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ORIGINAL PAPER
Extreme snow hazard and ground snow load for China
H. M. Mo
1,2
•L. Y. Dai
3
•F. Fan
2
•T. Che
3,4
•H. P. Hong
5
Received: 11 June 2016 / Accepted: 13 August 2016
ÓSpringer Science+Business Media Dordrecht 2016
Abstract The ground snow load is used as the reference snow load to estimate the design
snow load on roofs. The ground snow load is recommended in Chinese load code for the
design of building structures in the applicable jurisdiction; this load needs to be updated
regularly by integrating new available snow measurements and new analysis techniques.
This study is concentrated on the estimation of extreme snow depth and ground snow load
and on snow hazard mapping in China by using historical snow measurement data. A
probabilistic model of the snowpack bulk density was developed. For the extreme value
analysis of annual maximum snow depth, both the at-site analysis and region of influence
approach were applied. Also, several commonly used probabilistic models and distribution
fitting methods were considered for the extreme value analysis. For the annual maximum
snow depth, it was identified from the at-site analysis results that the number of sites where
the lognormal distribution is preferred is greater than that where the Gumbel distribution is
preferred. The 50-year return period value obtained from the ROI approach is insensitive to
whether the three-parameter lognormal distribution or the generalized extreme value
&H. P. Hong
Hongh@eng.uwo.ca
F. Fan
fanf@hit.edu.cn
T. Che
chetao@lzb.ac.cn
1
School of Civil Engineering, Harbin Institute of Technology, Harbin 150090, China
2
Key Lab of Structures Dynamics Behavior and Control of the Ministry of Education, Harbin
Institute of Technology, Harbin 150090, China
3
Heihe Remote Sensing Experimental Research Station, Cold and Arid Regions Environmental and
Engineering Research Institute, Chinese Academy of Sciences, Lanzhou 730000, China
4
Center for Excellence in Tibetan Plateau Earth Sciences, Chinese Academy of Sciences,
Beijing 100101, China
5
Department of Civil and Environmental Engineering, University of Western Ontario,
London N6A 5B9, Canada
123
Nat Hazards
DOI 10.1007/s11069-016-2536-1
distribution is adopted. Maps of annual maximum snow depth and ground snow load were
developed. Comparison of the estimated ground snow load to that recommended in the
design code was presented, and potential updating to the ground snow load in the design
code was suggested.
Keywords Extreme value analysis Snow depth Ground snow load Snowpack bulk
density Design code
1 Introduction
The snow load on a roof is calculated based on the ground snow load in codified design.
One of the main objectives in assessing snow hazard is to estimate a quantile of the
extreme ground snow load that is implemented in building codes (GB-50009 2012; NBCC
2010; ASCE-7 2010). The selection of the nonexceedance probability to estimate the
quantile follows well-established reliability-based design code calibration procedure
(Ellingwood et al. 1980; Madsen et al. 2006).
A review of the reliability-based building standards in the USA focused on the ground
snow load is presented in Sack (2015). The review was focused on the available historical
meteorological data, estimation of snowpack bulk density, extreme value analysis and
snow hazard mapping. It was indicated that the ASCE/SEI 7-10 was developed based on
the lognormal model, although both the Gumbel and lognormal distributions were used to
model the annual extreme snow water equivalent (Ellingwood and Redfield 1983; Lee and
Rosowsky 2005). An overview of the development of ground snow load for Canadian
design code can be found in Newark et al. (1989) and Hong and Ye (2014). These studies
carried out at-site analyses for the annual maximum snow depth and used the return period
value of annual maximum snow depth and the average snowpack bulk density to estimate
the ground snow load. In their analyses, the Gumbel distribution was used to model the
annual maximum snow depth. To reduce the small sample size effect on the estimated
return period value of the annual maximum snow depth, Ye et al. (2016) applied the
regional frequency analysis approach and the region of influence (ROI) approach (Acre-
man and Wiltshire 1987; Hosking and Wallis 1997; Burn 1990). They concluded that
among the lognormal distribution, Gumbel distribution and generalized extreme value
distribution (GEVD), the most preferred distribution model for the annual maximum snow
depth is the lognormal distribution. This is followed by the Gumbel distribution. It was
shown that the application of the regional frequency analysis and the ROI approach pro-
vide increased spatial smoothing for the snow load maps and that the ordinary kriging and
ordinary co-kriging with altitude as a covariate are the preferred spatial interpolation
techniques for snow hazard mapping.
A general description of the development of the Chinese load code for the design of
building structures (GB-50009 2012) is presented in Jin and Zhao (2012). The most recent
code updating used the snow measurements up to 2008. The basic ground snow load
specified by the code is based on the 50-year return period value of the annual maximum
ground snow load; the ground snow loads for return periods of 10 and 100 years are also
given. The 100-year return period value of the ground snow load is recommended for
design of snow load sensitive structures such as light-weight and large-span structures.
Although a detailed report on the analysis of extreme ground snow load leading to the
Nat Hazards
123
code-recommended values is unavailable, the commentary to the code indicated that the
return period value of ground snow load can be estimated using snow depth measurements,
and the ground snow load can be calculated using the estimated return period value of
annual maximum snow depth multiplying the regional average snowpack bulk density.
Moreover, the commentary to GB-50009 (2012) suggests that the ground snow load can be
modeled as a Gumbel variate, implying that the snow depth is Gumbel distributed, if the
snowpack bulk density is treated deterministically (see pages 99–101, GB-50009 (2012)).
The application of the ground snow load for design and the evaluation of the snow loads on
roofs can be found in Takahashi et al. (2001), O’Rourke and Wrenn (2007) and Zhou et al.
(2013,2015).
This study is focused on the mapping of snow hazard for China. The main objectives are
to: (1) develop a probabilistic model for the snowpack bulk density; (2) investigate the
preferred probability distribution model for the annual maximum snow depth by using the
snow depth records and the at-site analysis results; (3) map snow hazard in terms of the
return period value of the annual maximum snow depth or of the ground snow load; and (4)
make recommendations for potentially updating the ground snow load in Chinese design
code. It must be noted that the present study uses snow measurements from meteorological
stations that may not be applicable to microclimate regions (e.g., mountainous regions or
gorges), where the assessment of the snow hazard should consider local snow measure-
ments and topographic effects. Also, the assessment of potential temporal variation of the
extreme snow hazard that can be important is beyond the scope of this study.
