INTERNATIONAL JOURNAL OF CLIMATOLOGY, VOL. 12,625-636 (1992) 551.513.7551.577.36(94)

SO1 PHASE RELATIONSHIPS WITH RAINFALL IN EASTERN AUSTRALIA

ROGER STONE

Agricultural Produelion Svstems Research Unii. Queensland Department of Primary Industries and Commonwealth Scientific. Industrial and Research Organisation. P.O. Box 102, Toowoomba, Qld 4350, Australia,

AND

ANDRIS AULICIEMS

Applied Climate Research Unit, University of Queenslund, Qld 4072, Australia

Received 3 Septenzber 1991 Accepted 24 March 1992

ABSTRACT

Phases of the Southern Oscillation Index (SOI) have been identified using cluster analysis. The monthly phases are associated with amounts of rainfall expressed in terms of probability distributions for various locations in eastern Australia. The phase representing rapid rise in SO1 indicates above median rainfall during Southern Hemisphere autumn and spring at the locations analysed. The phase representing consistently positive SO1 generally corresponds with above median rainfall amounts while the phase representing consistently negative SO1 corresponds with below median rainfall amounts. As SO1 phases relate to actual rainfall amounts, expressed in terms of probability distributions, the phases have, for instance, direct application to agricultural decision support programmes.

KEY WORDS Southern Oscillation Index Phases Rainfall probability distributions Cluster analysis

INTRODUCTION

Pittock (1975) used principal components analysis (PCA) to identify mechanisms responsible for rainfall variation in Australia. The most important were the (annual) values of the Troup Southern Oscillation Index (SOI) (Troup, 1965) and the latitude of the subtropical high pressure belt. McBride and Nicholls (1983) constructed correlation maps that represented real-time and lag associations between seasonal rainfall in Australia and seasonal (absolute) values of the Troup SOI, Wright SOI, Darwin pressure, and Papeete pressure. The maps produced by McBride and Nicholls provided a benchmark indication of rainfall variability associated with different values of the SOL Lindesay (1988) and Ogallo (1988) provided similar correlation studies for South and East Africa, respectively, while Suppiah (1989) provided correlation results for Sri Lanka rainfall.

Yet an examination of the time series of the SO1 reveals recognizable peaks and troughs and similarly occurring ‘phases’ as the SO1 demonstrates a ‘life-cycle’ or progression in Southern Oscillation (SO) events. Troup’s SOT measures the value of pressure differences across the Pacific over a set period (a month) and does not represent any change in pressure values over time. In other words, the correlation results obtained by McBride and Nicholls (1983) and others may explain only part of the relationship between the SO1 and rainfall. Evidence in the literature that relates ‘phases’ with rainfall within a life-cycle of the SO1 is lacking. This paper attempts to establish a means of identifying various life-cycle phases of the SO1 and to relate these phases to rainfall amounts in eastern Australia.

Quinn et al. (1 978) demonstrated that fluctuations in SO index anomaly trends represented interannual fluctuations in south-east trade and equatorial easterly wind strength (as represented by the SO). Significant

0899-84 18/92/060625- 12$11 .OO 0 1992 by the Royal Meteorological Society

626 R. STONE A N D A. AULICIEMS

was the suggestion made that El Niiio activity (an extreme ‘phase’ of the SO) could involve a two or three- stage development within an overall life-cycle framework of the SO. Thus, within a continuum of the SO1 there exists not only ENSO (El Niiio/Southern Oscillation) and its opposite phase ‘anti-ENS0 but ‘pre- E N S 0 and ‘post-ENSO’ and possibly ‘non-ENS0 phases. A problem remains to correctly identify precisely what constitutes an ENSO, anti-ENSO, pre-ENSO, post-ENSO, or any other ‘phase’. Quinn et al. (1978) preferred the term El Niiio ‘type’ as a means of overcoming precisely what was and was not a ‘true’ El Niiio and used an appropriate subjective classification system to classify EI Niiio type events. Similarly, one could employ an objective classification system (e.g. cluster analysis) to define phases (or ‘types’) of the SO1 and thus identify the various stages of the SO in its progression and life-cycle. So far, little appears to have been done that has attempted to isolate phases within a time series of the SOI. Such work could provide a clearer understanding of relationships between the SO and rainfall and thus provide better input into farming decision support systems. Furthermore, published relationships between the SO and rainfall usually have been confined to correlation values. The employment of phases of the SO1 allows groupings of months (or seasons) into a typology. Probability values then may be attached to the receipt of rainfall amounts in a given month at a particular location. Forecast of a given SO1 phase would then allow a ‘prediction’ of probable rainfall values for a location based on previous similarly occuring rainfall amounts recorded during that phase and month.

