WATER & SOIL - These are not the droids you are looking for.

Draper and Smith 1966). ln our case, the true value of the
dependent variable Qnu. is not known with certainty. It is
subject to time sampling error and can only be estimated
from the period of flow record available. The relative magnitudes
of time sampling errors are indicated by the pooled
Cy values in Figures 4.8 and 4.12.
A second difficulty is that within a region some adjacent
catchments may be subject to the same storms of large areal
extent and a pair of series of annual maxima for such catchments
is likely to be cross correlated: errors in estimates of
Qo6, will therefore not be random, but will also be correlated.
In this situation regression analysis is still permissible,
but estimation of the prediction error is more complex.
An analysis of the situation is provided by Matalas
and Gilroy (1968), and some practical examples are given
by Hardison (1971).
When a regional regression of log'o Q is calculated on a
set of m catchment parameters, the standard deviation of
the log,o Q about the regression expressed in log,o Q units
is denoted by Sp. lf these deviations of the log,o Q from the
regression are normally distributed, then the coefficient of
variation of the untransformed Q about the regression, C¡,
is given by
Cä = .*p (2.303 SR)' - I 47
When Q is estimated from a flood record that is N years
long, and the coeffìcient of variation of the annual maxima
is denoted Cy, the estimate of Q will differ from the population
value (that would be obtained from a very long record)
with a coel'ficient of variation of CylN.l.
ci : ci /N"
When Q is predicted using the regional regression it will As the quantity Nu could provide a useful guide for using
differ from the population value, and the coefficient ofvar- the regression equations, it is evaluated for each of the reiation
of the difference averaged over k sites, denoted by gions together with Cp for each region (Table 4.12). Esti-
Table 4.12 Prediction errors and equivalent lengths of record.
Cp, is calculated from Equation 4.8, which is derived from
Hardison (1971).
c'p : ch(t - + -
tf+-, ) + ci (2q - r)/N6..... 4.8
where p is the average cross-correlation between annual
maxima series from pairs oi catchments in the region, and
Nç is the average length of record. When there are many
uncorrelated records such that sampling errors tend to cancel,
and the average record is short (k large, p small, N6
small), then Cþ can be less than CilNc, so rhat a better
estimate is obtained from the regression at a site than from
the record at a site.
Note that Cp is an average prediction error, and will
over-estimate errors on predictions for ungauged catchments
whose parameters are near the mean value used in
calculating the regression, and conversely. In New Zealand,
no estimates ofthe interstation correlation coefficient q are
available, but typical values may reasonably be expected
within the range 0.2 to 0.8. Three values of p (0.2, 0.5 and
0.8), were therefore used in evaluating Equation 4.8.
Given the coefficient of variation of the prediction error
Cp, ân estimate can be made of the length of record necessary
to estimate Q with the same degree of accuracy as is
given by the regression equation. If Nu is this equivalent
length of record, anLd the prediction error is expressed as a
percentage, then
49
Region S¡
(for regression
of logarithms)
Cvkm
(no. of (no. of
stations) regression
variables)
Nca
(av length
record)
cR cP
(Eqn 4.7) (Eqn 4.8)
l"l,\ l"/"1
N, Typical
(Eqn 4.9) Nu
(yrs)
(yrs)
Northland/
Coromandel/
East Cape
Pumice
East Coast Nl
Wairarapai
Manawatu/
Wellington
West Coast Nl
West Coast Sl
o.119
o.128
0.082
o.118
o.1 67
0.148
o.54
o.54
o.54
o.40
o.40
0.36
21
14
t1
19
25
t9
1 1.6
12.O
13.O
13.5
110
9.4
o.2
o.5
0.8
o.2
o.5
o.8
o.2
0.5
o.8
o.2
o.5
o.8
o.2
o.5
o.8
o-2
0.5
o.8
27.9
21 .9
27.9
29.9
29.9
29.9
1 9.1
1 9.1
1 9.1
27.7
27.7
27.7
39.9
39.9
39.9
35.O
35.O
3 5.O
27.5
30.1
32.5
33.7
35.8
37.7
19.4
22.6
25.4
30.0
31.1
32.3
41.6
42.6
43.6
37.1
38.1
39.3
39
3.2
2.3
2.6
2.3
2.1
7.8
5.7
4.5
1.8
1.7
1.5
o.9
o.9
0.9
0.9
o9
o.8
lnland
Marlborough/
Canterbury
East Coast Sl
Mackenzie/
lnland Otago/
Southland
o.1 09
o.1 05
o.146
0.66
o.98
o.66
13
1'l
'I 5
18.0 0.2
0.5
o.8
8.4 0.2
o.5
o.8
89 02
o5
o8
25.O
25.O
25.O
24.5
24.5
24.5
34.5
34.5
34.5
24.4
27.2
29.8
12.5
29.O
39.1
34.6
38.6
42.2
7.3
5.9
4.9
61.6
11 .4
6.3
3.7
2.9
2.4
Water & soil technical publication no. 20 (1982)
75