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INTERNATIONAL JOURNAL OF CLIMATOLOGY
Int. J. Climatol. 19: 405–421 (1999)
THE POTENTIAL LONG-RANGE PREDICTABILITY OF
PRECIPITATION OVER NEW ZEALAND
ROLAND A. MADDENa,*, DENNIS J. SHEAa, RICHARD W. KATZa and JOHN W. KIDSONb
a
National Center for Atmospheric Research, Boulder, CO 80307, USA
b
National Institute of Water and Atmospheric Research Ltd., Wellington, New Zealand
Recei6ed 9 September 1997
Re6ised 30 June 1998
Accepted 22 July 1998
ABSTRACT
It is assumed that the interannual variance of seasonal precipitation totals is made up of a component reflecting daily
weather variations which, as a result, is unpredictable beyond deterministic predictability limits of about 2 weeks. The
second component is any additional variance that is, at least, potentially predictable. The first component is
considered noise and is estimated using a statistical model whose parameters are determined from daily, within
season, precipitation. Estimates are compared with the total variance and where the total variance exceeds the
estimated noise it is concluded that there is potential for long-range prediction. Results indicate that only 30% or less
of the total variance at stations is potentially predictable. Countrywide totals do not improve the situation.
Persistence of the ENSO signal may be able to help realize a small fraction of the potential predictability or about
5% of the total variance. Copyright © 1999 Royal Meteorological Society.
KEY WORDS: long
range predictability; precipitation; ENSO; New Zealand; principal components
1. INTRODUCTION
The value of skillful long-range (beyond the limits of the deterministic predictability of weather variations
which is thought to be about 2 weeks) prediction of precipitation is unquestioned. In our efforts to attain
and improve such skill, we must recognize that some of the variance of precipitation totals is the result
of summing over finite time intervals of individual precipitation events accompanying daily weather.
Details of this weather and of the precipitation events are unpredictable at long-range. In this paper we
make first order estimates of what that unpredictable component is. We then conclude that any additional
variance is, at least, potentially predictable. In a companion paper (Madden and Kidson, 1997), we have
made parallel estimates of the potential long-range predictability of New Zealand temperature, and have
discussed ideas behind the analysis in somewhat more detail. Further references on the subject are
contained in that paper also.
2. BACKGROUND
Our estimates of the potential long-range predictability of precipitation are based on an idealized view of
the interannual variance of seasonal precipitation totals as consisting of two separable components. One
is the standard error of precipitation totals resulting from daily ‘weather’ variations. We view the weather
during a season to be a realization sampled from a process with constant statistical properties or an
unchanging climate. These standard errors are only predictable at lead times less than deterministic
predictability limits and therefore can be referred to as ‘climate noise.’ The second component is any
* Correspondence to: National Center for Atmospheric Research, Boulder, CO 80307, USA.
CCC 0899–8418/99/040405 – 17$17.50
Copyright © 1999 Royal Meteorological Society
406
R.A. MADDEN ET AL.
interannual variability that may exist in excess of the climate noise. We presume that it is, at least,
potentially predictable at long lead time. We estimate the size of the first component, climate noise, by
computing the variance of time-averaged data based on a statistical model whose parameters are derived
from observed daily data. The actual variance is viewed as made up of this unpredictable climate noise
and, if that is less than the actual variance, a second potentially predictable component.
It is often reported in the hydrology literature (Buishand, 1978; Wilks, 1989) that simple stochastic
models, as we will use here, fitted to daily precipitation data, tend to underestimate the observed variance
of monthly or seasonal total precipitation. The seasonal variability of simulated data determined from
more complex models fall short of reality too (Gregory et al., 1993). One interpretation, and the one
considered here, is that the model captures the daily variability, but cannot capture slower changes that
represent potentially predictable signals. Katz and Parlange (1993) fit chain-dependent processes conditioned on a time-averaged index of the large-scale atmospheric circulation, and the combined model
reduced the typical underestimate of the actual variance. The additional variance explained by the index
may reflect slower changes that are potentially predictable at the long range.
Klugman and Klugman (1981), Madden and Shea (1982), Klugman (1983), Singh and Kripalani (1986)
and Shea et al. (1995) have interpreted the underestimation of actual variance by stochastic models fit to
daily data as evidence of climate change or potential predictability. The first three mentioned studies deal
with precipitation from US stations and the last two from Indian stations. Here we extend the analysis
to precipitation data from New Zealand.
