International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Application of region of influence approach to estimate extreme snow load for a northeastern province… Mo, Huamei; Fan, Feng; Hong, H. P. Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 Application of Region of Influence Approach to Estimate Extreme Snow Load for a Northeastern Province in China Huamei Mo Graduate Student, School of Civil Engineering, Harbin Inst. of Tech., Harbin, China Feng Fan Professor, School of Civil Engineering, Harbin Inst. of Tech., Harbin, China H.P. Hong Professor, Dept. of Civil and Environmental Engineering, Univ. of Western Ontario, London, Canada ABSTRACT: The consideration of snow load is important for regions experiencing severe winter climate and snowfall, such as Heilongjiang Province located in northeastern China. In this study, annual maximum snow depth from 1981 to 2010 for 83 stations in Heilongjiang was used as the basis to carry out extreme value analysis and to estimate ground snow load. To reduce the sample size effect in the extreme value analysis, the region of influence (ROI) approach is considered and the results are compared with those obtained from at-site analysis as well as from the current Chinese load code. Maps are developed for the region based on the estimated return period value of the ground snow load.  1. INTRODUCTION The consideration of snow load is important for regions experiencing severe winter climate and snowfall, such as Heilongjiang Province located in northeast region in China. This province has an area of more than 470,000 km2, but the basic ground snow load for only 31 locations in the province are specified in the Chinese load code (GB-50009, 2012). To better understand the extreme snow depth and snow load distribution pattern in this region and to take advantage of more recent snow depth measurements in specifying the basic ground snow load, annual maximum snow depths from 1981 to 2010 for 83 stations in the region were obtained from Heilongjiang Meteorological Information Centre (HMIC) in Heilongjiang Bureau of Meteorology. The record lengths of annual maximum snow depth for these stations range from 5 to 30 years. The records are used as the basis to estimate the return period of annual maximum snow depth and snow load based on the at-site analysis (Mo et al. 2015), although it is known that an increased sample size could increase the confidence in the estimated extreme values. To potentially overcome this problem, a regionalization approach that pools the information from other stations in a homogeneous region could be considered. One of the regionalization approaches is called the region of influence (ROI) (Acreman and Wiltshire 1987; Acreman 1987; Burn 1990).  The ROI was developed for regional flood frequency analysis. In this approach, a region is formed for each station of interest, and the information of all the stations in this region is weighted and used in the extreme value analysis for the station of interest. The regions are not mutually exclusive, and a station may be included in different regions. This approach differs from the regional frequency analysis described by Hosking and Wallis (1997), where the stations are divided into several mutually exclusive regions. In this paper, the ROI approach described in Burn (1990) is applied to estimate the return period values of the annual maximum ground snow load for Heilongjiang Province using  1 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 measured annual maximum snow depths of up to 30 years for 83 meteorological stations. The results obtained using the ROI approach are compared with those obtained from the at-site analysis as well as the recommended procedure in the current Chinese load code (GB-50009, 2012). A map of snow load for this region is given based on the ground snow load obtained using the ROI approach. 2. DATA The data used in this study is the annual maximum snow depth from 1981 to 2010 for 83 meteorological stations in Heilongjiang Province, which was provided by HMIC. The record length, nS, is 30 years for 74 stations, between 20 to 29 years for 6 stations, and less than 10 years for the three remaining stations. The locations of the Heilongjiang Province and the stations with their corresponding record length (interval) are shown in Fig. 1.    