Assessing Rockfall

Classifying and Assessing the Geologic Contributionto Rockfall Hazard

CHRISTOPHER J. VANDEWATER

WILLIAM M. DUNNE

Department of Earth & Planetary Sciences, University of Tennessee–Knoxville,Knoxville, TN 37996

MATTHEW MAULDON

Department of Civil & Environmental Engineering,Virginia Tech, 200 Patton Hall, Mail Code 0105, Blacksburg, VA 24061

ERIC C. DRUMM

Department of Civil & Environmental Engineering,University of Tennessee–Knoxville, Knoxville, TN 37996

VANESSA BATEMAN

Geotechnical Engineering Section, Tennessee Department of Transportation,6601 Centennial Blvd., Nashville, TN 37243

Key Terms: Rockfall Hazard, Rockfall Modes, Geo-logic Character, Logistic Regression

ABSTRACT

Rockfalls from roadcuts are a major hazard andpose problems for transportation agencies across thecountry. In the context of rockfall hazard manage-ment, however, no consensus exists about the role ofgeology in assessing rockfall hazard. This studyinvestigates the geologic contribution to rockfallhazard through application of rockfall hazard ratingsystems to roadcuts in Tennessee and throughadditional data collection to reveal correlationsbetween hazard characteristics and geologic attrib-utes. The geologic character of 80 roadcuts in centraland eastern Tennessee was evaluated using theTennessee Rockfall Hazard Rating System (RHRS),which is a revision of the National Highway Institute(NHI) RHRS. Scores for both RHRSs were comparedto evaluate whether the improved reproducibility ofscoring for the Tennessee RHRS yielded unintendedlosses of scoring accuracy and sensitivity. Additionalgeologic attribute data beyond those used in the RHRSsystem were collected to determine with logisticregression analysis whether relationships among thegeologic attributes, rockfall type, and block size exist.Results indicate the revised geologic component of

Tennessee’s RHRS is more informative and permitsdescription of a wider spectrum of geologic conditionsthan does the NHI version. Logistic regression analysisindicates rockfall type correlates to lithologic varia-tion and the number of discontinuity sets; and blocksize correlates to structurally controlled rockfall,lithologic variation, mechanical layer thickness, andnumber of discontinuity sets. Consequently, roadcutscontaining potential rockfall modes with two or morediscontinuity sets, no lithologic variation, and me-chanical thicknesses that exceed 1.0 m are expected tohave greater Geologic Character scores.

INTRODUCTION

Rockfall occurrences along roadcuts create consider-able risk for human injury and property damage, posingproblems for transportation agencies across the country.Negative consequences of rockfall include impact damageto pavement from falling rocks, rocks on roads posinghazards to motorists, road closures, and environmentalimpact due to collisions with vehicles transporting toxicsubstances (Royster, 1978; Moore, 1986; and Wyllie andNorrish, 1996). Consequently, as the demand for rockfallprotection increases (Flatland, 1993), transportationagencies are expected to respond with practices thatminimize damage and increase driver safety.

Rockfalls occur when rock or debris is shed froma roadcut or nearby steep slope by processes such as planar

Environmental & Engineering Geoscience, Vol. XI, No. 2, May 2005, pp. 141–154 141

sliding, wedge failure, toppling, differential weathering,and raveling onto the catchment and/or road (Norrish andWyllie, 1996). Characterization of rockfall hazard alongroadcuts is necessary for identifying hazard level andprioritizing remediation activities. The characterizationincludes attributes such as vehicular traffic patterns,roadway geometry, and rock-slope geometry (Wyllieand Norrish, 1996). However, in the context of rockfallhazard management, a variety of methods are used tocharacterize the role of geology in the rockfall process.Developing an effective approach to characterizinggeology is important, because this factor and its relationto the roadway controls whether material is available to beshed as a rockfall. Approaches to incorporating geology inhazard assessment have included defining hazard byassociation with rock type (Anonymous, 1996; Hadjin,2002), ascertaining whether geologic discontinuities areoriented favorably or unfavorably with respect to pro-moting rockfall (Abbott et al., 1998), and describing therockfall process for rock and soil slopes (Lowell andMorin, 2000).

The goal of this study is to investigate the role ofgeology through two approaches: (1) building on thegeologic component of the National Highway Institute(NHI) Rockfall Hazard Rating System (RHRS) bymodifying it to explicitly evaluate rockfall modes andtheir salient characteristics, which is hereafter referred toas the Tennessee RHRS; and (2) collecting additionalgeologic attributes beyond the RHRS system to determineif these geologic attributes correlate with rockfall type andblock size. The correlation in the second approachcompares rockfall modes and attributes developed forthe first approach with additional geological data thatwould not typically be collected during roadcut charac-terization for rockfall hazard rating. A purpose of thesecond approach is to provide transportation departmentsand future investigators with correlations that could beused to consider likely rockfall modes and block sizes fora prospective roadcut if the geologic unit and its propertiesare already known.

Role of Geology in Existing Rockfall HazardRating Systems

The RHRS is a tool to systematically inventory andrank hazardous roadcuts. The system was originallydeveloped by Pierson et al. (1990) for the OregonDepartment of Transportation (ODOT) in a study fundedby Oregon, nine other states, and the Federal HighwayAdministration. This effort was preceded by earlier work,dating back to the 1970s, to develop systematic inventoryand ranking procedures for hazardous roadcuts (Fish andLane, 2002). The ODOT RHRS is based on a rock slopeinventory and maintenance program developed by Wyllie(1987). Since 1990, the Federal Highway Administration

has adopted the ODOT RHRS (Pierson and Van Vickle,1993), hereafter referred to as the NHI RHRS.

The NHI RHRS employs a two-phase slope categori-zation process (Pierson and Van Vickle, 1993). The firstphase is a preliminary rating, where slopes are assigneda rating of A, B, or C based first on the estimated potentialfor rock to reach roadway and second on historicalrockfall activity. A-rated slopes are most hazardous andare characterized with a detailed rating that considers thefollowing factors:

� Slope height� Roadway width� Ditch effectiveness� Average vehicle risk� Decision sight distance� Geologic character� Block size/volume of rockfall per event� Climate/presence of water� Rockfall history

The factors affected by the geologic conditions ata roadcut are Geologic Character and Block Size.Geologic Character identifies whether the rockfall typeis controlled by the geologic structure or differentialerosion. Block Size is controlled by rock type, structuralconditions such as joint length and spacing, and roadcutconstruction methods.

