IT NEVER RAINS ON SUNDAY: THE PREVALENCE AND IMPLICATIONS OF UNTAGGED MULTI-DAY RAINFALL ACCUMULATIONS IN THE AUSTRALIAN HIGH QUALITY DATA SET

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1 INTERNATIONAL JOURNAL OF CLIMATOLOGY Int. J. Climatol. 24: (24) Published online in Wiley InterScience ( DOI: 1.12/joc.153 IT NEVER RAINS ON SUNDAY: THE PREVALENCE AND IMPLICATIONS OF UNTAGGED MULTI-DAY RAINFALL ACCUMULATIONS IN THE AUSTRALIAN HIGH QUALITY DATA SET NEIL R. VINEY* and BRYSON C. BATES CSIRO Land and Water, Wembley, WA, Australia Received 13 November 23 Revised 18 March 24 Accepted 3 March 24 ABSTRACT The perception prevalent in the literature that Australian rainfall records are reasonably uncontaminated by untagged weekend accumulations is reassessed. An objective probabilistic test for untagged accumulations is developed and applied to 181 gauges that have previously been identified as having high-quality data suitable for long-term analyses of climate change. As many as 12 of these gauges are found to contain hidden, untagged accumulations, and the overall prevalence of untagged accumulations in the high-quality data set is shown to be only slightly less than that of tagged accumulations. A simple study simulating the effects of accumulations in the records of the high-quality data set shows that, in records (or parts of records) with frequent accumulations, rainfall probability, mean wet-spell length and mean dry-spell length can be underestimated by as much as 24%, 34% and 18% respectively, and that the magnitude of the potential prediction error in these variables (and also in indices of rainfall intensity extremes) at a site shows strong dependence on the rainfall probability. Selected published studies on climate change are reanalysed to account for the presence of untagged accumulations and to show that significant changes in long-term trends can be obtained for individual locations. Copyright 24 Royal Meteorological Society. KEY WORDS: Australia; rain gauges; accumulated rainfall; untagged accumulations; day of week; data quality; climate change analysis 1. INTRODUCTION In order to assess and quantify climate change and variability, consistent, long-term data sets of meteorological variables are required. To provide such a data set, Lavery et al. (1992) scrutinized records for 66 daily rainfall observation sites within Australia and selected those with records extending back to 191 and with consistency in observing practices, gauge exposure, gauge location and gauge type. Much of the information they based their assessments upon was taken from station documentation. This was complemented by a suite of statistical tests to illuminate inconsistencies and non-stationarity in exposure and observer diligence. This screening process resulted in a benchmark data set of 191 gauges. In the time that has elapsed since the publication of that data set, a review of updated station documentation has resulted in a further 1 gauges being removed from the list (Haylock and Nicholls, 2). This data set has come to be known as the high-quality data set and has been used in several published analyses of climate change (e.g. Suppiah and Hennessy, 1996, 1998; Groisman et al., 1999; Hennessy et al., 1999; Haylock and Nicholls, 2; Manton et al., 21). In Australia, rain gauges are read at 9 a.m. each day and the rainfall amount is recorded against the date of observation. Throughout this paper, any reference to, for example, Sunday rainfall, means that amount of * Correspondence to: Neil R. Viney, CSIRO Land and Water, Private Bag No. 5, Wembley, WA 6913, Australia; neil.viney@csiro.au Copyright 24 Royal Meteorological Society

2 1172 N. R. VINEY AND B. C. BATES rain that fell between 9 a.m. Saturday and 9 a.m. Sunday. Thus, the expectation is that most of the Sunday rainfall would in fact have fallen on Saturday. 2. OCCURRENCE OF ACCUMULATED DATA IN THE HIGH-QUALITY DATA SET One data quality issue that was not considered as a selection criterion by Lavery et al. (1992) was the presence and prevalence of accumulated rainfall totals. Such data elements arise when a gauge remains unread for 2 days or more, thus leading to an accumulation of rainfall that has possibly fallen on more than 1 day being present in the gauge when it is eventually read. In these cases, observers are instructed to note in their records the accumulated rainfall amount and the number of days since the gauge was last read. In the published rainfall records, all days that form part of a multi-day accumulation period are flagged with a quality tag that distinguishes them from days with 24 h accumulations. Throughout this paper, we will refer to such records as tagged accumulations and will distinguish them from what we call untagged accumulations (which we introduce in Section 4). These tagged accumulations are present, at varying degrees of prevalence, in almost all of the 181 station records. They are particularly prevalent at stations located in public work places, such as post offices, where observers may be absent on weekends and public holidays. At least one-third of the 181 stations are located at post offices, and although some have very few accumulations, several have more than 25 days with accumulation tags since 191. Figure 1 shows the occurrence of accumulations at one typical station, in this case a gauge located at a post office. This site has more than 23 days with accumulated data in the 91 year period from 191 to 2. Immediately evident in Figure 1 is the change of slope in The number of days associated with accumulation periods is reasonably constant at a rate of about 11 per year up until However, since then they have occurred at a rate of about 6 per year. The change in slope coincides with the year in which post offices in Australia switched from a 6 day week to a 5 day week. That is, they no longer opened for business on Saturdays. As a consequence, the rainfall observer was often not on-site to read the gauge on either Saturday or Sunday, so Monday s reading comprised a 3 day accumulation. At the post office in Figure 1, where rain falls on about 8 days per year on average, the fivefold increase in accumulated data after 1974 cannot be explained in full by a change from 2-day weekend accumulations to 3-day weekend accumulations Number of accumulations Figure 1. Cumulative occurrence of accumulated rainfall totals for a typical rainfall station, Copyright 24 Royal Meteorological Society Int. J. Climatol. 24: (24)