2 Data and ground snow load in Chinese design code
2.1 Considered datasets
A dataset containing the ground snow depth and ground snow pressure records is
assembled using the data employed in several studies (Che et al. 2008; Dai and Che
2010,2014; Dai et al. 2012; Mo et al. 2015a). The snow pressure records are converted to
snow water equivalent (SWE) and, for simplicity, referred to as SWE records throughout
this study. Also, the calculated SWE from the measured ground snow pressure is simply
referred to as SWE measurement.
Fig. 1 Spatial distribution of stations with its associated n
y
:alocation of the stations, bn
y
value for each
station
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The values of the SWE given in the dataset are for 733 stations from 1999 to 2008. The
locations of the stations are shown in Fig. 1a. The SWE at a station is reported beginning at
the day when the depth of the accumulated snow on the ground reaches 5 cm for the first
time in each snow season (July 1 to June 30 next year) until the depth of the accumulated
snow is less than 5 cm. The measurement is carried out daily when there is new snowfall;
otherwise, the measurements are carried out only on 5th, 10th, 15th, 20th, 25th and the last
day of the month (CMA 2007). The corresponding snow depth for the reported SWE is also
given.
Let n
y
denote the number of years for a given station that there is at least one nonzero
SWE value reported. The value of n
y
for the considered 733 stations is summarized in
Table 1. It shows that there are 91 stations with n
y
=9 (i.e., there are reported nonzero
SWE values every year during the period from 1999 to 2008) and 354 stations with n
y
=0.
The latter is likely due to that there is rarely snow accumulation with depth greater than
5 cm in the southern China and coastal region. To appreciate this, the spatial distribution of
the stations with n
y
larger than 0 is shown in Fig. 1b. The figure illustrates that the SWE is
reported most frequently for the stations in the northeast and northwest regions of China,
where severe winter weather occurs.
For the 91 stations with n
y
=9, the number of annually reported measurements of
nonzero SWE (i.e., calculated SWE using the reported ground snow pressure), n
SWE
, varies
in each year and in each station. The spatial trends of the mean of n
SWE
are presented in
Fig. 2a, indicating that the stations with mean of n
SWE
greater than 10 are located in the
northeast and northwest regions of China. To further appreciate the variation of n
SWE
, the
mean and standard deviation of n
SWE
per year and per station are plotted in Fig. 2b. It
shows that the mean and the standard deviation of n
SWE
per station and per year are 15 and
8, respectively. This suggests that one may take advantage of the available data to carry out
statistical analysis of the SWE at least for the sites with n
y
equals to 9, especially if they
could be considered to be stationary.
In general, the variation of the mean and standard deviation of n
SWE
for stations with
different n
y
values is presented in Fig. 3. It indicates that n
SWE
decreases drastically as n
y
decreases from 9. The average number of the SWE measurements per station for the 9-year
period is likely to be less than 20 for stations with n
y
\7.
Daily snow depth measurements from 1951 to 2010 are included in the dataset for 734
stations. The spatial distribution of stations is presented in Fig. 4a. Measurements for snow
depth are required each day when more than half of the surrounding ground in vision near
the observation field is covered by snow (CMA 2007). To ‘‘qualitatively’’ appreciate the
spatial trends of the snow depth from the available information, let n
D
denote the number
of years, where each year has at least one nonzero snow depth reported for a given station.
Since n
D
=0 for a station implies that the snow depth records for the station are always 0
or unavailable for all available snow years, such a station is considered to have a return
period value of snow depth equals to zero. The spatial variability of n
D
for the remaining
stations is shown in Fig. 4b, indicating that the ground snow depth measurement is less
Table 1 Statistics of n
y
for the considered stations
n
y
0 123456789Total
# of stations 354 39 21 31 36 48 36 41 36 91 733
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123
frequently required for the south according to the specified criteria for taking measure-
ments mentioned earlier. This is expected because severe winter climate is less likely to
occur in the south than in the north of China.
Fig. 2 Spatial distribution of stations with n
y
=9 and their associated statistics of n
SWE
:aannual mean of
n
SWE
and the location of the stations, bmean and standard deviation of n
SWE
per station
Fig. 3 Statistics of n
SWE
per
station for stations with different
n
y
Fig. 4 Geographical distribution of stations with snow depth records and their corresponding n
D
.
aGeographical distribution of stations, bn
D
for the considered stations
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123
Before carrying out the extreme value analysis for annual maximum snow depths, the
probable month with the (annual) maximum snow depth occurrence, referred to as PM-
AMS, is first analyzed. For the analysis, only the years with complete records from October
1 to May 31 are considered since the records outside of this period are unlikely to contain
the annual maximum snow depth. It was found that there are 628 stations where the PM-
AMS is identified. The obtained PM-AMS is shown in Fig. 5a. The plot indicates that the
PM-AMS is January for most of the stations. This is followed by February, March and
December. However, unlike the case for other countries such as Canada where the PM-
AMS changes from February to April as the latitude increases (Hong and Ye 2014), there
are no clearly marked spatial regions and trends where the probable months are found. An
extracted annual maximum snow depth measurement from a snow season is considered
useable if the snow depth measurement record is complete during the snow season (i.e.,
from October 1 to May 31), and there is no clear identifiable reporting error. The number of
the useable annual maximum snow depth measurements available for each station, n
AS
,is
shown in Fig. 5b. In general, n
AS
decreases from north to south and from east to west. It
was found that there are 39 stations with n
AS
\10; 259 stations with 10 Bn
AS
\20; and
330 stations with n
AS
C20. Note that for the stations in the southern China, 0 may be
included in the series of annual maximum snow depths; the consideration of such a value in
the extreme value analysis is coped by using Eq. (1)
The annual maximum snow depth data are extracted from the years having complete
records during the PM-AMS, and the corresponding n
AS
for each station is shown in
Fig. 5 Spatial distribution of stations with the PM-AMS identified and of n
AS
:amap showing PM-AMS,
bn
AS
for each station with complete records from October 1 to May 31, cn
AS
for each station with complete
records for the PM-AMS
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123
Fig. 5c, indicating that n
AS
is greater than or equal to 20 for majority of stations. In fact,
there are only 6 stations with n
AS
\10, and 52 stations with 10 Bn
AS
\20. For the
remaining 570 stations, n
AS
is greater than or equal to 20. The processed data are used in
Sects. 3 and 4 for assessing the annual maximum snow depth and ground snow load.
