Simulation of 3D Block Popula- - geomecanica.orggeomecanica.org/groupBIM/files/course2017Medley/docs/Haneberg… · Simulation of 3D Block Popula-tions to Characterize Outcrop Sampling

HANEBERG: SIMULATION OF 3D BLOCK POPULATIONS TO CHARACTERIZE OUTCROP SAMPLING BIAS IN BIMROCKS

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tions to Characterize OutcropSampling Bias in Bimrocks

By William C. Haneberg

Simulation von dreidimensionalenGesteinsvorkommen zur Charakterisierungder Aufschlussstichprobenverzerrungin Block-In-Matrix Gesteinen (Bimrocks)

Die geotechnische und geohydrologische Charakterisie-rung von Block-in-Matrix Gesteinen (sogenannte Bim-rocks) wie tektonische Melange, Störungsfurchen, Ge-schiebemergel und Erdrutschgeröll ist mit einigen Schwie-rigkeiten verbunden. Eine genaue und zuverlässige Cha-rakterisierung ist jedoch extrem wichtig, da bekannter-weise Blöcke oder die Größenverteilung der Gesteinsblöckedie Materialeigenschaften wie Durchlässigkeit, Scherfes-tigkeit und die Wahl des Bauverfahrens entscheidend be-einflussen. Geotechnische und hydrogeologische Metho-den zur Baugrunduntersuchung wie Bohrungen und Auf-schlusskartierung ergeben jedoch widersprüchliche Er-gebnisse, da von ein- und zweidimensionalen Stichprobenauf dreidimensionale Vorkommen geschlossen wird. Mon-te Carlo Computer Simulationen können zur Untersuchungder Verzerrungswerte verwendet werden, die bei der Ablei-tung der dreidimensionalen Gesteinsblockverteilung auszweidimensionalen Darstellungen wie Aufschlusskartenoder Fotoaufnahmen entstehen. Die Simulation von zwei-dimensionalen Aufschlussdarstellungen von dreidimensi-onalen Gesteinen zeigt, dass Aufschlusskartierungen dieDurchschnittsgesteinsgröße und das Gesamtblockvolumenum mehr als 10 % über- oder unterschätzen können, ob-wohl gleichförmige Gesteinsblöcke tendenziell unter-schätzt werden. Die durch zweidimensionale Stichprobeneinfließende Fehlergrößenordnung bewegt sich im Bereichvon ± 50 % für Durschnittsgesteinsblöcke und ± 80 % fürdas Gesamtblockvolumen. Ein gründlich geplantes statis-

tisches Probenentnahmeprogramm von Gesteinsgröße undAusrichtung in Verbindung mit numerischer Simulationbirgt demnach die größte Chance auf verwendbare Infor-mation bezüglich der Gesteinsverteilungsstatistik, die we-sentlichen Einfluss auf die Ingenieurplanung und Bauaus-führung haben kann.

Geotechnical and hydrogeological characterization ofblock-in-matrix rocks (bimrocks) such as melange, faultgouge, till, and landslide debris can be difficult, but accu-rate and reliable characterization is important becauseblock or block size distributions are known to influencefactors such as permeability, shear strength, and thechoice of construction methods. Geotechnical and hydro-geological exploration methods such as drilling and out-crop mapping, however, produce biased results becausethey yield 1D or 2D samples of 3D populations. Monte Carlocomputer simulations can be used to explore the amount ofbias introduced when 3D block distribution information isinferred from 2D projections such as outcrop maps or pho-tographs. Simulations of the 2D outcrop projections of 3Dblocks show that outcrop mapping has the potential tooverestimate or underestimate mean block sizes and totalblock volumes by tens of percent, although the tendencywill be towards underestimation for blocks that are nothighly elongated. The magnitudes of errors introduced by2D outcrop sampling can be on the order of ± 50 % for meanblock sizes and ± 80 % for total block volumes. Carefullydesigned statistical sampling of block sizes and orienta-tions combined with numerical simulations, however, hasthe potential to yield valuable information about the statis-tics of block distributions that may have significant effectson the design and construction of engineered works.