METHODS

Cluster analysis or a combination of principal components analysis and cluster analysis are methods available to determine a typology of observations and variables. A useful procedure developed to obtain objective indices in climatology has been the so-called ‘temporal index’, a method that uses both PCA and cluster analysis techniques to objectively group synoptic patterns over time (Kalstein and Corrigan, 1986). Stone ( 1 989) used a similar technique to temporally group synoptic weather observations.

Principal components analysis may be used to remove correlation among variables and to detect underlying linear relationships in the data. Principal components analysis has been used widely in classification exercises (e.g. Christensen and Bryson, 1966; Jones and Kelly, 1982; Whetton, 1988, White and Perry, 1989). More detailed discussion of principal components analysis may be found in Rummel (1967), Morrison (1976), Harman (1976), and Richman (1986). Briefly, the first component extracted provides the best linear explanation of combination of variables while at the same time extracting the greatest amount of variance in the data. The second component extracted becomes the second best linear component, and so on until a satisfactory amount of variance in the data has been explained (Kim, 1975).

Cluster analysis may be used to identify homogeneous groups or clusters. Thus, one may group component scores into homogeneous groups. Usually, clustering is achieved by use of agglomerative hierarchical methods in which cases are combined by grouping cases into larger clusters, step by step. Various rules may be used to decide which clusters should be combined at each step. One also may use methods known as iterative partitioning, which assign a case to a cluster depending on the distance between the case and the centre of the cluster (the centroid). A benefit of using iterative partitioning methods is that they can find good clusters with only a few passes over very large data sets and, as they are successful in minimizing variance within each cluster, are useful for identifying poorly separated clusters. Most iterative partitioning methods are based on MacQueen’s k-means algorithm (MacQueen, 1967). Both the SAS statistical package (SAS Institute, Inc., 1987) and the SPSS package (SPSS Inc., 1990) incorporate standard algorithms based on iterative partitioning methods. A useful reference for a more detailed background description of various clustering techniques may be found in Anderberg (1973).

In this exercise both agglomerative hierarchical and iterative partitioning clustering methods were applied to the orthogonal principal component scores. One may, of course, also group observations without first applying principal component analysis where it is believed the original observations and variables need not be rewritten into linearly independent components. In this instance both procedures were applied, although only the results of the more ‘successful’ procedure will be used in discussion of associated rainfall probability distributions.

SO1 AND EASTERN AUSTRALIAN RAINFALL 627

Principal components analysis and cluster analysis were applied to a SO1 data set obtained from The Bureau of Meteorology which was adjusted to contain the following variables;

(i) The ‘current’ (Troup) SOI; (ii) Lag 1 month SO1 (LISOI);

(iii) Lag 2 month SO1 (L2SOI); (iv) Difference (i)-(ii) (DlSOI); (v) Difference (ii) - (iii) (D2SOI)

Thus, not only are the various recent SO1 values used in the classification but an indication of change of values also may be represented. Monthly values of SO1 since 1882 were used rather than seasonal values as it was considered important to detect changes in SO1 that reflect changes in tropospheric mechanisms that otherwise may be hidden if seasonal values were used.

In the following both the SAS statistical package (SAS Institute Inc., 1987) and SPSS statistical package (SPSS Inc., 1990) were used to supply the relevant clustering algorithms. Each month since 1882 was able to be classified according to which phase of the SO1 it belonged.

Percentile values were attached to rainfall amounts corresponding to each month of each phase for each location chosen in the study. Box-plots were used to summarize the rainfall distributions allocated to each SO1 phase and location. The median value is located within the box-plot. The upper boundary of the box represents the 70th percentile while the lower boundary represents the 30th percentile. The length of the box signifies the spread or variability of the rainfall values (SPSS Inc., 1990). Rainfall probability values corresponding to key locations in eastern Australia that represent limits to the eastern Australian wheat belt are described here (location map Figure 1.). These locations also are within the boundaries suggested in correlation studies that represent relations between the Troup SO1 and rainfall in eastern Australia (e.g. McBride and Nicholls, 1983).

Tests were applied to determine whether the results of various rainfall probability distributions identified using the above procedures were significant statistically or whether the results were due solely to chance. As

Figure 1. Location map indicating rainfall sites illustrated in this study

628 R. STONE AND A. AULICIEMS

the rainfall probability distributions produced may not be necessarily normally distributed a non-parametric test was applied. The Kruskal-Wallis test was used to create the Kruskal-Wallis H statistic. The statistic has approximately a chi-square distribution. All cases from each of the groups were combined and ranked. For each group the ranks were summed and the H statistic computed from these sums.

