
c¡ = (var (Qr/Q) )k/(Qr/Q) 326
and is a measure of the type (b) variation above'
Regression analyses of C¡ against the return period T,
and also fnT, were carried out and it was found that the relationships
between C¡ and /nT were approximately linear.
Regression equations of the form
Cp=(c+m./nT)/100 321
were therefore obtained, where c and m are constants for a
reglon.
The regression equations for the various regions are summarised
in Table 3.8. The square of the correlation coefficient
R was in the range of 0.946 to 0.992 in all cases but one,
indicating that generally the equations explained at least
9490 of the variation in C¡. The exception was the equation
for the Bay of Plenty region where the R'z value was only
0.380. This low R' value was presumably the result of the
regional curve having a different trend from some of the
frequency curves for the individual stations where there was
historical flood information. Further, the Bay of Plenty region
was the only one where the equation for CF was not
statistically significant at the l9o level.
While the statistical properties listed in Table 3'8 indicate
that the equations would generally be satisfactory for estimating
C¡ for a regional curve, some of the regions did not
contain sufficient stations to enable truly representative
equations to be determined. For instance, in three ofthe regiòns
less than l0 stations were used. Since the equations
were only a measure of the type (b) variation and were intended
to convey only an order of magnitude ofthe standard
error associated with an estimate of Q'¡/Q obtained
from a regional curve, it was decided to pool together the
individual station estimates of Q1/Q for different regions
into larger groups and to derive a new C¡ equation for each
group. These grou¡r equations could then be taken as the
estimating equatiorrs for C¡ for the regions they represented.
However, prior to the pooling of estimates, it was
necessary to dividr: the individual station estimates of
Qr/Q bv the corresponding regional curve ordinates.
Group C¡ values were then calculated for the pooled estimates
for the return period concerned.
Two groups of regions were considered for each of the
North and South Islands, with one group representing the
western and the other the eastern regions. Grouping the regions
in this mannen is supported by the overall trend in the
value for the slope m of the regional regression equations
(Table 3.8), with the value generally being greater on the
east of an island than on the west. This trend indicates that
the variance of the estimates of the regional ordinates
Q-¡/Q is greater on the east than on the west, and it also reflects
the greater variability in the flood peak data for the
eastern regions (see also section 4.9). Further justification
for this grouping of regions is given by the trend in the
characteristics of ttre regional curves (section 3.3.2).
The way in which the individual station estimates for the
regions were pooled together into groups is shown in Table
3.9. The estimates for the Bay of Plenty region were not
used in the derivation of the group equations. The C¡ equation
for this region was not meaningful in view of the low
R2 value and it thurs seemed inappropriate to use the data
for this region in deriving a useful equation for its group for
estimating C¡. The combined North Island West Coast and
South Island West Coast regions are the only regions in
their respective groups, and thus their regional C¡ eQuations
(Table 3.8) automatically became the corresponding
group equations.
Table 3.9 gives thre regression equations that were derived
for the groups andt it also lists the regions to which each
equation applies. The tabulated statistical properties indicate
that the equa,tions are statistically acceptable, with
each being significant at the l9o level. Table 3.9 gives the
100-year C¡ value as computed from each group equation.
The 1OO-year values range from 0.13 to 0.19 for the western
groups and from O.24to 0.25 for the eastern groups. They
compare very favourably with the value of 0.32 as given by
the corresponding Cp equation recommended by NERC
(1975, p.183) for use with all its regional curves.
Each group equation was derived from station estimates
for return periods only up to 100 years, even though most
regional curves extend to the 2(Ð-year return period. Normally
it is recommended that an equation should not be applied
outside the range of data from which it was derived.
However, in this case it was preferred to allow each equation
to be applied up to the 2D-year return period, rather
Table 3.8 The regional regression equations for estimated CF'
REGION
Coefficient
c
S ope
m
R2
s.E.
Est.
No. ol
Stat¡ons
NORTH ISTAND
1. Combined N.l. West Coast
2. Bay of Plenty
3. North lsland East Coast
4. Central Hawke's Bay
o.94
6.19
-2.21
o.25
3.93
o.91
6.O1
5.38
0.997
0.617
o.972
o.989
o.993
0.380
o.946
o.979
27.O* O.OO145
1.75 0.Oo518
9.32* 0.00645
14.9* 0.00361
65
12
12
8
SOUTH ISI-AND
5. South lsland West Coast
6. South lsland East Coast
7. South Canterbury
8. Otago-Southland
2.46
-o.37
4.53
-o.40
2.25
4.59
5.90
4.19
0.996
o.996
0.981
o.982
o.992
0.992
0.963
o.965
25.2* o.oo244
24.6* O.OO508
1 1 .4* 0.10410
11.7* O.OO974
21
14
9
t)
NOTE:
The regression is of the form C¡ = (c + m.fnT)/lOO
* ¡ndicates significance at the 196 level
S.E. Est. is the standard error of est¡mate of C¡
Water & soil technical publication no. 20 (1982)
47