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Magnitude and frequency of rock falls and rockslides along the main transportation corridors ofsouthwestern British Columbia1
O. Hungr, S.G. Evans, and J. Hazzard
Abstract: The two main transportation corridors of southwestern British Columbia are subject to a range of rock slopemovements (rock falls, rock slides, and rock avalanches) that pose significant risks to road and rail traffic travellingthrough the region. Volumes of these landslides range from less than 1 m3 to over 4.0 × 107 m3. A database of rockfalls and slides was compiled for rail and highway routes in each transportation corridor using maintenance recordsspanning four decades. The records number approximately 3500, of which about one half includes information onvolume. Magnitude – cumulative frequency (MCF) relationships were derived for each corridor. A scaled samplingprocedure was used in part to reduce the effects of censoring. Both corridors yield MCF curves with significant linearsegments on log–log plots at magnitudes greater than 1 m3. The form of both railway and road plots for each corridorshows similarity over several orders of magnitude. The slope of the linear segments of the curves depends ongeological conditions in the corridors. Temporal histograms of the data show a trend towards reduction of rock fallfrequency as a result of rock slope stabilization measures, implemented during the 1980s and 1990s. A risk analysismethodology using the slope of the magnitude–frequency relationship is outlined. The major part of the risk to life inthe case examined results from rock falls in the intermediate-magnitude range (1–10 m3).
Key words: rock fall, rock slide, landslide hazard, risk, magnitude–frequency, British Columbia.
Résumé :Les deux principaux corridors de transport de la partie sud-ouest de la Colombie Britannique sont soumis àun ensemble de mouvements de pentes rocheuses (chutes de blocs, glissements rocheux et avalanches rocheuses) quientraînent des risques importants pour le trafic routier et ferroviaire dans la région. Les volumes de ces glissementsvont de moins de 1 m3 à plus de 4,0 × 107 m3. Une base de données concernant les chutes et les glissements rocheuxa été constituée pour les itinéraires routiers et ferroviaires le long de chaque corridor, à partir de relevés d’entretien surune période de quatre décades. On a ainsi quelque 3500 données dont environ la moitié donne une information entermes de volume. On a développé pour chaque corridor des relations volume-fréquence cumulée (MCF). Uneprocédure d’échantillonnage en échelle a été utilisée pour réduire les effets de censure. Les deux corridors ont fournides courbes MCF présentant d’importantes portions linéaires dans une représentation log-log et pour des volumessupérieurs à 1 m3. La forme des courbes relatives aux activités routières ou ferroviaires dans chaque corridor reste lamême sur plusieurs ordres de grandeur. La pente des portions linéaires des courbes dépend des conditions géologiquesdans les corridors. Des histogrammes temporels montrent une tendance à la réduction de la fréquence des chutes deblocs lorsque des mesures de stabilisation des pentes rocheuses sont prises, comme dans les années 1980 et 1990. Uneméthodologie d’analyse du risque fondée sur la pente de la relation volume-fréquence est proposée. La majeure partiedu risque vis-à-vis de la vie humaine se situe, pour les données examinées, dans la fourchette intermédiaire desvolumes (1 à 10 m3).
Mots clés: chute de blocs, glissement rocheux, risque, volume-fréquence, Colombie Britannique.
[Traduit par la Rédaction] Hungr et al. 238
Introduction
Transportation links in southwestern British Columbiacross rugged, mountainous terrain and are exposed to a largenumber of landslide hazards, particularly rock falls and rockslides (Evans and Hungr 1993). Continuing maintenance of
the highways and railways is required to reduce the risks tothe travelling public and minimize costs and traffic disrup-tions. The expenditures on rock slope maintenance by sev-eral agencies in British Columbia amount to approximatelyCan$10 million per year (unpublished data; Theodore 1986).
Total risk on a highway or railway arises from exposure toboth frequent, small rock falls and rare, large rock slideevents. One of the most difficult decisions faced by mainte-nance managers is to allocate the correct proportion of re-sources to the treatment of each group of hazards so as tomaximize the reduction of risks for a given level of expendi-ture (Mackay 1997). Quantitative risk analysis can be usedto aid this decision, but it requires data on the magnitudeand frequency distribution of the hazards. Magnitude of a
Can. Geotech. J.36: 224–238 (1999) © 1999 NRC Canada
224
Received April 8, 1998. Accepted November 19, 1998.
O. Hungr and J. Hazzard. Department of Earth and OceanSciences, University of British Columbia, 6339 Stores Road,Vancouver, BC V6T 1Z4, Canada.S.G. Evans.Geological Survey of Canada, 601 Booth Street,Ottawa, ON K1A OE8, Canada.
1Geological Survey of Canada Contribution 1999045.
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landslide is represented in this paper by the volume ofdeposits. The purpose of this work is to present magnitude–frequency curves for bedrock-derived mass movementsalong transportation corridors in the study area. Once theshape of the curves is known, it will be possible to assessrelative proportions of risk deriving from different magni-tude classes.
The present study was completed jointly by researchersfrom the University of British Columbia and the GeologicalSurvey of Canada. Four transportation agencies supportedthe study and provided the authors with records of rock falland rock slide occurrence in the study area. They are B.C.Ministry of Transportation and Highways (BCMOTH), CPRail Systems (CP), CN Rail (CN), and BC Rail (BCR).These records were compiled in a database and supplementedby data on large events from the authors’ files. The databasewas processed to reduce censoring caused by uneven record-ing frequencies as described below. Typical magnitude–frequency relationships have been constructed from thesedata.