Figure 1. Location of the 20 stations. Stations in bold are the subset of the 17 discussed in the section ‘Effect of Spatial Averging’
Copyright © 1999 Royal Meteorological Society
Int. J. Climatol. 19: 405 – 421 (1999)
LONG-RANGE PREDICTABILITY OF PRECIPITATION OVER NEW ZEALAND
407
Table I. The minimum number of seasonal segments available for each station
Station
No.
Latitude
Longitude
Year
No. seasons
Percent missing
Kaitaia
Glenbervie F
Albert PK
Rotorua
Hamilton
New Plymouth
Gisborne
Napier
Palmerston N
Kelburn
Karioi
Hokitika S
Milford Snd
Appleby
Molesworth
Lincoln
Waimate
Dunedin
Queenstown
Invercargill
A53021
A54631
A64871
B86124
C75731
C94011
D87692
D96591
E05363
E14272
E95451
F20791
F47691
G13211
G23021
H32641
H41701
I50951
I58061
I68433
−35.1
−35.6
−36.8
−38.0
−37.8
−39.0
−38.7
−39.5
−40.4
−41.3
−39.5
−42.9
−44.7
−41.3
−42.1
−43.6
−44.7
−45.9
−45.0
−46.4
173.3
174.4
174.8
176.3
175.3
174.2
178.0
176.9
175.6
174.8
175.5
171.0
167.9
173.1
173.3
172.5
171.0
170.5
168.6
168.3
1949
1947
1863
1899
1907
1944
1937
1870
1928
1928
1927
1866
1929
1932
1944
1881
1908
1918
1890
1939
36
37
120
80
79
46
65
102
64
67
52
79
61
61
39
104
82
75
100
52
0.0
1.9
4.5
0.3
7.1
B0.1
0.9
9.3
0.4
0.0
1.7
14.5
1.5
B0.1
1.1
0.6
1.1
0.2
1.4
3.1
Contains station names, identification numbers, latitude and longitude (degrees east), the first year of data,
minimum number of seasons used, and the percent missing daily precipitation totals. No season was
included if it had any missing data. That partially explains why Hokitika had a season with only 79
members while Waimate’s minimum was 82 members even though the data started at 1866 at the former and
1908 at the latter.
3. DATA
Daily precipitation totals from the 20 stations shown in Figure 1 were divided into 92-, 92-, 91-, and
90-day seasonal segments beginning on 1 March, 1 June, 1 September, and 1 December, respectively. If
a season had any missing values it was not included. The minimum number of seasonal segments available
for each station are listed in Table I. In nearly all cases other seasons had the same number available (as
the minimum listed in Table I), or up to three more.
4. METHOD
Most of the earlier studies used a chain-dependent process to model the daily precipitation data and
estimate the noise. Katz (1977a,b) proposes such a process for modeling both precipitation occurrence and
amount. The variance of precipitation totals of a segment of a chain-dependent process is given by (Katz,
1985; Katz and Parlange, 1993)
s 2T,a : T p(s*)2 +p(1 − p)
n
1 +d
(m*)2
1 −d
(1a)
where p is the unconditional probability of precipitation on a given day; (s*)2 and (m*)2 are the variance
and squared mean of the precipitation amounts using wet days only (a wet day is defined as one with
\ 0.1 mm of precipitation); d is a ‘persistence parameter’ given by p11 (probability of a wet day given that
the previous day was wet)− p01 (probability of a wet day given that the previous day was dry). These
parameters are estimated from the data at each station for each segment and then averaged over all
segments for a given season. It is important to note that the variance (s*)2 is computed within each
segment and then averaged over all available segments. This means that (s*)2 is based only on variability
Copyright © 1999 Royal Meteorological Society
Int. J. Climatol. 19: 405 – 421 (1999)
408
R.A. MADDEN ET AL.
within segments and is hopefully representative of that associated with daily weather and not of
interannual changes. T is the length of the segment, in our case a season. The first term in (1a) represents
the variance in wet day totals if the number of wet days were fixed, and the second term includes the
variance in the number of wet days.