3 MULTI-DAY RAINFALL ACCUMULATIONS IN AUSTRALIAN RAINFALL DATA SET 1173 Clearly, the increase in the reporting rate of accumulated data in 1974 must also have coincided with some other type of cultural change in the gauge reading practices at this station. The conversion from imperial to metric measurement units for rainfall observations, which also occurred in 1974, is unlikely to have had any impact on accumulation reporting rates. Figure 2(a) shows that almost all the days with accumulations at the same station were Saturdays, Sundays or Mondays. If we remove all accumulated data from the record, then the number of rainy days (defined here as days with non-zero 24 h rainfall totals) is substantially less on the three weekend days (Saturday to Monday) than on the four midweek days (Tuesday to Friday; Figure 2(b)). Interestingly, although Figure 2(a) suggests that accumulations affect Sundays and Mondays with roughly equal frequency, there remain far more rainy Mondays than rainy Sundays in Figure 2(b). Sundays and Mondays are, however, approximately equal in rainfall amount when the accumulations are neglected (Figure 2(c)), but Sundays have substantially heavier mean events (Figure 2(d)). This suggests that although the observers may not have been officially at work on Sunday mornings, they may have been more inclined to make a special effort to read the gauge on Sundays if they knew that there had been substantial rainfall in the previous 24 h. This is highlighted further by the post-1974 record, which shows just 1 rainy Saturdays and three rainy Sundays in 27 years. However, the mean intensity of those Saturday events is twice the midweek average, and that of the Sunday events is three times the midweek average. Another aspect of observer bias is shown in Figure 3, for a gauge with about 45 accumulated days in a 51 year period, almost all of which occurred on Sundays and Mondays. If we again neglect days with accumulated data, then we find that both the number of rainy days (Figure 3(b)) and the total rainfall amount (Figure 3(c)) for Sundays are comparable to those on other days. In contrast, both occurrence and amount are substantially reduced on Mondays. In fact, the shortfall in rainy days on Mondays Number of missing observations Missing days (a) Number of rainy days Rainy days (b) Total rain amount (mm) Rain amount (c) Mean rainfall per event (mm) Intensity (d) Figure 2. Rainfall statistics for the 94 year period 197 2, apportioned by day of the week for station 2515 showing (a) the number of days without 24 h rainfall observations, (b) the number of rainy days (excluding days with accumulations), (c) the total rainfall amount, and (d) the mean rainfall amount per rainy day Copyright 24 Royal Meteorological Society Int. J. Climatol. 24: (24)

4 1174 N. R. VINEY AND B. C. BATES Number of missing observations Missing days (a) Number of rainy days Rainy days (b) Total rain amount (mm) Rain amount (c) Mean rainfall per event (mm) Intensity (d) Figure 3. As for Figure 2, but for station 1622 over the period is approximately equal to the number of Monday accumulations. It would appear that at this site the observers read the gauge on Sundays only if there had been rain in the previous 24 h. On other weekends, including those with rain on Monday, the (empty) gauge was not read on Sunday, so Monday s rain was recorded as a 2 day accumulation. This hypothesis is further supported by the observation that of the 183 Mondays with 1 day accumulations in Figure 3(b), no fewer than 173 directly followed rainy Sundays. Although the observers appear to have acted entirely in accordance with the Bureau of Meteorology s instructions (and, indeed, have shown extra diligence in observing the gauge so frequently on rainy Sundays when they might have been excused from duty), they have introduced a selection bias into the Sunday observations. Similar observation patterns are evident at other stations prior to the 197s, particularly those in rainfall districts in northern and western South Australia. This observation practice has implications for the development of strategies for coping with accumulated data and will be discussed further in Section 3. The occurrence of days with accumulated data for the 181 stations of the high-quality data set is shown in Figure 4. The increase in prevalence of accumulated data in 1974 is clearly shown. Groisman et al. (1999) and Hennessy et al. (1999) attribute this increase to the cessation of Saturday trading at post offices in February There is also evidence of a decline in accumulations since 1992, which Hennessy et al. (1999) attribute to the increasing deployment of automated rain gauges. In the early part of the 2th century, the occurrence of accumulations was reasonably constant at around.5% until 1945, with a slightly increasing trend thereafter. Despite presenting some evidence to suggest that Sunday rainfall totals may be about 1% less than those on other days of the week, Hennessy et al. (1999) conclude that, since Monday rainfall totals appear satisfactory, this relative absence of accumulations prior to 1974 is probably due to greater observer diligence, rather than to failed reporting of accumulations. Copyright 24 Royal Meteorological Society Int. J. Climatol. 24: (24)