Since the development and implementation of the NHIRHRS (Pierson et al., 1990; Pierson and Van Vickle,1993), more than 17 state and provincial agencies haveadopted the RHRS for rockfall management. Mosttransportation agencies have approached roadcut geologicconditions using the RHRS without modification, butabout seven have modified the RHRS, most notablyColorado (Stover, 1992), Washington (Lowell and Morin,2000), New York (Anonymous, 1996), and Ontario,Canada (Senior, 1999).

The Colorado Department of Transportation (DOT)incorporated Slope Inclination and Launching Features,because their experience suggested that these factorssignificantly contributed to rockfall hazard (Stover, 1992).The Washington State system distinguishes soil and rockmaterials but does not explicitly incorporate rockfallmodes, because the DOT wished to utilize raters who wereneither geologists nor geotechnical engineers (Lowell andMorin, 2000). The New York DOT modified the RHRS byconsidering hazard associated with two rock categories,crystalline and sedimentary, based on their assumptionthat crystalline rocks tend to have structurally controlledrockfall, whereas sedimentary rocks tend to have rockfallcontrolled by differential erosion (Anonymous, 1996).Ontario’s Ministry of Transportation (MTO) modified theRHRS to deal with rockfall modes common in Ontarioroadcuts. For example, in northern Ontario, raveling,

Vandewater, Dunne, Mauldon, Drumm, and Bateman

Environmental & Engineering Geoscience, Vol. XI, No. 2, May 2005, pp. 141–154142

toppling, and ice-jacking are the dominant rockfallbehaviors because of well-controlled blasting methods,pronounced physical weathering, and the fact that roadcutrelief is typically less than 8 m (Senior, 1999). Additionalparameters used by Ontario include height of the watertable at the slope face and looseness of the face.

STUDY AREA

Location and Physiography

Rockfall hazard was evaluated in five counties ofTennessee during 2001–2002 in the first phase of

a statewide rockfall hazard inventory for the TennesseeDOT. The five counties were selected to cover a broadrange of physiographic and geologic settings. Data fromthe roadcuts in these five counties were used to evaluatethe role of geology in rockfall hazard and to identifyspecific geologic controls on rockfall type and block size.

Eighty roadcuts in eastern and central Tennessee wereevaluated with the Tennessee RHRS, and 77 of these 80cuts were subsequently investigated to examine theinfluence of geologic factors on the geologic attributesof the rockfall hazard rating (Figure 1). Physiographically,this region is composed of the Blue Ridge Province;Valley and Ridge; Cumberland Plateau; the Highland

Figure 1. Physiographic (A) and geologic maps (B) of central and eastern Tennessee showing location of investigated counties.

Geologic Contribution to Rockfall Hazard


Rim; and the Nashville Basin (Bingham and Helton,1999) (Figure 1).

Geologic Setting

The Blue Ridge Province is underlain by mostly EarlyCambrian rifted margin sedimentary and volcanic rocksthat were deposited on Grenville basement. The Valleyand Ridge Province consists of a Cambrian and Ordovi-cian platform and Ordovician to Pennsylvanian synoro-genic sedimentary rocks that were folded and faultedduring the Late Mississippian–Permian Alleghenianorogeny (Hatcher et al., 1989). Nearly flat-lying Devonianto Pennsylvanian sedimentary rocks form the CumberlandPlateau, and these are moderately to deeply dissected,creating significant local relief. Adjacent to the Cumber-land Plateau to the west, but at lower elevation, is theHighland Rim, containing Ordovician to Mississippiansedimentary rocks that are moderately to deeply dissected.The Nashville Basin is a topographic low consisting ofa structural dome of Ordovician to Mississippian sedi-mentary rocks that gently dip away from the geologic apex(Hardeman, 1966; Bingham and Helton, 1999).

The five counties present contrasting geologic con-ditions. Lithologic variations range from crystalline rocks,composed primarily of granite, orthogneiss and para-gneiss, amphibolite, and gabbro, which occur in the BlueRidge Province (parts of Carter County), to sedimentaryrocks in the other provinces, including mudstones,siltstones, sandstones, and carbonates that occur in partsof Carter, Anderson, Bledsoe, Grainger, and SmithCounties (Figure 1). Additionally, structural variationsoccur: horizontally bedded rocks in Smith, Bledsoe, andAnderson Counties; moderately inclined and foldedsedimentary rocks in Bledsoe, Anderson, Grainger, andCarter Counties; and igneous and foliated metamorphicrocks in Carter County. Accordingly, roadcuts in the studyarea contain a variety of lithologies, structural domains,and discontinuity characteristics (i.e., spacing and persis-tence). This variety potentially influences rockfall modes,

because lithologies have different weathering and erosioncharacteristics, the presence of rock structures such asfolds and faults changes overall rock geometry, and theabundance of discontinuities influences rock strength.

METHODOLOGY OF RHRS REVISION

Geological Revisions to NHI RHRS

The Geologic Character category in the NHI RHRS(Pierson and Van Vickle, 1993) evaluates the geologicconditions contributing to rockfall hazard at a roadcut.However, the NHI approach does not explicitly in-corporate rockfall modes and uses ambiguous terminol-ogy, as described later in this section. Consequently, theNHI RHRS produces scores that do not clearly in-corporate geologic conditions at a roadcut, and scores maybe operator dependent, because the RHRS terminologycan be interpreted differently by different raters, limitingreproducibility. Given these issues, we revised the NHIRHRS Geologic Character to explicitly incorporaterockfall modes, eliminate ambiguous jargon, and promotereproducibility among different raters.

The Geologic Character category in the NHI RHRS(Pierson and Van Vickle, 1993) considers two cases(Table 1). ‘‘Case 1’’ is for structurally controlled rockfallwhere key factors are discontinuity size, discontinuityorientation, and rock friction. ‘‘Case 2’’ is for differentialerosion rockfall, where the factors are differential erosionfeatures and differential erosion rates. The cases aremutually exclusive, because only the score for the rockfallcondition with the greater hazard is recorded.

‘‘Case 1’’ considers two factors: Structural Conditionand Rock Friction (Table 1). Structural Condition at-tempts to describe the relative orientation and length ofjoints in the roadcut (Pierson and Van Vickle, 1993).Discontinuous joints are defined as less than 3.3 m inlength, whereas continuous joints are defined as greaterthan 3.3 m in length. Rock Friction describes the surfacesmoothness of the joints. Clay-filled and slickensided

Table 1. NHI Geologic Character rating scheme. Case 1 is for structurally related rockfall, and Case 2 is for erosion-related rockfall.