Equation (1a) depends on the assumption that the precipitation amounts on consecutive wet days are
independent. Some dependence can be included by assuming that precipitation amounts are a randomly
stopped, first-order autoregressive process with a 1-day lag correlation of f. That is, just the precipitation
amounts within a given wet spell are modeled by an AR(1) process. Hereafter this process will be referred
to as a generalized chain-dependent process. Then the equivalent to (1a) is (Katz and Parlange, 1995)
s 2T,b : T
p(1+ p11f)
(1 + d)
(s*)2 +p(1 −p)
(m*)2
(1− p11f)
(1 − d)
(1b)
when f “ 0, s 2T,b “s 2T,a. Here we use (1b) to estimate the noise. For single stations f tends to be small,
averaging over all stations and seasons it is only 0.05. Even the largest value of f, 0.17 at Glenbervie
during March–May increases the noise estimate only 20% from that given by (1a). However, when totals
for 17 stations are considered, f is large (near 0.30) and s 2T,b is considerably bigger than s 2T,a.
The actual variance of precipitation totals for a given season is given by
s 2A =
1 N
% (S(i, j ) −S(·, j ))2
N i=1
(2)
Figure 2. Standard deviations in millimeters associated with estimates of climate noise for (a) Summer (December – January – February) and (b) Winter (June – July – August)
Copyright © 1999 Royal Meteorological Society
Int. J. Climatol. 19: 405 – 421 (1999)
LONG-RANGE PREDICTABILITY OF PRECIPITATION OVER NEW ZEALAND
409
Figure 2 (Continued)
where N is the number of years, S(i, j ) is the precipitation total from season j of year i, and S(·, j ) is the
grand average precipitation total over all N years. Both s 2T,b and s 2A were estimated separately for
March–May (MAM), June – August (JJA), September–November (SON), and for December–February
(DJF).
5. RESULTS
Figure 2 shows the noise standard deviations of seasonal precipitation totals during winter (JJA) and
summer (DJF) determined from (1b). We interpret these values as being estimates of the standard
deviation of seasonal precipitation values that would arise simply from the variability within seasons
which in turn we attribute to weather variations that are unpredictable at long range. Figure 2 gives us
some idea about the amount of interseasonal variability that we will never be able to predict at long
range.
The ratios of actual variance from (2) to noise variance are presented in Figure 3 for all seasons. The
bigger the ratios the more the actual variance exceeds that portion we estimate to be unpredictable and,
hence, the greater the potential predictability. We would conclude that a ratio of 2.0 indicates 50% of the
variance of seasonal totals is potentially predictable.
Sampling variability is to be expected, and even if at each station the precipitation process were driven
by a generalized chain-dependent process and its true variance given by (1b), we would not expect the
ratios to be identically unity. To get an idea how large a ratio needs to be for it to be unlikely that it
differs from unity only as a result of sampling variability we could compare it to an F-distribution.
Copyright © 1999 Royal Meteorological Society
Int. J. Climatol. 19: 405 – 421 (1999)
410
R.A. MADDEN ET AL.
Unfortunately, we do not know how many degrees of freedom our estimates of the noise, the
denominator of the ratios of Figure 3, have. Therefore, we used a Monte Carlo approach in order to put
the ratios in better perspective. Essentially, we used the generalized chain-dependent process and median
values of parameter estimates from the 20 stations to generate time series of precipitation. Then the
variance of actual seasonal totals was computed for differing time series lengths along with the variance
given by (1b). The expected value of their ratio is unity; that is, no potential predictability by our
definition. An approximate 95% value can be established by determining the value that only 5% of
estimates exceed. A further description is contained in the Appendix.
Figure 4 contains the Monte Carlo 95% line and all the ratios from Figure 3. There are 20 stations and
four seasons resulting in 80 total ratios, so even if ratios were truly unity, four ratios or 5% could be
expected to fall above the line. In fact, 41 ratios fall above the line suggesting considerable evidence for
long-range potential predictability of precipitation. There is a suggestion that summer has more potential
predictability than other seasons with ten, eight, nine and 14 ratios falling above the line for MAM, JJA,
SON, and DJF, respectively. Correspondingly the 20-station average ratios are 1.4, 1.3, and 1.3, and 1.5
for the same seasons (median values are presented in Table II(b). Figure 5 presents the distribution of
ratios for the 20 stations for each season. The potential predictability at a location is on average one third
of the total variance or less. Ratios for monthly totals (not shown) are smaller than those for seasonal
totals averaging 1.2, 1.2, 1.4, and 1.2 for April, July, October, and January, respectively. A conclusion of
less potential predictability for monthly data than for seasonal data is consistent with estimates of skill
reported in a series of prediction experiments by Mullan and Renwick (1996).