2.2 Ground snow load in Chinese design code
The ground snow load for structural design is specified in GB-50009 (2012) for China. The
code indicates that such a load is specified based on 50-year return period value of annual
maximum SWE and that the evaluation of the ground snow load in the code is based on the
statistical analysis of ground snow measurements (SWE or snow depths) up to 2008 from
672 meteorological stations. For the statistical analysis, the code recommends the sample
of annual maximum SWE to be modeled as a Gumbel variate and the parameters of the
distribution model to be estimated using the method of moments (MOM). In all cases, the
estimated values are rounded upward to the nearest 0.05 kPa (e.g., 0.05, 0.1, 0.15) and
tabulated in the code for many locations. Using the tabulated values, the spatially inter-
polated snow load contour maps using the ordinary kriging (Chile
`s and Delfiner 1999;
Johnston et al. 2003, Ye et al. 2015) are shown in Fig. 6a for nugget equal to zero and in
Fig. 6b for nugget not equal to zero. Due to the adopted interpolation techniques, the sharp
spatial variation and the smearing effect can be observed in Fig. 6a, b, respectively. The
figure shows that the snow hazard for the northeast provinces is greater than that for other
densely populated regions; the snow hazard for Xinjiang Uyghur Autonomous Region (in
northwestern China) and Xizang Autonomous Region (in southwestern China) is also
large, but the population density in these regions is low as compared to that in the coastal
region of China. Along the southern coastal region of China, the ground snow load is
negligible or zero. In the remaining part of this study, only the ordinary kriging with nugget
equal to zero will be considered for spatial interpolation so more detailed and sharper
spatial changes in the snow hazard can be appreciated.
It is not clear whether the measured snow depth or SWE is used to develop the code-
recommended ground snow loads. The code also states that for the locations where only
snow depths are measured, a regional average snowpack bulk density q
s
applicable to the
region can be used to estimate the snow load. The recommended regional average
snowpack bulk density is 150 kg/m
3
for northeastern China, north Xinjiang and southern
Fig. 6 Interpolated ground snow load using the ordinary kriging and the tabulated values in GB-50009
(2012): aspatial interpolated map by considering nugget equal to zero, and bspatial interpolated map by
considering nugget not equal to zero
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123
China, and 130 kg/m
3
for northern and northwestern China. In addition to this, it is
recommended to use q
s
=120 kg/m
3
for Qinghai Province (in northwestern China), and
q
s
=200 kg/m
3
for Jiangxi and Zhejiang provinces (in southern China).
3 Probabilistic models and estimation of ground snow load based directly
on SWE
3.1 Probabilistic models
For a given site, the annual maximum snow depth, X, in a given year, x, may be zero; its
occurrence depends on the climatic condition at the site. Given the probability distribution
of nonzero annual maximum snow depth, denoted as F(X), the probability distribution of X
including both zero and nonzero annual maximum snow depth, denoted as F
A
(x), could be
expressed (Newark et al. 1989),
FAxðÞ¼pþ1pðÞFxðÞ;ð1Þ
where pis the probability of occurrence of the annual maximum snow depth equal to zero
in a year, and F(x) is discussed below.
As mentioned in the Introduction, the Chinese code GB-50009 (2012) recommends that
the annual maximum ground snow load to be modeled as a Gumbel variate. The use of the
GEVD and the lognormal distribution is also considered in the literature (Ellingwood and
Redfield 1983; Newark et al. 1989; Hong and Ye 2014).
The Gumbel distribution, F
GU
(x), is given by (Coles 2001),
FGUðxÞ¼exp exp ðxuÞ=aðÞðÞ;ð2Þ
where xdenotes the value of X, representing the annual maximum snow depth or the
annual maximum ground snow load, aand uare the scale and location parameters. The
mean l
X
and the standard deviation r
X
of Xare equal to u?0.5772aand ap=ffiffiffi
6
p,
respectively; the coefficient of variation (cov) of X,v
X
, equals r
X
/l
X
. Given the distri-
bution parameters, the T-year return period value of X,x
T
, can be estimated by solving
Eq (1) with F
GU
(x)=1-1/T.
The GEVD, F
GE
(x), is given by (Coles 2001),
FGE xðÞ¼exp 1kxuðÞ=aðÞ
1=k
for k6¼ 0ð3Þ
where u,aand kare the model parameters. For k=0, Eq. (3) turns into the Gumbel
distribution. If k[0, an upper bound equal to u?a/kexists for x. Again, once the
distribution model parameters are given, x
T
can be calculated by solving Eq. (3).
The three-parameter lognormal (3p-LN) distribution, F
LN
(x), can be written as,
FLNðxÞ¼Uln xnðÞmln x
ðÞ=rln x
ðÞ;ð4Þ
where n,m
lnx
and r
lnx
are the distribution parameters and U() denotes the standard normal
distribution function. Equation (4) with n=0 is referred to as the 2p-LN distribution.
For the distribution fitting, the method of L-moments (MLM) (Hosking and Wallis
1997) and the method of maximum likelihood (MML) are considered. Both of these
methods are popular, and the former outperforms the MOM if the sample size is small,
while the adequacy of the latter is well known for large sample size. A comparison of the
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123
performance of these two methods and some other fitting methods are given in Hong et al.