B lock-in-matrix rocks (bimrocks) are definedas “a mixture of rocks, composed of geo-

technically significant blocks within a bondedmatrix of finer texture” (1), a definition that en-compasses a wide range of geologic materialsincluding melanges, fault rocks, landslide de-bris, and glacial till.

Figure 1 shows a typical bimrock, in this casean outcrop of Mesozoic Franciscan Complex me-lange near Mendocino, California, USA. In thisexample the blocks visible in the outcrop are onthe order of centimeters to decimeters in theirlongest dimension, but in other places they canrange in size from meters to hundreds of metersin length. The geotechnical significance of bim-rock blocks was illustrated by Lindquist (2) andLindquist and Goodman (3), who showed that inphysical models of melange, the angle of internal

Fig. 1 Typicalbimrock outcrop ofFranciscan Complexmelange near Men-docino, California,USA. The white disknear the center of thephotograph is a 2.3 cmdiameter coin. (Photocourtesy of Ed Medley,1994).Bild 1 Typischer„Bimrock“-Aufschlussin San FranciscanMelange in der Nähevon Mendocino,Kalifornien, USA.Die weiße Scheibein Fotomitte stellt eineMünze mit 2,3 cmDurchmesser dar(Foto Ed Medley, 1994).

friction increased and the cohesive strength de-creased as a function of block proportion in cas-es where block proportions exceeded 25 to 30 %.

The standard engineering practice is to usethe strength of the weakest component (the bim-

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rock matrix in this case), but information aboutbimrock block proportions can allow the use ofmore optimistic strength estimates (4). At thesame time, however, the practitioner must con-sider the uncertainty associated with block pro-portion estimates in order not to overestimatethe strength added by blocks (5) or risk seriousunderestimation of the volume of blocks thatmay be encountered during construction. An ac-curate prediction of the largest blocks to be en-countered during tunneling or excavation mayalso be critical in the selection of appropriatemethods and machinery, but the scale-indepen-dent fractal (or nearly fractal) distribution ofbimrock blocks makes this a difficult proposition(1, 5, 6).

Blocks with permeability that is appreciablyhigher or lower than that of the surroundingmatrix can also affect groundwater flow. Al-though he did not use the term “bimrock”, Hane-berg (7) used a series of analytical and numerical

Fig. 2 A) Obliqueview showing theintersection of anellipsoid (representinga bimrock block) inter-secting a plane (repre-senting an outcropface). B) View perpen-dicular to the outcropplane, illustrating the2D ellipse formed bythe intersection of theellipsoid and the plane.Bild 2 A) Schräg-ansicht eines Schnittsvon Ellipsoid (Dar-stellung des Bimrock-Blocks) und Ebene(Darstellung der Auf-schlussfläche).B) Lotrechte Ansichtder Aufschlussflächezeigt die zweidimen-sionale Ellipse, die beiVerschneidung vonEllipsoid und Ebeneentsteht.

groundwater flow models to show how zones ofanomalously high or low permeability couldcontrol the directions and magnitudes of see-page force vectors in potentially unstable slopes,in some cases creating localized zones of instab-ility. The size, spacing, and permeability con-trast of blocks should therefore be importantconsiderations in the hydrogeologic character-ization of bimrocks.

Most previous studies of bimrock samplingbias and block distribution characterizationhave been based on the application of stereolog-ical principles to measurements of blocks fromborehole core and logs, outcrop maps of realbimrocks in the field, and scale-model bimrockscreated in the laboratory (1, 6, 7, 8, 9). Medley(1) concluded that, although sampling can insome cases reliably replicate the mean or medi-an block size, there is in general little equiva-lence between 1D chord lengths (e.g. from theintersection of blocks and borings or from out-crop scanlines) measured on maps and the true3D block size distributions. As might be expectedfrom consideration of the geometry of the prob-lem, 1D sampling consistently underestimatesthe relative proportions of the largest blocks andoverestimates the relative proportions of thesmallest blocks.

