The Frequency Of Heavy Daily Rainfalls In New Zealand.
Method of Statistical Analysis: It has been established (Seelye, 1947) with particular reference to Wellington rainfalls that a daily rainfall of amount × or more occurring on the average once in N years can be satisfactorily expressed as × = u + k log N, where u and k are constants: Such an equation applies to the heavier rains and is suitable when N is at least unity. The values of u and k can be determined by considering the extreme daily rainfall for each calendar year covered by the available record for a station.
Gumbel. who has developed the theory of extreme values, has proved that the mean and the mean absolute deviation of the series of extreme values lead to a rather smaller probable error in the calculated coefficients of the distribution than the use of the mean and the mean–square deviation. The steps involved in computing u and k are described below, but the statistical theory is not repeated here.
Let a rainfall record cover n years. The rainfalls on the wettest day of each of these n years are arranged in order of increasing magnitude and let the rth rainfall in such an arrangement be denoted by x_{r}. Corresponding to n, find m′ = 0.36788n—0.63212 (Gumbel. 1943). [It is convenient to have a table showing the integers n and the associated m′ values.] Let m be the nearest integer less than m′ which in usually fractional. Estimate the rainfall corresponding to. this fractional serial number by interpolation between x_{m} and x_{m+1}. that is, take u = x_{m} + (m′−m) (x_{m+1}−X_{m}).
[The section below cannot be correctly rendered as it contains complex formatting. See the image of the page for a more accurate rendering.]
u then is the rainfall which occurs with the average frequency of once a year. Let the mean of the n rainfalls be x= 1/n ·Σ^{n}_{r=1}x^{r}
[The section below cannot be correctly rendered as it contains complex formatting. See the image of the page for a more accurate rendering.]
and their mean absolute deviation be θ= 1/n ·Σ^{n}_{r=1}|x^{r}−x|
[The section below cannot be correctly rendered as it contains complex formatting. See the image of the page for a more accurate rendering.]
The latter is best computed as θ= 2/n ·[px−Σ^{p}_{r=1}x^{r}]
[The section below cannot be correctly rendered as it contains complex formatting. See the image of the page for a more accurate rendering.]
where p is the serial number of the rainfall immediately smaller than x. A more-precise estimate of the mean deviation is given by θ= √n/n−1 ·θ=[1+½n] θ approximately
From θ we have 1/α = 1.01731 θ (Gumbel. 1942), which is one of the coefficients in the “distribution of floods.” In its present application this expresses the probability W(x) that × is the largest daily rainfall in the year as W(x) = exp [—exp—(x—u)/α]. Our k is related to α (Seelye, 1947) and finally we have k = 2.3425 θ.
Discussion of Results: The New Zealand Meteorological Office has tabulations of the wettest days by months an 1 years for most of the longer rainfall records. These were completed to the end of 1945 for 91 places with over 40 years of records, for 99 extending over 30 years and for another 99 of shorter duration. For each of these records × and θ were computed and u and k derived as described above. Common logarithms are used in the formula so u, u + k, u + 2k represent rainfalls which are likely to occur once in 1. 10, and 100 years respectively. Representative results are set out in the accompanying table, while the two maps summarise the position. The rainfalls considered are those measured each morning and are not necessarily the maximum for any 24 hours. However, it was found that the present results needed to be increased some 12 or 13 percent, in the case of Wellington to give the fall likely to be encountered in any 24 hour period. In the absence of other autographic records of sufficient length to give reliable information for other places one can only say that a similar increase seems reasonable for any locality.
[The section below cannot be correctly rendered as it contains complex formatting. See the image of the page for a more accurate rendering.]