(2013). In addition, the MOM is used for the Gumbel distribution since it is recommended
by the design code. In such a case, the mean plus one standard deviation of T-year return
period value is employed following the code procedure (GB-50009 2012; Mo et al.
2015b,c).
The use of the 3p-LN distribution is not considered for the at-site analysis since the
application of the MML with this distribution often does not lead to convergence in
distribution fitting (Martin and Stedinger 2000); if convergence is achieved, preliminary
results carried out in this study showed that the use of the 3p-LN and 2p-LN distributions
leads to very similar results. Moreover, the 3p-LN is applied in the ROI approach where
the fitting is carried out using the MLM.
3.2 Estimation of ground snow load based directly on SWE
Given the SWE, the ground snow load, L(kPa), can be calculated using,
L¼105qwghsð5Þ
where the density of water q
w
is taken equal to 1000 kg/m
3
,g=9.8 m/s
2
is the gravi-
tational acceleration, h
s
is the value of the SWE in cm, and the constant 10
-5
is used for
unit conversion.
Note that the Chinese code GB-50009 (2012) recommends that the Gumbel distribution
fitted by the MOM should be used to estimate 50-year return period value of snow load L,
l
50
, and that the estimated l
50
plus its one standard deviation, denoted as l
c50
, should be
considered to take into account the small sample size effect on or statistical uncertainty in
the estimated extreme snow load (Mo et al. 2015b,c). By following the code procedure,
l
c50
is estimated for the 91 stations (each with n
y
=9 as shown in Table 1) using annual
maximum SWE measurements.
The estimates are compared to the code-recommended values in Fig. 7a. It shows that
the estimated l
c50
is larger than that inferred from the code for most of the stations. Also,
the mean and the standard deviation of the ratio of estimated l
c50
to the code-recommended
value, R
Lc50
, are shown in Fig. 7a and Table 2, indicating that on average the estimated
value is 24 % greater than the code-recommended value and that there is significant
Fig. 7 Comparison of estimated extreme ground snow load versus the value inferred from the design code
for the 91 stations with n
y
=9: acomparison of estimated and code-recommended l
c50
,bcomparison of
estimated l
50
and code-recommended l
c50
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123
variability in R
Lc50
. The correlation coefficient between the estimated and the code-rec-
ommended l
c50
equals 0.75, indicating that there is a linear trend between the estimated and
code-recommended l
c50
. For a particular station which is located in the northern slope of
the Himalayas in Xizang Autonomous Region (station ID: 55655), while the code-rec-
ommended l
c50
equals 3.3 kPa, the estimated value is only 1.6 kPa. The ground snow load
for this station as well as its surrounding area will be investigated further in the following
sections by considering uncertainty in the annual maximum snow depth and snowpack bulk
density. The above observations indicate that the use of 9 years of SWE measurements to
estimate the extreme ground snow load may not be sufficient and adequate.
For completeness, a comparison of l
50
to the code-recommended value is shown in
Fig. 7b, and the mean and standard deviation of the ratio between the estimated l
50
and the
code-recommended value, denoted as R
L50
, are included in Table 2. Moreover, l
50
is also
estimated by fitting the Gumbel distribution, 2p-LN distribution and GEVD to annual
maximum SWE samples using the MLM and MML. For some of the stations, the fitting of
the GEVD to data using the MML encounters convergence problems; if this is the case, the
estimated l
50
is not reliable. Also, the value of Akaike information criterion (AIC) (Akaike
1974) is calculated if the fitting is carried out by using the MML. The results indicate that
the Gumbel distribution, 2p-LN distribution and GEVD are preferred for 32, 44 and 15
stations (35, 48 and 17 % of all considered station), respectively. The statistics of R
L50
by
using the two most preferred distribution models are summarized in Table 2. In all cases,
the mean of R
L50
presented in the table is near 1.0 but the standard deviation is greater than
0.25. The standard deviation of R
L50
obtained by using the lognormal distribution is greater
than that by using the Gumbel distribution. This is expected since the code-recommended
value is based on the Gumbel distribution, and the lognormal distribution is associated with
heavy upper tail.
Since the spatial coverage of the SWE historical records is limited as shown in the
previous section, the use of the measured SWE as basis to directly estimate and map the
ground snow load for China is not feasible with available data for this study. To overcome
this, the ground snow load could be estimated based on the probabilistic models of the
annual maximum snow depth and the snowpack bulk density. The assignment of the
probabilistic models is described in the following sections.
4 Annual maximum snow depth statistics
4.1 At-site analysis
The values of the annual maximum snow depth Scan be extracted from snow depth
records. It was indicated that the number of useable annual maximum snow depth data n
AS
Table 2 Statistics of the ratios
Model used to fit LR
Lc50
, MOM Gumbel distribution Lognormal distribution
Fitting method R
L50
, MOM R
L50
, MML R
L50
, MLM R
L50
, MML R
L50
, MLM
Mean 1.24 1.03 0.98 1.05 1.18 1.15
Standard deviation 0.35 0.28 0.26 0.28 0.38 0.35
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at each station is not uniform. To minimize the sample size effect, first, a probability
distribution fitting is carried out by considering the Gumbel distribution, 2p-LN distribu-
tion and GEVD and by using the MML and MLM for the annual maximum snow depth
data from each of the 330 stations, where each station has n
AS
C20. For these stations, the
mean of Sranges from about 15 to 30 cm for northeastern and northwestern regions; this
value is less than 15 cm for the remaining part of China. The cov of Sis less than 0.6
except for some sites in the coastal region and northwestern region of Inner Mongolia.
Based on the AIC, it is concluded that the 2p-LN distribution is preferred for 172 stations,
the Gumbel distribution is preferred for 114 stations, and the GEVD is preferred for the
remaining 44 stations.
By adopting the two most preferred distribution models (the 2p-LN distribution and the
Gumbel distribution), the estimated 50-year return period value of S,s
50
, based on the
MLM and MML is shown in Fig. 8. Note that the stations with snow depth always equal to
0 are also considered in the interpolation by setting s
50
equal to 0 at these stations. It can be
seen from the figure that the results estimated by using the MLM and MML are quite
similar for a given probabilistic model. For some regions, values of s
50
estimated by using
the 2p-LN distribution tend to be greater than those estimated by using the Gumbel
distribution.
