Annals f Glacilgy 18 1993 Internati n al Glaciigicai Sciety Snw avalanche runut frm tw Canadian muntain ranges D.J. NIXON AND D. M. MCCLUNG Department f Civil Engineering, University f British Clumbia, Vancuver, B.C., Canada V6T IZ4 ABSTRACT. Field measurements n maximum runut frm tw different muntain ranges in Canada are presented and cmpared: the Cast Muntains in British Clumbia and the Rcky Muntains. We include a statistical analysis f tpgraphic terrain parameters such as starting zne catchment area, hrizntal reach, vertical drp and relevant slpe angles. Fllwing McClung and Mears (1991), we derived a dimensinless parameter which is a measure f run ut fr each avalanche and we fund that the runut ratis (defined belw) fr a given muntain range bey a Gumbel distributin cnsistent with previus results. In additin, we fund that the runut ratis fr bth muntain ranges have a length-scale dependence which is ptentially very imprtant fr land-use planning prcedures: the mean value f the runut rati decreases significantly as the hrizntal reach increases. Tgether with data frm ther muntain ranges, ur results shw that path length effects will have t be incrprated when using statistical predictin methds fr engineering zning purpses. The runut rati is defined as the qutient f tw lengths, D.x/ Xfj, where D.x is the hrizntal distance frm the 10 pint t the maximum runut psitin, and Xfj is the hrizntal distance frm the start psitin t the pint where slpe angle first declines t 10. INTRODUCTION The engineering aspects f zning in snw avalanche terrain include: (I) predictin f runut distances and (2) specificatin f expected impact frces if structures are t be placed in danger znes. It is nt currently pssible t predict runut distances in a deterministic sense by selecting frictin cefficients in a dynamic mdel because the prblem is t cmplex and the errrs cannt be defined. The nly alternative yet prpsed is t specify runut distances by a statistical definitin f runut psitin fr a muntain range in terms f tpgraphic parameters. In this study, new data btained frm the suthern British Clumbia Cast range are analyzed with the regressin methds f tpgraphic features (Lied and Bakkehi, 1980) and the extreme-value runut rati mdel f McClung and Mears (1991). McClung and Mears fund that runut distances bey a Gumbel distributin which is frequently used fr describing extreme values fr ther natural hazards. In additin, the data frm tw Canadian muntain ranges, the Rcky Muntains and the British Clumbia Cast Muntains, are cmpared using the tw statistical methds. The benefit f a statistical mdel is that the errrs in runut distances are quantified in standard statistical terms. Fr example, when the cnsequences f an avalanche are unacceptable a cnservative margin f safety is desirable. We emply the methd f multiple regressin analayses f tpgraphic parameters (Lied and Bakkeh0i, 1980) t discver practical and definable predictrs f avalanche runut distances. As ther studies shw (McClung and Mears, 1991 ), terrain parameters such as aspect, reach, slpe length and start-zne angle carry little r n significance in predicting the respnse variable f the regressin mdel. The length-scale effect bserved in the Clrad Rckies data (McClung and Mears, 1991 ) is als detected in bth the Rcky Muntains and the British Clumbia Cast Muntains. Partitining the data prvides a methd f imprving the predictive equatin fr runut distance. Finally, the least squares regressin mdel is cmpared t the runut rati mdel. The cmparisn shws that the latter is superir fr predicting runut distances based n engineering criteria. DESCRIPTIVE STATISTICS Fr this study, we cllected new data frm the Cast Muntains f British Clumbia by the methds discussed by Lied and BakkehOi (1980) and McClung and thers (1989). All maximum runut lcatins are determined frm vegetatin damage limits in mature frests. Cre samples frm trees brdering the maximum runut zne ranged in age frm 50 t 250 years (apprximate average arund 100 years). The randm errr intrduced by this variatin shuld diminish as the number f avalanche paths surveyed increases, accrding t McClung and Mears (1991), and they fund that 30 avalanche paths
