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The contribution of geostatistics to thecharacterisation of some bimrock properties

ARTICLE in ENGINEERING GEOLOGY · SEPTEMBER 2012

Impact Factor: 1.74 · DOI: 10.1016/j.enggeo.2012.03.015

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Niccolò Coli

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Paolo Berry

University of Bologna


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Daniela Boldini

University of Bologna


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Roberto Bruno

Ministero degli Affari Esteri e della Coopera…


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The contribution of geostatistics to the characterisation of some bimrock properties

N. Coli, P. Berry, D. Boldini ⁎, R. Bruno

Department of Civil, Environmental and Material Engineering (DICAM), University of Bologna, Via Terracini 28, 40131, Bologna, Italy

a b s t r a c ta r t i c l e i n f o

Article history:

Received 15 October 2010

Received in revised form 28 December 2011

Accepted 24 March 2012

Available online 1 April 2012

Keywords:

Bimrock

Image analysisGeostatistical analysis

Variogram

Correlation

Santa Barbara mine

This paper introduces a geostatistical approach for characterising the morphological and spatial variability of rock inclusions in bimrocks and for attempting their direct mechanical characterization through an indirect

description deduced by image analysis of outcrop images.

The study was carried out on the Shale–

Limestone Chaotic Complex (SLCC) bimrock, which is affected byseveral slope stability problems in the disused Santa Barbara open-pit mine (Tuscany, Italy). Due to itscomplex heterogeneity, the mechanical characterization of SLCC was based on a limited number of non-

conventional shear tests (BimTests).Due to the relevant inuence of rock inclusions on bimrock strength, the study focused on identifying the

possible image parameters characterising content and variability of rock inclusions in each sampled picture.A two-step analysis was carried out for: a) identication of an image parameter (centimetric scale) that isable to grasp some property of rock fragment shape and size spatial distribution; b) veri cation of the

correlation at the outcrop scale (multimetric scale) between the bimrock strength parameters and the new

regionalised variables (ReV) describing image properties related to rock fragment distribution. Ageostatistical approach was adopted for both steps, but applied to two different problems, at two differentscales, with different variables.

An extensive sampling of outcrop pictures was carried out and a variogram analysis was performed on the

“rock fragment indicator variable”, I B ( x, y), dened at the image scale, and obtained through the digital

segmentation of the collected pictures. A very good link was observed between the geostatistical modelparameters and the content, size and spatial variability of rock fragments shown in every picture.

The values of the total sill of the block indicator variograms were then assumed as a new ReV, C tot ( x, y),dened all over the SLLC outcrop. A cross-covariance was computed between the variogram total sill and thefriction angle, φ ( x, y) and cohesion c ( x, y), at the scale of the whole outcrop area and a preliminary analysis

showed a likely correlation.© 2012 Elsevier B.V. All rights reserved.

1. Introduction

Bimrocks (Medley, 1994; Medley and Lindquist, 1995; Medley,2002, 2007) are complex formations characterised by competent rockinclusions oating in a weaker matrix. Due to their complex structure,they always represent a challenging engineering problem: the

mechanical behaviour of bimrocks, in fact, is governed by theinteraction and the geometrical properties of rock inclusions(Lindquist, 1994; Lindquist and Goodman, 1994; Goodman andAhlgren, 2000; Kim et al., 2004; Sonmez et al., 2004a, 2006a). For

this reason, the mechanical characterisation of bimrocks must bepreceded by the investigation of the morphological and spatialvariability of rock fragments.

Several methods can be employed for this purpose, from theinterpretation of scanlines and boreholes to the analysis of digital

images and physical sieving.1D method, based on the interpretation of scanlines and

boreholes, although being one of the most used techniques, allowsinvestigating mainly the volumetric percentage of rock inclusions, by

assuming it to be stereologically equivalent to its cumulative linearproportion. As a consequence, this approach is heavily affected by theamount of available sampling data (Haneberg, 2004).