Comparison of the results shown in Figs. 6and 8indicates that the spatial trends of s
50
shown in Fig. 8do not necessarily follow those shown in Fig. 6for l
50
. This is expected as
Fig. 8 Estimated s
50
by considering the 2p-LN and Gumbel distributions fitted to data for stations with
n
AS
C20: ausing the 2p-LN distribution fitted by the MLM, busing the 2p-LN distribution fitted by the
MML, cusing the Gumbel distribution fitted by the MLM, dusing the Gumbel distribution fitted by the
MML
Nat Hazards
123
the extreme ground snow load is affected by the snowpack bulk density that is spatially
varying.
To investigate the effect of the records from stations with n
AS
less than 20 on the
estimated return period value of the snow depth, the above analysis is repeated but con-
sidering records from stations with n
AS
C10. This resulted in the use of snow data from
589 stations for the estimation of the extreme snow depth. In this case, the distribution
fitting results indicate that the 2p-LN distribution, Gumbel distribution and GEVD are
preferred for 49, 36 and 15 % of all the considered stations. The maps of the estimated
50-year return period of annual maximum snow depth are shown in Fig. 9for the 2p-LN
and Gumbel distributions.
Again, it can be observed that the use of the MLM or MML does not affect significantly
the estimated snow hazards and that the snow depth estimates and spatial snow hazard
trends are affected by the selected distribution type.
Comparison of the results shown in Figs. 8and 9indicates that the spatial trend in
Fig. 9is smoother than that in Fig. 8. This is expected since results shown in Fig. 9include
estimated s
50
with increased small sample size effect as the minimum n
AS
equals 10.
Fig. 9 Estimated s
50
by considering the 2p-LN and Gumbel distributions fitted to data for stations with
n
AS
C10: ausing the 2p-LN distribution fitted by the MLM, busing the 2p-LN distribution fitted by the
MML, cusing the Gumbel distribution fitted by the MLM, dusing the Gumbel distribution fitted by the
MML
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4.2 ROI approach
The snow hazard maps shown in Fig. 9contain small sample size effect. The application of
the ROI approach (Burn 1990) that was originally developed to assess flood frequency
could be used to reduce the small sample size effect in estimating the snow hazard. The
approach was used to estimate snow hazards in Canada (Ye et al. 2016) and in a north-
eastern province in China (Mo et al. 2015c). The approach requires the selection of a
distance metric to measure the closeness of the stations in question. Given a site of interest,
annual maximum snow depth data from any site within a threshold of the distance metric
measured from the site of interest will be included, weighted and used to estimate the T-
year return period value of the snow depth. The distance metric suggested by Burn (1990)
is given by,
Dij ¼X
M
m¼1
wmðAi
mAj
mÞ2
!
1=2
ð6Þ
where D
ij
is the weighted distance between the i-th and j-th stations, Mis the number of
attributes used to measure the similarity of the stations, w
m
is the weight of the m-th
attribute, and A
m
i
is the value of attribute mfor station i. Following Ye et al. (2015) and Mo
et al. (2015c), the normalized latitude, longitude and L-cv (i.e., L-coefficient of variation)
associated with the stations are used as attributes for the ROI approach, where the nor-
malization is carried out by dividing the value of an original variable by its corresponding
range for all stations.
The threshold distance h
i
given by Burn (1990),
hi¼
hL;NLi [¼ND
hLþhUhL
ðÞ
NDNLi
ND
;NLi\ND
(ð7Þ
is used to accept the j-th station to form the ROI of the i-th station, R
i
,ifD
ij
^h
i
.In
Eq. (7), h
L
is a lower threshold value for the inclusion of stations into R
i
,N
Li
is the number
of stations included in R
i
if the threshold value is set at h
L
,N
D
is the desired number of
stations for R
i
, and h
U
is an upper threshold value for sites with N
Li
\N
D
.h
L
and h
U
are set
to the 20th and the 40th percentile of the ascendingly sorted (nonzero) value of D
ij
,
respectively; N
D
is taken as 50 in this study.
The L-moment ratios (1,
ti,
ti
3) for the ROI of site iare calculated using,
ti¼X
Ni
j¼1
njgijtj=X
Ni
j¼1
njgij;ð8Þ
and,
ti
3¼X
Ni
j¼1
njgijt3;j=X
Ni
j¼1
njgij;ð9Þ
where N
i
is the number of stations in the ROI for site i;t
j
=(l
2
/l
1
), and t
3,j
=(l
3
/l
2
), denote
the L-moment ratios for the j-th station; (l
1
,l
2
,l
3
)
j
are the estimated first three L-moments;
n
j
is the number of samples at the j-th station, and g
ij
is the weighting function given by
(Burn, 1990),
Nat Hazards
123
gij ¼1Dij=W
hð10Þ
in which Wis considered to be the 50th percentile of D
ij
and hequals to 2.5 as suggested by
Burn (1990). Both the 3p-LN distribution and GEVD are employed to fit the calculated L-
moment ratios (1,
ti,
ti
3) in the ROI approach, and the quantile of nonexceedance proba-
bility F(i.e., the T-year return period value with T=1/(1 -F)) at the j-th station, Q
j
(F)is
given by,
QjðFÞ¼ljqðFÞ;ð11Þ
where l
j
is the mean value of the annual maximum snow depth for the j-th station, and
q(F) is the regional quantile of nonexceedance probability F.
The estimated s
50
, based on the ROI approach for snow depth records from station with
n
AS
C10, is shown in Fig. 10. Comparison of maps shown in Fig. 10a, b indicates that
they are almost identical. This suggests that the ROI approach is insensitive to the two
considered distribution types for the considered snow depth data.
Also, a comparison of the results shown in Figs. 9and 10 indicates that the spatial
trends of s
50
estimated by using the at-site analysis and ROI approach are very similar, and
the differences between s
50
estimated by using the at-site analysis and ROI approaches are
not very large. Therefore, it is expected either of the analysis procedures could be used to
adequately map the extreme snow depth.
