Journal of Hydrology, 66 (1983) Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands

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1 Journal of Hydrology, 66 (1983) Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands [1] THE USE OF CORRELATION LINKAGES IN THE ASSESSMENT OF DAILY RAINFALL PATTERNS G.N. SUMNER Department of Geography, St. David's University College, Lampeter, Dyfed SA48 7ED ( G tea t Britain ) (Received August 25, 1982; revised and accepted December 8, 1982) ABSTRACT Sumner, G.N., The use of correlation linkages in the assessment of daily rainfall patterns. J. Hydrol., 66: A cartographic technique to indicate discrete rain areas and provide a visual estimate of the degree of correlation distance--decay functions over a wide area is illustrated using a large body of daily rainfall data for a part of Australia. Interpretation of the maps yields information on the nature of prevailing local processes involved in rainfall generation, and may be used to infer rainfall areas to be used in both hydrological and meteorological forecasting. INTRODUCTION The detailed study of spatial rainfall patterns and the time variation of rainfall intensities for discrete rainfall events provides vital hydrological information. In addition, it yields information on the precipitation processes themselves. The spatial character of detailed storm rainfall distributions and the degree to which they change and move through time, varies primarily according to changing meteorological conditions, but may also be influenced by local topographical features. Convectional storm rainfall is usually very localised in both time and space. Rainfall due to larger-scale atmospheric disturbances is on the other hand, generally widespread and of longer duration. Both types of rainfall are best investigated using dense networks of pluviographs. However, where such networks do not exist some idea of the spatial character of rainfall may be obtained by analysing daily rainfall patterns. Daffy rainfall distributions associated with days on which convectional outbreaks prevail will often be characteristically "spotty", as the rainfall associated with each convectional cell describes a limited track and composite storm paths yield a wide variation in daily rainfall depths. The most pronounced variation in rainfall depths will occur where storm paths are very short or where storms develop and decay "in situ",

2 170 with some areas experiencing high totals, but others escaping altogether. By contrast, rainfall associated with larger-scale atmospheric mechanisms will often yield a more uniform distribution of rainfall over considerable areas. In both cases, however, dominant topographic features may superimpose a quasi-permanent spatial trend. One of the major problems associated with the portrayal and interpretation of daily rainfall patterns is their extreme variability. No two patterns are ever identical and yet it is often desirable to attempt to generalise the patterns and delimit areas related to specific local rainfall producing or enhancing factors. These may in turn have an important hydrological impact. Case studies may form the most specific approach to the problem, but there are clear logistical limits to the extent to which this form of analysis may be taken, and it is difficult to isolate general rainfall distributions from them. Research is currently being carried out by the author into the spatial character of rainfall episodes over eastern Australia (eastern New South Wales and southeastern Queensland). This work is being carried out with the primary intention of delimiting rainfall regions in the area, within which more detailed hydrometeorological and climatological analyses of rainfall may be conducted. Delimitation of rainfall areas is commonly obtained statistically and is based on the degree of correlation between gauge sites (linkage analysis), using the product-moment correlation coefficient. Such analysis concentrates only on the strongest correlation links to define area boundaries, and has previously been widely used to delimit such rainfall areas. Such previous analyses have generally used monthly figures rather than those for shorter periods [e.g., Anderson (1970) for the pastoral zone of eastern Australia; and Jackson (1972) in Tanzania]. Work concentrating on shorter periods is limited by progressively increasing skewness as the timeresolution of the data is narrowed. Cornish et al. (1961), for example, used 6-day periods to define correlation fields in southern Australia. In such a dry area the dominance of completely dry days ensures that positive skew is extreme and also creates abnormally high levels of correlation over wide areas. Even after normalisation the data used by Cornish et al. (1961) still yielded higher correlations during the very dry summer months for this very reason, as recognised by them at the time. In part this problem may be overcome by calculating coefficients, using data only for pairs of sites where rain has occurred at either or both. This means that the correlation coefficients obtained more truly indicate the level of association for occasions on which rain actually occurs. Although data are still often markedly skewed, they are far less so than if dry days are included. Exclusion of dry days is also a far more useful approach in arid or semi-arid areas where the factors which control the occurrence and distribution of rainfall may be of prime concern. A consequence of data filtering in this way is of course that data populations vary considerably in proportion to the number of days on which rainfall was recorded.

