SOME CONSIDERATIONS ON EXTREME STATISTICS OF STORM SURGE HEIGHT

 

MASATAKA YAMAGUCHI ¹ and YOSHIO HATADA ²

 

Department of Civil and Environmental Engineering, Ehime University

Bunkyocho 3, Matsuyama 790-8577, Japan

¹ tel. +81-89-927-9832, fax. +81-89-927-9844, e-mail myamag@en2.ehime-u.ac.jp

² tel. +81-89-927-9838, fax. +81-89-927-9844, e-mail hatada@en2.ehime-u.ac.jp

 

 

ABSTRACT

The return values and their standard deviations for typhoon-generated storm surge height measured at Osaka for a period of 94 years from 1902 to 1995 are estimated by using an extended version of Goda's extreme analysis model based on the least square method, in which the candidate distributions are the Gumbel and Weibull distributions. Investigations are conducted as to the effects of factors such as the censoring rate and the number of years for the measurement on return storm surge height and its standard deviation. Main conclusions are that increase of sample size associated with lowering of the censoring point and extension of the measurement period are crucial to improving the accuracy and efficiency of the estimates and that an extreme analysis using the data stratified according to typhoon track does not necessarily produce more efficient estimates in this case.

 

Keywords: typhoon-generated storm surge height data, extreme statistics, least square method, sensitivity analysis, stratified sampling technique, Osaka Bay

 

INTRODUCTION

In the design of coastal structures such as breakwaters and sea dikes, reliable estimation for extremes of sea levels, particularly storm surge heights to be expected during a life time at the site is an indispensable step. Extreme statistics for storm surge height have been extensively studied, in which case the most important condition in extreme analysis is to use samples of large size over long years which have approximately stationary, independent and homogeneous properties. In Japan, it is nearly impossible to gather measurement data of storm surge height over more than 50 years, but we have fortunately been able to obtain the measurement data over 94 years of typhoon-generated peak storm surge heights at Osaka, where typhoon-generated storm surges are dominant.

This paper investigates some problems concerning the extreme statistical analysis from viewpoints of (1) censoring rate and year periods of the data, (2) annual maximum data and data of peak over threshold, (3) stability of the estimates of extremes and (4) homogeneity of the data, through a case study using the long-year data of storm surge height at Osaka.

 

DESCRIPTION OF STORM SURGE HEIGHT DATA

The data set of typhoon-generated peak storm surge height measured at Osaka is made through a wide survey of many reports published by the Japan Meteorological Agency and the others, in which the period is 94 years ranging from 1902 to 1995. The effect of storm surge heights generated by storms excluding typhoons on the estimates of return storm surge heights is negligible, because they are considerably smaller than the values during typhoons.

Figure 1 shows a sketch of Osaka Bay and the bay axis passing through Osaka and Tomogashima. Annual maximum series data with sample size of 72 are selected from the peak storm surge height data of 125 cases greater than 20 cm. The largest surge height is 292 cm generated by T3412 (the 12th typhoon in 1934) and the second, third and fourth largest surge heights are respectively 245 cm (T6118), 237 cm (T5028) and 216 cm (T6523). These are all records of surge height exceeding 200 cm at Osaka during a period of 94 years. As is well-known, typhoon winds have different characteristics in the right semi-circle and in the left semi-circle. In order to make the data homogeneous, annual maximum series data stratified into 2 groups according to typhoon track near Osaka Bay are re-selected from the original data set. In this case, each typhoon is classified into either a typhoon which passed through the western area to the bay axis or a typhoon which passed through the eastern area to the bay axis.

Figure 2 indicates yearly variation of annual maximum surge height. Anomalous surge heights occurred intensively over a period of 21 years from 1945 to 1965 and the maximum of the surge height since 1966 is only 134 cm generated by T7916. A decreasing trend of 0.4 cm per year on average can be observed, but the extreme analysis is proceeded under the assumption of an insignificant trend.

 

Figure 1. Sketch of Osaka Bay and bay axis passing through Osaka and

 

Figure 2. Yearly variation of annual maximum storm surge height.