Magnitude – cumulative frequency(MCF)relationships
Magnitude – cumulative frequency (MCF) relationshipshave received widespread use in natural-hazards science.The most well known example is the Gutenberg–Richter re-lationship for earthquakes (Gutenberg and Richter 1954).This shows that the logarithm of the numberN of earth-quakes exceeding a given magnitudeM in a region is lin-early related to the magnitude:
[1] log N = A + bM
The constantA is highly variable, as it depends on the ex-tent of the study area and the length of the selected time in-terval. The slope parameterb, on the other hand, is fairlyconstant between different data sets. It ranges from about–0.6 to –1.0 and possibly relates to the tectonic setting of theearthquakes. SinceM is itself a logarithmic quantity, eq. [1]
represents a linear relationship on a log–log scale. Recordsof actual earthquakes often exhibit linear MCF curves overseveral orders of magnitude. However, at small magnitudesthe curves bend towards the horizontal, as it is more difficultto obtain a complete sample for small events.
More recently, similar relationships, linear on a logarith-mic scale, have been derived for several types of massmovements such as rock falls (e.g., Gardner 1970, 1983) androck avalanches (Whitehouse and Griffiths 1983).
Study area
The study covers an area of southwestern British Colum-bia bounded by the Pacific Coast on the west, and extendingas far north as Cache Creek and as far east as Kamloops(Fig. 1). The northwest–southeast-oriented ridges of theCoast Ranges (altitude up to 2944 m) traverse most of thearea and are separated by deep glacial valleys trending in thesame direction. Many of the valleys contain lakes or fjordsat or near sea level. The main valleys are cut across by lessfrequent perpendicular valleys and passes that appear to fol-low prominent joint sets (Monger and Journeay 1994).
The geology of the Coast Mountains is dominated by in-trusive rocks of the Coast Plutonic Complex which aremainly quartz diorite to granodiorite. The plutonic rocks arecharacterized by stress-relief (sheeting) joints formed paral-lel to topographic surfaces. When slopes in these rocks areundercut by erosion or human activity, the sheeting jointsfrequently daylight in the cut slopes, creating conditionssuitable for sliding or toppling. A variety of metamorphicrocks occurs in what Monger and Journeay (1994) call “sub-ordinate variably metamorphosed septa” (e.g., GambierGroup) and “fault bound slivers of sedimentary rocks” (e.g.,Jackass Mountain Group) which are usually highly alteredand (or) fractured. These zones occur at all levels within thevalley slopes of the study area.
A string of Quaternary volcanic centres making up theGaribaldi Volcanic Belt, a northward extension of the Cas-cade Volcanic Belt of the northwest United States, runsnorth–south across the Coast Mountains in the western partof the study area.
The topography of the Coast Mountains is heavily influ-enced by the effects of Pleistocene glacial erosion, an impor-tant consideration in the development of valley-side slopesin the region. The major transportation corridors are situatedmostly at the foot of valley slopes, as the narrow valleyfloors are occupied by streams, lakes, or inlets. The routestherefore typically traverse steep side slopes of bedrock,coarse colluvium, or terraces of fluvial, glaciofluvial, orglaciolacustrine origin.
Landslides in southwestern BritishColumbia
A review of landslide occurrence in the study area wascarried out by Evans and Savigny (1994), who illustrate thewide variety of landslide types that exists in the area.
Rock falls have been a major problem along the transpor-tation corridors in the region and have been the subject of anintense mitigative effort by the operators of the transporta-tion systems utilizing the corridors (e.g., Peckover and Kerr
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Fig. 1. Study area in southwestern British Columbia.
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1977; Theodore 1986; Hungr and Evans 1988; Bunce et al.1997). Larger rock slides have also occurred in rock cuts atthe base of steep natural slopes (Leighton 1990). Typical ex-amples of a small-scale rock fall and a medium rock slideare shown in Figs. 2 and 3.
Rotational and compound slides are common in the Ter-tiary lavas of the Fraser and Interior plateaux (Evans 1984)and the weak tuffaceous sediments associated with them(Jubien and Abbott 1989). They are also common in the un-consolidated Quaternary materials of the study area.
Flexural toppling on the mountain slopes occurs at Hell’sGate in the Fraser Canyon, at Mount Currie, near Pemberton(Bovis and Evans 1995), and at other locations in the CoastMountains.
Large rock avalanches have been noted in the southernCoast Mountains by Piteau (1977), Ryder (1981), and Evansand Savigny (1994). Piteau noted a prehistoric rock ava-lanche that appears to have blocked and deflected the courseof the Fraser River, 9.7 km south of Lytton. Naumann andSavigny (1992) give radiocarbon dates of several major rockavalanches in the lower Fraser Valley.
Major volcanic debris flows and rock avalanches have oc-curred in the Pleistocene volcanic rocks of the GaribaldiVolcanic Belt. Landslides involving the failure of the edificeslopes themselves (Hungr and Rawlings 1995) and the col-lapse of oversteepened lava flow margins (Moore andMathews 1978; Hardy et al. 1978) have occurred in the
Mount Garibaldi Volcanic Complex, and debris involved inthese mass movements has reached the Howe Sound –Lillooet corridor as recently as 1855.
A summary of dated major rock avalanches that areknown to have occurred in the transportation corridors inpostglacial time is given in Table 1. Figure 4 shows a typicallarge prehistoric rock avalanche in the Howe Sound –Lillooet corridor, north of Whistler.
This paper is concerned with rock falls, rock slides, androck avalanches.
The transportation corridors
The study covers two main transportation corridors(Fig. 1). The Fraser–Thompson corridor extends from Van-couver to Kamloops via Hope and Lytton for a distance of420 km and is utilized by the Trans-Canada Highway andCP and CN main lines. The Howe Sound – Lillooet corridoris 250 km long and is occupied by British Columbia High-way 99 (the Squamish Highway) and the BCR track.