5 MULTI-DAY RAINFALL ACCUMULATIONS IN AUSTRALIAN RAINFALL DATA SET Accumulation prevalence Figure 4. Occurrence of accumulated data (mean number of days per year per operational station) in the 181 stations of the high-quality data set 3. APPROACHES TO COPING WITH ACCUMULATED DATA Despite the obvious potential for observer selectivity to bias some of the rainfall observations, particularly at those stations with large numbers of accumulations, it does not necessarily follow that Lavery et al. (1992) erred in including those stations in the high-quality data set. Firstly, it is not unreasonable to expect that total rainfall will be more or less preserved in such records. Secondly, given that the accumulations are tagged, it is relatively easy for the user to find them and devise simple ways of dealing with them. The treatment of accumulated data depends on the application they are being used for. For example, Suppiah and Hennessy (1996), in a study of the intensity and frequency of heavy summer rainfall in tropical Australia, distributed accumulated rainfall evenly over the accumulation period. Despite the obvious tendency to overestimate slightly the frequency of rainy days and to underestimate their intensity, this is a justifiable strategy for tropical stations, where there is a high conditional probability of a rainy day given rain on the previous day. As part of their study, Suppiah and Hennessy (1996) tested the effects of several other distribution patterns and found little evidence of significant differences in the intensity of heavy rainfall. Suppiah and Hennessy (1998) and Hennessy et al. (1999) later cited this test as justification for adopting the same distribution strategy in Australia-wide studies of rainfall frequency and intensity, although it is likely to be less valid for temperate regions, where conditional probabilities may be significantly less than in the tropics. In contrast, Haylock and Nicholls (2) used two separate distribution strategies. One was to treat all accumulations as missing data. However, in their analysis of the number of days with rainfall greater than 1 mm, they appear to have treated each accumulation period as comprising one rainy day amongst a sequence of dry days. Clearly, this latter strategy would lead to an underprediction of the number of rainy days. However, it should be noted that Suppiah and Hennessy (1998), Hennessy et al. (1999) and Haylock and Nicholls (2) all strived to minimize the potential for problems associated with the use of accumulated data by objective elimination from their analyses of stations with large numbers of accumulations. An alternative strategy is to use information from neighbouring stations to distribute accumulations. Potential techniques are similar to those used for spatial interpolation, and include Thiessen polygons, distanceweighted interpolation and geostatistical methods such as splining and kriging (Goovaerts, 2; Xia et al., 21). However, the accuracy of these techniques decreases with decreasing station density, a problem that Copyright 24 Royal Meteorological Society Int. J. Climatol. 24: (24)

6 1176 N. R. VINEY AND B. C. BATES is particularly acute in the relatively sparse Australian observational network. Spatial interpolation techniques can also be affected if the quality of observations at neighbouring stations is compromised. A non-distributary treatment was tested by Suppiah and Hennessy (1996). In their study of heavy tropical rainfall, the use of stations with large numbers of accumulations was unavoidable. Recognizing the potential implications, they compared long-term trends in 9th and 95th percentile rainfall intensities calculated from midweek data only (i.e. Tuesday to Friday) with those calculated using a 7 day week. They found some notable changes in trends for some stations with large numbers of accumulations; but, since similar changes were observed for stations with few accumulations, they concluded that the differences could not necessarily be attributed to the removal of weekend accumulations. They further noted that the reduction in sample size of 43% reduces the significance levels of any observed trends. In the light of Figure 3, it is worth noting that, as well as depending on the type of application, the treatment of accumulated data, especially the distributary treatments, should also be predicated on the observational practices that lead to the accumulations and on the biases inherent in those practices. For examples like Figure 3, a distribution strategy that apportions all the accumulated rainfall to Monday and leaves Sunday dry is likely to be more appropriate than a strategy that distributes rainfall evenly among the accumulation days. 4. UNTAGGED ACCUMULATIONS The statement by Hennessy et al. (1999), that the relatively low occurrence of rainfall accumulations in the first half of the 2th century was due to greater observer diligence, merits closer investigation. Consider Figure 5. This station, which is part of the high-quality data set, has fewer than 2 days with accumulated data in the period , almost all of them on Sunday and Monday. However, after removing those days with accumulated data from the analysis, there are only 87 rainy Sundays (Figure 5(b)), compared with Number of missing observations Missing days (a) Number of rainy days Rainy days (b) Total rain amount (mm) Rain amount (c) Mean rainfall per event (mm) Intensity (d) Figure 5. As for Figure 2, but for station 155 over the period Copyright 24 Royal Meteorological Society Int. J. Climatol. 24: (24)