Geologic Characteristics

Values of Hazard Assessment

Low Hazard ��! High Hazard

Case 1

Structural Condition Discontinuous joints,

favorable orientation

Discontinuous joints,

random orientation

Discontinuous joints,

adverse orientation

Continuous joints, adverse

orientation

Rock Friction Rough, irregular Undulating Planar Clay infilling or slickensided

Case 2

Structural Condition Few differential

erosion features

Occasional differential

erosion features

Many differential

erosion features

Major differential

erosion features

Difference in Erosion Rates Small difference Moderate difference Large difference Extreme difference



joints are assigned the highest hazard, because less shearstress is required to exceed the coefficient of friction.

‘‘Case 2’’ considers two factors: Structural Conditionand Difference in Erosion Rates (Table 1). StructuralCondition describes the surficial weathering features ofa roadcut, whereas Difference in Erosion Rates describesthe formation rate of surficial weathering features (Piersonand Van Vickle, 1993), as inferred from the amount ofweathering relief in a slope face.

Examination of the scoring factors for the NHIGeologic Character category during development of theTennessee RHRS, as well as field tests for consistency andrepeatability among several raters, highlighted the needfor an improved approach to resolve issues of assessment,field application, and terminology. Issues included:

(1) The NHI RHRS does not apply well-establishedgeotechnical or geologic terms (Turner and Schus-ter, 1996) or rockfall modes (e.g., plane, wedge,topple, etc.).

(2) The NHI RHRS does not assess the abundance ordegree to which a potential rockfall mode is presentin a rock slope. This factor often controls thevolume of rock that may be shed to the road.

(3) Only the most hazardous condition is considered,whereas a better assessment of risk is to consider allrockfall modes in a roadcut that could deliver rockto the road.

(4) The system does not consider raveling explicitly,even though it may be a prominent rockfall type, asOntario’s MTO (Senior, 1999) and New York(Anonymous, 1996; Hadjin, 2002) recognized.

(5) Use of ‘‘structural condition’’ for both structural andnonstructural rockfall creates confusion.

(6) Use of ‘‘random’’ for intermediate orientationhazard condition is problematic, because ‘‘random’’could include potentially very hazardous orienta-tions, and the term does not encompass the case ofparallel discontinuities as an intermediate-riskorientation.

(7) Use of ‘‘joints’’ is confusing to geologists, becausebedding surfaces, faults, and cleavage may providediscontinuities for structural cases.

(8) Use of ‘‘continuous’’ and ‘‘discontinuous’’ to de-scribe discontinuity persistence is confusing.

(9) Use of ‘‘favorable’’ and ‘‘adverse’’ for hazardconditions is ambiguous. ‘‘Favorable’’ means favor-able to stability, but it could be misinterpreted asfavorably disposed to rockfall.

(10) Differential erosion is more accurately described bythe term ‘‘differential weathering.’’

The NHI RHRS was revised for use in Tennessee toincorporate rockfall modes explicitly; to include raveling;to use measurable attributes, including abundance; to sum

the ratings where more than one rockfall mode is present;and to eliminate ambiguous terminology.

Tennessee Geologic Character Scoring System

The Tennessee RHRS considers five rockfall modes:plane, wedge, topple, differential weathering, and ravel-ing, using appropriate combinations of six characteristics(Tables 2 and 3). Characteristics common to all rockfallmodes are the relative Abundance of the rockfall modeand Block Size. The relative Abundance of a rockfall modeis expressed as a percentage of the total slope face surfacearea containing the mode (Figure 2 and Table 3). In theNHI RHRS, Block Size is treated separately fromGeologic Characteristics, but it is incorporated into theTennessee RHRS because block size is primarily afunction of geology.

Characteristics unique to planar and wedge rockfall areSteepness of the failure plane(s) and wedge intersection,respectively, and the Roughness Profiles of the failureplane(s). As rockfall hazard increases with steepness,frictional resistance, which is largely a function ofroughness, controls whether the rock mass fails. First-order (planar or undulating) and second-order (rough orsmooth) friction is evaluated in a profile parallel to thelikely movement direction of the rock mass (Figure 3).First-order (macroscale) roughness is considered to havethe greatest effect on shear strength, because substantialshear stress is required to overcome first-order asperities,whereas slip over the second-order asperities requires onlylocalized shearing (Patton and Deere, 1970; Barton, 1973).

The traditional topple rockfall mode requires disconti-nuities dipping steeply into a slope face (Norrish andWyllie, 1996). However, the degree of discontinuitysteepness usually has little effect on this rotational failuremechanism. Therefore, discontinuity steepness is notconsidered in the Tennessee RHRS for the toppling mode.Additionally, because interlayer slip is a very smallcomponent of the topple resistance, friction is notconsidered in the Tennessee RHRS for the toppling mode(Table 3).

Table 2. Rockfall modes and characteristics of the Tennessee Geologic

Character rating scheme. X indicates inclusion of rated criteria fora rockfall mode.

Characteristics

Rockfall

Modes Abundance

Block

Size Inclination Friction Relief

Block

Shape

Planar X X X X

Wedge X X X X

Topple X X

Differential

weathering X X X

Raveling X X X



In many parts of Tennessee, especially on theCumberland Plateau, adjacent sedimentary beds of thesame lithology dip into the roadcut. This geometryproduces rockfall because the blocks fail in tension acrossbedding, releasing from the bed above. The behavioroccurs without a difference in lithology between beds, andthus it is not the result of differential weathering, so a newdescriptive term is required. We refer to this failure modeas ‘‘bedding plane release.’’ Although the mechanism isnot classic topple failure, the failure is not governed bysliding, such as wedge or planar failure, but does involvediscontinuities dipping into the roadcut, and thus it isincluded in the topple category in the Tennessee RHRS.

Amount of Relief is an attribute unique to differentialweathering, and it is defined in this article as referring tothe extent that a rock mass overhangs the material directlyunderneath (Tables 2 and 3). Block Shape is an importantattribute for raveling, because of the greater runout

potential for spherical blocks than tabular or blockyblocks. Therefore, block shapes are described in order ofincreasing hazard as tabular, blocky, or spherical.

The NHI RHRS uses point values equal to powers of 3to score characteristics (Pearson and Van Vickle, 1993). Ascore of 31 ¼ 3 points is assigned for the lowest hazardcategory and 34 ¼ 81 points for the highest hazardcategory, with intermediate values of 32¼ 9 and 33¼ 27.The premise behind exponential scoring is to give addedweight to high-hazard roadcuts so they will be readilyidentifiable in the database.