Fraser–Thompson corridorThe Fraser–Thompson corridor begins on the wide
floodplain and glacial lowlands of the Lower Mainland andthe lower Fraser Valley. It extends into the steep-sided andnarrow Fraser Canyon associated with the Fraser River FaultSystem. Between Lytton and Spences Bridge, the corridortraverses a transition zone between the Coast Mountains and
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Fig. 2. A rock fall on a highway near Hope, in southwestern British Columbia. Note cratering damage to pavement. The rockfragments arrived from the top of the photograph (photograph courtesy of D.Gerraghty, BCMOTH).
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the Interior Plateau. Here the Thompson River runs alongthe northern edge of the Cascade Mountains and along thesouthern edge of the Fraser and Nicoamen plateaux. Theriver is deeply incised into highly fractured volcanicSpences Bridge Group rocks. Railway lines through this seg-ment of the corridor are protected by numerous rock sheds(see Figs. 18–23 in Wyllie and Norrish 1996).
At Spences Bridge exposures of the glaciolacustrine silts,typical of valley-fill deposits of the Interior Plateau, begin todominate along the corridor. Farther upstream, the Thomp-son River is incised into rocks of the Triassic Nicola Group(andesite, basalt, limestone, and argillite) the TertiaryKamloops Group volcanics, the Paleozoic Cache CreekGroup (limestone), and some Middle Jurassic sediments.
Howe Sound – Lillooet corridorThe Howe Sound – Lillooet corridor follows a fjord and
valleys scoured by glaciers in the igneous rocks of the CoastPlutonic Complex. The transportation routes cross bedrockslopes, coarse talus aprons, and alluvial and debris-flowfans. The landslide problems comprise rock falls, rockslides, and debris flows (Evans and Savigny 1994).
The Garibaldi Volcanic Centre, approached by the HoweSound – Lillooet corridor north of Squamish, is dominatedby Mount Garibaldi (2678 m), an extinct Pleistocene vol-cano with a record of large-scale slope instability. However,few volcanic rock outcrops appear in the immediate vicinityof the transportation routes.
Farther to the north, the corridor ascends the lowerSquamish valley to Cheakamus Canyon near the core of theCoast Plutonic Complex and proceeds over a low pass nearWhistler to join the Lillooet River valley at Pemberton. Itthen cuts across the Coast Mountains again to end atLillooet on the Fraser River.
Fig. 3. A rock slide with a deposit volume of approximately 60 000 m3 on British Columbia Highway 99 between Squamish andWhistler (photograph courtesy of D. Gerraghty, BCMOTH).
Transportationcorridora Rock avalanche Age (years BP) Dating method Source
H–L Mystery Creek 880±100 C-14 S.G. Evans, unpublished dataH–L Rubble Creek 143 (1855 A.D.) Dendrochronology Moore and Mathews 1978F–T Katz 3260±70 C-14 Naumann and Savigny 1992F–T Lake of the Woods 8260±70, 8430±60 C-14 Naumann and Savigny 1992F–T Cheam 5010±70, 4690±80, 4350±70 C-14 Naumann and Savigny 1992
aF–T, Fraser–Thompson; H–L, Howe Sound – Lillooet.
Table 1. Ages of large rock avalanches that cross the transportation corridors in the study area.
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Landslide database
The main attributes compiled in the database include loca-tion referenced to a common origin on each of the fiveroutes (in kilometres), date and time of occurrence, and thetotal volume deposited on the right-of-way in cubic metres,if available. All volumes referred to in this paper are depositvolumes which include any bulking associated with rockfailure. The database contains fields for a number of otherattributes, including number of fragments, width of the de-posit, deposit distribution, geology and geometry of the
slope above the line, precipitation, and other causal factorsas well as data on injuries and damage.
Data sources and quantity
The data sources were hard-copy and computer files madeavailable by the study sponsors, published and unpublisheddocuments, and personal files of the writers.
As may be expected, only a few of the records contain thefull range of attributes defined in the database. A large groupof records contains no more than mileage (expressed in kilo-metres) and date of occurrence. This is true especially forthe CN records, which are based on warning fence triggerreports. Because of the lack of volume information, the CNdata were not used in this phase of the study or in the analy-ses reported here.
A summary of the rock fall and rock slide records avail-able to date is given in Table 2. The majority of the recordsdate from the last four decades.
Temporal–spatial trends and censoring
Censoring of data may occur for three different reasons.First, data may be underreported, or the record may be in-complete (particularly by missing information on volume).Second, the sample time interval may be too short to ade-quately represent low-frequency events. Third, a “systemic”censoring may result from the character of the phenomenonitself. For example, censoring may be caused by the effectsof protective ditches and barriers in some parts of the routes,which are effective in intercepting some of the smallerevents.
Each data set has a different character, depending on themethods employed in its compilation. Therefore, differentstrategies were used in processing the data, with the aim toreduce nonsystemic censoring to the greatest possible de-gree.
CP and BCR dataBoth the CP and BCR sets consist of blocks of data ob-
tained in the course of systematically organized data-gathering programs, with fairly uniform distribution in time.This is illustrated in Fig. 5a, a map of the CP data set intime and space. As could be expected, the data isnonuniformly distributed in the spatial direction, showingthat some segments of the line are naturally more prone torock falls and rock slides than others. The data are separatedinto bands to acknowledge this spatial variation. A histo-gram of the data plotted on an annual basis is shown inFig. 5b. Although the time frequency of events is also non-
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Fig. 4. The Mystery Creek rock avalanche, approximately 4.0 ×107 m3, involving foliated intrusive rocks of the Coast PlutonicComplex, 20 km north of Whistler. The rock avalanche depositreaches British Columbia Highway 99 beyond the left-handmargin of the photograph. The minimum age of the avalanche is880 ± 100 years BP.