Figure 1: Locations of the targeted region and stations in this study  According to the criteria given by China Meteorological Administration (CMA) (CMA, 2007), snow depth should be measured when the snow covers more than half of the surrounding ground near the observation field (the coverage of snow cover is evaluated based on manual observation). For each measurement, at least 3 samples are required to calculate the average depth in the observation field, and the average is set to be the snow depth for that day. Measurement results are recorded in centimeters and rounded to the nearest integer. It is treated as 0 if the measured depth is less than 0.5cm. The annual maximum snow depth was extracted for each calendar year (i.e., Jan. 1st to Dec. 31st). The mean and coefficient of variation (cov) of the annual maximum snow depth S, denoted by mS and δS, respectively, were calculated and shown in Fig. 2. Also shown in Fig. 2 is the range of S for each station (whiskers in the figure). The results show that mS ranges from 6.9 to 35.3 cm with an overall mean of 17.9 cm; δS ranges from 0.26 to 0.80 with an average of 0.42. The overall mean of the coefficient of variation for Heilongjiang Province is similar to other countries that experience severe winter climate and snowfall such as Canada (e.g., Hong and Ye 2014a).   Figure 2: Mean and Cov of annual maximum snow depth for each station 3. ANALYSIS APPROACHES 3.1. Steps for ROI approach There are mainly 3 tasks in the ROI approach. First, a distance metric is to be selected to measure the closeness of the stations in question. It is suggested that this metric could be appropriately presented by the following weighted Euclidean distance (Burn, 1990): 0 10 20 30 40 50 60 70 800102030405060708090 mS for each station δS for each station Overall mean of mS Overall mean of δSStation No.Mean of S, mS (cm)0.00.10.20.30.40.50.60.70.80.9 Cov of S, δS 2 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 1/221( )Mi jij m m mmD W A A= = −  ∑  (1) where Dij is the weighted distance between the i-th and j-th stations, M is the number of attributes used to measure the similarity of the stations, Wm is the weight of the m-th attribute, and imA is the value of attribute m for station i.  The attributes can be selected either from the physical characteristics of the station, such as latitude and longitude, or statistics of the data at the station, for instance, mean and cov of S, or from both, so there could be many combinations of the attributes to be incorporated in the distance metric. However, there is no guideline in selecting the best set of attributes. Therefore, only mS and δS of the stations are selected as the attributes and equal weights are assigned to them in this study. Given the distance metric for each station, determination of threshold distance for inclusion of stations into the ROI of station i, θi, is required in the next step, so that the set of stations in the ROI of station i, Ri, is given by: Ri={j: Dij ≦θi} (2) The threshold distance defined in Option 1 in Burn (1990) is adopted in the following numerical analysis: ,( )( ),L Li Di D LiL U L Li DDN NN NN NNθθθ θ θ>== − + − < (3) where θL is a lower threshold value for the inclusion of stations into Ri, NLi is the number of stations included in Ri when the threshold value is set at θL; ND is the desired number of stations for Ri, and θU is an upper threshold value for sites with NLi <ND. The threshold distance showed in Eq. (3) implies that all the stations whose distance to site i is less than θL should be included in the ROI for site i; and if the number of stations in the ROI is less than the desired number, a less restrictive threshold should apply. θL and θU are set to the 25th and the 75th percentile of the ascendingly sorted, non-zero Dij, respectively, ND is taken as 1/4 of the total stations considered, i.e., 21 in this study. Once the ROI for site i is formed, a weighting function is used to pool the information from the included stations to the site of interest. This can be done by setting the weight for station j in the ROI, ηij, as (Burn, 1990): ηij=1-(Dij/C)n (4) where C and n are parameters of the weighting function, which are set to the 85th percentile of the sorted Dij and 2.5, respectively, in this study. 3.2. Probability model and fitting After the ROI is formed for a specified station, the information on all the stations in the ROI can be pooled together according to the weighting function defined in Eq. (4) and the return period value of annual maximum snow depth is estimated by assuming appropriate distribution model and fitting method. In the current Chinese load code (GB-50009, 2012), the annual maximum snow load is modeled as a Gumbel variate. For the stations where only snow depth is measured, the snow density is treated as a constant in the code, thus the annual maximum snow depth is also assumed to follow a Gumbel distribution. Although other distribution models such as log-normal (Ellingwood and Redfield, 1983; Durmaz and Daloğlu, 2006) and generalized extreme value (GEV) (Blanchet and Lehning, 2010; Hong and Ye, 2014a) are also considered in modeling annual maximum snow load (depth), to facilitate the comparison with the code recommended values, only the Gumbel distribution is considered since it is the suggested model in the Chinese load code.  