3 171 Varying sample size and the use of skewed data dictate that the assign.. ment of significance levels to correlations is rarely valid. This inevitably has an impact on the usefulness of areas determined by conventional linkage analysis. Such analysis not only suffers from the difficulty of assigning levels of statistical significance, it also gives no information on the degree of association between the sites within the area itself, and between these and those in adjacent areas. In this respect the areas defined can imply an order of demarkation and homogeneity which may be very misleading. One further very common use of correlations is to attempt to portray correlation distance--decay curves and correlation fields on maps (e.g., Sharon, 1974; Sumner, 1983), but this technique assumes again that the area in which it is being carried out is a discrete meteorological, hydrological or topographical unit. In a topographically varied area, such as eastern Australia, there is first a need to define the initial rainfall areas within which to carry out such analyses. The technique adopted in this paper permits such a general initial delimitation of rainfall areas to be made, but makes no assumptions as to statistical significance. The results demonstrate, cartographically, the degree of association of rainfall over a large area, provide a visual impression of daffy rainfall regions within the study area, and may be used to demonstrate the broader-scale impact which seasonal changes and topography have on their extent. THE STUDY AREA Work in progress entails the detailed mapping of daily rainfall areas associated with different synoptic patterns, and as a part of the initial stages of this research an analysis of inter-gauge daffy rainfall correlations has been carried out for a sample of sites in eastern New South Wales. The area has been selected partly because of the considerable amount of rainfall research already carried out in this part of Australia (e.g., Hobbs, 1971, 1972; Thompson, 1973), and also because the area possesses considerable relief, with the tablelands of the Great Divide separating the nearby Pacific Ocean coast from the interior. The eastern margin of the tablelands generally takes the form of an abrupt escarpment extending up to 1500 m. This escarpment forms a very marked topographic and climatic barrier between the coastal lowlands, including embayments of major rivers such as the Hunter (Fig. 1), the tablelands, and, further west, the interior slopes and plains. The area is subject to a variety of different rainfall-producing mechanisms at different times of the year. Because of the marked relief, relatively localised, heavy rainfall is often a dominant feature in the area. However, because of the area's location (between latitudes 29 and 35 S) it lies astride the general location of the belt of subtropical anticyclones. This means that rainfall is the exception rather than the rule, and drought, water shortage and general aridity are common.

4 172 QUEENSLAND.i i! i! Moree~ ~ ::i!!:ii:ii:i!!::ii!:i:i:!.!: imannmg R ~.~ : " ) Over ~--~ ~S)/\~yd.ey ~ ~ - ~/~ I.J : ( 0 km 100 ; J Gauge Sites, ~ Study Area Boundary Fig. 1. Map of the study area. L,/ In addition, there is a pronounced seasonal variation in rainfall occurrence, which also varies across the area, particularly from north to south. Northern parts tend towards a summer maximum of rainfall associated with tropical easterlies at a time when the main anticyclone belt is well to the south. A winter maximum of rainfall, however, tends to occur in the south associated with disturbances in the temperate westerlies which impinge from the south at this time. Daily rainfall data for a 25-yr. period from 1954 to 1978 inclusive for