Transportation route Corridor Total recordsRecords withvolume information
B.C. Highway 99 H–L 882 612Trans-Canada Highway F–T 395 297BCR Vancouver to Lillooet H–L 443 178CP Vancouver to Kamloops F–T 918 623CN Vancouver to Kamloops F–T 900 0
Total records 3538 1710
Table 2. Summary of available database records on rock falls and rock slides.
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uniform, probably as a result of climatic factors, there is noapparent trend in this data that could be used to detect cen-soring. The frequency of events was therefore derived on thebasis of a constant sample time interval of 22 years (1975–1996 inclusive). Similarly, in case of the BCR, a uniformsampling interval of 13 years was used (1984–1996). It is ofconcern that this relatively short sampling period may mis-represent the frequency of large, low-frequency rock slides.
British Columbia Ministry of Transportation andHighways (BCMOTH) data
The two highway records required more complex interpre-tation. British Columbia Highway 99 set is used as an exam-ple. Both the space–time map (Fig. 6a) and the timehistogram (Fig. 6b) contain strong indications of censoring.Clusters of points represent periods of relatively systematicrecord keeping, while the sparse areas represent periodswhen fewer records were being kept. For the route as awhole, very few records were kept prior to 1980 and noneprior to 1960. The histogram in Fig. 6b, plotted for band A
only, rises from 1980 to 1990 as more systematicobservations were gradually implemented. An especiallysharp increase coincides with the year 1992, when the high-way maintenance programs in British Columbia were privat-ized and the private contractors were instructed to beginkeeping systematic records of rock fall incidents.
The extent of censoring varies from one location to an-other, as the records were kept by different groups and indi-viduals. The first 26 km segment, along the southern portionof Howe Sound (band A), has the longest history of observa-tions, whereas record keeping was practically nonexistent inthe northern part of the route prior to 1992 (bands I–J).
One obvious way to eliminate the influence ofunderreporting would be to reject any data preceding 1992.However, such an approach would not provide a representa-tive sample of the large, low-frequency events which requirelonger sampling periods. Our approach is based on the as-sumption that the recording intensity is related to magnitude.Low-magnitude, high-frequency records are probably cen-
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Hungr et al. 229
Fig. 5. Summary of rock fall records from the CP Rail route,Fraser–Thompson corridor, bands A–F. (a) All records mappedin time and space. (b) Histogram of all recorded events.
Fig. 6. Summary of rock fall records from bands A–J of BritishColumbia Highway 99 (kilometres 0–26), Howe Sound–Lillooetcorridor. (a) All records mapped in time and space.(b) Histogram of all recorded events (plotted for band A only).
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230 Can. Geotech. J. Vol. 36, 1999
sored everywhere, except in the densest available clusterswhich are assumed to represent periods of optimal recordingintensity (i.e., all significant events are recorded). Events oflarger magnitude are more reliably recorded and the sam-pling period can be longer in certain bands, to make use ofthe maximum amount of available data. The largest eventshave sampling periods extending over the full time span ofthe database.
To test this hypothesis, separate histograms of occurrence
were plotted for band A in each of seven order of magnitudeclasses as shown in Fig. 7. On inspection of these histo-grams, it was decided that similar sampling patterns existedfor events between 0 and 1 m3, 1 and 1000 m3, and greaterthan 1000 m3. The data within each of these three classeswere therefore combined as shown in Fig. 8. The shape ofthe resulting histograms confirms that longer sampling peri-ods can be used for larger events.
It is interesting to note that combining events with un-
Fig. 7. Histograms of the temporal distribution of rock falls in band A of British Columbia Highway 99, plotted separately for eachmagnitude class.
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known magnitude with those less than 1 m3 makes the histo-gram more uniform during 1992–1996 (compare Fig. 8awith Figs. 7a and 7b). This suggests that most of the eventswith unknown magnitudes were small and that more quanti-tative volume observations were introduced during the givenyears. Similar trends were observed in all the BCMOTHdata and the same three size categories were utilizedthroughout.
It is nevertheless difficult to decide on the correct maxi-mum length of the sampling interval in each class. The sam-pling interval must be chosen to preserve as much data with
normal variation as possible, but at the same time excluderegions with abnormal variation which indicates the pres-ence of censoring. Use of subjective judgement is obviouslyrequired. However, to make the selection of the samplingboundaries more objective, we explored the skewness of thesampling regions. By inspection of the histograms it is ap-parent that the strongest clue for censoring is the presence ofa sharp and sustained increase in apparent frequency in time.Thus, the sampling region should not be strongly skewed inthe positive direction on the time axis, when reasonablesmoothing of its random variation is applied.
For the histograms shown in Fig. 8, the right-hand limitsof the sampling intervals were assumed to be at the end ofthe last year of complete observations (31 December 1995).Interim left-hand margins were first placed by judgement soas to minimize the apparent skewness of the sampling inter-val. Several intervals were then taken longer and shorter ofthe interim starting date. A line was fitted to the distributionof events in each interval using the least squares method.Figure 9 shows the slope of the line across each interval,plotted against the starting date of the interval. All the slopesare initially positive, indicating skewness to the right, buteventually approach zero or become negative due to naturalvariation of the data. The left margin of each sampling inter-val was placed at the point where the slope curve first ap-proaches the horizontal axis.