3D sieving can be easily performed at the laboratory scale, but for

larger volumes it becomes cumbersome and very expensive. More-over, this technique only investigates the volumetric content and thegrain-size distribution of rock inclusions, but not their morphological

and spatial variability.On the other hand, the use of 2D digital image processing for

characterising rock masses and ornamental stones is well known toprovide satisfactory results. This technique, in fact, ensures a

relatively fast, non-destructive and inexpensive way for studying avariety of morphological properties, which can be described and

Engineering Geology 137–138 (2012) 53–63

⁎ Corresponding author. Tel.: +39 0512090233; fax: +39 051 2090247.

E-mail address: [email protected] (D. Boldini).

0013-7952/$ – see front matter © 2012 Elsevier B.V. All rights reserved.

doi:10.1016/j.enggeo.2012.03.015

Contents lists available at SciVerse ScienceDirect

Engineering Geology

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / e n g g e o

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quantied through synthetic indexes. These indexes, when correlatedwith the results of laboratory or in-situ mechanical tests, can be veryhelpful for an indirect and quick estimation of specic rock massmechanical properties (Hagerman et al., 1980; Dougherty and Pelz,

1991; Horgan, 1998; Persson, 1998; Coster and Chermant, 2001; Yueet al., 2003; Chen et al., 2004; Lebourg et al., 2004; Bruno andFocaccia, 2009).

The present study was carried out on the Shale–Limestone Chaotic

Complex (SLCC) bimrock, outcropping over a wide slope in thedisused Santa Barbara (SB) open-pit mine (Tuscany, Italy). Theinvestigated domain is a surface S encompassing the SLCC outcrop

exposure at the SB mine, for a total extension of 345,000 m2.The mechanical characterization of SLCC was previously carried

out by a few in situ shear tests (BimTests).Subsequently, a photographic “sampling” of 2 m×2 m outcrop

pictures was carried out with the aim of extracting one or moresynthetic indexes to be correlated with strength experimentalvalues. Such potential correlation is possible only by a geostatisticalspatial cross-covariance analysis, due to the fact that the variables

are not referred to the same location on the outcrop, i.e. are notisotopic.

Through image analysis it is possible to investigate severalproperties of rock inclusions, for instance: statistical distribution of

the maximum observable dimensions (dmod) (Medley, 1994), statis-

tical shape distribution, and spatial variability of useful parameters.The present research rst introduces a geostatistical approach for

the characterization of the 2D spatial distribution of SLCC rock

fragments at the scale of the 4 m2 image, by studying the regionalized

variable “indicator” of the presence/absence of fragments, I B ( x, y),derived by digital processing of outcrop pictures. The 2D spatialvariability of rock inclusions was characterised and modelled, and avalue of a specic synthetic parameter deduced by the geostatistical

model, C tot, was assigned to each picture.

In the second geostatistical study the synthetic parameter wasconsidered a ReV over the SLCC eld, C tot ( x, y), and its spatialcorrelation with the SLCC strength parameters was studied.

In the following, a rst attempt to quantify and validate the

possible correlations between the scarce direct information from thein situ shear test data, and the large indirect information from thesynthetic image parameters is also presented.

2. The SLCC and the Santa Barbara open-pit mine

The SLCC is a mélange of the Northern Apennine orogenic chain.Its genesis is commonly associated with a gravitational accumulationof material derived from dismembered formations, and originatedfrom huge submarine mass movements (olistostromes) that took

place during the Oligo-Miocene era (Castellucci and Cornaggia, 1980;Abbate et al., 1981; Boccaletti et al., 1996). Recent advances in thegeological comprehension of the Northern Apennines mountainchain, however, interpret the SLCC as a tectonic mélange (i.e.

tectonosome), that originated from the in-place dismemberment of pre-existing formations due to the overthrusting of tectonic nappesduring the Apennine orogenetic phases (Boccaletti et al., 1996;

Bonini, 1999).The SLCC is characterised by a mixture of a highly tectonised dark

grey clayey matrix with calcareous fragment inclusions of varioussizes. The clayey matrix, a typically structured over-consolidated clay,

is subdivided into centimetre and millimetre thin lens-like laminaealong pervasive and polished ssility planes (Pini, 1999). Thecalcarenite and calcilutite fragments are distributed throughout the

clayey matrix without any apparent preferred orientation in a typicalblock-in-matrix fabric (Figure 1).