5 173 QUEENSLAND t'/- J,' e3 / ( ++,,'~ ; ~,, 21 \ ~, ~. 14 ~ ~ ($} r-~" MID NORTH e,,../, ~,,~ ]17 >" / ~-, / $22 ". J {N} C~ e;5 ~\\ ~ r ~ CENTRAL i HUNTER Y "~ ~ -18 / (_... T++0s,,,,,' / NORTH!,,," {E) //COAST. ET+OU,AN Gauge Locations 1. Walgett 13. Gloucester 2. Boggabilla 14. Wauchope 3. Moree 15. Muswellbrook 4. Boggabri 16. Newcastle 5. Tamworth 17. Cassilis 6. Inverell 18. Rylstone 7. Tenterfield 19. Katoomba 8. Wollomombi 20. Orange 9. Ballina 21. Coonabarabran 10. Grafton 22. Dubbo 11. Kyogle 23. Sydney 12. Coff's Harbour 24. Nowra 0 Km 100 L i Fig. 2. Location of monitoring sites used in the analysis and Australian Bureau of Meteorology forecasting areas. 24 locations in the area have been used in this analysis. The sites are shown in Fig. 2. In the selection of sites a major priority was attached to obtaining, as far as possible, an even spread of locations over the area. The separation of the gauges used has therefore been dictated by a relative paucity of gauge sites in the far west which possessed data of sufficient quality and consistency throughout the study period. This has meant that sites are on average ~ 100 km from their nearest neighbour. The use of daffy data for a 25-yr. period means that populations are consistently large even though a majority of days, the dry ones, are excluded from the analysis. METHODOLOGY Product-moment correlation coefficients based on daily rainfalls have been computed between each pair of data sites for the study period. Because of the length of this period in no case were correlation coefficients calculated from data populations (n) of less than 60. The minimum tolerated level of correlation (r) which is assumed to indicate a reasonable degree of association between sites was taken to be r > 0.4 or r < , as this implies a level of statistical explanation of only 16%, and is also close to the 5%

6 174 FEBRUAR / APR,~ ~ 7 : Tii i i.."" :; i/!1 ~~ii''i ;?'~./ DECEMBER ~ 3 /./" ~~ ~.~" /i -..." 16 i ~ TM 5 " '" ~9{:"~ ~. ~i 23 r> ~:r < ~;r < ~:r < km L i Fig. 3. Correlation linkages for daily rainfall,

7 175 rejection level of statistical significance, assuming a normally distributed data sample for n = 60 (r = ). Maps were then constructed which show the presence of correlation coefficients between sites for each month at or above a variety of correlation levels, the lowest of which is r Other levels used were at r = 0.5 (25% explanation), r--0.6 (36% explanation) and r (49% explanation). These appear in Fig. 3. The results show the month to month variation in daffy rainfall associations in a way which is both analytically useful and cartographicauy clear. The location of a relatively large number of linkages in one area as opposed to another, and the absence of any links between different areas, assist in the visual sub-division of daily rainfall zones within the study area. However, because the only data available for analysis were from sites within the study area links across the study area boundaries cannot be shown. The absence of links at the margins of the study area does not therefore indicate the absence of associations, and such "edge effects" must be noted in the interpretation of Fig. 3. THE USE OF LINKAGE MAPS The number of links, their strength in terms of the modulus value of the correlation coefficient and the presence of breaks, particularly at r < 0.4, are now used to indicate preferred areas of daffy rainfall occurrence, from which dominant influences on the nature of the rainfall in a given month may be inferred. Strong links (with high correlation coefficients) over the whole study area indicate a tendency to more widespread daffy rainfall. Non-existent or weak and scattered links indicate that daffy rainfall is sparse and highly variable (e.g., Fig. 3, November). The occurrence of areas containing numerous inter-correlated sites, but with each area separated from others by zones with an absence or very few links above the threshold, indicates that daily rainfall is generally confined to one area or another (Fig. 3, July). This may generally be taken to indicate that local controls have a marked impact on the occurrence of rainfall and on rainfall amount. The separation of rainfall areas reflects a tendency for rainfall to occur in different areas under different circulation conditions. Often, separation is afforded by uplift on the windward side of topographic barriers, or along local zones of convergence at sea breeze fronts. At other times, notably when large-scale phenomena, such as depressions, troughs or cyclones predominate, local effects are swamped out by generally good rainfall associations, and there will be numerous and extensive intra- and inter-area linkages, indicating the degree to which widespread rainfall occurs. At all times linkage maps also permit a crude estimate of correlation distance--decay rates, and, notably, indicate anisotropy in general correlation fields. For example, in May (Fig. 3) the strongest linkages at r > 0.7 occur on the coastal side of the Great Divide and show a pronounced