Using this procedure, the sampling interval for band Awas taken as 1990–1995 for events less than 1 m3, 1980–1995 for those ranging from 1 to 1000 m3, and 1956–1995for those ranging from 1000 to 100 000 m3 (see arrowedranges in Fig. 8). A similar procedure was used for the moresparse band B.
The data in bands C–J have not been used in the construc-tion of the magnitude–frequency curve for British ColumbiaHighway 99, due to the shortness of the sampling interval(3 years). Such data would distort the curve by under-representing the larger events. For the same reason, a 15 kmsegment (band F) on British Columbia Highway 1 was alsoexcluded from the analysis.
The temporal histograms for medium and large rock fallson Highway 99, shown in Figs. 8b and 8c, have a negativeskewness within the sampling interval, which probably rep-resents the effects of intense rock slope stabilization activityimplemented during the 1980s and 1990s. More recent stabi-lization work, including extensive meshing, was applied af-ter 1995. This probably would affect the temporal trends aswell, but it is not covered by the available data.
A special class was reserved for major rock avalanches.Approximately four major rock avalanches are known tohave taken place in each of the corridors in the 10 000 yearsof postglacial time (Table 1). The volume of these rock ava-lanches averages approximately 3.0 × 107 m3. The annualfrequency of occurrence is therefore 0.0004 per corridor.The value of 0.0001 was used in the case of the Highway 99data set, which was truncated to bands A and B only, or onequarter of the corridor length as described above. The fulldeposit volumes were used for the rock avalanches. This isnot incompatible with the volumes from the database, al-though the latter were collected from the road right-of-wayonly. The small- and medium-size events recorded in the da-tabase deposited almost exclusively within the right-of-way.
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Fig. 8. Histograms of the temporal distribution of rock falls inband A of Highway 99, plotted separately for magnitude rangeswith similar censoring characteristics. Arrowed ranges indicatethe sampling intervals.
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Derivation of the MCF relationships
Magnitude – cumulative frequency curves can be con-structed directly from observed data using a graphicalmethod similar to that employed by Gutenberg and Richter(1954) as described below.
Using the methods of the preceding section, a samplinginterval was defined for each data band and each of the threesize categories. It was then possible to associate an incre-mental annual frequency of occurrencefi with each recordedevent i:
[2] fT
ii
= 1
where Ti is the length in years of the sampling interval asappropriate, considering the size category of eventi and theband to which it belongs. This incremental frequency wasstored as an additional attribute with each record in the data-base.
The MCF relationship for any group of records is com-piled by sorting the records in the order of increasing magni-tude and accumulating the incremental frequencies from thelargest magnitude to the smallest. Withj being the magni-tude rank of a given record in the group, the cumulative fre-quency is given as
[3] F fji
j
i==∑
1
The resulting MCF plots are shown in Fig. 10 for theHowe Sound – Lillooet corridor and in Fig. 11 for theFraser–Thompson corridor.
Initial MCF plots were constructed using only data withvolume information. Each of the plots showed strong curva-ture at magnitudes less than 1 m3, as indicated by the thinsolid lines near the upper range of the curves in Figs. 10 and11. This curvature is ascribed to the effects of censoring andis a common feature of all similar plots for landslides orother phenomena, as mentioned earlier.
Subsequently, data from records with unknown magni-tudes were added into the magnitude range of 0.01–1.0 m3,making the assumption that most of the data with unknownmagnitudes represent small events. These added data weredistributed linearly in the log–log plots. In each case, thisprocedure led to a straightening of the plot, as shown by thedarker lines in Figs. 10 and 11. This, together with the moreuniform histogram in Fig. 8a as described earlier, indicatesthat inclusion of data with unknown magnitude into themagnitude range below 1 m3 appears justified.
The curves span six to seven orders of landslide magni-tude. The Highway 99 curve is linear over six orders ofmagnitude without the correction for unknown volumes andover seven orders of magnitude with it. Each of the othercurves has a linear segment over at least three orders ofmagnitude.
The MCF relationships for all four data sets are superim-posed in Fig. 12 and data from major rock avalanches areadded. With the exception of the BCR record, which is con-siderably flatter, the slopes of the curves at magnitudesabove 1 m3 are similar. Further, linear extensions of the
curves continue over eight to 10 orders of magnitude to-wards the rock avalanche records.
Curvature of the relationships is noted at magnitudes be-low 1 m3 (except in the case of British Columbia Highway99) and this is probably due to systemic censoring. It is pos-sible, however, that some residual effects of nonsystemiccensoring (underreporting) also contribute to the curvatureof the graphs.
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232 Can. Geotech. J. Vol. 36, 1999
Fig. 9. Plots of linear regression line slopes fitted to thehistograms of Fig. 8. In each case, regression analysis wascarried out over an interval beginning with a variable start dateand ending on 31 December 1995. The regression line slopes areplotted against the start date. A low value of the regression slopeindicates an interval of low skewness. Arrows indicate the leftmargin of each sampling interval.
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Fig. 10. Magnitude – cumulative frequency (MCF) curves forrock fall records in the Howe Sound – Lillooet corridor.(a) British Columbia Highway 99, bands A and B only (length75 km; 390 records). (b) BCR line (257 km; 403 records). Thelight line in the upper range of each diagram represents thecurve before the addition of records of unknown magnitude.
Fig. 11. Magnitude – cumulative frequency (MCF) curves forrock fall records in the Fraser–Thompson corridor. (a) BritishColumbia Highway 1 (length 385 km; 226 records). (b) CP line(325 km; 918 records). The light line shown in the upper rangeof each diagram represents the curve before the addition ofrecords of unknown magnitude.