Following the NHI approach, each characteristic in theTennessee system was scored using four categories (Table3). For Raveling, only three categories were used for theblock shape score, because it was felt that the use ofa fourth category overemphasized the hazard for thismode, which typically sheds only small blocks inTennessee. With this reduction for Raveling, each of the

Table 3. Scoring schemes for the Tennessee RHRS Geologic Character.

Planar Rockfall Mode

Abundance ,10% 10–20% 20–30% .30%

Score 3 9 27 81

Block size ,1 ft (,0.3 m) 1 to 3 ft (0.3–0.9 m) 3 to 6 ft (0.9–1.8 m) .6 ft (.1.8m)

Score 3 9 27 81

Steepness 0–208 20–408 40–608 .608

Score 2 5 14 41

Friction (micro/macro) Rough/undulating Smooth/undulating Rough/planar Smooth/planar

Score 2 5 14 41

Wedge Rockfall Mode

Abundance ,10% 10–20% 20–30% .30%

Score 3 9 27 81

Block size ,1 ft (,0.3 m) 1 to 3 ft (0.3–0.9 m) 3 to 6 ft (0.9–1.8 m) .6 ft (.1.8 m)

Score 3 9 27 81

Steepness 0–208 20–408 40–608 .608

Score 2 5 14 41

Friction (micro/macro) Rough/undulating Smooth/undulating Rough/planar Smooth/planar

Score 2 5 14 41

Topple Rockfall Mode

Abundance ,10% 10–20% 20–30% .30%

Score 5 4 41 122


Score 5 14 41 122

Differential Weathering Rockfall Mode

Abundance ,10% 10–20% 20–30% .30%

Score 3 9 27 81


Score 3 9 27 81

Relief ,1 ft (,0.3 m) 1 to 3 ft (0.3–0.9 m) 3 to 6 ft (0.9–1.8 m) .6 ft (.1.8 m)

Score 3 9 27 81

Raveling Rockfall Mode

Abundance ,10% 10–20% 20–30% .30%

Score 3 9 27 81


Score 3 9 27 81

Shape Tabular Blocky Round —

Score 3 9 27 —



Figure 2. Roadcuts showing different failure modes and example Abundance values for the failure modes. Abundance values reflect only the percentages

visible in the photograph. (A) Example of subhorizontal sedimentary rocks: differential weathering at the sandstone/coal contact with abundance of 10 to

20 percent, and bedding release at the cross-bed surfaces in the sandstones as toppling at .30 percent. (Percentages neglect portion of slope above bench,

which is not fully visible in the photograph. Vehicles and people for scale.) (B) Example of inclined sedimentary rocks: planar mode along inclined

bedding surfaces in the sandstone at .30 percent and raveling on the right in mudstone at ,10 percent. (C) Example of metamorphic rocks: raveling with

abundance .30 percent related to discontinuities created from cleavage surfaces, bedding surfaces, joints, and blasting fractures.



other four rockfall modes can yield scores of up to 244(Table 3). For Toppling, which has two scoring character-istics, each characteristic has a maximum score of 122.For Differential Weathering, with three scoring character-istics, the maximum characteristic score is 81. For planarand wedge rockfall, with four scoring characteristics,Abundance and Block Size have maximum scores of 81,but to maintain a total possible score of 244, Friction andSteepness are limited to 41 points each. This differencein maximum score between characteristics is justified onthe basis that the Friction term is the most difficult tojudge in the field (Barton, 1973), and so it was combinedwith Steepness to share 82 points.

METHODS FOR RELATING GEOLOGICATTRIBUTES TO RHRS CHARACTERISTICS

Geologic Attributes

Slope geometry, lithology, and discontinuity data for134 potential rockfall modes at 77 roadcuts werecollected. A statistical analysis using logistic regressionwas conducted to relate RHRS characteristics to geologic

attributes. Logistic regression requires categorical values(Table 4) for many geologic attributes (Hosmer andLemeshow, 1989). Slope orientation data were collected,because it was expected that north-facing roadcuts couldhave high abundance of differential weathering rockfall.Rock type, fissility/cleavage, and geomechanical layerthickness could affect block size, and lithologic variationcould affect rockfall mode and abundance. Discontinuitygeometric data were collected to investigate associationwith rockfall mode and block size.

Most data values in Table 4 are self-explanatory, buta few require comment. Layer thickness as defined in thisstudy is not actual bedding thickness in sedimentary rocks,but rather geomechanical thickness between upper andlower bounding discontinuities for a set of beds with thesame geomechanical and weathering characteristics. Inmetamorphic rocks, well-defined foliation surfaces androck layer boundaries were used to define the geo-mechanical thickness. A discontinuity set, as defined inthis study, is typically (but is not limited to) systematicfractures or geomechanical layer surfaces that haveparallel to subparallel orientation and measurable spacing.For analytical purposes, all data, including the number ofdiscontinuity sets, were collected with respect to eachidentified rockfall mode and not necessarily the entireroadcut.

Logistic Regression: Method Overview

We chose to investigate the presence or absence ofcorrelations between RHRS characteristics and geologicattributes using logistical regression, because TennesseeRHRS data are categorical, and the method is well suitedfor analyzing categorical data (Hosmer and Lemeshow,1989; Allison, 1999; Stokes et al., 2000; and Ott andLongnecker, 2001). Logistic regression is often used inthe social sciences (Cleary and Angel, 1984; Wang andFitzhugh, 1995; and Studenmund, 1997) and is now beingapplied to fields such as landslide hazard assessment (Daiand Lee, 2001; Apt et al., 2002), hydrology (Zain, 2001;Bent and Archfield, 2002), and resource exploration(Harris and Pan, 1999; Sahoo and Pandalai, 1999).

Figure 3. Visual scoring aid for friction. Terms indicate micro- and macrofriction profiles, respectively (modified from Barton, 1973).

Table 4. Geologic attributes collected for logistic regression.

Attribute Values

Slope orientation and failure mode information

Slope aspect Slope trend þ 90

Rockfall type Structural, nonstructural

Rockfall mode Planar, wedge, topple, differential

weathering, raveling

Lythology of potential rockfall

Rock type Based on field description; clastic

sedimentary, carbonate

sedimentary, crystalline

Lithologic variation Yes, no

Fissility/cleavage Yes, no

Geomechanical layer thickness ,0.2 m, 0.2 m–0.5 m,

0.5 m–1.0 m, .1.0 m

Discontinuity data

Number of discontinuity sets 0, 1, 2, .2



In logistic regression, the dependent ‘‘outcome’’ vari-able is restricted to two values, 1 and 0, which representthe occurrence or nonoccurrence, respectively, of anoutcome event. The analysis determines if the outcome ispredicted by one or more independent ‘‘explanatory’’variables and produces the coefficients of a predictionmodel formula, with their standard errors of estimate, andsignificance levels and odds ratios, with their 95 percentconfidence intervals (Allison, 1999; Stokes et al., 2000).Odds are defined as the probability of the event divided bythe probability of the nonevent. An odds ratio is the ratioof the odds of an outcome occurring, given a value for theexplanatory variable, compared to the odds of the sameoutcome occurring for a given reference value of theexplanatory variable.