8 176 anisotropy, approximately parallel to the trend of both the coast and the Divide crest. Clearly, interpretations made from such maps are valid only if the distribution of sites is even over the study area. If this were not the case, then stronger linkages would simply be associated with areas of high site densities, when the separation between sites is small. INTERPRETATION OF LINKAGE PATTERNS The sequence of maps in Fig. 3 shows that there is a marked seasonal variation in the nature of daily rainfall organisation over the study area. The more extreme cases have already been indicated above and these represent features of rainfall organisation which are associated with particular prevailing atmospheric conditions. Conventional Australian nomenclature defines the summer as the period from November to March inclusive, with the true winter months as June, July and August. This division conveniently matches the types of weather generally experienced rather than purely the nature of rainfall occurrence and distribution. The seasonal progression shown in Fig. 3 indicates that the seasonal division based on daily rainfall patterns differs slightly from the normally accepted seasons. There is a clear distinction between a summer and a winter season based on the seasonal variation in prevailing winds and the location of drier anticyclonic conditions. Because of the role of topography and the alignment of the Divide along the coast, but normal to both easterly and westerly winds, there is, in many months, a notable tendency for linkages, particularly on the coast itself, to parallel the trend of higher ground. This is most pronounced in the cooler part of the year between April and October. Throughout this period maps for most months consistently show very few linkages across the divide. Those which do occur are overwhelmingly at r ~ 0.5. More localised patterns also occur, notably in July and August. In the remaining 5 months of the year linkages are much fewer, generally weak, and where more extensive as in January and February, display an organisation which contrasts with the cooler season. Except in February there is a tendency for linkage groupings to occur in well-separated and relatively small areas. During January and February there is no break in linkages coincident with the main topographic barrier similar to that which occurs in the cooler season, and in January the organisation of linkages in the south of the study area centres on one site in the Hunter Valley (Fig. 1). This valley is by far the largest of all the coastal embayments, and its upper catchment margin marks a relatively low altitude breach in the Divide. Two distinct seasons have emerged: April--October and December-- February. March and November show no pattern consistent with either season and may be considered as transitional months. A more detailed analysis of seasonal changes now follows.

9 177 April--October This period has been identified as one in which there is, (1) a large number of strong linkages in the study area; and (2), one in which the role of the main topographic barrier is clearly important in determining the nature and degree of spatial organisation of daffy rainfall. The pattern, commenced in March, but then only weakly developed, intensifies in April, until in May there is a large number of strong linkages on both sides of the Divide. These are, however, stronger and more numerous on the eastward flanks, and the tendency for links to parallel the Divide too, is greater on its eastern flank. The second period, from June to October, sees the progressive disintegration of the east--west split into four well-defined areas. By November, neither this four-way subdivision nor that along the Divide is present. This cooler period of the year is marked mostly by westerly winds, although since these approach across a large continental interior they are frequently very dry, and cold fronts and disturbances in them deposit little or no rainfall along the western flanks of the Divide. Injection of moister air ahead of disturbances in the westerlies once they reach the coast, however, can induce rainfall along its coastal flank. Thus at all times through this season linkages are relatively poorly-developed on the western slopes and on the plains below. The frequency of active rain-producing disturbances in the westerlies reaches a peak in May, and thereafter declines through the remainder of the winter (Gentilli, 1971). This too emerges in the maps in Fig. 3. As the magnitude of rainfall activity associated with westerly disturbances declines through the remainder of the period, not only is the Divide retained as an important topographic barrier, but so too do other minor topographical features emerge further to subdivide the area. Thus the far northeast around the Richmond and Clarence Rivers (Fig. 1) becomes separated from the major embayments around Sydney and the Hunter Valley, and inland the northwest and southwest are also separated. Because of its latitude, however, the area is subject not only to incursions of westerly winds, but also to easterlies of one type or another even in the winter. At all times of the year these bring large quantities of moisture off the Pacific Ocean, and the presence of the very marked escarpment forces much of this to be deposited on the coastal flanks of the Divide itself. The incidence and severity of disturbances in the easterlies during the cooler part of the year also reaches a peak in April and May ("East Coast Cyclones"; Kraus, 1954), so that in these months there is a tendency for rainfall to occur on one flank of the Divide or the other, according to wind direction. This factor is represented in the relative weakness and lack of links across the Divide, compared to their strength along its flanks. The Divide is an important topographic barrier to rainfall throughout the cooler season, but is particularly so in April and May. Where larger-scale rain-producing disturbances are weaker, further topographically-defined