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Discussion
The linear ranges of the MCF plots in Figs. 10 and 11 canbe represented by a power-law relationship between magni-tude (M) and cumulative frequency (F). The form of theequation is similar to that of the Gutenberg–Richter relation-ship presented in eq. [1]:
[4] log F = A + b log M
The constantA of course depends on the length of corri-dor in question and on the relative propensity of various lo-cations to produce landslides. The curves themselves haveno universal significance, as each relates to the specific cor-ridor for which it was derived. The slopeb, on the otherhand, characterizes the distribution of rock fall frequenciesin the study area and within its subdomains. The slopes ofthe power relationship in the linear segments of the curveswere determined by linear regression and are listed in Ta-
ble 3. Highway 99 (bands A and B) appears to present themost robust set, linear over seven orders of magnitude(Fig. 10a).
The two Fraser–Thompson sets (Highway 1 and CP) arelinear only at magnitudes greater than 1 m3 and it is as-sumed that the data at lower magnitudes are distorted bycensoring (Fig. 11). It is probable that systemic censoringexists in this corridor as a result of the effectiveness ofditches along the highway and railway tracks, which inter-cept many of the smaller events.
The BCR relationship is highly curved, and suggests cen-soring effects (Fig. 10b). It is noted that this is the smallestdata sample, distributed over a great length of track and ashort period of time (12 years). The BCR track north ofPemberton was affected by a cluster of medium-scale rockslides in the 1980s (Leighton 1990). It is possible that thefrequency of the larger events is distorted upwards by theoccurrence of this cluster within the short sampling period.
The MCF slopes for each of the corridors are reasonablyconsistent, despite the fact that each corridor contains bothhighway and railway data and despite the presence of resid-ual censoring. The Fraser–Thompson corridor is character-ized by a slope of –0.65 to –0.70 at magnitudes above 1 m3
and –0.25 at lower magnitudes. The Howe Sound – Lillooetcorridor is best represented by the Highway 99 curve, with aslope of –0.43. The BCR curve, although apparently ofsomewhat poorer quality, gives a slope of –0.40 above the1 m3 magnitude. Thus, it appears that the Howe Sound –Lillooet lines experience a relatively greater frequency oflarger events, while small-magnitude rock fall is more domi-nant on the inland routes. This corresponds to the geologicalconditions, which place the former corridor closer to thecore of the Coast Plutonic Complex, whereas the latter tra-verses faulted and both structurally and lithologically com-plex interior regions. Further, the apparent systemiccensoring of the Fraser–Lillooet corridor data suggests agreater effectiveness of ditches and roadside barriers in thisarea.
The lines extending the MCF relationships from100 000 m3 to the points represented by the rock avalancherecords (Fig. 12) slope on average at –0.64, similar to theFraser–Thompson data. These extensions must be viewedwith caution, as the sample represented by the large-scaleevents is very small and their interaction with the road right-of-way is influenced by the width of the valleys. Also, inter-mediate records (magnitude 100 000 to 1.0 × 107 m3) aremissing, possibly as a result of nonrecognition. Neverthe-less, the relationships shown in Fig. 12 suggest that the lin-
© 1999 NRC Canada
234 Can. Geotech. J. Vol. 36, 1999
Route CorridorNo. ofrecords Intercepta Slopeb
Correlationcoefficient
Highway 99 (bands A and B)a H–L 389 0.773 –0.434 0.99BCRb H–L 123 0.121 –0.402 0.94Highway 1 (except band F)b F–T 64 1.358 –0.703 0.95CPb F–T 122 1.133 –0.646 0.99
aMagnitude range 0.01 to 10 000 m3.bMagnitude range 1.0 to 10 000 m3.
Table 3. Least squares regression line fits to the linear parts of the magnitude – cumulative frequency relationships (eq. [3]).
Fig. 12. All magnitude – cumulative frequency curves (seeFigs. 10 and 11), superimposed and extended by the addition ofmajor rock avalanche records.
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ear MCF curves may extend over many higher degrees ofmagnitude, although they are difficult to sample.
For comparison, Fig. 13 shows data from detailed obser-vations by Gardner (1970) in the Lake Louise area, RockyMountains, Alberta. These data represent direct observationof 409 mostly small rock falls over two summer periodsfrom natural calcareous and quartzitic cliffs. The curvatureof the MCF diagrams again suggests censoring, but thenearly linear segments in the magnitude range of 0.1 to10 m3 have a slope of –0.72, remarkably close to that of theslopes of the Fraser–Thompson data above the censoringlimit in the present study.
In summary, the results of this study indicate that a slopein the order of –0.7 is characteristic of the MCF relation-ships in moderately to highly jointed metamorphic, igneous,and strong sedimentary rocks. A flatter slope, of the order of–0.4, appears appropriate for massive felsic intrusive rockswhich possibly produce a relatively greater proportion oflarge-magnitude, structurally controlled failures. The above-mentioned comparison with Gardner’s (1970) data suggeststhat the slope of the relationships is quite independent of themode of origin of the slopes, i.e., constructed road cuts orslopes generated by natural denudation.
Example risk analysis
As an example of the potential use of the MCF curves, arisk analysis was carried out along a hypothetical segment ofa two-lane highway, placed in the Howe Sound – Lillooetcorridor. It is assumed that the total number of rock falls
larger than 0.001 m3 recorded along this segment is 100 peryear.
To derive the frequency,fh, of the rock falls it is necessaryto construct an MCF curve specific to the road segment un-der consideration. This is based on the assumption that onlythe constantA in eq. [4] depends on the length and activityof the road segment, whereas the slope parameterb can betaken from the relationship derived for the corridor as awhole. In other words, the MCF curve is shifted vertically toa position appropriate to the segment, while maintaining itsshape. The curve is anchored vertically so as to obtain therequisite total number of events per year, i.e., 100.