7 MULTI-DAY RAINFALL ACCUMULATIONS IN AUSTRALIAN RAINFALL DATA SET rainy Mondays and an average of 889 rainy days for the remaining days of the week. Clearly, even if all 96 Sundays with accumulated data (Figure 5(a)) are added to this total, rainy Sundays are still vastly underrepresented. The discrepancy is too large to have been caused by chance. The number of rainy Mondays is substantially higher than the Tuesday Saturday average, despite having 97 Mondays with accumulated data removed from the analysis. Clearly, rainy Mondays are overrepresented in Figure 5(b). In Figure 5(c), the rainfall totals on Sundays and Mondays are respectively significantly less and significantly greater than the average of the other days. The mean intensities for both Sunday and Monday are slightly greater than the Tuesday Saturday average (Figure 5(d)). Clearly, the gauge depicted in Figure 5 includes a very large number of untagged weekend accumulations. In other words, while recording the rainfall depth of weekend accumulations as Monday rainfall, the observers have failed to flag most of the accumulations in their records. For a station where the untagged accumulations are overwhelmingly 2 day accumulations (Sunday and Monday), we can obtain a first approximation of the number of untagged accumulations from the difference between the number of rainy Mondays and the number of rainy Sundays. In the case depicted in Figure 5, we may surmise that about 1 weekends with untagged accumulations are concealed within the 56 year record. Another example from the high-quality data set of a gauge with untagged accumulations is shown in Figure 6. Here, during the 7 year period there were no tagged accumulations and no missing data. However, rainfall was recorded on only one Sunday during the period out of a total of 74 rainy days. That one Sunday event was, however, a significant fall of nearly 12 mm. As was the case with the data in Figure 5, Monday rainfall also appears to be over-represented here, both in frequency and amount. Based on the difference between the recorded occurrences of rainy Mondays and rainy Sundays in Figure 6, we may estimate that the data from this gauge contains about 17 weekends with untagged accumulations during the 7 year period. Number of missing observations Missing days (a) Number of rainy days Rainy days (b) Total rain amount (mm) Rain amount (c) Mean rainfall per event (mm) Intensity (d) Figure 6. As for Figure 2, but for station 9557 over the period Copyright 24 Royal Meteorological Society Int. J. Climatol. 24: (24)

8 1178 N. R. VINEY AND B. C. BATES 5. AN OBJECTIVE TEST FOR UNTAGGED ACCUMULATIONS In this section we develop an objective test for the presence of untagged accumulations and apply it to each of the 181 high-quality gauges for the 111 year period, We first make the assumption that untagged accumulations overwhelmingly include one Sunday on which the rain gauge remains unobserved and that data recorded between Tuesday and Friday (hereafter referred to as weekdays ) are relatively uncontaminated by untagged accumulations. In making this assumption we note that, apart from weekends, the other times that may be prone to accumulations are public holidays. In Australia, most public holidays fall on Monday, although some (e.g. Anzac Day (25 April) and the Christmas and New Year public holidays in late December and early January) occur on fixed dates and can, therefore, fall on weekdays. We also assume that for any accumulation, tagged or untagged, the probabilities that part of the rain fell during any particular day of the accumulation period are equal. This assumption will obviously not hold for gauges like that in Figure 3, but the effect of such observational biases will act conservatively on the prediction of untagged accumulations. The test involves calculating the probability that the number of rainy Sundays in any given year could be as few as was observed. In order to do this we need an unbiased estimate of the probability p 1 of rain on any given Sunday in a particular year. For this we used the weekday rainfall probability, i.e. the number of rainy weekdays divided by the total number of weekdays during the year. Weekdays that form part of tagged accumulations were ignored. In some years, some of the gauges, particularly those in drier areas or those that were closed for part of the year, had few non-zero weekday rainfall observations. This leads to considerable uncertainty in the estimation of annual rainfall probabilities. In order to reduce this uncertainty, these annual rainfall probabilities were smoothed using a 5-year weighted average. Five years was deemed large enough to increase sample sizes sufficiently, yet was small enough to retain information about short- to medium-term changes in rainfall climate or in observer practice. There is some evidence of systematic variations in rainfall amount on different days of the week, particularly in large cities. For example, Simmonds and Keay (1997) found that in Melbourne, Australia, during the period , average daily rainfall totals on weekdays (which they defined as Monday to Friday) were 1% greater than on Saturdays and Sundays. They attributed this difference to the greater anthropogenic heat emissions on weekdays. In developing the high-quality data set, Lavery et al. (1992) were cognizant of the potential for urbanization to mask climatic trends and, consequently, eliminated stations in urban areas from the data set. As a result, effects such as those described by Simmonds and Keay (1997) are unlikely to have significant impact on the high-quality data set or to compromise the assumption made here, that weekday rainfall probability is an adequate predictor of Sunday rainfall probability. As already noted, the midweek rainfall probability can potentially be affected by untagged accumulations associated with public holidays. For example, many public holidays in Australia occur on Mondays, so some rain gauges may remain unread until Tuesday. If these accumulations are not tagged, then it is likely that the number of rainy Tuesdays will be over-observed, since some of these records will include rain that fell on one or more days prior to a dry Tuesday. However, this effect is compensated by the possibility that during other untagged public-holiday accumulations, some weekday rainfall events will be unobserved. Analysis of the potential overall effect of untagged public-holiday accumulations for a typical cycle of public holidays (1 days per year falling on various days of the week) shows that the expected number of rainy Sundays can be overpredicted slightly for sites with p 1 <.39 and underpredicted slightly for sites with p 1 >.39. The maximum magnitude of this overprediction is.25 rainy Sundays per year in some of the drier sites. This amount is unlikely to have a significant impact on the detection of untagged Sunday accumulations. Furthermore, since it is highly likely that any gauge with a preponderance of untagged publicholiday accumulations will also have significant numbers of weekend accumulations, the possibility of false identification of a gauge with weekend accumulations is remote. On this basis, p 1 was accepted without modification as the predictor of the Sunday rainfall probability. One problem remains: to take account of any tagged accumulations involving Sundays. We note that, for an accumulation of a days duration, the probability p S that part of this rain fell on Sunday is equal to the probability of Sunday rainfall divided by the probability of rain during the accumulation period. Then, Copyright 24 Royal Meteorological Society Int. J. Climatol. 24: (24)