RESULTS

Various analyses using cluster analysis (CA) on comprehensive data sets (e.g. SOI, LlSOI, L2SOI) with and without differences in SO1 values were run. Clusters were created using a combination of PCA and cluster analysis and also by simply clustering the original observations. The PCA analysis created a series of orthogonal principal component scores suitable for later clustering. Using the restricted variable set, PCA revealed the factor pattern shown in Table 1. Factor 1 may be described as ‘persistence’ in SO1 values. Factor 2 may be described as ‘change’ in SOI. In other words the SO1 life-cycle, using these criteria, may be described in terms of ‘persistence’ and ‘change’.

The most ‘successful’ clustering procedure used was the k-means method. The cluster pattern produced using k-means closely represented the time-series of the SOT. Furthermore, there were similar numbers of cases per cluster using this iterative partitioning method compared with, say, the average-linkage method where 1217 cases were grouped into the one cluster and no more than 15 cases were grouped into any of the remaining clusters, making comparison of resulting rainfall probability distribution analysis unfeasible. It should be emphasized that there is no necessarily ‘correct’ clustering result. Each clustering result has value depending on its purpose.

Using the ‘cubic clustering criteria’ (Mojena, 1977) to detect significant ‘break-points’ in the clusters, demarcation points at the five and 1 1 clustering levels were identified in the k-means solution. The cluster means at the five cluster level (the most significant demarcation) are described in terms of factor scores in Table 11.

Table I . SO1 factor profile”

Factor 1 Factor 2

so1 0.9059 1 0.42347 LlSOI 0.90444 - 0.4266 1 DlSOl 0~0022 1 1 ~OOOOO

*Where SO1 is the ‘current’ value of the Southern Oscillation Index, LlSOI the lag one month SO1 and DlSOI the difference in SO1 current minus previous month.

Table 11. Cluster means of significant clusters in terms of factor scores

Cluster means

Cluster Factor 1 Factor 2

1 - 1.52625 0.28033 2 1.15294 -0.20739 3 - 0.42287 - 1’52863 4 010809 1.32264 5 - 0 13258 -0.15793

SO1 AND EASTERN AUSTRALIAN RAINFALL 629

Table 111. Cluster means at the five cluster level

Cluster means

Cluster SO1 SD LlSOI SD DlSOI SD

1 -12.1 6.5 - 14.5 6.2 2.3 5.7 2 9.5 5.2 11.2 5.1 - 1.7 5.0 3 -10.0 6.8 2.7 6.5 -12.7 5.1 4 6.6 6.1 - 4.4 6 4 11.0 5.2 5 - 1.7 3.6 - 0.3 3.7 - 1.3 3.3

Table IV. Cluster means of clusters produced using iterative clustering procedure on raw SO1 data

Cluster so1 SD DlSOI SD

1 -12.08 4.63 -0.94 6.12 2 12.54 4.39 2.05 4.70 3 7.96 7.38 15.73 4.72 4 -16’73 8.37 - 16.79 5.82 5 0.43 428 -1.60 5.85

Cluster 1 in Table I1 reflects deep negative SO1 with little change. Cluster 2 reflects high positive SO1 with little change. Falling SO1 and rapidly rising SO1 are reflected in clusters 3 and 4 respectively, while cluster 5 reflects minimal value in SO1 with little change in value over the previous month.

Table 111 describes the five clusters in terms of mean value of SOI, lag one month SO1 and difference in SOI. The results of PCA and cluster analyses solutions using SOI, LlSOI, L2S01, DlSOI, and D2SOI produced results similar to those produced using the reduced data set except that the SO1 phases produced did not ‘fit’ the time-series of the original SO1 data set as well as the SO1 phases produced using the reduced data set.

A description of clusters obtained when cluster analysis was applied to data not first subject to principal components analysis is contained in Table IV. Although both agglomerative hierarchical methods (e.g. Ward’s Method, average-linkage) and iterative partitioning methods (e.g. k-means, nearest centroid sorting) were run, only the results of the iterative partitioning method are shown in Table IV.

The results of the analysis using only cluster analysis indicated some agreement between the results obtained using PCA first to remove correlation in the data (Table 111) and the results obtained using just SO1 and DlSOI without first applying PCA (Table IV). However, when an examination of the ability of each method to closely model the original SO1 time-series was made it was apparent that the results obtained by first applying PCA most closely ‘fitted’ the SO1 time series. This was especially so where change of SO1 was involved. When the SO1 made a sudden drop the SO1 ‘phase’ produced by first applying PCA to the data more closely portrayed the actual movements of the SOT than the ‘phase’ produced without first applying PCA. This may be because the use of PCA removed correlation in the SO1 group structure and allowed better group definition, suitable for clustering.

Rainfall percentile values (30 percentile, 50 percentile, and 70 percentile) corresponding to each phase produced using PCA and cluster analysis for each month since 1882 for each location in Australia were calculated. Examples of results for Emerald (May), Goondiwindi (June), Inverell (November), and Bendigo (May) are presented in Figure 2. These probability values (related to SO phases) are based on percentile calculations. They also may be described in terms of the percentage probability of exceeding a required rainfall amount (Figure 3).