Figure 3. Ratios of actual variance to the climate-noise variance for each season: (a) MAM; (b) JJA; (c) SON; (d) DJF. Ratios of
2.0 are interpreted to mean that 50% of the actual variance exceeds the noise and is, at least, potentially predictable
Copyright © 1999 Royal Meteorological Society
Int. J. Climatol. 19: 405 – 421 (1999)
LONG-RANGE PREDICTABILITY OF PRECIPITATION OVER NEW ZEALAND
411
Figure 3 (Continued)
6. EFFECT OF SPATIAL AVERAGING
In spite of the fact that the mountain range on the South Island can affect rainfall totals oppositely on
its windward and leeward sides (Garnier, 1958), empirical orthogonal function (EOF) analysis of
monthly-averaged precipitation totals shows that the leading EOF, explaining more than 40% of the
variance, is in phase across the entire country (Kidson and Gordon, 1986). This suggests that a total of
all available stations might have more potential predictability than totals from individual stations. All
station totals would tend to minimize the noise and maximize the signal associated with this leading, in
phase, EOF if the spatial scale of the climate noise was smaller than the countrywide scale of the EOF.
Daily, countrywide, totals were determined to see if they do exhibit more potential predictability than
individual stations. A subset of overlapping data beginning on 1 March 1949 and running to 28 February
1986 (13879 days) for the 17 stations indicated in Figure 1 was used to determine these totals. The climate
noise was estimated with (1b) for this single time series and compared with the actual variance. Table II(a)
contains the estimated model parameters, the noise, actual variance and their ratio for each season. For
comparison, the median values from the 20 individual station results are presented in Table II(b).
We note that p and p11 approach 1.0 for the 17-station totals. Precipitation falls somewhere nearly every
day. Substituting p = 1 into (1b) gives:
s 2T,b : T
1+ f
(s*)2
1− f
Copyright © 1999 Royal Meteorological Society
(3)
Int. J. Climatol. 19: 405 – 421 (1999)
412
R.A. MADDEN ET AL.
Figure 3 (Continued)
which can be shown to be equal to the variance of a sum of T values for large T from a first order
autoregressive process with lag-one correlation f and variance (s*)2 (see Cryer, 1986; p. 26, Equation
3.6). Indeed, using (3) gives sT,b =57, 51, 44, and 54 mm for MAM, JJA, SON, and DJF which compare
well with values from (1b) in Table II(a).
Country totals do not appear to have more potential predictability than ones from individual stations.
In fact the ratio for country totals is less than the average ratio of station totals during MAM, JJA, and
SON. During DJF the country total ratio is 1.5 (33% of the variance is potentially predictable) and equal
to the average and median ratio of the stations for that season. A possible explanation for the lack of
improvement is that the spatial scale of the noise is as large as that of the interannual variations. That is,
the EOFs of daily data are similar to those of the monthly or seasonal means. Because of the many zeroes
in daily station data we decided against checking this by computing EOFs based on daily precipitation
totals.
7. DISCUSSION
Gordon (1986) found relatively large correlations between the Southern Oscillation Index (SOI) and
contemporaneous precipitation at New Zealand stations, and speculated that the relations might be able
to provide skill in seasonal predictions. Mullan and Renwick (1996) showed that persistence in the SOI
and in local sea surface temperature anomalies actually were contributors to some apparent skill in
precipitation prediction for the North Island, especially from Autumn to Winter and from Winter to
Copyright © 1999 Royal Meteorological Society
Int. J. Climatol. 19: 405 – 421 (1999)
LONG-RANGE PREDICTABILITY OF PRECIPITATION OVER NEW ZEALAND
413
Figure 3 (Continued)
Spring. Those predictors could explain some of the potential predictability evident at North Island
stations during JJA and SON.
It is interesting to try to relate possibly skillful methods of predicting seasonal rainfall totals with our
estimates of potential predictability. The values in Figure 3 can be expressed as the percent of variance
potentially predictable, %pp, by:
%pp =[1−s 2T/s 2A] 100%
(4)
In principle, real skill cannot exceed %pp.