Years of Record | k | Daily reached 1 yr. u | Rainfall on average 10 yr. u+k | (inches) once in 100 yr. u+2k | Largest Daily Rainfalls of the Record | ||
---|---|---|---|---|---|---|---|
Whangarei | 36 | 3,31 | 2.93 | 6.24 | 9.55 | 8.52 | 11.41 |
Auckland | 82 | 2.04 | 2.00 | 4.04 | 6.08 | 6.38 | 6.39 |
Waihi | 46 | 5.10 | 4.96 | 10.06 | 15.16 | 12.15 | 16.50 |
Hamilton | 46 | 1.44 | 2.02 | 3.46 | 4.90 | 4.69 | 5.12 |
Tauranga | 44 | 2.77 | 2.92 | 5.69 | 8.46 | 6.45 | 9.41 |
Rotorua | 60 | 2.58 | 2.55 | 5.13 | 7.71 | 7.78 | 8.80 |
Taumarunui | 31 | 1.65 | 2.01 | 3.66 | 5.31 | 4.29 | 5.34 |
“Riversdale,” Inglewood | 58 | 3.93 | 3.97 | 7.90 | 11.83 | 10.90 | 12.16 |
Pakihiroa | 32 | 2.30 | 5.06 | 7.36 | 9.66 | 9.51 | 9.82 |
Gisborne | 44 | 2.79 | 2.28 | 5.07 | 7.86 | 7.50 | 7.68 |
Tutira | 46 | 4.63 | 3.20 | 8.83 | 11.46 | 12.76 | 13.56 |
Napier | 67 | 2.58 | 2.28 | 4.86 | 7.44 | 7.16 | 8.03 |
Masterton | 57 | 1.85 | 1.94 | 3.13 | 5.64 | 5.45 | 7.06 |
Martinborough | 33 | 1.35 | 1.51 | 2.86 | 4.21 | 3.14 | 3.80 |
New Plymouth | 70 | 1.80 | 2.43 | 4.24 | 6.03 | 4.86 | 7.29 |
Stratford | 40 | 2.31 | 3.24 | 5.55 | 7.86 | 8.43 | 9.56 |
Wanganui | 60 | 1.14 | 1.63 | 2.77 | 3.91 | 3.52 | 3.70 |
Feilding | 59 | 0.99 | 1.58 | 2.57 | 3.56 | 3.44 | 3.45 |
Palmerston North | 53 | 1.25 | 1.59 | 2.84 | 4.09 | 2.92 | 6.34 |
Wellington | 83 | 1.84 | 2.29 | 4.13 | 5.97 | 6.00 | 6.32 |
Westport | 53 | 1.77 | 2.59 | 4.36 | 6.13 | 5.80 | 6.86 |
Greymouth | 46 | 2.32 | 3.12 | 5.44 | 7.76 | 6.55 | 12.50 |
Otira | 40 | 3.29 | 7.12 | 10.41 | 13.70 | 11.30 | 11.81 |
Hokitika | 67 | 2.32 | 3.71 | 6.03 | 8.35 | 8.16 | 9.17 |
Nelson | 49 | 1.58 | 2.16 | 3.74 | 5.32 | 4.50 | 4.83 |
Spring Creek, Bm. | 37 | 1.25 | 1.95 | 3.20 | 4.45 | 4.50 | 4.95 |
“Emscote,” Stag and Spey | 23 | 5.14 | 2.94 | 8.08 | 13.22 | 7.30 | 19.69 |
Arthurs Pass | 27 | 3.76 | 6.83 | 10.59 | 14.35 | 11.32 | 12.70 |
Christchurch | 71 | 1.34 | 1.59 | 2.93 | 4.27 | 4.00 | 4.71 |
Timaru | 53 | 1.63 | 1.45 | 3.08 | 4.71 | 4.37 | 5.79 |
Benmore Station | 39 | 1.14 | 1.71 | 2.85 | 3.99 | 3.28 | 3.98 |
Clyde | 49 | 0.83 | 1.06 | 1.89 | 2.72 | 2.14 | 2.65 |
Dunedin | 93 | 1.83 | 1.82 | 3.65 | 5.48 | 5.42 | 6.81 |
Roxburgh | 49 | 0.64 | 1.13 | 1.77 | 2.41 | 2.28 | 2.28 |
Gore | 37 | 0.73 | 1.28 | 2.01 | 2.74 | 2.12 | 2.53 |
Invercargill | 49 | 0.90 | 1.30 | 2.20 | 3.10 | 2.78 | 3.25 |
Half Moon Bay (Stewart Island) | 31 | 1.10 | 1.52 | 2.62 | 3.72 | 2.90 | 3.18 |
As regards the rainfall likely to be attained once a year on the average, it is seen that for most of the North Island low country the amount is about 2 in. In the lower Manawatu values are as small as 1.5 in. This is a region where steady frontal rains with orographic reinforcement can rarely occur and the strong westerly winds to which it is at times subject usually bring only showery precipitation. The east coast, though having a low average annual rainfall, shows higher values on the present map than does the west coast. The meteorological situation with an active depression to the north and onshore easterly winds is not uncommon and favours steady rains on the east coast. The high values 4.96 in. for Waihi and over 5 in. at Wairongomai and Pakihiroa Stations towards East Cape are noteworthy. For the South Island, amounts are under 2 in. for the majority of places east of the main ranges. The lowest figure found was 0.93 for Alexandra in Central Otago. The chief exceptions are Banks Peninsula and the Kaikoura coast. In Westland values are more than 3 in., for Otira the figure is 7 in., and for Milford Sound 9 in. It does not appear likely that values appreciably exceed the last amount over any great extent of country.