Nixn and McClung: Avalanche runut frm Canadian muntain ranges H / Start psitin " "" "".;::-.;::-.,;:--,,,,,,,,,,,,,', ', ', ', ",,,, y ~ ~,, Extreme runut, psitin ;, / - ----xp-----!- -6x-1 Fig. 1. Definitin f gemetry and angles t determine extreme runut. were the minimum required t perfrm an analysis. As with previus studies, we defined terrain parameters frm ur data. The angle alpha (0) is defined by sighting frm the extreme avalanche runut pint (0') t the tp f the avalanche start zne. The angle beta ({3) is defined by sighting frm the psitin where the slpe angle first declines t 10 (f3.) t the avalanche start zne. The beta pint ({n is a reference pint frm which runut distance (~x) is calculated. The angle delta (c) is defined by sighting between the and the f3. The field data measured cnsist f tw angles: alpha (0) and beta (f3) (see Figure 1) and a slpe prfile surveyed frm the beta pint un t the alpha pint (0'), The runut distance ( ~x) is the hrizntal distance between the f3. and the 0 '. The vertical displacement (H) is measured between the {3. and the avalanche start zne. The surveyed slpe prfile allws the calculatin f ~x and angle C. The reach (X{J) is the hrizntal distance between the start zne and the f3. The runut rati is a dimensinless parameter (McClung and thers, 1989) defined as: ~x tanf3 - tan X{J = tan 0 - tan 15. The mean, standard deviatin and range f the variables used are presented in Table 1. THE REGRESSION MODELS Data frm the tw muntain ranges were analyzed with least squares regressin techniques t btain mdels f the frm 0 = C. f3 + Cl (Lied and BakkehOi, 1980). The fllwing equatins are btained ( dentes predicted value): Rcky Muntains: cl = -0.784 + 0.956 f3 (2) (R2 = 0.75, N = 126, Se = 1.75) where R2 is the cefficient f determinatin and Se is the standard errr. (1) Cast Muntains: cl = -1.395 + 0.954 {3 (3) (R2 = 0.74, ){ = 31, Se = 1.70). In bth cases the cnstant Cl is nt significant: the t-statistics are -0.52 and -0.45, respectively, cmpared with t-statistics fr f3 f 19.1 and 9.1, respectively. These results are in agreement with the results f McClung and Mears (1991) indicating that f3 is usually the nly significant parameter. A statistical test f the mean 0 angles frm the tw muntain ranges results in acceptance f the null hypthesis that the tw means are equal. This implies that the mean runut distances, calculated frm the least squares equatin, are als equal by the therem that "any Table 1. Descriptive statistics fr the Canadian Rcky Muntains and the Cast Muntains f British Clumbia Mean Standard deviatin Range f values Number f paths mzmmum maxzmum Canadian Rckies and Purcells 0 27.8 3.5 20.5 40.0 127 f3 29.8 3.1 23.0 42.0 126 8 5.5 5.3-21.5 20.6 125 H(m ) 869 268 350 1960 124 ~x 168 131-190 524 124 ~x/x{j 0.114 0.100-0.185 0.404 125 Cast Muntains f British Clumbia 0 26.8 3.3 20.4 32.5 31 {3 29.5 3.0 22.8 34.0 31 8 5.5 3.6-5.0 14.1 31 H(m ) 903 313 426 1915 31 ~x 229 202 0 1150 31 ~x/x{j 0.159 0.115 0.000 0.559 31 2
Nixn and McClung: Avalanche runutjrm Canadian muntain ranges linear cmbinatin f independent nrmal randm variables is als nrmally distributed" (N eter and thers, 1982). A similar statistical test perfrmed n the means f t1x fr bth ranges verifies that they are als similar with high significance. This suggests that the regressin mdels fr bth ranges are interchangeable and that there are n significant differences in terrain features. Hwever, when the prblem is analyzed by fitting runut distances t an extreme-value distributin, this cnclusin must be altered. Other tpgraphic features were added t assess their significance in a multiple regressin mdel. Sme f the variables used are: area f catchment (AC), aspect ({3), slpe distance (S2), start zne angle (8), vertical displacement (H), and hrizntal reach (Xp). Slpe distance is defined as X (3 /sinj3. The regressin mdel tested fr bth muntain ranges was f the frm a =!