 

OUTLINE OF THE SYSTEM FOR EXTREME ANALYSIS

The system using the least square method for the parameter estimation in the probability distribution is applied, which was originally constructed by Goda (1988, 1990) and extended by the authors (1997). The system makes use of the Gumbel distribution and the 3-parameter Weibull distribution with a fine resolution of the shape parameters varying from 0.5 to 100 as the candidate distributions, and the criterion of the largest correlation coefficient between the ordered data and its reduced variates for the selection of the optimum distribution. The plotting position is based on Goda's empirical formula proper to each distribution. The standard deviation of return value is evaluated by the application of a jackknife method (Miller, 1974). The system becomes consistently available for not only annual maximum data but also data of peak over threshold irrespective of the presence of data censoring, by introducing two factors of the mean occurrence rate and the censoring rate , where N_T is the total number of events, K the year period of data and N the number of data used in the analysis.

In the case of data grouped by typhoon track, the overall probability distribution F(x) and R-year return value x_R are evaluated on the basis of theoretical formula for compounding the optimum probability distribution for each stratified data F_j (x) and the estimation of the overall variance relies on Goda's empirical formula (1990) for compounding the variance for each stratified data. These are respectively expressed as:

(1), (2)

where n (=2) is the number of stratum and N_j the sample size of stratified data. Eq. (2) indicates an average of each variance weighted with both the sample size and the exceedance probability of R-year return value x_R.

 

ESTIMATION OF RETURN STORM SURGE HEIGHT

 

EFFECTS OF DATA CENSORING RATE AND YEAR PERIOD OF DATA

Table 1 indicates the results of extreme analysis for the storm surge height data of annual maximum and peak over threshold in 94 years from 1902 to 1995, in which the listed values are the year period of measurement K, the number of data used in the analysis N, the shape parameter k of the optimum Weibull distribution, the correlation coefficient , the R-year (R=100, 200, 500) return values and their standard deviations , and the measured maximum storm surge height . The analysis is separately made for case of all gathered data and cases of data over threshold values of =50, 75 and 100 cm. The total number of events N_T required in the analysis for data of peak over threshold is roughly estimated as the number of typhoons which passed through the area within a radius of hundreds of kilometers centered at Osaka Bay, that is, the average number per year =3.33 multiplied by the year period of data K=94. This is based on the fact that not all of the typhoon tracks before 1940s are identified and Goda's suggestion (1990) that an accurate estimation for N_T is not required with an allowable magnitude factor of 2. The feature to be pointed out firstly is that the difference between the results of analysis using both extreme data is small and practically negligible. From this fact, only the results estimated using the annual maximum data are hereafter discussed. Another indication is that the return values change little with the reduction of the censoring rate, whereas the standard deviations gradually increase, meaning less efficient estimates of return values. According to the results estimated using all of the gathered data, the R-year return value and its standard deviation is 270 ± 32 cm for R=100 and 309 ± 37 cm for R=200 respectively, and the return period for the largest storm surge height of 292 cm associated with T3412 is 148 years.

 

Table 1. Effect of censoring rate in annual maximum storm surge height data and peak storm surge height data on return value and standard deviation.

 

The effect of year period for measurement on the results of extreme analysis is indicated in Table 2, in which the data are sub-grouped by 3 year periods. The maximum of the difference among the 500-year return value is only 28 cm which is less than the standard deviation. But, the standard deviation takes a larger value with reduction of measurement period, which indicates less efficient estimate of the return value. The return value and its standard deviation tends to become larger in cases where anomalous high values are included in the data of a shorter period. This leads to the conclusion that an increase of the time span of measurements is an essential condition for improving the reliability of the estimates of return values.

 

Table 2. Effect of number of years for measurement on return storm surge height and standard deviation.