For a first order estimate of the percent variance that might be predicted by the El Niño–Southern
Oscillation (ENSO) we use the correlation between Darwin, Australia pressures from one season and
rainfall totals at our stations for the following season. Darwin pressure is one component of the SOI. A
positive anomaly in Darwin pressure usually goes with a negative anomaly in SOI. The lagged
correlations, r, are given by
1 N
% [P(i, j ) − P(·, j )][S(i, j +1)− S(·, j+ 1)]
N i=1
rP( j),S( j + 1) =
sP( j)sS( j + 1)
(5)
where P(i, j ) is the Darwin pressure averaged over season j of the ith year and S(i, j+ 1) is the
precipitation total at a station during the following season ( j+ 1). The sP( j) and sS( j + 1) are the standard
deviations of Darwin pressure for season j and the station’s total precipitation for season j +1,
respectively. Again N is the number of seasons available, and
Copyright © 1999 Royal Meteorological Society
Int. J. Climatol. 19: 405 – 421 (1999)
414
R.A. MADDEN ET AL.
P(·, j )=
1 N
% P(i, j );
N i=1
S(·, j+ 1) =
1 N
% S(i, j+ 1)
N i=1
(6)
We used a 43-year record of seasonally averaged Darwin pressure resulting in overlapping time series
with precipitation totals ranging from 31 to 43 years (except for Hokitika South which was not included
because it only had eight overlapping years). Assuming independence from one year to the next the 95%
level for estimated correlations whose true value is zero is from about 0.31 to 0.36 (e.g., Panofsky and
Brier, 1968; p. 92).
Assuming (5) were to give true values of the lag correlations, then the percent variance of precipitation
total that could be predicted by previous Darwin pressures would be:
%pr = r 2P,S 100%
(7)
The corresponding range of the 95% significance level for %pr is 10–13%. Table III lists %pp and %pr
for each station and season. Ratios from Figure 3 that are less than unity result in negative values of %pp.
They have been set to zero in Table III. There are some inconsistent values in the table with %pr \%pp.
For example, Kaitaia in SON has %pr=20% and %pp = 9% (s 2A/s 2T = 1.1). These inconsistencies can
result from the fact that data from different time periods were used to estimate s 2A/s 2T and rP,S, or from
sampling variations in the estimates, or because our model of a separable potentially predictable variance
and noise component is not valid. Assuming the latter is not the problem, we conclude that on average
20–30% of the variance is potentially predictable and about 5% might be predicted by ENSO. For
country totals, that conclusion only applies to DJF precipitation. These poor results are consistent with
a recent regression based assessment of predictability of New Zealand precipitation by Francis and
Figure 4. Ratios from Figure 3 plotted versus degrees of freedom. Solid line is the Monte Carlo estimate of the 95% significance
level (see text)
Copyright © 1999 Royal Meteorological Society
Int. J. Climatol. 19: 405 – 421 (1999)
LONG-RANGE PREDICTABILITY OF PRECIPITATION OVER NEW ZEALAND
415
Table II. (a) Estimated parameters and variances for country total precipitation; (b)
median parameters and variances of the 20 station values
MAM
JJA
SON
DJF
(a)
p
p11
p01
(s*)2
m*
f
sT,a
sT,b
sA
s 2A/s 2T,b
0.93
0.94
0.82
20
4
0.28
44
56
56
1.0
0.96
0.96
0.82
15
4
0.31
37
50
50
1.0
0.95
0.95
0.90
12
4
0.27
34
44
39
0.8
0.93
0.94
0.83
19
4
0.29
41
54
65
1.5
(b)
p
p11
p01
(s*)2
m*
f
sT,a
sT,b
sA
s 2A/s 2T,b
0.42
0.60
0.29
126
8
0.06
86
88
99
1.3
0.52
0.67
0.34
96
7
0.06
81
83
95
1.3
0.48
0.62
0.35
74
6
0.04
66
67
77
1.3
0.36
0.54
0.26
136
8
0.02
77
77
88
1.5
Units for m*, sT,a, sT,b and sA are mm, and for (s*)2 mm2.
Figure 5. Histograms of the ratios s 2A/s aT,b from each of the 20 stations for the indicated seasons
Copyright © 1999 Royal Meteorological Society
Int. J. Climatol. 19: 405 – 421 (1999)
416
R.A. MADDEN ET AL.
Renwick (‘A regression-based assessment of the predictability of New Zealand climate anomalies’,
manuscript submitted to Theor. Appl. Climatol.). For other seasons there is no evidence of potential
predictability for country totals. Getting more of the potentially predictable variance will require a better
understanding of the low-frequency variability.