The Gumbel distribution is given by: ( )( )GU ( ) exp exp ( ) /F x x u a= − − −  (5) where u and a are location and scale parameter, respectively. To estimate the model parameters, the method of L-Moments (MLM) presented in the next section is considered in this study since it is widely used and performs well in  3 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 regionalization analysis (e.g., Pearson, 1992; Fowler and Kilsby, 2003; Hong and Ye, 2014b).  3.3. Method of L-Moments For the ordered sample x1:n≦x2:n≦…≦xn:n, the r-th L-moment, λr, is defined as (Hosking, 1990): 1 *10( ) ( )r rx F P F dFλ −= ∫ , r=1,2,… (6) where F(x) is the cumulative distribution of x, x(F) is the quantile function,  * *,0( )rkr r kkP F p F==∑  (7) and *, ( 1)r kr kr r kpk k− +  = −      (8) The unbiased estimator of the first 4 L-moments (λr), l1,l2,l3 and l4, can be calculated by (Hosking and Wallis, 1997): l1=b0 (9) l2=2b1-b0 (10) l3=6b2-6b1+b0 (11) l4=20b3-30b2+12b1-b0 (12) where 1:1( 1)( 2)...( )( 1)( 2)...( )nr i ni ri i i rb n xn n n r−= +− − −=− − −∑  (13) l1 equals the mean of x. Rather than using l2, l3 and l4, the nondimensional quantities, τ, τ3 and τ4 could be considered, where τ = λ2 /λ1 is called L-cv since it’s analogous to cov; and τr=λr /λ2, r=3,4,… (14) in which τ3 is called L-skewness and τ4 is called L-kurtosis. The estimators of the first 3 L-moment ratios are denoted as t, t3, and t4, respectively. The L-moment ratios for the ROI of station i are derived from: ( ), , ,1 1/i iN Nir S j ij r j S jj jt n t nη= ==∑ ∑  (15) where Ni is the number of stations in the ROI for station i, nS,j is the record length for the j-th station in the ROI, ηij is the weight given in Eq. (4) and tr, j is the L-moment ratios for station j. Once irt  is calculated, the model parameters for the Gumbel distribution showed in Eq. (5) is given by (Hosking and Wallis, 1997): a=it /ln(2) (16) u=1-0.5772a (17) and the T year return period value of S for station i, ST,i, is given by: ST,i=l1,i{u-aln[-ln(1-1/T)]} (18) where l1,i is the mean of S for station i (i.e., l1 for the i-th station). 3.4. Estimation of extreme snow load The current Chinese load code (GB-50009, 2012) recommends a constant snow density of 150kg/m3 for Heilongjiang Province. The adequacy of this density is discussed in Mo et al. (2015). Given the T-year return period value of snow depth ST, the T-year return period value of ground snow load, PT (in kPa) can be estimated using: PT=ρsgST/1E5 (19) where ρs=150kg/m3 is the snow density, and g=9.8m/s2 is the gravitational acceleration.  4. PERFORMANCE OF THE CONSIDERED APPROACHES 4.1. Experiment setup Before developing the snow load contour map, it is desirable to assess the adequacy of the performance of the ROI for extreme snow load. For the assessment, a Monte Carlo experiment is carried out. For the analysis, the L-moments of the measured annual maximum snow depth and the fitted model parameters by the at-site analysis (fitting the data to Gumbel distribution using MLM), as well as the 50-year return period value derived by the at-site analysis are assumed to be the actual or “true” quantities for the stations. For the experiment, NC simulation cycles are carried out; in each cycle, samples of annual maximum snow depth are generated according to the “real” model parameters of each station; the sample size for each station is set equal to its actual record length. Three approaches are used to estimate the 50-year snow  4 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 load, P50, using the simulated samples. The three approaches are 1) ROI approach presented above; 2) at-site analysis by fitting data to Gumbel distribution using MLM; and 3) the at-site approach recommended by the Chinese load code (GB-50009, 2012) which is presented below. In the Chinese load code (GB-50009, 2012), the T-year return period value are to be estimated by fitting the data to Gumbel distribution using the method of moments (MOM): 1 /a C σ=  (20) 2 /u x C a= −  (21) ST=u-aln[-ln(1-1/T)] (22) where C1 and C2 depend on