The logistic regression equation has the log-linearform (Allison, 1999):

lnðp=½1� p�Þ ¼ aþ BX þ errorB, ð1Þ

where p is the probability that ‘‘outcome’’ event Y occurs(yes ¼ 1), ln( p/[1 � p]) is the natural-log odds, a is theintercept coefficient, B is the coefficient of the in-dependent ‘‘explanatory’’ variable X, and errorB is thestandard error of B (Figure 4). The method uses theindependent variable, X, values and the probabilities ofthe dependent variable, p, to calculate values for thecoefficients by maximum-likelihood estimates, and co-efficient significance is indicated by the significanceprobability value ( pw value) determined from the Waldstatistic with significance level of .05 (Hosmer andLemeshow, 1989).

Evaluation of model parameters is accomplished byconsidering: (1) the likelihood ratio (chi-square statistic)to determine if the overall model is statistically significantby testing the global null hypothesis that the coefficientsof all independent variables are equal to 0; (2) theHosmer-Lemeshow goodness-of-fit statistic to test theglobal null hypothesis that the model fits the data,rejecting the hypothesis if the model does not fit the data;and (3) the c-statistic, which measures the logisticequation discriminatory power. The c-statistic varies from0.5 (predictions no better than chance) to 1.0 (predictionsare always correct) (Hosmer and Lemeshow, 1989).

Odds Ratios

The slope coefficient in logistic regression is inter-preted as the rate of change in the log-odds of Y as Xchanges (Figure 4). However, a more intuitive interpre-tation of the coefficient utilizes the odds ratio (Hosmerand Lemeshow, 1989; Allison, 1999; and Stokes et al.,2000). The graph in Figure 4 shows that the coefficient, B,is equal to the change in log odds as follows:

lnðodds jX ¼ 1Þ � lnðodds jX ¼ 0Þ

¼ lnodds jX ¼ 1

odds jX ¼ 0

� �¼ B ð2Þ

where the vertical bar means ‘‘given that.’’ Taking bothsides of Eq. 2 as exponents of e yields the odds ratio,

odds ratio ¼ odds jX ¼ 1

odds jX ¼ 0¼ eB ð3Þ

For example, if coefficient B¼ 1.6, then the odds ratio(e1.6) equals 5, meaning that when the independentvariable, X, increases 1 unit, the odds that Y will equal 1increase by a factor of 5. Odds ratios of 1 indicate a 50/50chance the event will occur with a change in X. If the 95percent confidence interval for eB does not include thevalue 1.0, a change in X from 0 to 1 produces a statisticallysignificant change in the odds for Y.

Data Models

For this study, the software package SAS (Stokes et al.,2000) was used to perform logistic regression, with thedependent variables being the two major factors in boththe Tennessee RHRS and the NHI RHRS: Rockfall Typesand Block Size (Table 5). The dependent variableROCKFALL is equal to 1 if rockfall type is nonstructuraland 0 if structural. Nonstructural rockfall types are

Figure 4. Graphical relationship between the log odds of Y, where py is

the probability that event Y occurs, and the independent variable, X, in

the logistic regression equation.



differential weathering and raveling modes, and structuralrockfall types are plane, wedge, and topple modes. Thedependent variable BLOCKSIZE is equal to 1 if blocklength is greater than 0.3 m and 0 otherwise.

Independent geologic or physical variables wereselected to test for correlation with the dependent variableof ROCKFALL type and BLOCKSIZE (Tables 4 and 5).

Independent variables are categorical or ordinal, and onevalue for a variable becomes the reference value (Hosmerand Lemeshow, 1989; Kleinbaum et al., 1998; and Stokeset al., 2000).

RESULTS

Comparison of Tennessee RHRS Scores toNHI Scores for Geologic Character

Roadcuts were scored using both the Tennessee RHRSand the NHI Geologic Character schemes (Figure 5).Both systems yielded the same score at roadcuts whereDifferential Weathering is the only mode. Yet, theaverage Geologic Character score of 84 for the Tennesseesystem is higher than the average NHI GeologicCharacter/Block Size score of 66 (Figure 5). One reasonthat the Tennessee system scores are higher is becausescores are cumulative for all potential rockfall modes ata roadcut, whereas the NHI RHRS records only thehighest-case score. Of the 80 roadcuts rated, 62 percenthave multiple rockfall modes, with Differential Weather-ing and Raveling being most common (Figures 5 and 6).

Table 5. Variables and interactions for logistic regression models.

Logistic Regression

Model

Rockfall

Model

Block Size

Model

Dependent

variable, YROCKFALL BLOCKSIZE

Modeled category

( p j Y ¼ 1)

Nonstructural Block size . 0.3 m

Independent

variables, X(geologic

attributes)

Slope aspect Rockfall type

Rock type Geomechanical

layer thickness

Lithologic variation Lithologic variation

Number of

discontinuity sets

Fissility/cleavage

Rock type

Number of

discontinuity sets

Figure 5. Comparison of Tennessee Geologic Character scores with

NHI Geologic Character and Block Size scores. Note: Points without

lettering indicate only a single rockfall mode at that roadcut. P¼ plane;

W ¼ wedge; T ¼ topple; D ¼ differential weathering; R ¼ raveling.

Some points represent identical scores for multiple roadcuts.

Figure 6. Number of rockfall modes at a roadcut (A) and rockfall

mode occurrence at roadcuts (B), showing that multiple modes

may be present at a roadcut.



Another reason for the higher scores is the largepercentage of roadcuts that have Raveling, which wouldnot typically be scored for the 80 roadcuts with the NHIRHRS, because Raveling is not typically the greatestscoring rockfall mode.

Abundance is another factor that is not evaluated in theNHI RHRS. However, Abundance causes a relativedecrease for some Tennessee scores versus NHI scoresfor two reasons. First, where multiple rockfall modes arepresent, the scores for each individual mode tend to belower for the Tennessee system than for the NHI system,because mode abundances tend to be smaller wheremultiple rockfall modes are present. Second, NHI scorestend to be higher than the Tennessee scores where onlystructural modes are present and in Low Abundance,which occurs at five roadcuts.