In this paper, the 2D projection or expressionof 3D blocks is referred to as the outcrop effect.One way to mathematically estimate the bias in-troduced by inferring bimrock block sizes froman outcrop face is to use an apparent radius,which is the radius of the circle formed by asphere (representing a bimrock block) intersect-ing a plane (representing an outcrop face). Themean apparent radius of a group of spheres thathave a true radius r and are randomly orientedrelative to the outcrop plane can be found bysolving the equation for a circle

x y r2 2 2+ = .................................................... [1]

for y and then evaluating the expression

rr

y dxr

r x dxr

D

r r

10

2 2

0

1 14

= = − =∫ ∫ π............ [2]

The mean apparent radius for randomly lo-cated spheres cut by a planar outcrop face istherefore approximately 3/4 r. If the underlyingblock population can be approximated as agroup of uniform spheres, then the calculation ofthe true block radius from a population of appar-ent block radii is trivial and the outcrop effect isconsistent because apparent radii observed inoutcrops will always be less than the true radii ofthe spheres.

Real bimrock blocks, however, are in generalneither spherical nor uniformly sized and naïveinferences based on an assumption of uniformlysized spherical boulders can significantly over-or under-estimate effective radii and block sizedistributions of non-spherical blocks. Because ofthese complications, there exist no direct solu-



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tions to the inverse problem of reconstructing 3Dblock size distributions from 1D borehole or 2Doutcrop measurements. If something is knownabout block shape and orientation, and perhapsthe true block size distribution, then several in-direct methods can be used to infer 3D distribu-tion statistics from borehole or outcrop mea-surements (10, 11). These methods use numeri-cal Monte Carlo simulations to generate popula-tions of 3D blocks, from which apparent blocksize distributions are calculated and comparedto the distributions of block sizes observed inoutcrops or thin sections. There does not appearto exist, however, a simple but universal empiri-cal conversion factor that will allow the practic-ing engineer or geologist to reliably infer 3Dblock size distributions from 1D or 2D samples ofnon-spherical blocks. Indeed: as shown below, amajor conclusion of this paper is that the practi-tioner should be wary of using any such conver-sion factors without understanding the underly-ing assumptions.

Method

The results described in this paper were ob-tained using Monte Carlo simulations of ellip-soids (representing bimrock blocks or, as theyare referred to in this paper, blocks) cut byplanes (representing outcrop faces) as illustrat-ed in Figure 2. For each simulation, 100 random-ly oriented ellipsoids were generated. Then, theellipse formed by the intersection of each ellip-soid with a planar outcrop face was determined,apparent equivalent radii were calculated in or-der to represent each block size with a singlescalar, and summary statistics such as the meanapparent block size and the apparent block vol-ume were calculated. In contrast to the apparentblock size used in this paper, Medley (8) andMedley and Lindquist (12) used the maximumobserved dimension to characterize the sizes ofbimrock blocks. Haneberg (13) gives details ofthe derivations and Monte Carlo simulations thatare summarized below, and an analogous exer-cise can be performed to evaluate the apparentblock size distributions produced by the inter-section of linear boreholes with ellipsoidalblocks (W.C. Haneberg, 2004, unpublished data).

The general expression for an ellipsoid witharbitrary orientation and semi-major axes a, b,and c is (14):

X U R V R X U−( ) ⋅ ⋅ ⋅ ⋅ −( ) =T 1 ........................ [3]

in which X = {x, y, z} is a vector containing thethree Cartesian coordinate directions, U = {∆x,∆y, ∆z} is a displacement vector describing thecenter of the ellipsoid relative to the coordinatesystem origin.

The shape matrix V is a diagonal matrix con-taining the lengths of the three semi-major axesof the ellipsoid

V =

−

−

−

a

b

c

2

2

2

0 0

0 0

0 0........................................ [4]

Equation [3] can be written for the specialcase of a sphere by letting a = b = c in equa-tion [4].