5 Estimation of snowpack bulk density and ground snow load
5.1 Estimation of the snowpack bulk density
The snowpack bulk density q
s
(kg/m
3
) can be calculated by dividing SWE by snow depth
(i.e., q
s
=q
w
h
s
/s). The snowpack bulk density q
s
at a site is uncertain with significant
variability (Ellingwood and Redfield 1983). It is a complex function of snow depth, snow
temperature, snow deposition history and the initial density of the individual snow layers
(Tobiasson and Greatorex 1997; Sturm et al. 2010).
To potentially develop an empirical relation to predict q
s
similar to those given in
Tobiasson and Greatorex (1997), Sturm et al. (2010) or Sack (2015) but applicable to
Fig. 10 Estimated s
50
based on the ROI approach: aconsidering the 3p-LN distribution, and bconsidering
the GEVD
Nat Hazards
123
China, first, a statistical analysis of q
s
is carried out to verify the adequacy of the q
s
values
recommended in the Chinese design code mentioned earlier. For the simple analysis, it is
assumed that q
s
is independent of the snow depth, and a distribution fitting exercise using
the MML is carried out for q
s
at each station with n
y
=9, by considering several com-
monly used candidate distributions. The results suggest that the lognormal distribution is
adequate. The mean and cov of q
s
are calculated for each station and are shown in Fig. 11a,
b, respectively. A somewhat arbitrary limit based on the location of the stations is imposed
for the plots shown in Fig. 11 since the 91 stations do not provide adequate spatial cov-
erage for the entire country. For comparison purpose, the code-suggested values of q
s
are
also included in Fig. 11c. The comparison suggests that the code-recommended mean
value of q
s
is greater than the estimated q
s
for northeastern China but smaller than the
estimated q
s
for northwestern Xinjiang Uyghur Autonomous Region. These differences
may be due to that the code is focused on the extreme snow load and are investigated
further in the following. The cov of q
s
varies within 0.2–0.4 for most regions.
Since the number of the stations considered for the results shown in Fig. 11 is small, to
increase the spatial coverage, estimation of the statistics of q
s
is carried out by considering
data from stations with n
y
C5 and from all stations with n
y
C1. The former ensures that,
on average, there are 15 samples from each station (see Fig. 3), while the latter basically
takes into account all available samples. The estimated statistics of q
s
are shown in
Fig. 12a, b for the case with n
y
C5 and in Fig. 12c, d for the case with n
y
C1. Com-
parison of the mean of q
s
shown in Figs. 11 and 12 indicates that the spatial trends of the
estimated mean of q
s
for n
y
C5 is consistent with those for n
y
C9 and that the spatial
Fig. 11 Interpolated mean and cov of q
s
based on data for the 91 stations with n
y
=9: amean of q
s
,bcov
of q
s
, and ccode-suggested values of q
s
Nat Hazards
123
resolution of q
s
increases as n
y
limit decreases. The mean of q
s
for majority of sites is less
than about 140 kg/m
3
. The mean of q
s
for sites near the boundary between Hunan, Jiangxi
and Zhejiang provinces and for sites near the boundary between Hebei and Shandong
provinces is approximately equal to or exceeds 200 kg/m
3
. These observations are con-
sistent with those made in Dai and Che (2010) and Ma and Qin (2012). Also, the obser-
vations made on the mean of q
s
shown in Fig. 11 are still applicable to the results presented
in Fig. 12. The range of cov values of q
s
presented in Figs. 11 and 12 is similar. In all
cases, the cov of q
s
varies within 0.25 and 0.4 for most locations. Although the estimated
values of the cov of q
s
shown in Fig. 12d for some locations cannot be used with confi-
dence because of the small sample size, they are shown for completeness.
To possibly develop an empirical model to predict q
s
as a function of snow depth and
other environmental parameters (Tobiasson and Greatorex 1997; Sturm et al. 2010), first,
values of q
s
and their corresponding snow depths for each of the 91 stations (each with
n
y
=9) are considered. Since the present study is focused on the extreme ground snow
depth, a regression analysis of q
s
is carried out by considering q
s
values with their cor-
responding snow depth values that are greater than f9s
50
at the considered stations,
where fis a selected constant within 0.7–1.0. For the regression analysis, the following two
models are considered:
qs¼a1sa2;ð12Þ
and
Fig. 12 Statistics of q
s
:amean of q
s
based on stations with n
y
C5, bcov of q
s
based on stations with
n
y
C5, and cmean of q
s
based on stations with n
y
C1, dcov of q
s
based on stations with n
y
C1
Nat Hazards
123
qs¼q0þqmax q0
ðÞ1exp bsðÞðÞ;ð13Þ
where a
1
,a
2
,q
0
,q
max
and bare model parameters to be determined through nonlinear
regression analysis. The functional form shown in Eq. (12) is employed in Tobiasson and
Greatorex (1997) to relate the 50-year return period values of ground snow load and the
snow depth, while that shown in Eq. (13) is considered in Sturm et al. (2010). In almost all
cases, the available data are scarce or limited, and the results of fitting exercise indicate
that the data do not follow the considered models.
An effort in identifying clusters or regions where values of q
s
for the stations within the
same cluster can be grouped and analyzed is carried out. The consideration of clusters is
consistent with current code approach (see Fig. 11c). The cluster identification is per-
formed using the k-means clustering (MacQueen 1967; Jain et al. 1999) and by varying the
number of clusters. An example of the identified four clusters is shown in Fig. 13a. For the
results shown in Fig. 13a, q
s
values with their corresponding snow depth greater than
0.8 9s
50
for each station are considered, and the normalized latitude, longitude and the
mean of q
s
are used for the k-means clustering. These clusters obtained do not necessarily
follow the four regions recommended by the code (see Fig. 11c). Again, the results of
fitting exercise carried out for q
s
within each cluster do not provide any convince argument
and guidance on selecting spatially varying probabilistic model for q
s
.