8. CONCLUSIONS
By modeling the variance of seasonal precipitation totals with a generalized chain-dependent process we
make estimates of variance that are unpredictable at long range. Comparison of those estimates with the
true interannual variance of seasonal precipitation totals indicates that only 30% or less of the total
variance is potentially predictable. Countrywide totals show no potential predictability except for the DJF
period when 33% of the variance is potentially predictable. One ‘signal’ that may provide some predictive
skill is persistence of the ENSO. Although there are variations from station to station and from season
to season, this skill is a small fraction of the potential predictability. More work is needed to be able to
realize a larger portion of the indicated potential predictability.
Before closing, it should be noted that our results are approximate. We do not know for sure what
model is the best for estimating the noise. In Table II we see differences in noise estimates resulting from
the two models of the noise that are considered here. Furthermore, there are uncertainties in the
parameters estimated for whatever model we choose. Nevertheless, we think the approach provides a
Table III. Percent of variance in station precipitation totals predictable by one season
lag correlations with prior Darwin pressures indicated in %pr columns, and the
estimated percent potentially predictable in %pp column for each season
Station
MAM
JJA
SON
DJF
%pr %pp %pr %pp %pr %pp %pr %pp
Kaitaia
Glenbervie F
Albert PK
Rotorua
Hamilton
New Plymouth
Gisborne
Napier
Palmerston N
Kelburn
Karioi
Hokitika S
Milford Snd
Appleby
Molesworth
Lincoln
Waimate
Dunedin
Queenstown
Invercargill
A53021
A54631
A64871
B86124
C75731
C94011
D87692
D96591
E05363
E14272
E95451
F20791
F47691
G13211
G23021
H32641
H41701
I50951
I58061
I68433
All stations average
North Island average
South Island average
Country totals
7
26
7
8
0
13
8
5
0
0
0
—
12
10
1
0
0
1
7
4
17
50
23
29
17
17
17
23
0
9
23
—
47
57
33
33
38
23
17
38
4
1
0
1
4
9
6
6
10
16
2
—
1
5
13
5
1
1
2
0
23
0
33
0
0
29
17
23
41
9
41
—
9
0
0
44
41
33
23
29
20
12
9
8
4
5
3
2
0
0
0
—
4
2
1
1
2
0
2
2
9
38
23
9
9
0
47
29
29
0
23
—
50
17
0
17
29
0
29
38
3
4
0
8
0
5
7
6
0
1
1
—
24
4
0
0
0
6
5
3
52
41
38
38
23
9
9
44
9
33
0
—
57
23
33
29
33
33
33
23
6
7
4
0
27
20
36
0
5
5
3
9
21
20
22
0
4
6
2
0
21
20
22
0
4
3
5
5
30
27
33
33
All stations, North Island, and South Island averages are simply the corresponding arithmetic
averages of values in the table. Country totals are based on the country totals discussed in the
section ‘Effect of Spatial Averging’. Hokitika South was not included because there were only eight
overlapping years of data.
Copyright © 1999 Royal Meteorological Society
Int. J. Climatol. 19: 405 – 421 (1999)
LONG-RANGE PREDICTABILITY OF PRECIPITATION OVER NEW ZEALAND
417
framework from which to consider the problem of long-range predictability, and the results provided
first-order estimates of the potential for that predictability.
ACKNOWLEDGEMENTS
This work began while Roland A. Madden was a visitor to the National Institute of Water and
Atmospheric Research Ltd., Wellington, New Zealand and partially supported by the New Zealand
Foundation for Research Science and Technology (Contract C01227). D. Wilks provided advice concerning the simulated data. Roland A. Madden, Dennis J. Shea, and Richard W. Katz are members of
NCAR’s Geophysical Statistics Project sponsored by the National Science Foundation (NSF grant
DMS-93-12686). The National Center for Atmospheric Research is sponsored by the National Science
Foundation.
APPENDIX A
In order to get estimates of the 95% limits for the ratios depicted in Figure 3, time series of precipitation
were simulated using a chain-dependent process. Observed daily precipitation data were used to calculate
each parameter necessary to define this statistical model. The distributions, including all stations and all
Figure A1. Distributions of the estimated parameters for four seasons at 20 stations. The parameters are: (a) the unconditional
probability of precipitation; (b) the variance of precipitation on wet days; (c) the probability of a wet day following a dry day; (d)
mean of precipitation on wet days; (e) the probability of a wet day following a wet day; and (f) the 1-day lag correlation on wet
days. Median values used in the simulation are indicated in each panel
Copyright © 1999 Royal Meteorological Society
Int. J. Climatol. 19: 405 – 421 (1999)
418
R.A. MADDEN ET AL.