The rotation matrix R contains informationabout the angular orientation of the ellipsoid rel-ative to the coordinate system and, as illustratedbelow, is the product of three component rota-tion matrices that can be defined using severaldifferent conventions. This paper will use roll,pitch, and yaw angles that have physical mean-ings and will be familiar to readers who havesailed a boat or flown an airplane. Roll (ψ) is theamount of rotation of the ellipsoid about the xaxis, pitch (φ) is that about the z axis, and yaw (θ)is that about the y axis. R can be written as thedot product of roll, pitch, and yaw matrices

R R R R= ⋅ ⋅roll pitch yaw ..................................... [5]

in which

Rroll =−

1 0 0

0

0

cos sin

sin cos

ψ ψψ ψ

.............................. [6]

Rpitch = −

cos sin

sin cos

φ φφ φ

0

0

0 0 1............................. [7]

R yaw =−

cos sin

sin cos

θ θ

θ θ

0

0 1 0

0.............................. [8]

The outcrop effect can be simulated by settingeither x, y, or z = 0 in equation [3], which chang-es the equation from one describing an ellipsoidto one describing an ellipse lying in the plane ofthe other two coordinate axes. In the examplesthat follow, the outcrop effect was accomplishedby setting z = 0 to obtain the equation for an el-lipse in the x-y plane representing an outcropface. Then, the offset ∆z was varied randomly tosimulate the effect of blocks located different dis-tances from the outcrop plane. The equation foran inclined ellipse that is produced by setting, forexample, z = 0 has the form

C x C xy C y12

2 32 1+ + = ................................. [9]

in which C1, C2, and C3 are real coefficients. Atthis point, equation [9] can be plotted for eachellipse to produce a graphical representation ofthe apparent block size distribution arising as aconsequence of the outcrop effect. Calculation ofthe size of each ellipse requires additional workto determine its semi-axes, from which its areaand apparent block size can be calculated. Thetwo semi-axes of the ellipse are given by

ˆˆa

b

= 1

λ..................................................... [10]



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in which λ is a vector containing the two eigen-values of the matrix

C C

C C1 2

2 3

2

2

/

/

............................................. [11]

It is convenient to represent the volume of theblocks using a scalar that has the same units forboth the original ellipsoids and the ellipsesformed on the outcrop plane. This can be accom-plished using an equivalent radius, which is theradius of a circle having the same area as theellipse or the radius of a sphere having the samevolume as the ellipsoid. The equivalent radius ofan ellipse in the outcrop plane, which is also bydefinition an apparent radius, is

r a bD2 = ˆ ˆ .................................................... [12]

whereas the equivalent radius of the original el-lipsoid is given by

r a b cD33= .................................................. [13]

The equivalent radius of an ellipse formed bythe intersection of an ellipsoid with a plane is anapparent equivalent radius because its value willdepend on the orientation and position of the el-lipsoid relative to the outcrop plane. The equiva-lent radius of the ellipsoid, however, is not appar-ent because it is calculated using the actual valuesof a, b, and c. Once equivalent radii have beencalculated for all of the ellipsoids and ellipses in a

Table 1 Geometric variables for the four Monte Carlo simulations.Tabelle 1 Geometrische Variablen für die vier Monte Carlo Simulationen.

Case Axial ratio Equivalent radius Outcrop offset Roll Pitch Yawa : b : c size distribution ψ φ θ

I 3 : 2 : 1 constant 0 ≤ ∆z ≤ 1 0° 0° 0°II 3 : 2 : 1 constant 0 ≤ ∆z ≤ 1 ± 10° ± 10° ± 10°III 3 : 2 : 1 constant 0 ≤ ∆z ≤ 1 ± 45° ± 45° ± 45°IV 3 : 2 : 1 Pareto 0 ≤ ∆z ≤ 1 ± 45° ± 45° ± 45°

simulation, the mean block size, block sorting,and other standard size distribution measurescan be calculated and compared to evaluate thedegree of bias introduced by the outcrop effect.