The frequencyfh is derived from the MCF curve by sub-tracting the cumulative frequencies for each of a series ofsuccessive magnitude classes. This procedure is shown inthe first three columns of Table 4. The cumulative frequen-cies in column 2 of the table are determined by parsing thetotal annual population of rock falls and rock slides (100)into magnitude categories from the smallest to the largest,according to the slope of the MCF relationship. A slope of–0.434 was used in Table 4, corresponding to the Highway99 data.
The cumulative frequency in each magnitude category isgiven by the equation
[5] log Fi = log Fi–1 + b(log Mi – log Mi–1)
whereMi is the magnitude in each category. The cumulativefrequencyFi was taken as 100 in the lowest magnitude class.
Bunce et al. (1997) used the binomial theorem to calculateaccident probability as a function of rock-fall frequency.Hungr and Beckie (1998) proposed that simple multiplica-tion of conditional probabilities yields equally accurate re-sults, provided that the accident probabilities are low. Thissimpler procedure was used in the present work.
The risk of a single fatal accident involving an occupantof a passenger vehicle is estimated by the following formula(based on Bunce et al. 1997 and Hungr and Beckie 1998):
[6] P(A) = fhP(S:H)P(T:S)P(I:S)P(L:I)
where fh is the annual frequency of landslides within thegiven sector of road, in a single magnitude category;P(A) isthe annual probability of an accident involving the death ofat least one occupant of a vehicle; andP(S:H) is the longitu-dinal encounter probability, defined as the probability of avehicle being present within the length of the highway im-pacted by the landslide at the time the landslide occurs (thedamage corridor), as calculated below. Since vehicle flowwith average intensity is assumed to be continuous in time,the temporal probability,P(T:S) equals 1.0.P(I:S) is the lat-eral impact probability, the probability of a vehicle beingimpacted by the debris given that it is present within thedamage corridor.P(L:I) is the probability of death of at leastone occupant, given an impact on the vehicle (vulnerability).
All of the individual factors in eq. [4] are related to themagnitude of the rock fall as follows. The longitudinalspatial–temporal probability of encounter,P(S:H) is esti-mated using a formula based on the geometrical relation-ships shown in Fig. 14. This formula is derived by assuminguniform average spacing between vehicles:
© 1999 NRC Canada
Hungr et al. 235
Fig. 13. Range of magnitude – cumulative frequency (MCF)curves compiled from Gardner’s (1970) data on small-scale rockfalls from natural slopes in the Lake Louise area, RockyMountains, Alberta.
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[7] P S HL L
Lv( : ) = +
2 l
d
whereLv is the length of a vehicle (taken as 5.4 m), andLdis the average spacing between vehicles in each lane, de-pendent on the average velocity and the daily traffic volume.For 5000 vehicles per day (vpd) and a speed of 80 km/h, thevehicle spacing isLd = 80 000 / [(5000/2)/24] = 768 m ineach lane.
The width of the landslide damage corridor,Ll, is esti-mated relative to the magnitude class of the landslide (rockfall) as given in column 4 of Table 4. The values of encoun-ter probabilityP(S:H), calculated from eq. [6], are given incolumn 5 of Table 4.
The lateral impact probability,P(I:S), is the probabilitythat the vehicle will be impacted, given that it is within thelandslide damage corridor when the rock fall occurs. Forsmall rock falls and a two-lane highway it will be close to0.5, as only one lane of the highway will be impacted. Forvery small rock falls it will be less than 0.5. For rock fallsand rock slides larger than 100 m3 it is assumed to equal 1.0,as any vehicle present in the corridor, in either lane, is likely
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236 Can. Geotech. J. Vol. 36, 1999
Fig. 14. Derivation of eq. [7] to express the probability of anencounter between a landslide and a vehicle as a function ofaverage vehicle spacing and the width of the landslide damagecorridor.
Ro
ckfa
llm
ag
nitu
de
cla
ssa
(m3 )
An
nu
al
cum
ula
tive
fre
qu
en
cyF
i
An
nu
al
incr
em
en
tal
fre
qu
en
cyf h
Co
rrid
or
wid
thL
l
(m)
En
cou
nte
rp
rob
ab
ility
P(S
:H)
(Fig
.1
4)
La
tera
lim
pa
ctp
rob
ab
ility
P(I
:S)
Pro
ba
bili
tyo
fd
ea
thP
(L:I
)
Pro
ba
bili
tyo
ffa
tal
acc
ide
nt
P(A
)
Re
turn
pe
rio
dR
(ye
ars
)
0.0
01
10
0.0
00
——
——
——
—0
.01
36
.81
36
3.1
87
0.1
0.0
10
.10
.05
0.0
05
22
10
.11
3.5
52
23
.26
10
.10
.01
0.2
0.1
0.0
07
15
01
4.9
89
8.5
63
10
.02
0.4
0.2
0.0
11
88
10
1.8
37
3.1
52
20
.02
0.6
0.5
0.0
18
55
10
00
.67
61
.16
05
0.0
30
.80
.80
.02
05
01
00
00
.24
90
.42
71
00
.04
11
0.0
17
58
10
00
00
.09
20
.15
73
00
.09
11
0.0
14
69
0.0
92
50
0.1
41
10
.01
37
6T
ota
l0
.10
69
Not
e:S
lope
bof
the
MC
Fre
latio
nshi
pin
eq.
[3]
isas
sum
edas
–0.4
34fo
ral
lm
agni
tude
clas
ses.
a Low
erlim
itof
the
mag
nitu
decl
ass.
