Data and parameter estimates

Following CIO2007, the sample estimates ( ˆf) in (6.8) were calculated for each calen- dar month in the series, by pooling all available data for the month over the 60-year record. For each month, the parameter estimates were obtained using the three fit- ting procedures: fit-o, fit-c and fit-c-pd. The resulting estimates are shown in Table 6.2.

Eleven parameter estimates (ˆλ1,βˆ1,ξˆ1,γˆ1,ηˆ1,λˆ2,βˆ2,ξˆ2,ˆγ2,ηˆ2,θ) are listed by columnsˆ

from left to right in Table 6.2. There are three sets of parameter estimates (i.e. three rows, one for each of the three model specifications in the order: fit-o, fit-c and

Table 6.2: Parameter estimates of three BLP model specifications for the Kelburn rainfall series. The units are hour−1 _{for all estimates exceptθwhich has units mm.}

m λ1ˆ β1ˆ ξ1ˆ γ1ˆ η1ˆ λ2ˆ β2ˆ ξ2ˆ ˆγ2 η2ˆ θˆ 1 0.00674 0.119 491 0.0421 0.709 0.000242 0.938 4960 0.181 6.46 0.00729 0.00573 0.232 875 0.0469 0.500 0.00571 2.66 5610 1.64 5.38 0.00186 0.0211 0.424 1770 0.244 0.366 0.000112 0.840 14600∗ _{0.00983 6.36 0.000760} 2 0.00798 0.531 181 0.110 1.26 0.000167 3.94 1300 0.308 8.84 0.0183 0.00776 1.06 256 0.107 1.37 0.000262 13.7 1240 0.299 12.5 0.00685 0.0308 0.192 262 0.283 0.103 0.00117 1.32 1310∗ _{0.0656 5.04 0.00487} 3 0.0117 0.344 263 0.113 0.885 0.000681 2.80 1960 0.429 7.00 0.0104 0.0109 0.668 272 0.107 0.888 0.00117 6.18 1450 0.389 6.83 0.00494 0.0270 0.400 210 0.268 0.258 0.000866 0.646 1590∗ _{0.0499 4.76 0.00556} 4 0.0123 0.233 276 0.0794 0.877 0.000443 1.21 2400 0.155 6.24 0.0104 0.0131 0.348 366 0.0758 0.844 0.000743 2.87 2010 0.130 6.51 0.00422 0.0329 0.117 503 0.112 0.417 0.000963 1.78 2760∗ _{0.0748 5.70 0.00279} 5 0.0180 0.146 195 0.0716 0.464 0.000986 0.564 1910 0.105 3.60 0.00933 0.0211 0.451 178 0.114 0.598 0.000977 1.16 1150 0.0712 3.44 0.00577 0.0340 0.213 178 0.134 0.320 0.00129 1.03 1160∗ _{0.0627 3.36 0.00532} 6 0.0186 0.075 90.8 0.0444 0.252 0.00151 0.434 1290 0.186 2.78 0.0162 0.0186 0.332 80.8 0.0719 0.366 0.00156 0.552 858 0.0946 2.45 0.00952 0.0393 0.388 56.8 0.176 0.139 0.000626 0.255 853∗ _{0.0171 2.28 0.00943} 7 0.0133 0.203 399 0.0516 0.503 0.00185 1.96 3140 0.188 5.70 0.00345 0.0126 0.419 209 0.0509 0.592 0.00218 4.23 1120 0.174 5.25 0.00370 0.0535 0.223 145 0.211 0.112 0.000771 0.846 1050∗ _{0.0255 4.02 0.00517} 8 0.0390 0.550 546 0.225 0.794 0.000915 2.19 4280 0.108 6.05 0.00231 0.0342 1.58 281 0.233 1.19 0.000826 5.41 1550 0.107 5.90 0.00261 0.0312 2.16 225 0.239 1.37 0.000688 5.21 1350∗ _{0.103 5.75 0.00313} 9 0.0159 0.337 198 0.0932 0.925 0.000889 2.59 1230 0.428 5.62 0.0101 0.0152 0.690 242 0.0937 0.995 0.00117 5.79 1080 0.401 5.79 0.00444 0.0307 0.361 203 0.179 0.377 0.00113 0.612 1140∗ _{0.0361 4.01 0.00442} 10 0.0138 0.351 430 0.0850 0.847 0.00103 2.59 2680 0.250 6.56 0.00448 0.0190 0.265 328 0.107 0.333 0.00181 0.527 1800 0.0440 4.63 0.00314 0.0369 0.260 395 0.203 0.212 0.00129 1.05 2170∗ _{0.0448 4.78 0.00233} 11 0.0136 0.196 133 0.0742 0.931 0.000140 3.43 1580 0.601 7.89 0.0245 0.0128 0.406 149 0.0712 0.916 0.000440 7.15 1140 0.542 7.80 0.0102 0.0356 0.0850 134 0.0857 0.389 0.000158 0.333 1110∗ _{0.0104 4.46 0.0108} 12 0.0132 0.730 470 0.166 1.51 0.000166 2.42 3810 0.160 6.66 0.00605 0.0135 1.37 483 0.165 1.59 0.000194 6.58 2750 0.162 7.07 0.00319 0.0214 1.27 1610 0.239 1.21 0.000850 5.03 9350∗ _{0.151 5.79 0.000628} ∗_{Forfit-c-pd,ξ}