With reference to the 10-year expectation, the largest amount calculated for the North Island was 10.06 in. at Waihi. Values of 5 in. or more are common in the eastern portions of Auckland Province and in the more exposed parts of Hawkes Bay. Less than 3 in. is shown for the Taihape-Manawatu region. In the Southern Alps and Southern Sounds 10–14 in. appears to be the range, with the latter figure only in the most exposed positions. An appreciable part of Central
It may be remarked that “Emscote,” Stag and Spey received 19.69 in. on the 6th May, 1923, and 10.83 in. the day following. The heaviest measured fall for the South Island, however, was 22 in. recorded at the P.W.D. Camp, Milford Sound, on 17th April, 1939, the Hostel receiving 18.39 in. at the time. For the North Island the record is held by “Riverbank,” Rissington, H.B., which experienced 20 14 in. in the space of 10 hours on the 11th March, 1924. The world record for a 24-hour fall is the 46 in. which fell at Baguio, Luzon (Philippines), between the 14th–15th July, 1911.
[The section below cannot be correctly rendered as it contains complex formatting. See the image of the page for a more accurate rendering.]
The Accuracy of the Results: Assuming the distribution of floods is the one followed by rainfall there will be an error in the statistical coefficient calculated from the results of a finite number of years. From Gumbel's work (1942) it is possible to assign a standard error to u and k as calculated here from records extending over n years. If the standard errors are denoted by σ_{u} and σ_{k} respectively, it can be shown that σ_{u} = .57k/√n σ_{k} = .87k/√n
Examples are given hereunder. These include the results for “Emscote” using 23 years' records and also omitting the year with the phenomenal 19.69 in. fall.
[The section below cannot be correctly rendered as it contains complex formatting. See the image of the page for a more accurate rendering.]
u ± σ_{u} | k ± σ_{k} | |
---|---|---|
Auckland | 2.00 ± .13 | 2.04 ± .18 |
Waihi | 4.96 ± .44 | 5.10 ± .64 |
Invercargill | 1.30 ± .07 | 0.90 ± .11 |
“Emscote” (23 years) | 2.94 ± .61 | 5.14 ± .89 |
(22 years) | 2.93 ± .44 | 3.58 ± .63 |
To give some indication of the accuracy of the method, the following analysis was made. After the constants were calculated for a station the number of years equal to the length of the record was introduced into the formula to give the rainfall that was likely to have been exceeded once during the record. Some records (32) received no rainfall as heavy as this, others received 1, 2, or 3 such falls. The distribution over the 289 records handled is tabulated.
[The section below cannot be correctly rendered as it contains complex formatting. See the image of the page for a more accurate rendering.]
0 | 1 | 2 | 3 | 4 or more | Total | |
---|---|---|---|---|---|---|
Actual No. of Records | 32 | 133 | 106 | 18 | 0 | 289 |
Poisson Distribution | 106 | 106 | 53 | 18 | 6 | 289 |
The average chance of one such rainfall at a station is [ unclear: ] per year. Poisson's law of rare events gives the likely number of records having 0, 1, 2, 3, 4 or more such rare falls. From the table it is apparent that there are more cases of 1 and 2 rainfalls occurring in the record than the Poisson theory would expect with a noticeable reduction in the cases which escape such a rainfall. Whereas we should expect to average one such heavy rainfall per record, the average from the records is 1.4. Thus the tendency is for the calculated frequency of occurrence to be somewhat too low. However, in spite of such a deficiency, the method, which is easy to apply, proves to be at least as accurate as more elaborate methods dealing with a more bulky volume of observational data.
Acknowledgment: I have to thank Dr. M. A. F. Barnett, Director of Meteorological Services, in whose Office this work was commenced, for kindly affording facilities for the completion of this study.
References.
Gumbel, E. J., 1942. Statistical Control-curves for Flood Discharges. Trans. Amer. Geophys. Union, 489–500.
—— 1943. On the Plotting of Flood Discharges. Trans. Amer. Geophys. Union, 699–713.
Seelye, C. J., 1947. Rainfall Intensities, at Wellington, N.Z. Proc. N.Z. Inst. Eng., 33, 452.