Tabl
e4.
Exa
mp
leca
lcu
latio
no
fth
eri
sko
fa
fata
la
ccid
en
tfo
ra
seg
me
nt
of
ah
igh
wa
yin
the
Ho
we
So
un
d–
Lill
oo
et
corr
ido
rw
ith1
00
rock
falls
pe
rye
ar,
atr
affi
cin
ten
sity
of
50
00
veh
icle
sp
er
da
y,a
nd
ad
esi
gn
spe
ed
of
80
km/h
.
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Hungr et al. 237
to be impacted. For all magnitude classes, the value ofP(I:S) is estimated as given in column 6 of Table 4.
The vulnerability of an occupant of a vehicle given an im-pact,P(L:I) is difficult to estimate, although damage statis-tics compiled in the database will be used in the future toanalyse this problem. For the present purposes, we startfrom an estimate that the vulnerability for small rocks is ofthe order of 0.1 (Cruden 1997). For events larger than1000 m3 it is 1.0. The other values are estimated in column7 of Table 4.
The calculations are completed in column 8 of Table 4 fora traffic intensity of 5000 vpd and plotted in Fig. 15, whichshows the incremental risk resulting from the various magni-tude classes. The greatest risk appears to derive from rockfalls in the intermediate magnitude range of 1–10 m3. Thetotal risk is obtained by summing all the incremental riskvalues and amounts to 0.106 per year (1 in 9).
The above procedure accounts only for accidents causedby a direct impact of the landslide on a moving vehicle. Sep-arate calculation based on similar principles would be re-quired to consider accidents caused by collisions of vehicleswith rock on the roadway (see Bunce et al. 1997).
The calculation presented here is merely an examplewhich does not purport to represent an actual situation.However, the selected cumulative frequency of landslides of100 per year corresponds approximately to the frequency ob-served in bands A and B of Highway 99 (Fig. 10a). Also,the traffic intensity of 5000 vpd is not far from the averagerecorded on Highway 99 over the last few decades. There-fore, the calculated return period of 9 years can be roughlycompared with the actual fatality records from the southern-most 75 km of British Columbia Highway 99. This includesthree fatal accidents caused by direct impact between 1960and 1996 (return period 12 years) and two accidents during1980–1996 (return period 8 years). Thus, the estimate of to-tal risk using the proposed method appears realistic. The es-timate does not take into account changes in traffic intensityover time, or the effects of stabilization measures in reduc-ing rock fall frequency.
Conclusions
Transportation corridors of southwestern British Columbiaare subject to a range of rock slope movements that pose
risks to road and rail traffic. Rock falls (<10 000 m3) occurfrequently from natural and excavated rock slopes. Largermagnitude rock slides (<100 000 m3) also occur, often inrock cuts at the foot of steep natural slopes. At the upper endof the magnitude spectrum, large rock avalanches (>1.0 ×107 m3) have occurred in the present corridors in the 10 000years of postglacial time.
A database of high-frequency, low-magnitude rock slopemovements along the corridors showed spatial clusteringdue to slope geometry and geological factors. Temporal pat-terns were also highly clustered, partly due to natural varia-tion, but also as a result of data censoring.
Magnitude – cumulative frequency (MCF) relationshipshave been derived from the database for rock falls and rockslides in two highway and railway corridors in the studyarea. The presence of censoring necessitated the use ofscaled sampling in the case of the two highway records,while direct sampling was used for the railway records. De-spite this fundamentally different approach to sampling, thetwo relationships derived for each corridor are quite similar(see Figs. 10 and 11).
Temporal histograms of the data show a certain trend to-wards reduction of rock fall frequency as a result of rockslope stabilization measures, implemented during the 1980sand 1990s (see Figs. 8b and 8c).
Both corridors yield MCF curves with significant linearsegments at magnitudes greater than 1 m3. The Fraser–Thompson data show strong censoring at magnitudes lowerthan 1 m3. This may be partly residual censoring resultingfrom incomplete records and partly systemic censoring re-sulting from the effectiveness of roadside ditches and barri-ers with regard to low-volume rock falls.
The slope of the linear segments of the MCF curves de-pends on geological conditions. Steeper curves (approx. –0.7)have been derived from the Fraser–Thompson corridor, char-acterized by fault zones and metamorphic rocks. Flattercurves result from the massive plutonic rocks of the HoweSound – Lilooet area (approx. –0.4). The shapes of the for-mer curves compare with the uncensored part of an MCF re-lationship derived by Gardner (1970) from natural slopes incarbonate rocks near Lake Louise, Alberta.
A risk-analysis methodology using the MCF relationshipis proposed. In applying it to road traffic on British Colum-bia Highway 99 it appears that the greatest risks to vehiclesderive from rock falls in the intermediate-magnitude rangeof 1–10 m3 (Table 4; Fig. 15).
Acknowledgements
Mr. Clive Mackay of CP Rail Systems, Inc. encouragedthe authors to undertake this study. Financial and in kindsupport by the Geological Survey of Canada, B.C. Ministryof Transportation and Highways (Mr. D. Gerraghty), CP RailSystems, Inc. (Mr. Clive Mackay), B.C. Rail Ltd. (Mr. BrianAbbott), CN Rail Ltd. (Mr. Doug Allen) and EmergencyPreparedness, Canada are gratefully acknowledged. Dr. G.Brooks and two anonymous referees provided useful com-ments.
Fig. 15. Annual risk of accidents involving one or more deathson a two-lane highway in the Howe Sound – Lillooet corridor,assuming an average traffic intensity of 5000 vehicles per day.See calculations in Table 4.
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