The exhausted SB mine, owned and operated by ENEL S.p.A until1994, is located in Tuscany, Italy, on the Western side of the Valdarno

Superiore Basin, approximately 30 air km South-East of the city of

Firenze. The mine, which encompasses an area of 15 km2, is currentlyundergoing a full reclamation programme. The mining of the lignite

produced a wide slope in the SLCC, with a maximum height of about180 m. During mining, the slope underwent several mass movementsand rotational landslides, which are in part still active today ( D'Eliaet al., 1984, 1986, 1988; Esu and D'Elia, 1990; D'Elia, 1993; D'Elia

et al., 1993).

3. The bimrocks

The term “bimrock” was introduced by Medley (1994) in order to

describe, from a geomechanical point of view, structurally complexformations made up of a mixture of competent and multi-size rock

fragments embedded in a weaker matrix. The term bimrock has nogenetic signicance and it refers to a wide variety of geologicalformations like mélanges, fault gouges, debris and glacial tills

(Medley, 2002; Medley and Wakabayashi, 2004).

3.1. Block/matrix threshold

The possible inuence of a rock fragment of size r on the overallmechanical behaviour of a bimrock is strictly controlled by the scaleof the engineering problem under investigation. Given a scale of

interest, some rock fragments, in fact, can be so small compared to thescale of the investigated problem that they have no practical

inuence on the strength of the bimrock, and they can be consideredas part of the matrix itself.

The threshold below which rock fragments are assumed to havenegligible inuence on the overall bimrock strength is dened as the“block/matrix threshold” (B/M) (Medley, 1994, 2001, 2002).

Medley (1994, 2002) and Medley and Lindquist (1995) proposed

to estimate the B/M as 5% of Lc, where Lc is a linear dimensioncorresponding to a characteristic dimension of the engineering

problem under investigation. Lc can be the average thickness of asliding body (Medley and Sanz Rehermann, 2004), the extension of adam foundation (Goodman and Ahlgren, 2000), the dimension of anin-situ test specimen (Xu et al., 2007) or a fraction of a tunnel

diameter (Button et al., 2004).It is important to keep in mind that in the present paper the term

rock inclusions and rock fragments are used to indicate all the

Fig. 1. Typical aspect of the SLCC bimrock. The light-grey calcareous blocks areembedded in the clayey matrix according to a matrix-supported fabric.

54 N. Coli et al. / Engineering Geology 137 –138 (2012) 53–63

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calcareous fragments regardless of their size, while blocks indicatesonly the rock fragments above the B/M.

At the scale of the 2 m×2 m image window adopted in the rstpart of the present study. The rule of 5% Lc yields a B/M of 0.1 m (5% of

2 m edge size), while it yields a B/M of 0.01 m at the scale of theBimTests performed in the SB mine, and briey described later inSection 6.1 (Coli et al., 2011).

In the present study, however, the geostatistical analysis at the

scale of images was performed on the whole population of rockfragments visible on the sampling picture and, therefore, the resultsare not affected by the choice of different B/Ms.

3.2. Properties of rock inclusions

Several studies on Franciscan complex mélanges (California)indicated that the size distribution of the maximum observabledimension of rock fragments (dmod) follows a negative power lawgiven by Eq. (1) (Medley, 1994; Medley and Lindquist, 1995;

Riedmuller et al., 2001; Medley, 2002):

N ¼ r −D

ð1Þ

where N is the relative frequency of fragments with a lineardimension greater than r and the exponent D is the fractal dimension(Mandelbrot, 1983; Turcotte, 1997). The latter is equal to 1.2 for the

Franciscan Mélange (Medley, 1994).One of the most important properties of a power law distribution

is self similarity, i.e. the invariance of fractal dimension D regardlessof the scale of observation.