To illustrate the calculation of equivalent ra-dii, consider an ellipsoid with semi-axes a = 5,b = 3, and c = 1 that is rotated 20° about the z-axis and intersects the x-y plane at z = 0. Theellipse formed by the intersection of the ellipsoidand the outcrop plane is

1 0 0483 0 0457 0 1032 2= − +. . .x xy y ............ [14]

which has the coefficient matrix eigenvaluesλ ={0.111, 0.040} and, from equation [10], semi-axes â = 5 and b̂ = 3. Note that in this case â =aand b̂ = b because there was no rotation aroundthe x or y axes. In general, this will not be true.The apparent equivalent radius in the plane ofthe outcrop is, using equation [12], r2D = (5 · 3)0.5 =3.87, which is larger than the ellipsoid equivalentradius of r3D = (5 · 3 · 1)0.33 = 2.47 obtained from


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equation [13]. In this case, therefore, the outcropeffect would lead one to underestimate the larg-est dimension of the ellipsoid but overestimatethe equivalent block radius of the ellipsoid.

Results

The results of four test cases illustrate the degreeof bias that can be introduced by the outcrop effect(Table 1). Block orientation was allowed to varybut block size was held constant in the first threecases. In the fourth case, block sizes were speci-fied using a fractal-like Pareto distribution of equi-valent radii to simulate the apparent “well-grad-ed” block size distribution that might arise fromthe fragmentation and shearing leading to the for-mation of a melange, fault rock, or perhaps land-slide debris. Ellipsoids with semi-axis ratios of3 : 2 : 1 were used in all of the simulations describ-ed in this paper, with the long and intermediatesemi-axes of the ellipsoids originally parallel tothe x-y outcrop plane. This corresponds to a bim-rock with bullet- or torpedo-shaped ellipsoidalblocks. The outcrop offset for each ellipsoid waschosen at random from a uniform distributionwith 0 ≤ ∆z ≤ 1. Haneberg (13) describes the re-sults of a similar set of simulations using ellipsoidswith 5 : 3 : 1 axial ratios.

Fig. 3 Apparentcumulative size distri-

bution curves for 2Dellipses produced bythe outcrop effect for

case I, II, and III MonteCarlo simulations of

3D ellipsoids with uni-form size located at

random distances fromthe outcrop face. Redline: maximum possi-ble apparent equiva-

lent radius of an ellipseformed by the inter-section of a 3 : 2 : 1

ellipsoid with the out-crop plane. Blue line:

True equivalent radiusof a 3 : 2 : 1 ellipsoid.

Green line: Mean appa-rent equivalent radius

of ellipses in each sim-ulation. Variables as

described in Table 1.Bild 3 Summen-größenverteilungs-

kurven für zwei-dimensionale Ellipsen,entstanden durch den

Aufschlusseffekt fürdie Fälle I, II und III der

Monte Carlo Simu-lationen von drei-

dimensionalen Ellipso-iden gleicher Größe

mit underschiedlicherEntfernung von der

Aufschlussfläche. RoteLinie: größtmöglicheräquivalenter Radius,

der bei der Verschnei-dung eines 3 : 2 : 1

Ellipsoids mit der Auf-schlussfläche entsteht.

Grüne Linie: Mittlereräquivalenter Radiusfür Ellipsen in jeder

der Simulationen. Vari-ablen siehe Tabelle 1.

The ellipsoids generated in Case I were offsetfrom the outcrop face but not rotated (ψ = φ = θ= 0). As shown in Figure 3, the apparent cumula-tive block size distribution curve inferred fromthe ellipses in the outcrop plane overestimatesthe mean equivalent radius by approximately5 % and overestimates the total block volume by45 %, which is consistent with the concave-up-ward cumulative distribution curve that asymp-totically approaches the maximum possibleequivalent block size of (3 · 2)0.5 = 2.45 lengthunits (red lines in Figure 3). The blue lines in Fig-ure 3 show the equivalent radius of the identical-ly sized 3D ellipsoids, which is (3 · 2 · 1)0.33 = 1.82.

