1:45 PM - 3:15 PM
[HDS08-P08] Modeling the return period-return level relationship of maximum snow depth using compound process and copula.
Keywords:Maximum snow depth, Copula, Stochastic process, Compound process
Big pile of snow brings about various adversities and estimating its return period/return level relationship is one of the essentials in disaster managements. The maximum snow depth (MSD) is the consequence of several different processes and is not the block maximum considered in the extreme value theory, so the MSD does not necessarily follow the generalized extreme value (GEV) distribution. For example, Izumi et al. (1988) refer to papers using GEV Type I, II, III, normal, and lognormal distributions to fit the MSD, they state that even at one observatory, data seemed to be from different distributions, and proposed that use only highest 1/3 data to calculate the 100-year return level, with their own model. In their work and cited research, the return periods of the MSD are estimated from only the seasonal MSD data. We thought if we include in the model the days and amounts of snowfall and decreasing of the snow pile etc., we might do a better job.
Many factors affect the MSD, and we first thought that we would have to consider the temperature, humidity, wind flow, etc. However, it turned out that there is a simple proportional relationship between the MSD and the cumulative snow fall before the maximum depth (CSF). For example, at Kanazawa observation station, the ratio of MSD to the CSF, nu, is 0.2933 (std. 0.015) with the coefficient of determination R-squared = 0.867. We made use of this proportionality to model the return period of MSD.
For each year, let N be the number of days of snow fall before the maximum depth, Xbar be the mean seasonal daily snow fall, and nu be the ratio of MSD to CSF.
Then the seasonal MSD, S, is expressed as nu Xbar N, where we make use of the law of large numbers.
We analyzed the data from Kanazawa, Niigata, and Sapporo observation stations, and found that N follows the negative binomial distribution, and Xbar follows the lognormal distribution. The dependence between N and Xbar are different for different observation points, at Kanazawa, N and Xbar are practically independent, at Niigata, the dependence follows the Clayton copula, and at Sapporo it follows the Frank copula.
Figure shows the return period-return level relationships. The assumed dependence between N and Xbar remarkably affected the results. Assuming independence causes huge overestimation of return levels. On the other hand, using the linear regression relationship resulted in the underestimation. Using the Clayton copula, with alpha = -0.248 obtained by fitting to the marginal cumulative distribution function of N and Xbar, successfully reproduced the return period-return level relationship.
Izumi M, Mihashi H, Takahashi T, Statistical properties of the annual maximum snow depth and a new approach to estimate the return period values, Journal of Structural and Construction Engineering, No. 90, 49—58 (1988).
Many factors affect the MSD, and we first thought that we would have to consider the temperature, humidity, wind flow, etc. However, it turned out that there is a simple proportional relationship between the MSD and the cumulative snow fall before the maximum depth (CSF). For example, at Kanazawa observation station, the ratio of MSD to the CSF, nu, is 0.2933 (std. 0.015) with the coefficient of determination R-squared = 0.867. We made use of this proportionality to model the return period of MSD.
For each year, let N be the number of days of snow fall before the maximum depth, Xbar be the mean seasonal daily snow fall, and nu be the ratio of MSD to CSF.
Then the seasonal MSD, S, is expressed as nu Xbar N, where we make use of the law of large numbers.
We analyzed the data from Kanazawa, Niigata, and Sapporo observation stations, and found that N follows the negative binomial distribution, and Xbar follows the lognormal distribution. The dependence between N and Xbar are different for different observation points, at Kanazawa, N and Xbar are practically independent, at Niigata, the dependence follows the Clayton copula, and at Sapporo it follows the Frank copula.
Figure shows the return period-return level relationships. The assumed dependence between N and Xbar remarkably affected the results. Assuming independence causes huge overestimation of return levels. On the other hand, using the linear regression relationship resulted in the underestimation. Using the Clayton copula, with alpha = -0.248 obtained by fitting to the marginal cumulative distribution function of N and Xbar, successfully reproduced the return period-return level relationship.
Izumi M, Mihashi H, Takahashi T, Statistical properties of the annual maximum snow depth and a new approach to estimate the return period values, Journal of Structural and Construction Engineering, No. 90, 49—58 (1988).