(CI, AC, aspect, {3, 82, 8, Xp, H). When cmparing the multiple variable mdel t the single variable mdel it was fund that additinal tpgraphic variables were nt significant (Table 2): the angle {3 is the nly variable with a highly significant t-statistic. Fr the Rcky Muntains, aspect is a significant, but nt highly significant, variable, and it is nt significant at all in the Cast Muntain analysis. Further analysis f ther ranges is required t determine whether aspect shuld be included as a significant predictr variable. Since the accuracy f field measurements is ±0.5 and the imprvement in the predicted angle is less than this errr, there is n benefit in including additinal tpgraphic variables. This cnclusin is strengthened by examinatin f the marginal imprvement in the cefficients f determinatin and the standard errr f estimate. Other cmbinatins f parameters, such as thse used by Lied and Bakkehi (1980), prduce similar results: the additinal parameters nly serve t prvide ver-fitted equatins withut significantly reducing the standard errr r imprving the predictin. Table 2. Multiple regressin statistics. Se is the standard errr f estimate; R2 is the cefficient f determinatin t-statistics Variable Cast Muntains Rcky Muntains R2 = 0.78 R2 = 0.78 Se = 1.65 Se = 1.64 Cl - 1.4 0.8 AC -0.8-1.2 aspect 0.7 2.2 j3 3.8 5.0 S2-1.4 1.6 8-0.6 0.05 XfJ 1.4-1.5 H -0.1 0.3 EXTREME-VALUE MODELS McClung and Mears (1991) analyzed extreme runu t frm mre than 500 avalanche paths frm five different muntain ranges. Their results shwed that a set f extreme runut ratis frm a muntain range may be assumed t bey a Gumbel distributin. Assuming that the runut distance shuld cnfrm t an - extreme-value distributin, McClung and thers (1989) prpsed fitting a dimensinless runut rati (Equatin (1» t a Gumbel distributin. In simplest frm, a least squares fit t the data takes the frm Xp= u+ b Y, (4) where [P] is the nn-exceedance prbability, Y is the reduced variate (Y = -In [-In[p]), u and b are lcatin and scale parameters, respectively, and Xp is the runut rati t1x/ Xp fr the chsen nn-exceedance prbability. Table 3. Lcatin (u) and scale (b) parameters u b Se R2 N Rcky Muntains 0.079 0.070 0.012 0.98 79 (censred) Cast Muntains 0.096 0.092 0.021 0.96 20 (censred) Cast Muntains 0_107 0.088 0.020 0.97 31 (uncensred) Lcatin and scale parameters fr the tw muntain ranges are given in Table 3. T illustrate the differences in muntain ranges, the data and the regressin lines fr the tw ranges are presented in Figure 2. In Table 3, the data fr the Rckies were censred at p = e- 1 t prvide a least squares fit n the tail f the distributin where engineering applicatins predminate. The Cast Muntains runut ratis are presented bth fr censred data (p = e- 1 ) and uncensred data_ Fr the Cast range, the resulting differences in Ax are n mre than 1-2% fr bth relatinships. Therefre, the effect f censring the Cast Muntains data (reducing the sample size t 20 paths) is minimal. A cmparisn f the runut ratis frm six muntain ranges is cntained in Table 4 fr p = 0.99 and p = 0.80 based upn the Gumbel parameters prvided by McClung and Mears (1991 ). The runut ratis fr Nrway and the Cast Muntains f British Clumbia are the mst similar, whereas the data frm the Canadian Rckies give the lwest ratis. This trend als applies t the mean values fax fr these three ranges. The effect f climate regime is nt apparent in the data represented in Table 3 (McClung and thers, 1989). Fr example, the Clrad Rckies and the Canadian Rckies are bth in cntinental climates, yet the frmer have the lngest runut distances while the latter have the shrtest. 3