It is important to note that in the cumulativeplots for cases I, II, and III there is no correspond-ing cumulative distribution for the 3D ellipsoidsbecause the ellipsoids are all the same size; the ap-parent size distribution curve for the blocks issolely a result of the outcrop effect. The over-pre-diction of mean block size and total block volume isin part an artifact of the initial ellipsoid orientationrelative to the outcrop plane. Had the intermediateand short axes been placed parallel to the outcrop,the ellipses in the outcrop plane likely would haveunderestimated the mean block size and volume.

If the roll, pitch, and yaw angles are all al-lowed to vary uniformly over a range of ± 10° asin case II, however, the cumulative distributioncurve approaches a straight line up to the theo-retical limit of 2.45 (see Figure 3). The outcropeffect leads to an under-estimation of the meanequivalent radius by 35 % and under-estimationof the total block volume by 44 % in this simula-tion. As above, different results would have beenobtained had the initial orientation of the ellip-soids or their axial ratios been different.

Rotation of the ellipsoids with roll, pitch, andyaw angles drawn from uniform distributionsranging over ± 45° (case III) leads to a similar un-der-estimation of 54 % for the mean equivalentradius and 76 % for the total block volume. Thethree apparent block size distribution curves inFigure 3 show that the mean block size and totalblock volume errors increase as block orientationbecomes more irregular even if the underlying3D block size does not change.

Results from cases I, II, and III show the com-plexity that can arise due solely to the orientationand position of a population of identically sizedellipsoidal blocks relative to an outcrop face. Theblock size distribution of bimrocks such as melangeand fault rocks, though, can be simulated usingPareto distributions that are closely related to frac-tal size distributions (15). Medley and Lindquist(12), in particular, discussed the engineering impli-cations of fractal bimrock block size distributions.

The probability density function (PDF) of a Pa-reto distribution as implemented in the softwareused to perform the calculations in this paper is,using an approach described by Haneberg (13):

PDF r k r r kD D D3 3 3

1( ) = ≥− +( )

α α α............... [15]



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in which r3D is the equivalent radius of an ellip-soid being simulated, k > 0 is the minimum possi-ble radius, and α is a shape parameter that isequivalent to the fractal dimension of the distri-bution. This formulation is slightly different thanthe Pareto distribution given in some references,which assume a minimum value of zero and shiftthe abscissa by an increment k. The shape of thePDF curve and the geometric significance of α,however, are the same in both versions.

Published estimates of fractal dimensions forartificially and naturally produced rock fragmentsrange from about 1.9 to 3.5, with a mean value inthe range of 2.5 to 2.6 (15). Medley and Lindquist12) estimated a fractal dimension of 2.3 for Fran-ciscan Complex melanges in northern California.In this study, a value of α = 2.5 was chosen as atypical fractal dimension for the case IV simula-tion. In order to generate a Pareto distribution withan average block size similar to those in cases I, II,and III, the mean equivalent radius of the Pareto-distributed blocks was set equal to the equivalentradii of the blocks used in the first three simula-tions –r3D = (3 · 2 · 1)0.33 = 1.82. The analytical ex-pression for the mean of the Pareto distribution is

rk

D3 1=

−α

α.................................................... [16]

Assuming a value of α = 2.5, setting equation[16] equal to 1.82, and solving for k yields thevalue of k = 1.12, which was used in the case IVsimulation.