However, it is important to remark that for the size distribution of natural objects there are lower and upper limits to the validity of scale independency (Turcotte, 1986, 1997). In recent studies, in fact,some authors indicate that a power law distribution may not be valid

across the entire range of particle sizes (Filgueira et al., 2006), and

that, within the same grain size distribution, multiple scale domainsmay exist with different fractal dimensions (Millan et al., 2003).

The dmod size distribution for the SLCC rock inclusions, at the 4 m2

scale of the sampling window, also resulted in following a negativepower-law distribution with an average fractal dimension D of 1.55

(Figure 2) (Coli et al., 2008, 2009).Studies carried out on outcrop pictures of the same SLCC

formation, but in different locations in the Northern Apennines,indicated that, at the scale range of 2–9 m, the spatial arrangement of

rock fragments was well described by a positional dependant power-law, based on the cantor and the Sierpinsky fractal models, with afractal dimension DH varying between 1.737 and 1.745 (Catani andFanti, 1996; Catani, 1999).

4. The use of image analysis for bimrock characterisation

The 2D image analysis of bimrock inclusions, although notaccurate, is widely used in the geoengineering practice.

Medley (1994) and Medley and Lindquist (1995), in order tocalculate the dmod size distribution of rock fragments, used simplemethods of manual contouring on a variety of two-dimensional

sources (images of bimrock outcrop exposures, geological maps).

Sonmez et al. (2004a,b, 2006a,b), in order to investigate thevolumetric proportions of andesite inclusions in the Ankara Agglomerateperformed image classications and node counting on greyscale and

colour pictures of outcrop exposures. They used a supervised imageclassication procedure consisting in a training stage, where thepixel intensity rangeof each lithotype wasdetermined,followedby aclassication stage where the pixels were categorized into the

constituents. They also used the node-point-counting technique,where a regular square mesh was overlaid on the pictures and theunderlying material at each node was then visually classiedaccording to its lithotype. The percentage of each constituent was

then determined by dividing the number of nodes belonging to eachconstituent by the total number of intersections of the mesh. Thevolumetric proportion obtained from the two methods resulted invery similar estimations.

Yue et al. (2004) developed a specic digital image processingtechnique for the analysis of material inhomogeneities aiming atmodelling inhomogeneous rock failures through numerical codes.These methods estimate the structures of geomaterials by applying a

multi-threshold algorithm based on edge detection lters. The sameprocedure, through contrast enhancement, edge detection andthreshold ltering, was used by Xu et al. (2007) in order to isolatethe blocks above the B/M on cross-section pictures of a soil–rock

mixture in the Leaping Tiger Gorge (China).Catani and co-workers performed the fractal analysis of the rock

fragment indicator variable from binary images of SLCC outcropsthrough a box-counting procedure implemented in the GIS ArcInfo©

software (Catani and Fanti, 1996; Catani, 1999).However, the use of 2D images to estimate bimrock 3D

geometrical properties of rock fragments inevitably introduces anoutcrop sampling bias (Haneberg, 2004). The shape and orientation

of the rock inclusions relative to the outcrop exposure, in fact,determine whether the analysis of the 2D image underestimates oroverestimates the real 3D geometrical parameters.

Nevertheless, this issue is not relevant to our study, which onlyaims at verifying whether or not some parameters deduced by imageanalysis show a statistically meaningful correlation with the physicaland strength rock properties as measured by the in-situ shear tests.

5. Identication of image parameters related to rock fragments

spatial variability

The dmod approach and the image analysis methods previouslydescribed are very effective in investigating equivalent size-distribution and volumetric proportion of rock fragments, but theydisregard information such as shape and spatial variability. Thespatial information is, instead, taken properly into account by the

position-dependant Cantor and Sierpinsky fractal models by assum-ing as a descriptive index the Hausdorff –Besicovitch fractal dimen-sion DH (Catani and Fanti, 1996; Catani, 1999).

