TULLE ROUGE.

The seasonal patterns and diurnal patterns of rainfall depths and proportion of dry periods (PD) can be examined numerically from Table 5.3 and 5.4. The rainfall depth is defined by the total rainfall (mm) recorded in 5-minute intervals. The seasonal pattern is examined by the variation from month to month within a year. The diurnal pattern is examined by the variation from hour to hour within a day and the hour of a day starting from midnight. Note that, in Table 5.3, the two highest proportion dry months and the two highest depth months happened together in January and February, two of New Zealand’s summer months. For the winter season, the two lowest proportion dry months were June and July while the two lowest depth months were August and September. This implies that it was likely to have the most intensive rainfalls in January and February but the rainfall occurrences were less frequent than in the other months. On the other hand, although the rainfall occurrences were most frequent in June and July, the least intensive rainfalls were most likely to happen in August and September. We must be cautious about the interpretation of the diurnal proportion dry pattern. Both dry spells and wet spells are grouped by their starting times, but a dry or wet spell which started in this hour may well last for more than

Table 5.3: The seasonal patterns of mean depth and proportion dry

month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

depth 0.175 0.171 0.167 0.166 0.153 0.152 0.144 0.137 0.138 0.148 0.151 0.158 PD 0.951 0.947 0.940 0.936 0.911 0.894 0.898 0.902 0.921 0.918 0.928 0.942

Table 5.4: The diurnal patterns of mean depth and proportion dry

hour 0 1 2 3 4 5 6 7 8 9 10 11 depth 0.155 0.150 0.147 0.148 0.145 0.147 0.146 0.143 0.156 0.167 0.165 0.167 PD 0.921 0.920 0.920 0.922 0.934 0.940 0.940 0.948 0.941 0.917 0.936 0.938 hour 12 13 14 15 16 17 18 19 20 21 22 23 depth 0.162 0.162 0.158 0.157 0.155 0.155 0.146 0.152 0.151 0.150 0.145 0.147 PD 0.938 0.918 0.906 0.914 0.914 0.912 0.900 0.893 0.923 0.919 0.918 0.915

one hour. Therefore, the diurnal proportion dry patterns should only be interpreted in the sense of the dry spells starting times. Table 5.4 shows that dry spells started more often (i.e. the corresponding wet spells ended) in the early morning hours and started least often in the late afternoon hours. Since, on average, a wet spell lasts about one hour (Table 5.5), this implies that early morning hours are wetter and the late afternoon hours are drier which by and large match the conclusions made from examining Figure 5.6. The highest depth period appeared in the late morning hours.

We examine the empirical distributions of rainfall depths and the dry spell and wet spell durations in Table 5.5. The depth is defined as rainfall total (mm) per 5-minute given a wet 5-minute interval. Dry spell and wet spell durations are measured in the number of 5-minute intervals, e.g. the median of dry spells durations is 14.0 means that the median duration is 5×14.0 = 70 minutes. The distributions of rainfall depths, dry spell durations, and wet spell durations are all right-skewed. In particular, the distributions of depths and dry spell durations had very long right tails (mean >>

median). On the other hand, the wet spell durations distribution was skewed to the right to a less extent.

Table 5.6 shows the lag 1 to lag 6 autocorrelations among dry and wet spell se- quences in terms of their durations. Note that all correlation coefficients are small; the highest one is the lag 1 correlation for the pooled wet spell sequence (0.1160). In a rainfall process, dry spells and wet spells form an alternating sequence. From the cor-

Table 5.5: Empirical distribution of rainfall depths and dry or wet spell durations