9 MULTI-DAY RAINFALL ACCUMULATIONS IN AUSTRALIAN RAINFALL DATA SET 1179 assuming the rainfall sequence may be approximated as a first-order Markov process, p S = p 1 1 (1 p 1 )(1 p 1 ) a 1 where p 1 is the conditional probability of a wet day occuring immediately after a dry day and, like p 1,is calculated from weekday data. Thus, using this equation for each tagged accumulation during the year, we may obtain an estimate of the expected number of rainy Sundays associated with the tagged accumulations. This is rounded upwards and added to the observed number of rainy Sundays to give the total number of rainy Sundays during the year N o. The probability that the true number of rainy Sundays N does not exceed N o in any given year is p(n N o ) = N o i= n! i!(n i)! pi 1 (1 p 1) n i (1) where n is the number of number of Sundays in the record. This number includes those Sundays with rainfall observations (whether zero or non-zero) and those Sundays that are part of tagged accumulations, but it does not include periods during which the gauge was not operating at all. In most years n = 52. Whenever p(n N o ) is less than some critical threshold, we may conclude that the number of rainy Sundays appearing in the record for that year is too few to have been caused by chance. These are the years for which it is reasonable to suspect the presence of untagged weekend accumulations. A Monte Carlo analysis was used to establish the threshold value, which was set at the level that we would expect to be breached by chance in just 1 year of a 1 year period by no more than 5% of gauges. This threshold probability p c was found to be equal to.8 and to be approximately invariant with n and with p 1. Figure 7 shows annual values of p(n N o ) for one of the gauges in the high-quality data set. It clearly shows a preponderance of untagged weekend accumulations in 1948 and between 1957 and In the years up to about 1945, the annual non-exceedence probabilities are scattered evenly about.5 (the expected value) with a perfectly acceptable minimum of.5. However, from the late 194s to the early 197s, there p ( N N obs ) e 5 1e 6 1e 7 1e 8 1e Figure 7. Annual non-exceedence probability for rainy Sundays at station 1525, The solid horizontal lines show the expected mean (.5) and the critical probability (.8). Years with non-exceedence probabilities of less than p c are marked with a solid circle Copyright 24 Royal Meteorological Society Int. J. Climatol. 24: (24)

10 118 N. R. VINEY AND B. C. BATES is a noticeable decrease in non-exceedence probabilities, with 9 years falling below p c. Non-exceedence probabilities recover somewhat from the early 197s onwards, although there appear to be periods in the late 198s and the late 199s with sustained low values. During the periods and , while the gauge consistently has probability values of less than.5, these values never drop below p c. Nonetheless, the persistence of these low values of p(n N o ) is suggestive of a systematic under-observation of Sunday rainfall during this period. For example, the likelihood of p(n N o ) dipping below.5 for 3 years in a row (e.g ) must clearly be much less than.5, and may well have been less than we could reasonably expect by chance. To test this, the probabilities given by Equation (1) were repeated using 3 year and 5 year sequences of Sundays. Given that p c is approximately invariant with n, we may plot these probabilities on the same graph (Figure 8). It is now evident that the entire sequence of years between 1947 and 1973 contains untagged accumulations. Furthermore, the period from 1997 to 2 is now also identified as having too few Sunday rainfall events, but the questionable period in the late 198s remains credible. By way of contrast, the probabilities shown in Figure 9 do not indicate the presence of untagged accumulations at any time during that gauge s record. Annual non-exceedence probabilities remain evenly scattered about.5 throughout the record, with no sustained periods of low probabilities. The analysis shown in Figures 8 and 9 was carried out for all 181 gauges in the high-quality data set. Untagged accumulations were assumed to occur in years where the 1 year non-exceedence probability was less than p c, or either of the 3 year or 5 year non-exceedence probabilities centred on that year were less than p c. Untagged accumulations were also assumed where both the 3 year probabilities on either side of a particular year were less than p c or where at least two of the 5 year probabilities on either side of a particular year were less than p c. Figure 1 indicates that 12 of the 181 gauges show evidence of untagged accumulations during the period 189 to 2. Of these, 63 have sequences of at least five successive years of untagged accumulations, with one gauge having as many as 84 years of untagged accumulations. The occurrence of untagged accumulations appears to have declined significantly since the mid-197s. A significant proportion of the incidences of untagged accumulations since 198 occur around 1983, particularly in southeastern Australia. Examination of exceedence probability plots for many of the sites in Victoria and Tasmania and some sites in southern New South Wales and eastern South Australia suggests p ( N N obs ) e 5 1e 6 1e 7 1e 8 One year Three year Five year 1e Figure 8. As for Figure 7, and for the same station, but with the 3 year and 5 year non-exceedence probabilities included. Years with 3 year or 5 year non-exceedence probabilities of less than p c are marked with solid diamonds and triangles respectively. Other years that are flanked on both sides by solid markers are indicated with an open diamond or triangle Copyright 24 Royal Meteorological Society Int. J. Climatol. 24: (24)