In this study, a variographic analysis of the rock fragment indicator

variable I B ( x, y) was performed with the purpose of describing andquantifying the 2D morphological and spatial variability of the rockinclusions. The block indicator, I B, was obtained through the digitalsegmentation of the sampled pictures, and it expresses the presence

or absence of the calcareous lithology in the two-dimensional domain

of the image.Fig. 2. dmod size distribution of the SLCC bimrock at the 4 m2

scale of investigation.

55N. Coli et al. / Engineering Geology 137 –138 (2012) 53–63

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The stationary indicator variable I ( x, y) of a random set A isexpressed as (Matheron, 1973; Journel and Huijbregts, 1978; Chilèsand Delner, 1999):

I x; yð Þ ¼ 0 if x∈ A

1 if x∈ A :

ð2Þ

For our purpose, set A represents the calcareous lithology, i.e. thepresence of calcareous rock fragments in the investigated bimrock.

In a binary image each pixel can only assume the values of 0 (black) or 255 (white). The pixel value is a regionalized variable,which is used to dene an indicator variable, i ( x, y), assumed as therealisation of a random function I B ( x, y). The I B ( x, y) is here denedon a punctual support (i.e. the image pixels).

The working scale of this rst geostatistical analysis is the 4 m2

image area. The area is comparable with the size of the specimenssubjected to a series of non-conventional in situ shear tests (0.8 m by0.8 m by 0.5 m), which had previously been performed by the authors

at the SB mine (further description of the test is given in Section 6.1).A 2 m×2 m sampling window was considered a good compromisebetween the necessity to encompass the area representative of the insitu shearspecimen and at the same time to account for a slight larger

surface in order to avoid edge-biases and to ensure an easy shootingof the pictures from the ground.

5.1. Sampling technique

The rst step of the study was the selection of an appropriatephotographic sampling methodology of the SLCC outcrop surface S . In

general, the best sampling technique is the regular mesh sampling,but, due to the impossibility of physically reaching some areas of theinvestigated domain, the stratied sampling technique (SST) wasadopted. In fact, for a given number of samples, the SST represents the

best approximation of the regular mesh (optimal) sampling. The SSTworks by subdividing the investigated domain into several non-overlapping subsets (strata), and by random choosing the sample

location inside each stratum.The investigated domain was subdivided into 68 square cells, each

encompassing an area of 5000 m2, which represents about 1.5% of theoutcrop surface s. Inside each cell, a picture encompassing an area of 4 m2 of the SLCC outcrop was taken at a random position (Figure 3).

Only 40 of the 68 cells were sampled, because 18 cells were

discarded due to the inaccessibility of the area or excessive vegetationcover.

The pictures were shot by using a Sony DSC-800 digital reexcamera with 8 megapixel resolution, which ensures a high image

resolution and a quick transfer of pictures from camera to PC for theimage processing.

In order to account for lens and angular distortion, an aluminiumsquare frame of 2 m×2 m was assembled, placed on the ground and

centred in the camera's eld of view. Pictures were taken asperpendicularly as possible to the exposed surfaces and then digitallyrectied and cropped to the 2 m×2 m frame in order to restore theexact angular and linear proportions (Figure 4).

5.2. Digital image processing

The term digital image processing (DIP) refers to the modica-

tions that can be made to a digital image (DI) using a computer. A DIcan be expressed as a discrete two-dimensional function f ( x, y) where

x and y are spatial coordinates and the value of f at any pair of

coordinates ( x, y) is the light intensity at that point. For amonochromatic grey-scale image, the intensity values are integerswhich dene the grey level for each pixel by assuming integer values

between 0≤ f ( x, y)≤255.

Fig. 3. Simplied geological map of the investigated area at the SB mine. The map

shows the strati

ed sampling grid, the location of pictures and of the BimTests.

Fig. 4. Cropped and orthorectied picture of an SLCC outcrop exposure.