Based on the above observations, the four regions suggested in the code to assign q
s
(see
Fig. 11c) and the results shown in Figs. 11 and 12, six clusters shown in Fig. 13b are
subjectively assigned to assess the probabilistic model of q
s
in the following. The
Fig. 13 Clusters: aidentified 4 clusters by using the k-means clustering considering q
s
with corresponding
snow depth [0.8 9s
50
at each station, bsubjectively assigned 6 clusters; csuggested zone for code-
making
Nat Hazards
123
corresponding zone for each cluster that could be used for code-making is presented in
Fig. 13c. For each cluster, values of q
s
are shown in Fig. 14. Results of regression analysis
indicate that the models shown in Eqs. (12) and (13) are inadequate (i.e., R
2
value is less
than 0.05 in all cases). By considering values of q
s
for sgreater than 20 cm for Clusters 1
and 4, and 10 cm for the other four clusters, the estimated standard deviation of q
s
as a
function of snow depth sand the overall mean of q
s
are also shown in Fig. 14. The
consideration of the lower limits on the snow depth is to avoid potential biases on the
estimated mean of q
s
caused by the measured q
s
values for very small snow depth that do
not affect the extreme ground snow load. The use of mean of q
s
that is independent of snow
depth is consistent with current code-suggested approach (GB-50009 2012). The mean of
q
s
,m
qs
, shown in Fig. 14 indicates that m
qs
equals 149, 121, 130, 170, 133 and 173 kg/m
3
for Clusters 1–6, respectively. m
qs
for Cluster 1 is very close to the code-suggested value of
Fig. 14 Snowpack bulk density versus snow depth for each cluster
Nat Hazards
123
150 kg/m
3
. For other clusters, m
qs
differs from that suggested in the code. In particular, m
qs
for Cluster 4 (part of Xinjiang Uyghur Autonomous Region) is 20 kg/m
3
higher than the
code-recommended value and m
qs
for Cluster 6 is 27 kg/m
3
less than the code-suggested
value.
The standard deviations shown in Fig. 14 are estimated by using variable bin width to
ensure that there are at least 50 samples within a considered bin. These values suggest that
as an approximation the standard deviation of q
s
for each cluster can be considered to be
independent of s. By adopting this assumption, the estimated standard deviation equals 33,
36, 46, 39, 51 and 70 kg/m
3
, for Clusters 1–6, respectively. This results in the coefficient of
variation of q
s
ranging from 0.22 to 0.40 for the six clusters. The lower values of the
coefficient of variation of q
s
are associated with the northeast and northwest regions of
China (Clusters 1 and 4); the highest values are associated with Clusters 5 and 6.
Fig. 15 q
s
presented in the lognormal probability paper for each cluster
Nat Hazards
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To develop a probabilistic model of q
s
, samples of q
s
are shown in the lognormal
probability paper in Fig. 15. The use of Kolmogorov–Smirnov test indicates that the
lognormal model for q
s
cannot be rejected at 5 % significance level for Clusters 5 and 6
and at 1 % significance level for Cluster 3. Figure 15 shows that the inadequacy of the fit
for Clusters 1, 2 and 4 occurred at the lower tail of the distribution. An inspection of the
data indicates that the values of q
s
in the lower tail region are associated with small snow
depth which is unlikely to impact the extreme ground snow load. Therefore, q
s
is modeled
as a lognormal variate in the following.
5.2 Mapping the ground snow load
A very simple approach to estimate the T-year return period value of the ground snow load
L,l
T
, is to use the code-suggested procedure and value of q
s
(kg/m
3
) (GB-50009 2012),
lT¼105qsgsTð14Þ
where l
T
(kPa) is the T-year return period value of the ground snow load, q
s
is taken equal
to the code-suggested value, and s
T
is the estimated T-year return period value of S(cm)
such as those shown in Figs. 8,9and 10. Examples of the estimated l
T
by using s
T
shown in
Figs. 9a, c, and 10a are presented in Fig. 16a–c. The spatial trends shown in these maps are
similar to those presented in Fig. 6. Rather than using the code-recommended q
s
(presented
in Fig. 11c), if the suggested q
s
values shown in Fig. 13c are employed, the corresponding
maps are presented in Fig. 16d–f. As expected, a comparison of the results shown in
Fig. 16a–c and in Figs. 16d–f indicates that the differences between these two sets of
results are due to the adopted q
s
values shown in Figs. 11c and 13c.
To better appreciate the differences between the estimated l
50
shown in Fig. 16 and the
code-recommended value, their ratio, denoted as R
E/C
, is calculated and presented in
Figs. 17. The figures show that for most regions R
E/C
ranges from 0.7 to 1.3 for results
shown in Fig. 17a and from 0.8 to 1.2 for results shown in Fig. 17b, c. This is expected
since results presented in Fig. 17a are based on s
50
estimated using the 2p-LN distribution
and the at-site analysis. Values of R
E/C
for many regions shown in Fig. 17b are near 1.0; it
may be partly due to that the code-suggested Gumbel distribution type is used. The
relatively narrow range and smooth spatial variation of R
E/C
shown in Fig. 17c could be
attributed to that the estimated s
50
by using the ROI approach is relatively smooth and not
very sensitive to the considered distribution type (see Fig. 10). Figure 17d–f is similar to
Fig. 17a–c except that the ratios for northwestern Xinjiang (the far northwest of China)
have significantly increased from about 1.0 to 1.2 to higher than 1.2, and this is due to the
increase in the regional average snow density in this region. For this region, the code-
suggested snow density is 150 or 130 kg/m
3
while the one estimated by the current study is
170 kg/m
3
.
Note that the mapped ground snow load shown in Fig. 16 neglects the uncertainty in q
s
.
This uncertainty affects the estimated l
T
(Mo et al. 2015c). To take this uncertainty into
account to estimate l
T
, samples of L,
L¼105qsgS ð15Þ
can be generated based on simple Monte Carlo technique and the probability distributions
of q
s
and S. These samples are then used to estimate l
T
from the empirical distribution of
L. It is noteworthy that, in particular, if q
s
and Sare considered to be independent
Nat Hazards
123
lognormal variates, the calculation of l
T
is simplified since in such a case Lis also a
lognormal variate, and analytical solution can be obtained.