11 MULTI-DAY RAINFALL ACCUMULATIONS IN AUSTRALIAN RAINFALL DATA SET p ( N N obs ) One year Three year Five year Figure 9. As for Figure 8, but for a gauge (station 865) without any apparent untagged accumulations that Sunday rainfall was consistently underrecorded in the early 198s throughout the region. Figure 9 is an example of a gauge in Victoria with a low (but not subcritical) exceedence probability in Given the widespread prevalence of low exceedence probabilities during this period, it is possible that this reflects a natural phenomenon (i.e. that Sunday rainfall was unusually infrequent that year in comparison with other days of the week) and may be associated with the record low values of the southern oscillation index that were recorded at the same time. Further analysis suggests that the low exceedence probabilities during this period are not statistical artefacts associated with the use of 5 year weighted averages for annual rainfall probabilities. Thus, if this phenomenon is indeed natural, we might reprieve many of the gauges from the suspicion of having untagged accumulations in the early 198s, especially those which do not show signs of untagged accumulations in other years. For each station, we may estimate the number of untagged accumulations by the difference between the expected and observed numbers of rainy Sundays. Using this procedure, but only for years that have been identified by their non-exceedence probabilities as deficient in rainy Sundays, we may compare the prevalence of untagged accumulations with the prevalence of tagged accumulations. Figure 11 shows that the occurrence of untagged accumulations had a slightly increasing trend of 1. to 2.5 Sundays per year per station between 19 and 1961, before declining abruptly in 1962 and again in In contrast, the occurrence of tagged accumulations increased during the 194s and 195s from a steady base of about.75 to 1.5, stabilized briefly and then increased abruptly in There also appears to have been a reduction in the number of tagged accumulations since It should be noted that this analysis is likely to underestimate the number of untagged accumulations for two reasons. Firstly, it is only for years that are bad enough to be identified as having subcritical probabilities. For many of the gauges, there are other years with suspected untagged accumulations, but where the prevalence of those accumulations is insufficient to reduce p(n N o ) below p c. Secondly, the analysis presented here only seeks to identify untagged accumulations that involve an underrecording of Sunday rainfall. It is likely that any station with untagged weekend accumulations will also contain untagged midweek accumulations corresponding to midweek public holidays. When the number of tagged and untagged accumulations are taken together (Figure 11), it is seen that the abrupt changes in each in 1974 caused only a barely perceptible change in the total number of Sunday accumulations, with the decrease in untagged accumulations being offset by the coincident increase in tagged accumulations. To a lesser extent, the decrease in untagged accumulations in 1962 is also offset Copyright 24 Royal Meteorological Society Int. J. Climatol. 24: (24)

12 1182 N. R. VINEY AND B. C. BATES Tas Vic NSW Qld SA NT WA Figure 1. Occurrence of untagged accumulations (circles) for each of the 181 gauges of the high-quality data set for the period Each horizontal line represents one station, with the thin lines indicating the years of operation for each station. Stations are grouped by state and territory (WA: Western Australia; NT: Northern Territory; SA: South Australia; Qld: Queensland; NSW: New South Wales; Vic: Victoria; Tas: Tasmania). There are no stations from the Australian Capital Territory or from offshore territories in the high-quality data set by an accompanying increase in tagged accumulations. Over the period 191 to 1992 there is a reasonably consistent increasing trend in total Sunday accumulations. Of course, given the increasing prevalence of 3 day accumulations after 1974, the total number of days involved in accumulations is likely to have increased significantly since then. Copyright 24 Royal Meteorological Society Int. J. Climatol. 24: (24)

13 MULTI-DAY RAINFALL ACCUMULATIONS IN AUSTRALIAN RAINFALL DATA SET Days with Sunday accumulations Untagged Tagged Total Figure 11. Approximate number of Sundays with tagged and untagged accumulations (mean number per year per operational station) Table I. Mean values of rainfall probability and the mean number of tagged and untagged Sunday accumulations per year of record for each state and territory State or territory No. of stations Rainfall probability Sunday accumulations Tagged Untagged Western Australia Northern Territory South Australia Queensland New South Wales Victoria Tasmania All The total number of Sundays with untagged accumulations across the entire 181 high-quality stations is about 22 8, an average of 126 per station. This total is of similar magnitude to the total number of Sundays with tagged accumulations, i.e. 26 8, or 148 per station. The greatest number of Sundays with untagged accumulations at any one station is 114, which compares with the maximum number of tagged Sunday accumulations (1292). The maximum combined total of Sunday accumulations at any station is 236, and 16 stations have more than 1 Sunday accumulations. Table I lists the mean number of tagged and untagged Sunday accumulations per year of record for each state and territory. Comparisons between states are complicated by the likelihood that regions with higher rainfall probability (and, therefore, more potential Sundays with rainfall) will have greater numbers of accumulations. Despite this, it would appear from Table I that stations in South Australia have significantly more tagged accumulations than the national average, whereas those in New South Wales have significantly more untagged accumulations. Three states (Western Australia, New South Wales and Victoria) have more untagged than tagged accumulations. The large disparity between tagged and untagged accumulations in South Australia reflects, in part, the preponderance of gauges with Copyright 24 Royal Meteorological Society Int. J. Climatol. 24: (24)