The Case IV simulation results show theamount of complexity that is added to outcroppatterns when real block sizes are of random di-mensions as well as the blocks having randomorientation and position relative to the outcropface (Figure 4). This situation is more typical ofthe disorder common in many bimrocks than arecases I, II, and III. In this simulation, the meanapparent equivalent radius was underestimatedby 54 % and the total block volume was underes-timated by 89 %. Comparison of the ellipsoid andellipse cumulative distribution curves also showsthat 21 % of the Pareto-distributed ellipsoids arelikely to have equivalent radii greater than thelargest value calculated from the 100 ellipsesseen in the simulated outcrop. The probabilitythat the largest ellipsoid in the underlying Paretodistribution has an equivalent radius more thantwice as great as the largest block inferred fromthe ellipses in the outcrop plane is approximately8 %. This result has significant economic andconstructability implications for earthwork, ex-cavation, or tunneling projects in which the sizeof the largest block expected to be encounteredduring excavation is inferred solely from uncor-rected outcrop or borehole data.

Conclusions

The inference of 3D bimrock block size distribu-tions from 2D outcrop maps and, although not

Fig. 4 Apparentcumulative size distri-bution curves for 2Dellipses produced bythe outcrop effect forcase IV Monte Carlosimulation of 3D ellip-soids with Pareto-distributed equivalentradii located at ran-dom distances fromthe outcrop face. Vari-ables as described inTable 1.Bild 4 Summen-größenverteilungs-kurven für zweidimen-sionale Ellipsen, ent-standen durch denAufschlusseffekt fürFall IV der Monte CarloSimulationen von drei-dimensionalen Ellipso-iden mit nach Pareto-verteilten äquivalentenRadien mit unter-schiedlicher Entfer-nung von der Auf-schlussfläche (Varia-blen siehe Tabelle 1).

described here, 1D borehole logs without supple-mentary 3D rock mass fabric data, has the poten-tial to produce significant errors in estimatedmean block sizes, maximum block sizes, and totalblock volumes. Results of the simple Monte Carlosimulations described in this paper suggest thatinferred mean block sizes may be in error by asmuch as 50 % and block volumes by as much as90 %. Perhaps most importantly for design andconstruction planning, outcrop or boreholesampling will almost inevitably underestimate thelargest block likely to be encountered during aproject. The sizes and volumes of spherical blockswill in all cases be under-estimated by outcropsampling. The sizes and volumes of mildly elon-gated ellipsoidal blocks will tend to follow the


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same pattern of under-estimation, although ex-ceptions are possible if the two longest axes ofthe ellipsoids are nearly parallel to the outcropface. Perhaps most importantly, simple empiricalmeasures relating factors such as apparent 1D or2D block dimensions with actual 3D dimensionsare likely to be of very limited, if any, utility tothe practicing engineer or engineering geologist.These factors will depend very heavily on theblock geometry and orientation, and will thereforediffer from project to project.

The orientation of blocks relative to the out-crop face and block shape determine whether2D sampling underestimates or overestimates 3Dparameters. Therefore, an understanding of the3D fabric of bimrocks should be considered anessential supplement to 2D outcrop or 1D bore-hole characterization projects. Knowledge of thelocal structural geology and regional tectonic set-ting of a project area, for example, can be used toestimate block orientation variability by applyingspherical statistics to equal-area projections ofrock fabric data. Sampling of alluvium derivedfrom bimrocks may provide important constraintson block dimensions in cases where 3D exposuresdo not exist. In situations where the bimrockmatrix is friable or otherwise unlithified, excava-tion and careful measurement of complete blocksshould be considered an essential aspect of sitecharacterization. Once this kind of information isobtained, practicing engineers and geologists canuse indirect comparison methods such as thosedescribed by Sahagian and Proussevitch (10) toback-calculate true block size distributions.

References1. Medley, E.W.: Estimating block size distributions of me-langes and similar block-in-matrix rocks (bimrocks). In: Ham-mah, R. ; Bawden, W. ; Curran, J. ; Telesnicki, M. (eds.):Proceedings of the 5th North American Rock Mechanics Sym-posium (NARMS), pp. 599-606, 2002.2. Lindquist, E.S.: The mechanical properties of a physicalmodel melange. In: Olivera, R. ; Rodrigues, L.F. ; Coelho,A.G. ; Cunha, A.P. (eds.): Proceedings, 7th International Con-gress, International Association of Engineering Geology: Lis-bon, Portugal, pp. 819-826, 1994.3. Lindquist, E.S. ; Goodman, R.E.: Strength and deformationproperties of a physical model melange. In: Nelson, P ; Lau-bach, S.E. (eds.): Proceedings of the 1st North American RockMechanics Symposium, Austin, Texas, USA, pp. 843-850,1994.