10 178 divisions emerge. The boundaries of these bear a resemblance to the boundaries of combinations of official forecasting areas shown in Fig. 2. No ve rn b er--ma rch Although the months of November and March should more correctly be considered as transitional between the two types of linkage pattern, they are included in this period as they are more usually classified as being in the summer part of the year. Linkages are very scattered and confined to clearly defined areas in November (Fig. 3). This pattern develops in December when daily rainfall is apparently better "organised" in the southeast and north along a part of the coast including the Hunter Valley and parts of the Divide. Linkages into the Hunter Valley become more extensive still during January and February, indicating that for this area, daily rainfall becomes better organised with a progressively wider distribution as summer progresses. In November and December rainfall is of a relatively localised character, but by January the area is the centre of a wider zone extending across the southwest of the study area and across the Divide. A majority of links, however, lie in the range 0.4 ~ r ~ 0.5. Also during January the far northeast emerges as a separate area of better rainfall organisation with more, and generally stronger, links. February is characterised by more general links, in particular connecting the far northeast of the study area across the Divide and then south again to the Hunter Valley. In this month only southeastern parts remain largely separate. By March it is only in this southeastern portion that rainfall linkages are in any way strong or frequent. During the November--March period this part of Australia is well to the north of temperate westerly winds which dominate the area in the winter months, and initially, in November, is not subject to larger-scale rainfallproducing mechanisms in the tropical easterlies. These become frequent only in the latter part of the summer. The main rainfall features are thunderstorms so that the "spotty" character of daily rainfall associated with these is illustrated by poor and few links between sites. As the summer progresses, however, there is a steady increase in the incidence of more widespread rainfall from disturbances in the tropical easterlies and occasionally, cyclones. Such disturbances reach a peak incidence in February and are reflected in the maps of linkages for this month. The radial nature of links from the Hunter probably represents a tendency towards common, but local, rainfall generation processes in the area at this time. The Hunter Valley is the largest of a number of coastal embayments in the study area. Its upper reaches breach the Divide across a relatively low saddle of land (Fig. 1). The pattern of linkages developed therefore probably represents extreme convectional activity concentrated by orographic effects. There is still evidence of this January linkage pattern in February but its form is masked by more general linkages over a wider area. By this time, in the