By considering the probabilistic models presented in Fig. 15 (i.e., the clusters shown in
Fig. 13c and the distribution model shown in Fig. 15), the calculated l
T
values are shown in
Fig. 18a–c if the probabilistic models leading to 9a, 9c, and 10a for s
50
are employed.
Comparison of the results shown in Figs. 16 and 18 indicates that they are consistent and
similar for most sites. The maximum and minimum differences between l
50
shown in
Fig. 18 and in Fig. 16 are equal to 0.4 and -0.1 kPa, respectively.
Fig. 16 Estimated l
50
based on Eq. (14) for stations with n
AS
C10. Plots a–care based on q
s
recommended
by the code, and plots d–fare based on q
s
estimated in this study: a,dare based on s
50
estimated by fitting
2p-LN distribution using MLM, band eare based on s
50
estimated by fitting Gumbel distribution using
MLM, and cand fare based on s
50
estimated by the ROI approach considering 3p-LN distribution
Nat Hazards
123
The consideration of uncertainty in q
s
leads to less than 5 % of increase in l
50
for most
sites. This relatively small increase in l
50
for most sites can be explained by noting that the
probabilistic model for q
s
(see Figs. 14,15) has a mean of q
s
smaller than the code-
suggested value. It is noteworthy that the estimated l
50
shown in Fig. 18a–c and s
50
shown
in Figs. 9a, c and 10a can be used to calculate the required values of q
s
,q
s-u
, such that
l50 ¼105qsugs50 ð16Þ
The calculated q
s-u
values are about 162, 135, 157 189, 155 and 203 kg/m
3
for Zones
1–6 shown in Fig. 13c, respectively. As expected these values are greater than the means of
q
s
shown in Fig. 13c. The values of q
s-u
for Zones 1, 5 and 6 are close to those suggested
Fig. 17 Ratio of estimated l
50
shown in Fig. 16 to the code-suggested value: a–fcorrespond to Fig. 16a–f,
respectively
Nat Hazards
123
in the current code for similar geographical regions shown in Fig. 11c. This implies that
the uncertainty in q
s
may be considered in assigning the values in the current design code.
It also indicates that if only statistics on Sare available, l
50
could be estimated by using
Eq. (16) which included the effect of uncertainty in q
s
.
To see the differences between the estimated l
50
shown in Fig. 18 and the code-rec-
ommended ground snow load, values of R
E/C
are calculated and depicted in Fig. 19a–c,
indicating that the spatial trends of R
E/C
are similar for all three plots. The values of R
E/C
range from 0.8 to 1.2 for most sites, except for the northwestern region. The increases in l
50
for Xinjiang Uyghur Autonomous Region and northwestern Inner Mongolia can be higher
than 20 %.
Overall, the results presented in Figs. 16,17 18 and 19 indicate that the derived values
of l
50
based on up-to-date snow measurements differ from those given in the current code
for many sites. It must be emphasized that the ground snow load hazard maps developed
herein are based on measurements from meteorological stations that may not be applicable
to special locations such as mountainous regions or gorges. For such sites, the ground snow
load should be developed using local snow measurements considering the orientation,
elevation and topographic effects.
Fig. 18 Estimated ground snow load based on Eq. (15) considering the uncertainty in q
s
(see Fig. 15):
ausing the 2p-LN distribution and MLM; busing the Gumbel distribution and MLM; cusing the ROI
approach and 3p-LN distribution
Nat Hazards
123
6 Conclusions
The present study focused on the estimation and mapping of extreme snow depth and
ground snow load in China. The extreme value analysis of the snow depth and ground
snow load is carried out using historical ground snow measurements. The analysis
employed both the at-site analysis and region of influence (ROI) approach by considering
several commonly employed probabilistic models.
The results of the at-site extreme value analysis of annual maximum snow depth
Sindicate that the 2p-LN distribution, Gumbel distribution and GEVD are preferred dis-
tribution of Sfor 49, 36 and 15 % of 589 stations, each with at least 10 years of useable
data. By adopting the 2p-LN distribution or Gumbel distribution, the snow hazard maps
developed based on the estimated 50-year return period values of S,s
50
, are not very
sensitive to whether the maximum likelihood method or the method of L-moments are used
for the fitting. However, the snow hazard maps are affected by the adopted probability
distribution of S.
It is shown that the snow hazard maps developed based on s
50
estimated using the ROI
approach are not very sensitive to whether the 3p-LN distribution or the GEVD is con-
sidered. Such maps are smoother than those based on the at-site analysis, although the
spatial trends of the snow hazard maps are consistent.
It is shown that the estimated mean of the snowpack bulk density q
s
differs from that
recommended in the current design code for several regions in China. It is suggested that q
s
can be modeled as a lognormal variate.
Fig. 19 Ratio of estimated l
50
shown in Fig. 18 to the code-suggested value: a–ccorrespond to Fig. 18a–c,
respectively
Nat Hazards
123
By adopting the code-suggested mean values of the snowpack bulk density q
s
and using
the estimated s
50
, the 50-year return period value of the ground snow load, l
50
, is devel-
oped. The spatial trends of the estimated l
50
are similar to those inferred from the values
suggested in the design code, although there are differences. Furthermore, values of l
50
are
also estimated by considering uncertainty in q
s
using the developed probabilistic models
for q
s
and Sin the present study. Again, the spatial trends of the estimated l
50
are similar to
those inferred from the code-recommended values. In all cases, the ratio of the estimated
l
50
to the code-suggested value varies spatially and is within 0.8–1.2 for most sites but can
be greater than 1.5, indicating the need to update the ground snow load recommended in
the code using most current snow data.
Acknowledgments Financial support received from National Natural Science Foundation of China (No.
51478147 and No. 41271087), National Science and Engineering Research Council of Canada (RGPIN-
2016-04814) and the University of Western Ontario is much acknowledged. We thank two reviewers for
their constructive comments which helped us to improve the manuscript
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