14 1184 N. R. VINEY AND B. C. BATES observational characteristics like those of Figure 3, where Monday rainfall is underrepresented. An objective test similar to Equation (1) could easily be developed to detect such stations, but this has not been pursued here. 6. IMPLICATIONS OF UNTAGGED ACCUMULATIONS: POTENTIAL IMPACT ON CLIMATE INDICES The magnitude of the potential impact of untagged accumulations on a variety of commonly used climatic indices may be evaluated by simulating the occurrence of weekend accumulations in the rainfall records. For each of the 181 stations the following long-term climatic averages were calculated: rainfall probability p 1, 9th (P 9 ) and 95th (P 95 ) percentiles of rainfall intensity, mean wet-spell length t w and mean dry-spell length t d. This procedure was repeated after accumulating rainfall totals from two successive days of the week into a single event. In this case, all Thursday rainfall was transferred to Friday s total. In this way, the effects of 2 day accumulations could be assessed. The procedure was repeated again after all Wednesday and Thursday rainfall was transferred to Friday s total in order to test the effects of 3 day accumulations. Wednesday, Thursday and Friday were used in this simulation because, in general, they contain far fewer existing accumulations than the weekend days. The impacts on the various climatic indices, averaged over all 181 sites, are shown in Table II. The presence of accumulated data leads to decreases in p 1,P 9, t w and t d and to increases in P 95. Interestingly, the impact on t d is greater for 2 day accumulations than for 3 day accumulations. The information in Table II, however, because it is averaged over all 181 sites, tells only part of the story. For each of the climatic indices, the magnitude of the response is strongly dependent on the frequency of rainy days (Figures 12 14). The proportional impact on rainfall probability becomes greater as the probability increases (Figure 12). The 3 day accumulations can decrease p 1, and hence the observed number of rainy days, by more than 2% at sites with high rainfall probability. Further detailed analysis for selected gauges indicates that, in addition to the gross ratios depicted in Figure 12, both seasonal and interannual ratios of rainfall probability follow the same trend. Figure 13 shows that, for both 2 day and 3 day accumulations, the 9th and 95th percentiles of daily rainfall totals are slightly greater than for the unaccumulated case for sites with high rainfall probability, but are substantially less for sites with low rainfall probability. For both percentiles, the departure from the unaccumulated case is greater for the 3 day accumulations. The point at which both 9th percentiles cross the 1 : 1 line is at a rainfall probability of about.4, whereas the 95th percentiles cross at about.2. This explains the observation in Table II of the average P 9 decreasing as the level of accumulation increases, while the average P 95 increases. Accumulations affect the cumulative distribution function of rainfall amounts in two ways. Firstly, any accumulation of rainfall days will decrease the number of rainy days and, therefore, increase the level of the highest percentile corresponding to zero rain and decrease the intensity of adjacent percentiles. Secondly, accumulation will generally increase the peak intensities and, therefore, the very highest percentile values. Sites with low rainfall probability are likely to have their P 9 and P 95 values dominated by the first Table II. Mean values of various climatic indices under three simulated accumulation schemes Climatic index Accumulation scheme None 2day 3day Rainfall probability p th percentile P 9 (mm) th percentile P 95 (mm) Mean wet-spell length t w (days) Mean dry-spell length t d (days) Copyright 24 Royal Meteorological Society Int. J. Climatol. 24: (24)

15 MULTI-DAY RAINFALL ACCUMULATIONS IN AUSTRALIAN RAINFALL DATA SET Proportional rainfall probability Rainfall probability Figure 12. Ratios of long-term daily rainfall probabilities for data with simulated 2 day (triangles) and 3 day (circles) accumulations to those of the raw rainfall series, plotted as a function of the raw rainfall probabilities. The linear trend lines for the 2 day (solid line) and 3 day (broken line) accumulations are significant at the 1% level effect, because those percentiles are close to the highest zero-valued percentile. On the other hand, sites with high rainfall probabilities are likely to have their P 9 and P 95 values dominated by the second effect, thus leading to increases in intensity. Ratios of average wet- and dry-spell length are shown in Figure 14. Average wet-spell lengths decrease substantially for both accumulation strategies, and the decrease is greatest for gauges with high rainfall probabilities. Average dry-spell length decreases for all 2 day accumulations and for some 3 day accumulations, but it increases for other 3 day accumulations, particularly for sites with high rainfall probability. Detailed analysis of wet and dry spells shows that the fragmentation of spells (i.e. the total number of spells) increases for both accumulation strategies, but that the increase is greater for 2 day accumulations than for 3 day accumulations. This is because many 3 day accumulations actually result in decreased fragmentation, especially in wet locations. For example, the 5 day sequence dry wet dry dry wet becomes dry dry dry wet wet when the middle 3 days are accumulated, and this results in a reduction in the total number of spells. In contrast, only one potential 4 day sequence (dry wet dry wet) results in a 2 day accumulation with decreased fragmentation. However, its occurrence probability is so low that its impact on moderating the increased fragmentation caused by other sequences is negligible. As shown in Figure 12, the total number of dry days increases as the level of accumulation increases, especially for sites with high rainfall probability. For 3 day accumulations at these wet sites, this increase in dry days more than offsets the moderate increase in the total number of spells, and thus leads to an increase in average dry spell length. Figures indicate the responses that can be expected in the various climatic indices if the gauges are never read on one or two particular days of the week. As such, they represent the maximum potential impact of accumulated rainfall totals, regardless of whether the accumulations are tagged or not. However, since interannual trends are similar in pattern and magnitude, we could reasonably expect similar responses to individual annual climatic indices as a result of sustained accumulations. This, coupled with the results shown in Figure 1, in which many stations suffer from untagged accumulations in some, but not all, years, suggests that any assessment of long-term trends in annual values of these indices could be severely compromised by the presence of accumulated data. Copyright 24 Royal Meteorological Society Int. J. Climatol. 24: (24)