the sample size nS (or record length) and tend to 1.2826 and 0.5772 as nS tends to infinite, x and σ are the sample mean and standard deviation of the data, respectively. C1 and C2 are tabulated in the code; they are used to reduce the small sample size effect. Although information on how C1 and C2 are derived is not provided by the code, simple calculation showed that the results obtained by Eqs. (20) – (22) are practically identical to TC STCS σ+ , where STC is the T-year return period value obtained by conventional MOM (i.e., C1=1.2826 and  C2=0.5772 are fixed), σSTC is the standard deviation of STC. 4.2. Comparison To compare the performance of the considered three approaches, the Root Mean Square Error (RMSE) and Mean Bias (MB) at each station are calculated: 1/2250 5011( )CNiiCRMSE P PN = = −  ∑  (23) and, 50 501 501 CN iiCP PMBN P= −=   ∑  (24) where NC is the simulation cycle as mentioned earlier, 50iP  is the 50-year return period value of ground snow load estimated in the i-th simulation cycle, P50 is the adopted “real” 50-year return period value of ground snow load for the station. The simulation is repeated 10,000 times since preliminary analysis shows that the RMSE and MB change negligibly by increasing simulation cycles. The derived RMSE and MB are shown in Fig. 3 and Fig. 4. It can be observed from Fig. 3 that the RMSE for Approach (1) is lower than Approach (2) for most of the stations, the ratio of the RMSE for Approach (1) to that for Approach (2) range from 0.65 to 1.20 with an average of 0.81, showing that Approach (1) can lead to a decreased RMSE of about 20 % when compared to Approach (2). In contrast, the RMSE for Approach (3) is always about 40% greater than Approach (2), this can be explained by noting that the MOM is used and the estimate is based on TC STCS σ+ , rather than STC. Thus, in terms of the RMSE, Approach (1) outperforms the other two approaches for the considered case.  0 10 20 30 40 50 60 70 800.000.050.100.150.200.250.300.350.40 RMSE for approach 2) Ratios of RMSE for approach 1) to that for approach 2) Ratios of RMSE for approach 3) to that for approach 2)Station No.RMSE (kPa)0.20.40.60.81.01.21.41.61.8Ratios Figure 3: RMSE for the considered 3 approaches in the Monte Carlo experiment  It is found that the MB is very close to 0 for all the stations when Approach (2) is applied. This is expected since the samples are generated by assuming that they follow the Gumbel distribution fitted by the MLM, reflecting unbiasness of the fitting method. Although Approach (1) also estimates the extreme value by fitting data using the Gumbel distribution and the MLM, it utilizes information from other stations in the ROI of the station, so the MB for this  5 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 approach is quite scattered, it ranges from -0.11 to 0.10 while the average is 0.00. For Approach (3), again, the use of σSTC leads to a conservative estimation of the extreme value, the average MB is 0.08 for this approach. Note that the three stations with large bias are associated with the three stations with nS less than 10.  0 10 20 30 40 50 60 70 80-0.15-0.10-0.050.000.050.100.150.20 Mean BiasStation No. Approach 1) Approach 2) Approach 3) Figure 4: Mean bias for the considered 3 approaches in the Monte Carlo experiment  To further inspect the bias, the 5th and 95th percentile of the bias are shown in Fig. 5. It can be seen from Fig. 5 that the 5th and 95th percentile for Approach (1) are smaller (absolute value) than Approach (2) for most of the stations, this implies that Approach (1) is more stable in estimating the extreme value than Approach (2). On the other hand, comparison between Approaches (2) and (3) showed that, as expected, the percentiles of bias for Approach (3) are almost parallel shifted upward from that for Approach (2). Like the case for the RMSE and MB, this is also caused by the σSTC considered by Approach (3). Note again that there are three distinct peaks in Fig. 5, they are corresponding to the stations whose record length is 9, 5 and 8 years, respectively, it can be seen that in these stations, the estimation of extreme value by Approach (1) is much better than the other two Approaches, showing that it is advantageous to use Approach (1), especially when the sample size for a station is small.  0.00.20.40.60.80 10 20 30 40 50 60 70 80-0.4-0.3-0.2-0.10.0  Bias Approach 1) Approach 2) Approach 3)95th percentile 5th percentile  Station No.  