Results of Logistic Regression: ROCKFALLDependent Variable

The results from the ROCKFALL logistic regressionanalysis (Table 6) indicate that rockfall type is influencedby the occurrence of lithologic variation and by thenumber of discontinuity sets. Based on the odds ratio,lithologic variation increases the odds of nonstructuralrockfall by a factor of 6. Odds for nonstructural rockfallare: 10 times greater where no discontinuity sets arepresent, as compared with slopes containing three or morediscontinuity sets; 6 times greater where there is onediscontinuity set as compared with slopes containing threeor more; and 4 times greater when there are twodiscontinuity sets as opposed to three or more. Theremaining variables are not statistically significant pre-dictors of rockfall type and were not included in the model.The overall model is significant at the .05 level accordingto the likelihood chi-square statistic and predicts 79

percent of the responses correctly according to the c-statistic. The goodness-of-fit statistic indicates that thelogistic regression model fits the data.

Results of Logistic Regression: BLOCKSIZEDependent Variable

The results from the BLOCKSIZE logistic regressionanalysis (Table 6) indicate that block size is influenced byrockfall type, lithologic variation, geomechanical thick-ness, and number of discontinuity sets. The odds for blocksizes larger than 0.3 m in the longest dimension are 19times greater for structural than for nonstructural rockfalland 6 times greater where a lithologic variation is presentthan where absent. However, the odds for block sizeslarger than 0.3 m in longest dimension are decreased by4.5 times (eB ¼ 0.22) where there are two discontinuitysets as compared to the presence of three or more sets, andodds are decreased by 8 times (eB ¼ 0.13) when thegeomechanical thickness is less than 0.2 m than where it isgreater than 1.0 m. The remaining variables are notstatistically significant predictors of block size and werenot included in the model. Overall, the model is significantat the .05 level according to the likelihood chi-squarestatistic, and the goodness-of-fit statistic indicates that thelogistic regression model fits the data and predicts 82percent of responses as indicated by the c-statistic.

DISCUSSION

Tennessee RHRS

As demonstrated by the greater geologic characterscores where raveling is present (Figure 5), the TennesseeRHRS captures the significance of raveling with respect tothe production of rockfall material. Though raveling

Table 6. Logistic regression results.

Variable

Reference

Value B ErrorB

Wald

Statistic

Wald

pw Value

Odds

Ratio

95% Confidence

Interval

ROCKFALL dependent variable

Lithologic variation: yes No 1.8 0.7 6.7 .01 5.8 1.5, 22.0

Number of discontinuity sets: 0 .2 sets 2.3 0.7 11.8 ,.01 9.7 2.6, 35.4

Number of discontinuity sets: 1 .2 sets 1.8 0.8 5.6 .02 6.0 1.4, 26.4

Number of discontinuity sets: 2 .2 sets 1.4 0.6 5.3 .02 4.0 1.2, 13.3

BLOCKSIZE dependent variable

Failure type: structural Weathering 2.6 0.6 15.4 ,.01 19.3 4.4, 84.5

Lithologic variation: yes No 1.72 0.52 11.0 ,.01 5.6 2.0, 15.4

Number of discontinuity sets: 2 .2 sets �1.5 0.7 5.2 .02 0.22 0.06, 0.8

Mechanical thickness: ,0.2 m .1.0 m �2.0 0.7 8.0 ,.01 0.13 0.03, 0.5

For dependent variable ROCKFALL, modeled category ¼ weathering. The likelihood chi-square value was 25.7 (df ¼ 4, p , .01). The Hosmer-

Lemeshow goodness-of-fit value was 0.91 (df ¼ 5, p¼ .97). The c-statistic value was 0.788. For dependent variable BLOCKSIZE, modeled cate-

gory ¼ block size . 0.3 m. The likelihood chi-square value was 48.8 (df ¼ 8, p , .01). The Hosmer-Lemeshow goodness-of-fit value was 6.7

(df ¼ 8, p¼ .57). The c-statistic value was 0.824.



occurs at 70 percent of the roadcuts in the study area(Figure 6), the mode typically involves a small block size.However, raveling can be a traffic threat or a source ofroadway damage if these small blocks are shed from largeheights; if large masses of raveled blocks are shedsimultaneously; or if the blocks roll or launch onto theroadway, necessitating raveling characterization foreffective rockfall management.

The cumulative scoring of potential rockfall modes inthe Tennessee RHRS yields greater total GeologicCharacter scores compared to the NHI RHRS. Onepossible concern is that the score increase would over-emphasize the role of Geologic Character in total hazardscore, but the following points counterbalance this. First,although the importance of an accurate geologic evalua-tion is the focus here, the Tennessee system also evaluatestraffic volume and roadway conditions with the sameproportion of possible score as the NHI RHRS. Forexample, a roadcut could score very high on GeologicCharacter, but if catchment is sufficient to minimizerockfall affecting the roadway, the overall difference intotal scores will not be large, despite the GeologicCharacter score.

Second, considering all potential rockfall modes, notjust the rockfall type with the greatest Geologic Characterscore, as occurs using the NHI RHRS, gives insight to thestructural and weathering condition of a roadcut. Thisassessment is critical because, for example, anotherrockfall mode with a lower Geologic Character scoremay cause greater roadway damage after failure becauseof block size. Similarly, the possibility also exists thatrockfall by one mode can trigger rockfall by anothermode. Therefore, recognition of all modes more com-pletely defines the portion of a roadcut that is prone tofailure, and scoring all modes characterizes the cumulativecontribution of Geologic Character to the RHRS rating.

Third, scoring all potential rockfall modes at a roadcutprovides an indication of likely successful remediationtechniques (Wyllie and Norrish, 1996). Furthermore, byevaluating all potential rockfall modes at a roadcut, theneed to separate the roadcut into segments where differentmodes are present is obviated, thereby reducing datacollection complexity. Overall, the methodology ofscoring all rockfall modes captures useful informationomitted by the single-case methodology in the NHI RHRSand does not overemphasize the role of geology in thehazard assessment.