11 179 late summer, disturbances in the easterlies reach a maximum, so that whilst well-organised convectional outbreaks still occur, their influence on linkage patterns is overshadowed. By March, the incidence of widespread convectional rainfall and of disturbances in the tropical easterlies is decreased, but rainfall associated with westerly disturbances has not yet attained a high frequency, so that no pronounced patterns emerge for this month. CONCLUSIONS The analysis has permitted a general appraisal to be made of the monthto-month variation in the spatial organisation of daily rainfall over a part of New South Wales, and has demonstrated the use of a technique to indicate both the regionalisation of daily rainfall and the general nature of rainfall organisation. General conclusions may be drawn relating: (1) to the technique; and (2) to its use in permitting the interpretation of seasonal trends in rainfall patterns over the study area. The technique provides a ready and speedy means by which general rainfall patterns may be indicated for a wide area. It does not permit statistical significance levels to be assigned to the relationships obtained. However, for daily rainfall, such assignments are frequently invalid since the data are generally highly positively skewed. However, the technique: (1) provides a convenient, visual and informative indication of general rainfall associations; (2) permits a ready visual comparison of month to month changes permitting an "at-a-glance" overview of seasonal trends; (3) permits the delimitation of approximate rainfall areas based on the presence or absence, frequency and magnitude, of correlation linkages; and, (4) provides a crude assessment of the nature of general correlation fields, indicating anisotropic trends. For the study area the technique has highlighted the presence of areas of daily rainfall, whose boundaries are similar to those derived for monthly and annual rainfall (Hobbs, 1971; Edwards, 1979) and agree closely with Australian Bureau of Meteorology forecasting areas, or combinations of them. It further demonstrates a significant seasonal variation in the character and organisation of rainfall. At times when larger-scale rainfall-generating activity is more pronounced, during the period of dominance of winter westerlies and during the briefer period of summer tropical disturbances, local factors which influence rainfall distributions are shown to be masked by the more general and widespread nature of rainfall occurrence. However, even at these times a degree of regionalisation of rainfall patterns may be distinguished, based on linkage patterns and areas of stronger correlations bounded by zones of poorer correlation, such as occur for example, in the June--September period. At higher levels of correlation (r ~ 0.5) there occurs a number of discrete rain areas.

12 180 (1) Between April and October the Divide emerges as an important barrier to rainfall occurrence. (2) This has a further consequence in that correlation distance--decay rates are small measured parallel to the Divide and on its coastal flank in particular. (3) In the warmer part of the year, dominated mostly by convectional activity, rainfall patterns are less constrained by the presence of the Divide itself. Early in the summer such activity is very limited, yielding only very poor linkages and associated rainfall areas. The strength of such associations increases as the summer progresses and areas are concentrated by the occurrence of large valleys and embayments such as the Hunter. (4) Smaller topographically-controlled rainfall areas emerge also in the winter months, but, (5) At all times of the year, if larger-scale atmospheric rainfall-producing features are frequent and well-organised, the impact of local topographic units is masked. ACKNOWLEDGEMENTS The bulk of the work for this paper was carried out whilst the author was a Visiting Lecturer in the Department of Geography, University of New England, Armidale, N.S.W., Australia, between July and November 1980, and its production was helped considerably by the award of a small grant made by the University towards research expenses. The author is particularly grateful to Mr. Robin Ormerod, at that time a member of that department, for consenting to the use of his research data in this exercise. REFERENCES Anderson, J.R., Rainfall correlations in the pastoral zone of eastern Australia. Aust. Meteorol. Mag., 18(2): Cornish, E.A., Hill, G.W. and Evans, M.J., Inter~station correlation of rainfall in southern Australia. Commonw. Sci. Ind. Res. Org. (CSIRO), Div. Math. Stat. Tech. Pap. No. 10. Edwards, K., Rainfall in New South Wales. Soil Conserv. Serv. N.S.W., Tech. Handb. No. 3. Gentilli, J. (Editor), World Survey of Climatology Vol. 13, Australia and New Zealand. Elsevier, Amsterdam. Hobbs, J., Rainfall regimes of northeastern New South Wales. Aust. Meteorol. Mag., 19: Hobbs, J., An appraisal of rainfall trends in northeastern New South Wales. Aust. Geogr. Stud., 10: Jackson, I., The spatial correlation of fluctuations of rainfall over Tanzania: A preliminary analysis. Arch. Meteorol. Geophys. Bioklimatol. Set. B, 20: Krans, E.B., Secular changes in the rainfall regime of southeastern Australia. Q.J.R. Meteorol. Soc., 80: 591.

13 Sharon, D., The spatial pattern of convective rainfall in Sukumaland, Tanzania: a statistical analysis. Arch. Meteorol. Geophys. Bioklimatol. Ser. B., 22: Sumner, G.N., Daily rainfall variability in coastal Tanzania. Geogr. Ann., Ser. A (in press). Thompson, R.D., The contribution of airflow circulations to local temperatures and rainfall in the New England area, New South Wales, Australia. Arch. Meteorol. Geophys. Bioklimatol. Set. B, 21:

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