2is not a free parameter. ξ2[i] =ξ1[i]×ratio[i],i= 1, ..., 12 (month),

whereratio is a constant vector: ratio= (8.2, 5.0, 7.6, 5.5, 6.5, 15.0, 7.2, 6.0, 5.6, 5.5, 8.3, 5.8).

Since a BLP model is conceptually a physical process based rainfall model, some intended physical meanings may be attached to model parameters for interpretation of what we may learn from the estimation results. Table 6.2 shows that ˆλ1 >λˆ2 and

ˆ

ξ1�ξˆ2for all three models. This implies that process 2 has a lower storm generation

rate with a much more frequent pulse generation than process 1. Because the mean depth ˆθ, is the same for both processes, more frequent pulse generation is equivalent to higher rainfall intensity. Furthermore, in most months, process 1 storms last longer (on average) than process 2 storms (i.e. ˆγ1−1>γˆ2−1) with fit-o. This seems to imply that

we may match the meteorological definition of stratiform rainfalls with process 1 and convective rainfalls with process 2. However, if we look at thefit-c-pdresults, process 1 storms last consistently shorter (on average) than process 2 storms (i.e. ˆγ1−1<γˆ−21).

fit-c sits somewhere in between fit-o and fit-c-pd. Therefore, the two distinct

processes specified in a BLP model are rather arbitrary in a meteorological sense. The parameter estimates ˆλiand ˆγi(i = 1,2) seem significantly correlated (Table 6.4). Note that fit-c-pd consistently has higher ˆλ1 values and lower ˆγ1−1 values than fit-

o and fit-c. This implies that fit-c-pd describes process 1 as storms that occurs

more often but where each storm has a shorter mean duration. Although the resulting parameter estimates with fit-c-pdare very different from fit-oand fit-c, it can be shown that the mean storm hours per month are reasonably close for all three models. Furthermore, they all have the same pattern in that the mean storm hours per month of process 1 is much larger than process 2 and the minimum of ˆλ1 occurs in January.

The parameter estimates ˆβi and ˆηi (i = 1,2) show strong positive correlations (>

0.8). This implies that a higher cell generation rate (larger ˆβi) tends to be accompanied by a shorter cell duration (smaller ˆηi−1). The parameters ˆη1and ˆη2are highly positively

correlated and ˆη1−1 � ηˆ−21. This means that process 1 has a much longer mean cell

duration than process 2. The patterns are similar for all three model specifications but

fit-c-pd has a much bigger range of variation in terms of the ratio ˆη2/ηˆ1.

Since ˆγ−1 _{is intended to account for the mean storm lifetime and ˆη}−1 _{for mean cell}

duration, intuitively we expect to see that ˆγi−1 > ηˆ−i1 (i = 1,2). However, we notice that ˆγ1−1 < ηˆ1−1 for February, March, June, and July with fit-c-pd. A BLP model

assumes that the process of pulses terminates with the cell or storm lifetime, whichever is the sooner. This setting has the effect of introducing dependence between cells within storms. Therefore, when ˆγ−1

1 < ηˆ1−1 occurs, termination of pulses process before the

The parameter estimates ˆξ1 and ˆξ2 are so strongly positively correlated (> 0.93)

that it seems that we should assume ˆξ2 = constant×ξˆ1 in model estimation. The

lowest rainfall intensity occurs in June (New Zealand winter) and the highest intensity occurs in January (New Zealand summer) for both processes.

By looking at the ˆθvalues, which represent the mean depths associated with pulses, they are very small (less than 0.01 mm in most months) which suggests that BLP models may be able to capture a rainfall process even at droplets level. The values of ˆ

θwith fit-candfit-c-pd are consistently much less than fit-o.

In the model estimation process, the convergence status can be achieved in minimis- ing (6.8) with different parameter sets. Different initial values or changes in parameter space limits can all result in different parameter estimates. Over-parametrisation oc- curs when the number of independent estimated parameters is more than the number of independent sample properties, and is likely to be the cause for having more than one sets of ‘optimal’ parameter estimates for the same objective function.

It is worth noting that there are substantial differences between the parameter estimates obtained fromfit-oandfit-c. But the trade-offbetween parameters means that these do not necessarily reflect big changes in the model properties as will be shown in the following sections.