Compared to the NHI RHRS, the contribution ofindividual rockfall modes to the Geologic Character scoreis appropriately lower in the Tennessee system when themode is less abundant. The advantage of this result is thatthe use of Abundance as a geologic characteristic in theTennessee RHRS differentiates roadcuts that would scoreidentically in the NHI RHRS, which does not evaluateAbundance. Additionally, the use of Abundance over-

comes a potential problem for the NHI RHRS, which isthat the rockfall type with the greatest score for GeologicCharacter, as identified by the NHI RHRS, may occupyonly a small portion of the roadcut and may be lessabundant than another mode with a smaller score.However, second mode, which is not recorded for theNHI system, increases the cumulative risk because ofgreater abundance along the roadcut. Therefore, the use ofAbundance in the Tennessee RHRS acts as a sensitivityindicator to the overall role of potential rockfall mode.Finally, the use of rockfall modes and distinctivecharacteristics eliminates some of the terminology prob-lems of the NHI RHRS.

Predictors of Rockfall Type and Block Size

The logistic regression results (Table 6) indicate thatboth lithologic variation and number of discontinuities aresignificant predictors of rockfall type. Intuitively, litho-logic variation should affect rockfall type through itseffect on differential weathering. Similarly, as the numberof discontinuity sets increases, structural conditions arecreated that promote planar, wedge, and topple rockfall,because planar and topple modes require at least one setof systematic discontinuities, and wedges require at leasttwo sets.

The logistic regression results (Table 6) indicate thatrockfall type, number of discontinuities, geomechanicalthickness, and lithologic variation are significant predic-tors of block sizes. Structural rockfall strongly favors blocksizes larger than 0.3 m, and conversely, nonstructuralrockfall favors block sizes smaller than 0.3 m. Lithologicvariation also favors larger block size, because roadcutswith such variation have nearly 5.5 times greater odds toproduce large blocks, suggesting that lithologic contrastfavors the creation of overhangs greater than 0.3 m.

Rock units with geomechanical thicknesses smallerthan 0.2 m favor the formation of blocks with length lessthan 0.3 m. As intuitively expected, rock units withsmaller geomechanical thickness have greater odds forshedding small blocks, whereas units with larger geo-mechanical thickness have greater odds for shedding largeblocks.

Interestingly, rockfall modes with more than twodiscontinuity sets are more likely to produce block sizesgreater than 0.3 m than those with just two discontinuitysets. One might expect that as the number of discontinuitysets increases, a multiplicity of intersecting surfaces iscreated, and therefore, smaller block sizes should beobserved. However, this relationship is not observed.Rather, as the number of discontinuity sets increases,a multiplicity of intersecting surfaces is created, generat-ing discontinuity geometries that define structural rockfallmodes (Norrish and Wyllie, 1996), typically yieldinglarger blocks.



Implications

The logistic regression results highlight geologicalattributes of roadcuts that correlate to greater or lesserrockfall block size or with particular rockfall modes.Lithologic variation increases the odds for nonstructuralrockfall, because roadcuts without lithologic variationinclude a large majority of the structural rockfall modes.Furthermore, the Tennessee RHRS average GeologicCharacter score for structural modes is 84, as comparedwith 43 for nonstructural modes.

Additionally, as the number of discontinuity setsincreases, the odds of structural rockfall increase.Similarly, the odds of block sizes greater than 0.3 m aregreatest with the existence of more than two discontinuitysets. In the Tennessee RHRS, structural rockfall modeshave an average Block Size score of 26, which translatesapproximately to the third hazard category (1–2 m).

Therefore, if a roadcut contains rocks without lithologicvariation and two or more discontinuity sets, a greaterGeologic Character score is expected because of theincreased odds of large intact rock blocks shed bystructural rockfall modes. This expectation is even greaterwhere geomechanical layer thicknesses exceed 1.0 m,because of the increased odds of block sizes greater than0.3 m.

CONCLUSIONS

The Geologic Character category utilized in theTennessee RHRS gives consistent descriptions of a widerspectrum of geologic conditions than the NHI versionbecause it:

� Explicitly identifies potential rockfall modes, includingraveling

� Accumulates hazard scores for all potential modes ata roadcut

� Considers the abundance of occurrence for particularmodes

� Avoids ambiguous terminology

The Tennessee RHRS does yield greater GeologicCharacter scores, in general, than the NHI RHRS, butthis difference reflects the importance of scoring allrockfall modes, including raveling, that have the potentialfor delivering rock to the roadway.

An analysis of relationships between geologic attrib-utes and RHRS characteristics shows:

� Lithologic variation and number of discontinuity setsare significant predictors of rockfall type, becauselithologic variation coincides with differential weath-ering, and a greater number of discontinuity setsincreases the odds of structural rockfall modes.

� Block sizes greater than 0.3 m are predicted by theoccurrence of structural rockfall modes, the presenceof more than two discontinuity sets, lithologicvariation, and geomechanical thicknesses greater than1.0 m.

With respect to the Tennessee RHRS, roadcuts withoutlithologic variation but with two or more discontinuitysets are expected to have relatively high geologiccharacter scores because of the increased odds of largeintact rock blocks shed by structural rockfall modes. Thisexpectation is even greater where geomechanical thick-nesses exceed 1.0 m because of the increased odds ofblock sizes greater than 0.3 m.

ACKNOWLEDGMENTS

The authors would like to acknowledge Bill Trolinger,Len Oliver, Harry Moore, and the Tennessee Departmentof Transportation for their contributions and funding forthis research. The manuscript was improved by reviewsby Tom Badger, Martin Woodward, and Robert Watters.

REFERENCES

ABBOTT, B.; BRUCE, I.; SAVIGNY, W.; KEEGAN, T.; AND OBONI, F., 1998,

Application of a new methodology for the management of

rockfall risk along a railway. In Moore, D. P., Hungr, O.

(Editors), 8th Congress of the International Association ofEngineering Geologists: Vancouver, BC, A. A. Balkema

Publishers, Netherlands, pp. 1201–1208.

ALLISON, P. D., 1999, Logistic Regression Using the SAS System:Theory and Application: BBU Press, Cary, NC, 288 p.

ANONYMOUS, 1996, GEM-15: Rock Slope Rating Procedure: State of

New York Department of Transportation Geotechnical Engineer-

ing Bureau, 41 p.

APT, J.; SKAUGSET, A. V.; AND PYLES, M., 2002, Discriminating

between landslide sites and adjacent terrain using topographic

variables: Geological Society America Abstracts with Programs,Vol. 34, No. 6, p. A40.

BARTON, N. R., 1973, Review of a new shear strength criterion for rock

joints: Engineering Geology, Vol. 7, pp. 287–332.

BENT, G. C. AND ARCHFIELD, S. A., 2002, A Logistic RegressionEquation for Estimating the Probability of a Stream FlowingPerennially in Massachusetts: U.S. Geological Survey Water-

Resources Investigations Report 02-4043, 51 p.

