S n o w

C o v e r

S t a b i l i t y,

A v a l a n c h e

I n i t ia t i o n

a n d

F o r e c a s t i n g

TheStochasticModelofSnowCoverStabilityonMountainSlopes PavelA.Chernouss1andYuryV.Fedorenko2 1 Centre of Avalanche Safety of "Apatit" JSC, 33a, 50 years of October st, Kirovsk, Murmansk Region 184230, Russia Tel. 78153196230, Fax. 4778914124 e-mail: master@apatit.murmansk.su 2 Institute of Ecology, Kola Scientific Centre of Academy of Science, f. 39, 35 Stroitelei st. Apatity, Murmansk Region 184210, Russia. Tel. 78155541452, Fax. 4778914117, e-mail: yura@alphais.inep.ksc.ru

Key words: avalanche, stochastic model, snowpack stability

--------^{ð (hsðs)}
ð s¹¹
) _{+ --------}(h
ðs ^{12+ r g ( e}
1^.eg^)h-Ffr ^{cos a}2 ⁼⁰
12
--------^{ð (hs}
ð¹²
) _{+ --------}ð (hs)
sðs^{22+ r g ( e}
1^.eg^)h-Ffr ^{cos a}2 ⁼⁰
12

N-h
s
11^k
F= c+fN^{1-hs22k 2+rg(en.eg )h=0}
fr

ABSTRACT

Three-demensionaldeterministicmodelofthinelastic
shell on a rigid underlying surface of arbitrary configura-
tion isused as approach to a snow slab on the mountain
slope.Thefinitedifferencemethodisusedforcalcula-
tion of stress distributionsin the snow cover.Spatial dis-
tributionsofsnowcovercharacteristicsarerepresented
as stochastic fields which realisations are simulated with
Monte Carlo method. Such characteristics as snow thick-
ness,densityand cohesionwere simulated on abase of
information about spatial statistical structure of these pa-
rameters to obtain a stress field overa slope.

INTRODUCTION

Theforecastingofavalanche releasecan be made using
estimates of current snowpack stress field. Such estimates
may be obtained using an information of snow thickness,
snow density,shearand tensile strength and dry friction
coefficient. If values of these parameters are known at any
point of the snowpack, one able to compute stress field by
any numericalmethod and determinepotentially danger-
ous zones where the stress exceeds some threshold level
of stress.
Such simple scheme rarely may be applied to predict
an avalanche release ortodetermine a dangerous zones
indeter ministicmanner.Thespatialvariabilityof
snowpack parameters is significant and can not be deter-
minedinpracticewithsufficientresolution.Thisfact
stimulatetheusing ofprobabilisticmethods,where the
probabilitydensityandcovariationsofparameterswill
be used instead of exact values of them.

PRESENTATION OF THE PROBLEM

Westudy a stationary deterministic andstochastic problem
of snowpack balanceon arbitrary shaped mountain slope.
The problem of calculating the stress field in a snowpack
lying on a mountain slope of arbitrary shape is in fact a 3-
dimensionalproblem. Beingsolved numericallyas 3D it is a
very time-consuming task. As has been shown in previous
studies (Nye J.K.,1959,Nefed'ev V.O and Bozhinsky A.N.,
1989) the 3D problem may be reducedto 2D if the param-
eters of snow dependweakly on snow depth.
The mostly appropriate coordinatesystem for this prob-
lemis a local orthogonal basise ,e₂ ,e
3 ^,wheree1 ^,e
the unit vectorsto the two¹vature line at^2are
tangentialcurany
pointofsurface,ande_n=e₁ xe₂ .Here weassumethatall
points of consideredsurface are non-umbilic, henceat any
pointtwodifferentcurvature lineexist.Undersuchas-
sumptions the stress field governed by the simple partial
differentialequations(PDE)ofbalanceofathinelastic
non-moment shell (Novojilov V.V, 1962).

( 1 )

Heree_g,isaunitvectorrepresented gravitationalforce
direction;s
n^{1s 2 -curvaturecoordinatesofcurrentpoint}
(KorG.A., Korn T.M,1968);h=h(s₁ ,s₂) is a snow depth,
measured perpendicularlyto slope surface;s_ij- stress ten-
sori, j=1,2;p-snowdensity;g-gravitationalaccelera-
tion, cosa₁and cosa₂- directional cosines of displasement
vectoru=(u₁ ,u
2 ^{) in local basise}
icti^{1 , e2, cos a1= cosa2 =0 if and}
onlyif | u
|=0,F_fr- fron force between the snowpack
and underlying surface, c-coefficient of cohesion, f - coef-
ficient of friction.
System( 1) should be completed by linearequations
which couple the strains and stresses:

s=----^Eu
11_{1- n}^2(------ðu
ðs^1+n-----ð
ðs²⁾
12
s₁₂
= ----
E_(------
ðu_------ðu
1- nðs¹⁺ðs²⁾
21
s₂₂
= ----
E_(------ðuðu
1- n²ðs^2+n------
2^ðs1)
1

( 2 )

where E isaYoung's modulus,nisa Poissonratio.Ac-
cording to (NyeJ.K.,1959,Nefed'ev V.Oand Bozhinsky
A.N., 1989) this system of equation should be solved with
Dirichlet boundary conditions u_T =0, where is a boundary
of considered surface.
In order to solve this set of equations the knowledge of
all snow parameters is required. Many field experiments
demonstratelarge spatialvariabilityanduncertaintyof
such parameters as snow depth,coefficient ofcohesion,
coefficient of friction and snow density. This fact strongly
motivates a stochastic description of the snowpack prop-
erties.Thus,otherphysical quantities in the model,the
displacement vectoruand stresstensors _ijalsobecome
stochastic. In the present study we assume thath,r , care
distributed as a Gaussian random field with a prescribed
expectations, variancesand covariation functions, that are
found previously by a field measurements.Ouraim is to
find different statistical moments, for example, such use-
fulstatisticalestimatesasprobabilitytoexceedsome
threshold value ofstress at every point ofslope orprob-
ability density function of stress.
The stochastic solution of the problemin this study is
obtained by the Monte Carlosimulation method.In this
method, equations ( 1 ) - ( 2 ) are solved for a large number
of realizations ofh,r ,c.Fromthe large numberof deter-

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