}{ixn and McClung: Avalanche run ut frm Canadian muntain ranges 0.60 0.54 * 0.5 - - - -- 0.48 0.42 0.36 ~ 0.30 0.24 0.18 0.12 0.06 * * 0.00 2 3 4 5 - In [-In p [ Fig. 2. Data frm the Rcky Muntains and Cast Muntains f British Clumbia censred at p = e- 1. dentes Cast Muntains; 0 dentes Rcky Muntains. 6 0.4 0 0.3 ~ 0.2 0.1 [J!J IO-. 0.0 500 --------------- 0 0 0... ' 0 -' ' 0 ".. 0 DD 0 0000 0 00 OD 0 0 0 ~B 0 1000 1500 2000 X, (m) 2500 3000 Fig. 3. Data frm the Canadian Rcky Muntains partitined at XfJ = 1500 m. The three lines represent three values f the nn-exceedance prbability (- - -, p = 0.99), (- - -, p = 0.90), (-...., p = 0.80). LENGTH-8CALE EFFECTS McClung and Mears (1991) highlighted the imprtance f length-scale effects fr data frm the Clrad and Sierra Nevada ranges. Similar effects were als present fr bth Canadian ranges. T investigate the effect flengthscale n runut ratis, runut rati mdels were fitted t the arbitrarily partitined data (see McClung and Mears, 1991 ). Tw Gumbel relatinships were derived by partitining at XfJ = 1500 m fr the Rcky Muntains and XfJ = 2100 m fr the Cast Muntains data. The plt f run ut rati vs XfJ illustrates the length-scale effect fr the Rckies (Fig. 3). With partitining and p = 0.99, the Rcky Muntains runut distances are 15% higher than the nn-partitined mdel if XfJ < 1500 m and 28% lwer if XfJ > 1500 m. This result is very imprtant fr planning and land use decisins. The length-scale effect als appears in the Cast Muntain data; hwever, mre data are needed t quantify the relatinships. Table 4. Runut ratis fr nn-exceedance prbabilities Muntain range p 0.99 P 0.80 Canadian Rckies 0.401 0.184 and Purcells (censred) British Clumbia 0.512 0.239 Cast Muntains Western Nrway 0.497 0.258 Castal Alaska 0.682 0.347 Clrad Rckies 1.217 0.591 Sierra Nevada 1.32 0.683 The data in Figure 3 suggest that runut rati is inversely prprtinal t Xfj. Hwever, a least squares analysis f the frm tlxj Xfj = Cl + C Xfj -1 has a lw cefficient f determinatin (R 2 = 0.044) and the cnstant Cl has the nly significant t-statistic (5.4). Mre sphisticated methds f regressin might reveal a better relatinship fr the data in Figure 3; hwever, partitining the data and using the Gumbel distributin are simple and effective methds f imprving the estimates fr avalanche zning. COMPARISON OF REGRESSION AND GUMBEL DISTRmUTION METHODS Fllwing the wrk f Lied and BakkehOi (1980) many peple still use the least squares apprach fr runut psitins. In this sectin, we prvide a cmparisn f the least squares and Gumbel statistics methds. The regressin mdel (least squares) cntains the assumptin that the runut distances are apprximately nrmally distributed. A least squares fit f the data allws the predictin f an Cl! angle as a functin f a nn-exceedance prbability fr cmparisn with the Gumbel predictins. The data in Table 5 allw cmparisn f the tw mdels fr seven avalanche paths chsen randmly frm the Rcky Muntains. Fr the regressin mdel, runut distance (tlx) is calculated frm Cl!P' {3, H, and XfJ. The runut rati mdel cntains the assumptin that runut distances fit a Gumbel distributin. A least squares fit f pltting psitins (McClung and Mears, 1991) gives a linear equatin which predicts a dimensinless runut rati (Equatin (4)) fr a given nn-exceedance prbability. Runut distance (tlx) is calculated by multiplying Xp and XfJ. When bth mdels are c<?mpared at the median 4
Nixn and McClung; Avalanche runut frm Canadian muntain ranges Table 5. Cmparisn f runut distances ( ~x) fr seven avalanche paths selected at randm frm the Rcky Muntains fr the regressin mdel Q'p = -0.784 + 0.956 {3 andfr the runut rati mdel Xp = 0.07 + 0.076(-ln[-ln pj) Regressin mdel Runut rati mdel Avalanche path p Q'p Predicted ~x Xp Predicted ~x number m m 41 0.99 29.7 0.90 31.7 0.80 32.4 0.50 33.8 35 0.99 20.0 0.90 22.0 0.80 22.7 0.50 24.1 69 0.99 22.9 0.90 24.9 0.80 25.6 0.50 26.9 70 0.99 23.3 0.90 25.3 0.80 26.0 0.50 27.4 71 0.99 20.9 0.90 22.9 0.80 23.6 0.50 25.0 87 0.99 21.4 0.90 23.4 0.80 24.1 0.50 25.5 100 0.99 24.3 0.90 26.3 0.80 ' 27.0 0.50 28.4 371 0.420 436 237 0.241 251 195 0.184 191 117 0.098 102 732 0.420 699 437 0.241 402 349 0.184 307 192 0.098 163 343 0.420 475 204 0.241 273 161 0.184 208 107 0.098 III 375 0.420 429 226 0.241 246 182 0.184 188 104 0.098 100 461 0.420 642 287 0.241 369 232 0.184 281 130 0.098 150 1095 0.420 923 645 0.241 530 513 0.184 405 278 0.098 215 306 0.420 428 193 0.241 246 157 0.184 187 90 0.098 100 (p = 0.5) f