16 1186 N. R. VINEY AND B. C. BATES 1.2 Proportional 9th percentiles a Rainfall probability 1.2 Proportional 95th percentiles b Rainfall probability Figure 13. As for Figure 12, but showing ratios of (a) the long-term 9th percentile and (b) the long-term 95th percentile of daily rainfall totals for the two accumulation strategies to the respective percentiles of the raw rainfall series. The non-linear trend curves for the 2 day (solid line) and 3 day (broken line) accumulations are significant at the 1% level 7. IMPLICATIONS OF UNTAGGED ACCUMULATIONS: REANALYSIS OF CLIMATE CHANGE STUDIES 7.1. Nicholls and Kariko (1993) Nicholls and Kariko (1993) assessed the number, average length (in days) and average intensity (rain amount per rainy day) of rain events at five stations in eastern Australia between 191 and Two of the main selection criteria were that the stations have few days with missing data and few tagged accumulations. The five stations have a maximum of 61 missing days in the 79 year record and a maximum of 13 days with tagged accumulations. For tagged accumulations, Nicholls and Kariko (1993) appear to have assumed the entire accumulation to have fallen on the final day of the accumulation period. This strategy is likely to result in an underestimation of average length and an overestimation of average intensity for sites with a large number of accumulations. Three of the five stations that Nicholls and Kariko (1993) used also contain untagged weekend rainfall accumulations. One record (Peak Hill) contains about 3 Sundays as part of untagged accumulations and is also the record with the most tagged accumulations. Using the technique of Suppiah and Hennessy (1996) to sample only midweek days (Tuesday to Friday), it is straightforward to reanalyse average intensities. Furthermore, given that the average wet-spell length is equal to the inverse of the conditional probability that a dry day follows a wet day, it is also possible to reanalyse the number and duration of events. This reanalysis, when compared with Nicholls and Kariko (1993: table 4), indicates that the correlation between number and length of events changes from being positive and significant at the 5% level to being negative Copyright 24 Royal Meteorological Society Int. J. Climatol. 24: (24)

17 MULTI-DAY RAINFALL ACCUMULATIONS IN AUSTRALIAN RAINFALL DATA SET Proportional wet spell length a Rainfall probability 1.1 Proportional dry spell length b Rainfall probability Figure 14. As for Figure 12, but showing ratios of (a) the long-term mean wet-spell length and (b) the long-term mean dry-spell length for the two accumulation strategies to the respective spell lengths of the raw rainfall series. The linear trend lines for the 2 day (solid line) and 3 day (broken line) accumulations are significant at the 1% level and non-significant. Comparison with Nicholls and Kariko (1993: table 7) shows that the correlations with annual total, number and intensity have increased slightly, whereas that for length has decreased substantially and is no longer significant. At Peak Hill there is a strong declining trend in the occurrence of accumulated data (tagged and untagged) over the period 191 to As a result, we might expect that an analysis that includes weekend data would have a stronger tendency to overestimate average intensity and underestimate event duration earlier in the period compared with later. Consequently, any reanalysis using weekday data only is likely to result in increased (more positive) trends in average intensity and decreased (less positive) trends in event length. Peak Hill s midweek rainfall probability is about.19. According to Figures 12 and 14, we could expect 2 day and 3 day accumulations to lead to maximum underestimations of the occurrence of wet days of 6% and 14% respectively, and to maximum underestimations in averagewet-spelllength of 15% and 19% respectively. These combine to yield maximum overpredictions in the respective number of events (spells) of 1% and 6%. The actual impact on these variables would have been a little less than these maximum estimates, because not all Sundays were unread. During the period , where the untagged accumulations occur, about 8% of the expected number of rainy Sundays are missing. On the assumption that these untagged accumulations involve only 2 days each, the actual underestimation of wet-spell length during this period might, therefore, be about 12%, with an overprediction in the number of events of about 8%. Again assuming a declining trend in accumulations, Nicholls and Kariko s (1993) analysis would be expected to have overestimated the magnitude of the positive trend in wet-spell length, but to have underestimated the magnitude of the positive trend in the number of events. Copyright 24 Royal Meteorological Society Int. J. Climatol. 24: (24)