4. Goodman, R.E. ; Ahlgren, C.S.: Evaluating safety of con-crete gravity dam on weak rock: Scott Dam. J. Geotechnicaland Geoenvironmental Eng., Vol. 126 (2000), pp. 429-442.5. Medley, E.: Uncertainty in estimates of block volumetricproportions in melange bimrocks. In: Marinos, P.G. ; Koukis,G.C. ; Tsiambaos, G.C. ; Stournaras, G.C. (eds.): Proc. Inter-national Congress, International Association of EngineeringGeologists, Engineering Geology and the Environment, Ath-ens, Greece, 1997.6. Medley, E. ; Goodman, R.E.: Estimating the block volu-metric proportions of melanges and similar block-in-matrixrocks (bimrocks). In: Nelson, P.P. ; Laubach, S.E. (eds.): Pro-ceedings of the 1st North American Rock Mechanics Sympo-sium, Austin, Texas, USA, pp. 851-858, 1994.7. Haneberg, W.C.: Groundwater flow and the stability ofheterogeneous infinite slopes underlain by impervious sub-strata. In: Haneberg, W.C. ; Anderson, S.A., (eds.): Clay andShale Slope Instability. Geological Society of America Re-views in Engineering Geology 10, pp. 63-78, 1995.8. Medley, E.W.: Engineering characterization of melangesand similar block-in-matrix rocks (bimrocks). Ph.D. disserta-tion, Department of Civil Engineering, University of Californiaat Berkeley, 1994.9. Medley, E.W.: Using stereological methods to estimatethe volumetric proportions of blocks in melanges and similarblock-in-matrix rocks (bimrocks). In: Olivera, R.; Rodrigues,L.F. ; Coelho, A.G. ; Cunha, A.P. (eds.): Proceedings 7th In-ternational Congress, International Association of Engineer-ing Geology: Lisbon, Portugal, pp. 1031-1040, 1994.10. Sahagian, D.L. ; Proussevitch, A.A.: 3D particle size dis-tributions from 2D observations; stereology for natural appli-cations. Journal of Volcanology and Geothermal Research,Vol. 84 (1998), pp. 173-196.11. Higgins, M.D.: Measurement of crystal size distributions.American Mineralogist, Vol. 85 (2000), pp. 1105-1116.12. Medley, E. ; Lindquist, E.S.: The engineering significanceof the scale-independence of some Franciscan melanges inCalifornia, USA. In: Daemen, J.J.K. ; Schultz, R.A. (eds.):Rock Mechanics, Proceedings of the 35th U.S. Symposium,Reno, Nevada, pp. 907-914, 1995.13. Haneberg, W.C.: Computational Geosciences with Mathe-matica. Berlin, Heidelberg, New York: Springer-Verlag, 2004.14. Herbison-Evans, D.: Animated Cartoons by ComputerUsing Ellipsoids. University of Sydney, Basser Department ofComputer Science, Technical Report 94, (online version)http://linus.it.uts.edu.au/~don/pubs/cartoon.html (2002).15. Turcotte, D.L.: Fractals and Chaos in Geology and Geo-physics. 2nd ed. Cambridge: University Press, 1997.

AcknowledgementsMy interest in bimrock sampling and simulation was motivatedby many conversations with Ed Medley, whose contributionsand references are greatly appreciated. Wolfram Research,Inc. donated a copy of the Mathematica software used in thisresearch. Thanks also to Isabelle Pawlik, PE, of Jacobs Asso-ciates, San Francisco for the German translations.

AuthorWilliam C. Haneberg, Haneberg Geoscience, 10208 39th Ave-nue SW, Seattle WA 98146 USA, E-Mail bill@haneberg. com


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