Figure 5: 95th and 5th percentile of the bias for the considered 3 approach in the Monte Carlo experiment 5. EXTREME SNOW LOAD ESTIMATED BY ROI APPROACH The 50-year return period value of the ground snow load estimated by following the ROI approach presented in Section 3 for Heilongjiang Province is shown in Fig. 6a. It can be seen from Fig. 6a that the ground snow load for this region ranges from 0.21kPa to 1.07kPa and increase almost gradually from west to east and the peak value is at the far east of this region. When compared to the ground snow load estimated by the procedure recommended by the Chinese load code (GB-50009, 2012) (see Fig. 6b), it is found that they follow the similar pattern except that the ground snow load showed in Fig. 6b are in general higher than that showed in Fig. 6a, a feature may be caused by the use of TC STCS σ+ . Simple statistical analysis shows that the minimum, maximum and mean of the ratio of the ground snow load showed in Fig. 6a to that showed in Fig. 6b are 0.77, 1.03 and 0.92,  6 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 respectively, indicating that the code recommended approach leads to about 8% increase in the estimated snow load.    Figure 6: P50 derived from a) ROI approach and b) from the procedure recommended by the Chinese load code 6. CONCLUSIONS The ROI approach is applied to estimate extreme ground snow load using annual maximum snow depth data from 1981 to 2010 for 83 stations in Heilongjiang Province in China. The main findings of this study are as follows: 1) The ROI approach outperform the at-site analysis in terms of the RMSE and bias in estimating extreme ground snow load, especially when the sample size is small; 2) The 50-year return period value of ground snow load for Heilongjiang Province ranges from 0.21kPa to 1.07kPa; there is a clear spatial trend in the distribution of ground snow load in this region, i.e., lower in the west but higher in the east and increase gradually from west to east; 3) The use of the procedure recommended in the current Chinese load code (GB-50009, 2012) leads to about 8% increase in the estimated snow load when compared to those estimated by ROI approach presented in this study. 7. REFERENCES Acreman, M. C. (1987). “Regional flood frequency analysis in the U.K.: Recent research – new ideas”. Report, Inst. of Hydrol., Wallingford, United Kingdom.  Acreman, M. C., and S. E. Wiltshire (1987). “Identification of regions for regional flood frequency analysis” (abstract). Eos Trans. AGU, 68(44), 1262. Blanchet J, Lehning M (2010) Mapping snow depth return levels: smooth spatial modeling versus station interpolation. Hydrology and Earth System Sciences, 14(12): 2527-2544. Burn, D. H. (1990). “Evaluation of regional flood frequency analysis with a region of influence approach”. Water Resources Research, 26(10): 2257-2265. CMA (2007). Specifications for Surface Meteorological Observation. China Meteorological Press, Beijing (in Chinese). Durmaz M, Daloğlu AT (2006) Frequency analysis of ground snow data and production of the snow load map using geographic information system for the Eastern Black Sea region of Turkey. Journal of Structural Engineering 132(7): 1166-1177. Ellingwood B, Redfield RK (1983) Ground snow loads for structural design. J. Struct. Eng. 109(4): 950-964. Fowler H J, Kilsby C G. (2003). A regional frequency analysis of United Kingdom extreme rainfall from 1961 to 2000. International Journal of Climatology, 23(11): 1313-1334. GB-50009 (2012). “Load code for the design of building structures”. Ministry of Housing and Urban-Rural Development of the People’s  7 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 Republic of China, Beijing: China Architecture & Building Press (in Chinese). Hong HP and Ye W (2014a) Analysis of extreme of ground snow loads for Canada using snow depth records. Natural Hazards, 73(2): 355-371. Hong H P, Ye W. (2014b). Estimating extreme wind speed based on regional frequency analysis. Structural Safety, 47(0): 67-77. Hosking J R M. (1990). L-Moments: Analysis and Estimation of Distributions Using Linear Combinations of Order Statistics. Journal of the Royal Statistical Society. Series B (Methodological), 52(1): 105-124. Hosking, J.R.M. and Wallis, J.R. (1997). Regional frequency analysis: an approach based on L-moments. Cambridge University Press, Cambridge, UK. Mo H M, Fan F, Hong H P. (2015) Snow hazard estimation and mapping for a province in northeast China. Natural Hazards. DOI: 10.1007/s11069-014-1566-9: 1-16. Pearson C. (1992). New Zealand regional flood frequency analysis using L-moments. Journal of Hydrology, 30(2): 53-64.    8 

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