seasons (80 values), of the six required parameters are shown in Figure A1. The median values from these
distributions were used and they are indicated in Figure A1. The simulation process proceeded as follows:
(i) a random number, r, from 0 to 1 is selected from a uniform distribution,
(ii) if r is less than p, the unconditional probability of precipitation, then a wet day is declared,
(iii) the amount of precipitation is determined from a gamma probability density function
fx =
(x/b)a − 1 exp ( −x/b)
bG(a)
(A1)
where a is the shape parameter =(m*)2/(s*)2, b is the scale parameter= (s*)2/m*, and G(a) is a
gamma function (see Wilks, 1995; p. 86). A routine written by Wilks (personnel communication) was
used to actually select a precipitation amount from the distribution (1a).
(iv) if r is greater than p, a dry day is declared,
(v) having established if the first day is wet or dry, the simulation continues selecting a random number
as in (i), then testing its size against p11, or p01 accordingly. If this test results in a wet day, a
precipitation amount is then selected as in step (iii), but, in addition an amount determined by the
1-day lag correlation, f, is added to account for possible persistence in precipitation amounts.
Step (v) is repeated over and over to generate a seasonal time series of 92 values.
Figure A2. Distribution of one 67-member, 92-value, simulated time series based on the median parameters (bottom), and, for
comparison, the distribution of observed daily precipitation amounts during March – April – May at Kelburn for 67 years (top) are
shown. Parameters determined from the data are listed. n is the number of wet days
Copyright © 1999 Royal Meteorological Society
Int. J. Climatol. 19: 405 – 421 (1999)
LONG-RANGE PREDICTABILITY OF PRECIPITATION OVER NEW ZEALAND
419
Figure A3. Distributions of 1000 ratios (s 2A/s 2T,b) determined from the
simulated data for the case of 20 members (bottom), 60 members (middle), and 100 members (top). Number of members
corresponds to the number of years in the observed data. The means and standard deviations are indicated
Copyright © 1999 Royal Meteorological Society
Int. J. Climatol. 19: 405 – 421 (1999)
420
R.A. MADDEN ET AL.
To demonstrate that the method gives realistic precipitation time series, Figure A2 shows the
distribution of simulated data generated by using the median values of the estimated parameters from all
stations and seasons, and for comparison, the distribution of actual March–April–May precipitation at
Kelburn is included in Figure A2. These data were chosen because their corresponding parameters were
roughly similar to the median values used for the simulated data. Both the Kelburn and the simulated
values consisted of 67 92-day seasons or 6164 days. It is clear that the simulated data are realistic. The
fact that there are slightly more days with precipitation of 30 mm or more at Kelburn than in the
simulated data probably reflects the 128 mm2 variance at Kelburn and the smaller 101 mm2 median
variance used for the simulated data. Similarily, the greater number of days with the smallest amount of
precipitation in the simulated data may result from the smaller mean for wet days (7.0 mm versus 8.0
mm).
To establish Monte Carlo estimates for critical probability levels a series of simulated data sets were
generated. First, 1000 20-member, 92-value series were generated and subjected to the analysis described
in the ‘Method’ section. The result was 1000 ratios whose distribution is shown in Figure A3. The
20-members and 92 values correspond to 20-year observed records of 92-day seasonal segments. Similarly,
1000 ratios were determined from 40-, 60-, 80-, 100-, and 120-member 92-value series. The distributions
of the resulting ratios for the 60- and 100-member series are also shown in Figure A3. All the distributions
are roughly normal although some positive skewness is evident. The averages of the ratios, which range
between 1.01 and 1.03, are taken as the null hypothesis for our actual ratios; that is, we consider that the
chain-dependent process captures all of the actual variability and that there is no potentially predictable
component. Then we place the 95% confidence limit for rejecting the null hypothesis and concluding that
there is evidence for potential long-range predictability at
G0.05 =R( N +1.96sN
(A2)
where R( N and sN are the mean and standard deviation for N member series, N= 20, 40, 60, 80, 100, 120.
The line in Figure 4 is drawn through the six points determined from (A2) using the median values from
Figure A1. A full test of the sensitivity of G0.05 to varying parameters is beyond the scope of this paper;
however, when we changed the parameters by plus and minus one standard deviation about the medians
from Figure A1 the largest changes in G0.05 were less than 6%.
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