BINGHAM, E. AND HELTON, W. L., 1999, Physiographic Map ofTennessee: State of Tennessee Division of Geology, scale 1:

500,000.

CLEARY, P. D. AND ANGEL, R., 1984, The analysis of relationships in-

volving dichotomous dependent variables: Journal Health SocialBehavior, Vol. 25, No. 3, pp. 334–348.

DAI, F. C. AND LEE, C. F., 2001, Terrain-based mapping of landslide

susceptibility using a geographical information system: A case

study: Canadian Geotechnical Journal, Vol. 38, No. 5, pp.

911–923.

FISH, M. AND LANE, R., 2002, Linking New Hampshire’s rock cut

management system with a geographic information system:

Transportation Research Record, No. 1786, pp. 51–59.

FLATLAND, R., 1993, Application of the Rockfall Hazard Rating System



to the Rock Slopes Adjacent to US50 and State Route 28 in theEast Side of Lake Tahoe, Nevada: Unpublished M.S. Thesis,

University of Nevada, Mackay School of Mines, Reno, 316 p.

HADJIN, D. J., 2002, New York State Department of Transportation

rock slope rating procedure and rockfall assessment: Trans-portation Research Record, No. 1786, pp. 60–68.

HARDEMAN, W. D., 1966, Geologic Map of Tennessee: State of

Tennessee Department of Conservation, Division of Geology,

4 sheets, scale 1:250,000.

HARRIS, D. AND PAN, G., 1999, Mineral favorability mapping: A

comparison of artificial neural networks, logistic regression,

and discriminant analysis: Natural Resources Research (In-ternational Association Mathematical Geology), Vol. 8, No. 2,

pp. 93–109.

HATCHER, R. D., JR.; THOMAS, W. A.; AND VIELE, G. W., 1989, TheAppalachian-Ouachita Orogen in the United States: The Decade

of North American Geology Project Series, Vol. F-2, Geological

Society of America, Boulder, CO, 767 p., 12 plates.

HOSMER, D. W. AND LEMESHOW, S., 1989, Applied Logistic Regression:John Wiley and Sons, New York, 307 p.

KLEINBAUM, D. G.; MULLER, K.; KUPPER, L.; AND NIZATI, A., 1998,

Applied Regression Analysis & Multivariable Methods, 3rd ed.:

Brooks/Cole Publishing Company, Pacific Grove, CA, 736 p.

LOWELL, S. AND MORIN, P., 2000, Unstable slope management:

Washington State: TR News, No. 207, pp. 11–15.

MOORE, H. L., 1986, Wedge rockfalls along Tennessee highways

in the Appalachian region: Their occurrence and correction:

Bulletin Association Engineering Geologists, Vol. 23, No. 4, pp.

441–460.

NORRISH, N. I. AND WYLLIE, D. C., 1996, Rock slope stability analysis.

In Turner, A. K. and Schuster, R. L. (Editors), Landslides:Investigation and Mitigation: Transportation Research Board

Special Report 247, National Research Council, Washington, DC,

pp. 391–425.

OTT, R. L. AND LONGNECKER, M., 2001, An Introduction to StatisticalMethods and Data Analysis, 5th ed.: Duxbury, Pacific Grove,

CA, 1152 p.

PATTON, F. D. AND DEERE, D. U., 1970, Significant geologic factors in

rock slope stability. In Planning Open Pit Mines: A. A. Balkema,

Capetown, South Africa, pp. 143–151.

PIERSON, L. A.; DAVIS, S. A.; AND VAN VICKLE, R., 1990, RockfallHazard Rating System: Implementation Manual: Federal High-

way Administration Report OR-EG-90-01, 172 p.

PIERSON, L. A. AND VAN VICKLE, R., 1993, Rockfall Hazard RatingSystem: Participant’s Manual: Federal Highway Administration

Publication SA-93-057, 104 p.

ROYSTER, D. L., 1978, Landslide Remedial Measures: Tennessee

Department of Transportation, Prepared for the 37th Annual

Southeastern Association of State Highway and Transportation

Officials Convention, Nashville, TN, 62 p.

SAHOO, N. R. AND PANDALAI, H. S., 1999, Integration of sparse

geologic information in gold targeting using logistic regression

analysis in the Hutti-Maski schist belt, Raichur, Karnataka, India:

A case study: Natural Resources Research (InternationalAssociation Mathematical Geology), Vol. 8, No. 3, pp. 233–250.

SENIOR, S. A., 1999, Rockfall hazard remediation along Ontario

highways: Proceedings, 50th Highway Geology Symposium,

Roanoke, VA, May 20–23, p. 276–286.

STOKES, M. E.; DAVIS, C. S.; AND KOCH, G. G., 2000, Categorical DataAnalysis Using the SAS System, 2nd ed.: BBU Press, Cary, NC,

626 p.

STOVER, B. K., 1992, Highway Rockfall Research Report: Colorado

Geological Survey Special Publication, Denver, CO, 27 p.

STUDENMUND, A. H., 1997, Using Econometrics: A Practical Guide,3rd ed.: Addison-Wesley, Reading, MA, 670 p.

TURNER, A. K. AND SCHUSTER, R. L., 1996, Landslides: Investigationand Mitigation: Transportation Research Board Special Report

247, National Research Council, Washington, DC, 673 p.

WANG, E. J. M. AND FITZHUGH, E. C., 1995, Application of odds ratio

and logistic models in epidemiology and health research: HealthValues, Vol. 19, No. 1, pp. 59–62.

WYLLIE, D. C., 1987, Rock slope inventory: Proceedings, FederalHighway Administration Rockfall Mitigation Seminar, FHWA

Region 10, Portland, OR. 25 p.

WYLLIE, D. C. AND NORRISH, N. I., 1996, Stabilization of rock slopes.

In Turner, A. K. and Schuster, R. L. (Editors), Landslides:Investigation and Mitigation: Transportation Research Board

Special Report 247, National Research Council, Washington, DC,

pp. 474–504.

ZAIN, H. H. M., 2001, Assessing ground-water vulnerability using

logistic regression. In Hayes, D. F. and McKee, M. (Editors),

Proceedings, American Water Resources Association (AWRA)Summer Specialty Conference and Universities Council on WaterResources (UCOWR) Annual Conference. Decision Support

Systems for Water Resources Management, American Water

Resources Association, Middleburg, VA, pp. 265–271.



Assessing Rockfall

Documents