Application of the Kruskal-Wallis H statistic demonstrated that the results of the above procedures were significant statistically ( P = 0.05) in most instances. Where a test indicated the distribution of all five phases

630 R. STONE AND A. AULICIEMS

Figure 2. (a) Percentile values for (a) Emerald, Queensland and (b) Bendigo, Victoria for May, and for (c) Inverell, New South Wale for November, and (d) Goondiwindi, Queensland for June

might not be different significantly, other non-parametric tests (e.g. k-sample test) produced chi-square statistics that demonstrated significance between various phases depending on time of year and location. For example, phase 4 (rapid rise in SOI) produces significantly different rainfall probability distributions from other rainfall distributions corresponding to other SO1 phases for most locations in eastern Australia during May and June.

Medians of cluster 1 or phase 1 rainfall values indicate below median rainfall amounts for each location for each month and agree with a concept of an El Niiio type rainfall amount, perhaps suggested in correlation studies. Medians of cluster 2 or phase 2 rainfall values indicate generally above median rainfall amounts for each location for each month and agree with a concept of an anti-ENS0 type rainfall distribution somewhat

SO1 AND EASTERN AUSTRALIAN RAINFALL 63 1

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5 10 15 20 25 30 35 40 45 50 55 60 65 70

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- Phase 1 -f Phase 2 * Phase 3 * Phase 4 * Phase 5

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‘Phase 1 i- Phase 2 * Phase 3 * Phase 4 * Phase 5

% probability exceeding

10 20 30 40 50 60 70 80 90 100 110 120

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-Phase 1 + Phase 2 * Phase 3 * Phase 4 * Phase 5

Figure 3. Rainfall probability charts for (a) Emerald and (b) Goondiwindi for June, for (c) Inverell for July, and for (d) Bendigo for August

suggested in correlation studies. Interesting results are obtained in clusters 3 and 4, which reflect falling and rising SO1 values respectively. Cluster 3 values generally indicate below median rainfall amounts while Cluster 4 values indicate well above median rainfall values. This is despite the fact the ‘current’ SO1 value may not be highly positive as is suggested needs to be the case in correlation studies (e.g. McBride and Nicholls, 1983; Allan, 1985).

Figure 3 describes examples of percentile values in terms of probability values according to which phase is existing at the time. Direct application to decision support may be obtained here. The required amount of rainfall needed in order to apply, for example, fertilizer can be estimated in probability terms according to whichever SO1 phase has been predicted. The Australian Bureau of Meteorology (1991) and other groups (e.g. Auliciems et a/., 1990) currently provide a seasonal forecasting system for eastern Australia that predicts SO1 values. The appendix contains a detailed month-by-month description of the SO1 phase type.

During the March-May period limited usefulness seems to be obtained from correlation results of SO1 with rainfall in many eastern districts of Australia. The above results indicate better relationships possibly are obtained once change of SO1 value is used. November relationships also may be better delineated once change of SO1 value is used.

CONCLUSIONS

Correlation studies so far have provided a benchmark level of understanding of relationships of rainfall with SOI. This paper further elucidates the relationship between SO1 and rainfall by providing an indication of a means of extracting life-cycle phases of the SOI. The phases have provided insight into percentile rainfall values that vary according to type of phase produced. The median of the phase representing persistently low SO1 values generally indicates rainfall values below the long-term median value (depending on the location of

632 R. STONE AND A. AULICIEMS

the rainfall station). The median of the phase representing persistently high SO1 values generally indicates rainfall values above long-term median values. Significantly, (rapid) rise of the SO1 value also produces rainfall values above the long-term median, even though the ‘current’ or latest SO1 value may not necessarily be highly positive. The response of rainfall to rapid rise of SO1 may vary spatially. Most significant effects produced by rapid rise of SO1 appear to occur during southern hemisphere autumn and late spring. Corresponding falling SO1 values appear to produce below average rainfall values in some instances. The resulting rainfall probability distributions may have application in decision support programmes (e.g. farming decision support models derived by Hammer et al., 1987). As application of the SO1 in rainfall studies is not restricted to eastern Australia the phases identified may have application in world regions associated, or suspected of being associated, with SO1 variability. The resulting percentile rainfall values may have use in farming decision support systems.

ACKNOWLEDGMENTS

This research was primarily completed at The Applied Climate Research Unit, University of Queensland partly as a result of research funding supplied by The Vice Chancellor, University of Queensland and Rex Falls, Regional Director, Bureau of Meteorology. Thanks also are extended to the following: The Bureau of Meteorology for supplying SO1 and rainfall data, Gerd Dowideit, Geoff Smith, Dave Butler and Alison Kelly for supplying useful comments and programming advice.

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