 

STABILITY OF THE ESTIMATE

In order to investigate the stability of the estimates of return storm surge heights, the extreme analysis is made by using 3 sub-grouped data sets. These are (1) the data set which excludes the largest value of 292 cm, (2) the data set selected every even year, which includes the largest value of 292 cm but does not include the second largest value of 245 cm and (3) the data set selected every odd year, which includes the second largest value of 245 cm but does not include the largest value of 292 cm. Table 3 summarizes the results of extreme analysis. Exclusion of the largest value in the analysis gives rise to slightly larger estimates of the return values and their standard deviations, but the estimates may be said to be relatively stable, because the difference of the return value is not significant, if the standard deviation is taken into account. On the other hand, the return values and their standard deviations estimated using the data of even years are much greater than those estimated using the data of odd years. The analyzed results are accompanied with large variations, depending on the presence of anomalous data, as the occurrences of extraordinary storm surges are rare even in a year period for measurement of almost 100 years.

 

Table 3. Stability of estimated return storm surge height and standard deviation.

 

Similar variations can be observed more clearly in Table 4, in which case the analysis is separately conducted using the data of 47 years sub-grouped by the periods of (1) 1902 to 1948, (2) 1926 to 1972 and (3) 1949 to 1995. The largest and second largest surge heights occurred in 1934 and 1961 respectively. The greater the number of anomalous values in the data set is, the larger the return value is, and the difference among return values estimated using the 3 data sets comes to significant magnitude comparable to the minimum of the standard deviations. The standard deviations estimated using the data of 1902 to 1948 take a fairly large value by reason of smaller size of the sample and smaller shape parameter of the optimum distribution. As was mentioned above, the estimates of return values might vary significantly, depending on the year period of data acquisition. Therefore, increase of the time span of measurements is an indispensable condition for obtaining stable estimates of return values.

 

Table 4. Change of return storm surge height and standard deviation with 3

 

HOMOGENEITY OF EXTREME DATA

Table 5 illustrates the results of extreme analysis estimated using the 2 data sets for a period of 63 years from 1933 to 1995 which are stratified by the typhoon track. The last column gives the return values and their standard deviations aggregated by eqs. (1) and (2), and index 'G' in the third column means the Gumbel distribution. The return values of surge heights associated with the typhoons which passed through the western area to the bay axis are rather greater than those with the typhoons which took the eastern area track, and the resulting aggregated return values are almost identical to those for the typhoons of the eastern area track and/or those estimated using all of the gathered data. These are true for the standard deviations, but the correlation coefficient in the case of stratified data is slightly lower compared to that in the case of unstratified data. Advantages of a stratified sampling technique over the usual method cannot be found in the extreme analysis for the storm surge data at Osaka, probably due to the augment of statistical variability of the data associated with the stratification.

 

Table 5. Return storm surge height and standard deviation estimated from data stratified by typhoon track.

 

CONCLUSIONS

Some problems concerning the extreme data analysis are investigated through the case study that analyzes the typhoon-generated maximum storm surge height data measured at Osaka during a 94 year period of 1902 to1995 from variety of viewpoints. Main results are summarized as follows.

(1) Increase of sample size associated with lowering of censoring point and extension of a year period for measurement are essential to improving the accuracy and efficiency for the estimates of the return values.

(2) Data selection method such as annual maximum method or peak over threshold method produces little difference in the estimates of return values and their standard deviations for the measurement data at Osaka.

(3) A data stratification method does not necessarily yield better estimates of return values compared to the conventional method, probably due to sample variability in the present case.

 

REFERENCES

Goda, Y. (1988): On the methodology of selecting design wave height, Proc. 21st ICCE, Vol.1, pp.899-913.

Goda, Y. (1990): Design of Harbour Structures against Random Sea Waves-Introduction to Wave Engineering (2nd edit.), Kajima Pub., p.333 (in Japanese).

Miller, R.G. (1974): The jackknife-a review, Biometrica, Vol.61, No.1, pp.1-15.

Yamaguchi, M. and Y. Hatada (1997): An extremal analysis system and its application to the estimation of meteorological and oceanographic elements around the coasts of Japan, Ocean Wave Measurement and Analysis (WAVES97), Vol.2, pp.932-946.