Minimum 1stQuartile Median Mean 3rdQuartile Maximum

depth(mm) 0.0100 0.0400 0.0900 0.1525 0.1800 9.100

dryspell(_×5 min) 1.0 5.0 14.0 159.2 63.0 7302.0

wetspell(_×5 min) 1.00 3.00 6.00 13.14 12.00 495.00

Table 5.6: Autocorrelations among dry and wet spell sequences

lag 1 lag 2 lag 3 lag 4 lag 5 lag 6

dry and wet spell alternating sequence

-0.04881 0.09129 -0.04815 0.08245 -0.04788 0.07036 dry spell sequence 0.04263 0.03527 0.02064 0.03986 0.03717 0.02176 wet spell sequence 0.1160 0.08273 0.06900 0.06053 0.05664 0.05584

relation coefficient values in the first row, we note that all odd lags are negative values and all even lags are positive values. The negative correlation at odd lags implies that after a long dry spell we may expect a relatively shorter wet spell, or alternatively, after a long wet spell we may expect a relatively shorter dry spell. On the other hand, the positive correlation at even lags implies that a long dry (wet) spell is more likely to be followed by a relatively longer dry (wet) spell. The positive correlation patterns are highlighted by pooling the dry (wet) spells in one sequence group as exhibited by the correlation coefficient values in row two (pooled dry spell sequence) and row three (pooled wet spell sequence). Note that the correlations among the consecutive wet spells are stronger than the dry spells and it seems that these low level, but significant positive correlations, decay at a very slow rate (i.e. exhibit long memory).

We denote the skewness and lag 1 autocorrelation statistics calculated at the 5- minute, 1 hour, and 24 hours aggregation levels by skew5m, skew1h, skew24h, ac5m, ac1h, and ac24h, respectively.1 _{The results given by Table 5.7 confirm the results in}

the literature (e.g. Buishand, 1977), i.e. rainfall observations over longer intervals have less autocorrelation. By pooling the observations of the same month together for each of the 12 months over the 60-year sampling period, weak stationarity is assumed within each subsample. The skewness and lag 1 autocorrelation statistics are calculated for each subsample at different aggregation levels for comparison. From the table, rainfall observations at the 5-minute level consistently exhibit a much stronger autocorrela-

Table 5.7: Sample statistics of skewness and lag 1 autocorrelation for pooled subsamples by calendar month

month skew5m skew1h skew24h ac5m ac1h ac24h Jan 23.55 12.10 5.030 0.8647 0.6123 0.2783 Feb 17.83 11.18 5.234 0.8356 0.6197 0.1854 Mar 19.50 11.56 4.436 0.8453 0.6164 0.1598 Apr 18.92 10.85 4.885 0.8338 0.5875 0.2146 May 15.78 10.82 4.182 0.8597 0.5926 0.2312 Jun 17.45 12.79 3.645 0.8464 0.6123 0.2415 Jul 12.74 8.064 3.602 0.8525 0.6647 0.2554 Aug 12.29 7.932 5.168 0.8596 0.6398 0.1786 Sep 13.02 8.538 3.397 0.8587 0.6250 0.1977 Oct 12.83 8.072 3.734 0.8637 0.6512 0.2090 Nov 22.46 12.40 4.130 0.8119 0.5986 0.2175 Dec 19.06 12.04 6.921 0.8674 0.6008 0.1434

tion than the hourly observations and likewise the autocorrelation of the hourly data is stronger than the daily data. The observed autocorrelation among rainfall series probably results from clustering of rainfall events.

While a significant seasonal pattern can be identified with the skewness statistics (low in the winter and high in the summer), the seasonal pattern with the lag 1 au- tocorrelation statistics is much less obvious. Note that, in the ‘skew24h’ column of Table 5.7, there is a big sudden rise for the August value compared with the June, July, September, and October observations. This makes the 24-hour skewness seasonal pattern unexpectedly different from those of 5-min skewness and 1-hour skewness sea- sonal patterns. By checking the raw data records, we found that there was a 116.88 mm daily rainfall recorded for the date of 7th August, 1963. This is the second highest daily rainfall record over the 60-year sample period. This is very unusual because intensive rains are more likely to occur in summer season; for example, the highest daily rain- fall, 147.8 mm, was recorded in December. Exploratory analysis shows that without including this single day’s record, the historical 24-hour skewness for August should have been 4.335 instead of 5.168. Such a high influence observation can have a signifi- cant impact on the model parameter estimation as the empirical study results show in Chapter 6 and Chapter 8.

5.4

Seasonality within the pooled calendar month