bth distributins (Fig. 4) the runut rati mdel predicts shr,ter runut distances. Hwever, at higher values f p the ppsite is true: the Gumbel mdel is cnsistently mre cnservative. SUMMARY AND DISCUSSION In this paper, we examine and cmpare tw methds f predicting runut distances: least squares regressin f tpgraphic parameters and the runut ratis fitted t a Gumbel distributin. Analysis f the British Clumbia Cast Muntains data with the least squares methd shwed that the angle {3 is the nly significant parameter fr predicting the Q' angle. Other tpgraphic terrain parameters were nt significant and d nt imprve the predicted angle mre than the data errr. In additin, analysis f the data frm the British Clumbia Cast range shws that runut ratis bey a Gumbe1 distributin. Statistical tests cnfirm that the least. squares regressin parameters fr the Cast range and the Rcky Muntains are similar while thse fr the runut rati mdels are significantly different. As a result, when Gumbel statistics are emplyed it is imprtant that the runut rati parameters fr ne range nt be used fr anther. Our mst imprtant result cmes frm the cmparisn f the tw mdels using the same data. Specifically, the 5
Nixn and McClung: Avalanche runut frm Canadian muntain ranges VI G)..... 700 600 ~ 500 tt en : 400 G)..J I : 300.. G) u ; 200 VI C 100 100 200 300 400 500 600 700 Distance (m)-gumbel Distributin Fig. 4. Cmparisn f runut predictins fr the Rcky Muntains fr the least squares (Equatin (2)) and Gumbel distributin (see Table 3) at p = 0.5. arbitrarily chsen. In additin, a larger data set is rquired when length-scale effects are present t avid sample size effects. Althugh the methd f partitining is simple t emply, perhaps in the future mre sphisticated methds can be explred. We believe, based n data analysis frm ver 500 avalanche paths, that the runut rati mdel is superir t the least squares regressin f tpgraphic parameters fr estimating safe runut distances. The implicatin is that lng runut distances bey a Gumbel distributin rather than a Gaussian distributin (implied by chice f a least squares mdel). The chice f a Gumbel distributin has the benefit f a mre cnservative predictin f runut distances nce a nn-exceedance prbability is chsen. ACKNOWLEDGEMENTS We wish t thank the Snw Avalanche Sectin f the British Clumbia Ministry f Transprtatin and Highways fr their supprt in funding this prject. Als, special thanks g t the vlunteer field assistants: B. Britten, B. Guld and P. Weckwrth. runut rati mdel predicts lnger runut distances than the regressin mdel in the zne f engineering applicatins. The differences in runut distances between mdels increase as the prbability f nn-exceedance increases. Furthermre, the errrs fr the regressin mdel (and hence runut predictins) are assumed t be apprximately nrmally distributed in a least squares mdel. Hwever, ur analysis shws that the runut ratis (and distances) bey a Gumbel distributin. Length-scale effects have nw appeared in runut data frm several muntain ranges (including the Rcky Muntains and Castal Muntains in British Clumbia). Unless it is prperly accunted fr, this cnditin may lead t large errrs in estimating runut distances. One methd, which we have illustrated, is partitining at significant divisins in the data with separate derivatins fr data abve and belw the dividing hrizntal reach. This is nt entirely satisfying since the dividing line is REFERENCES Lied, K. and S. Bakkeh0i. 1980. Empirical calculatins f snw- avalanche run-ut distance based n tpgraphic parameters. ]. Glacil., 26(94), 165-177. McClung, D. M. and A. I. Mears. 1991. Extreme value predictin f snw avalanche runut. Cld Reg. Sci. Technl., 19(2), 163-175. McClung, D. M., A. I. Mears and P. Schaerer. 1989. Extreme avalanche run-ut: data frm fur muntain ranges. Ann. Glacil., 13, 180-184. Neter, ]., W. Wasserman and G. A. Whitmre. 1982. Applied statistics. Secnd editin. Bstn, MA, Allyn and Bacn. The accuracy f references in the text and in this list is the respnsibility f the authrs, t whm queries shuld be addressed. 6