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Since the main difference between calcareous fragments and theclayey matrix is related to variations of grey levels (Figure 5), theSLCC pictures were converted to monochrome 8-bit greyscale imagesand processed through contrast enhancement lters (histogram

equalization and histogram stretching).An image segmentation procedure was then used in order to

obtain binary images where calcareous fragments (foreground) areisolated from the matrix (background).

Segmentation is performed by studying the variations of greyintensity in the input image and by applying a threshold (T ) to theimage histogram: all pixels with an intensity lower than thethreshold are set at 255 (white) while the others at 0 (black)

(Sahoo et al., 1988; Glasbey, 1993; Nikhil and Sankar, 1993; Costerand Chermant, 2001; Gonzalez and Woods, 2001):

g x; yð Þ ¼ 0 f x; yð Þ≥T ⇒ x; yð Þ∉ A255 f x; yð ÞbT ⇒ x; yð Þ∉ A

: ð3Þ

As a consequence, the binary image can be interpreted as arealisation of an indicator variable describing the presence or absenceof calcareous fragments constituting set A: I B( x, y) (Figure 6).

A supervised maximum entropy threshold algorithm was used for

the image segmentation (Kapur et al., 1985; Sahoo et al., 1988). Thealgorithm performs the histogram thresholding on the basis of

entropy considerations: the optimal threshold T is dened as the

grey level which maximizes the sum of the entropies of the two sets U and V :

T ¼ ArgMaxT ∈ 0;255½

H U ð Þ þ H V ð Þf g

¼ ArgMax −XT i¼0

piP T

ln piP T

!þ þ

X255 j¼1þT

p jP T

ln p jP T

0@

1A

8<:

9=; ð4Þ

where H (U ) and H (V ) are the entropy equations associated with eachdistribution, p is the probability distribution of grey levels, T the

threshold value and P T ¼PT i¼0

pi.

This method is very effective on randomly distributed objects overa rather uniform background.

Some artefacts introduced by segmentation were successivelyremoved by applying mathematical morphology lters (Serra, 1982)

to remove isolated pixels and lling holes.The open-source software ImageJ (NIH, 2004) was used for the

entire digital image processing.

5.3. Structural analysis of block indicator

In order to investigate the spatial variability of the I B ( x, y),experimental variograms were calculated on the binary images along

the directions of 0° and 90°, assuming lags (h) ranging between0.005 m and 1 m (Figure 7). The open-source GIS software ILWIS(©NorthPole) was used for the calculation of experimentalvariograms.

The experimental variograms were tted by nested structuremodels by means of a Visual Basic macro library of DICAM, developedfor Microsoft™ Of ce®. The nested models take into account both anomnidirectional component of spatial variability and/or a geometrical

anisotropy, when they assume different ranges along differentdirections maintaining a constant sill.

The tted structures of Fig. 8, for example, show a cleargeometrical anisotropy: the N–S component (90°) has a larger range

compared to the E–W one (0°) and, in fact, the black features of thebinary image are elongated towards the direction between 45° and

70°. In Fig. 9, instead, a substantial isotropy can be observed.

Fig. 5. Linear plot of grey intensity values (a) along the A–B scanline of the image (b).

The high intensity contrast at the edges of the block produces abrupt variations of grey

levels.

Fig. 6. Binary image representing the fragment indicator variable I B ( x, y).


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Variogram models are described by the sum of several elementarystructures, each one being identied by a different function type andby specic parameters (sill, range, period).

The I B ( x, y) variograms show a very interesting feature: everyvariogram is always characterised by three common elementary

structures, with constant range a and variable sill C (Figure 10):

1. A spherical model at a small scale, always isotropic, with a constantrange of about 8 mm;

2. A rst exponential model for medium scale variability, always

isotropic, with a range between 11 and 15 mm;3. A second exponential model for large scale variability, with a range

between 30 and 300 mm. This model sometimes shows ageometric anisotropy.

Moreover, some of the nested variograms also show a slightperiodicity, which, when relevant, was modelled with a hole effect ora periodical model.

The characteristic parameters of the elementary models for all

analysed images are listed in Table 1.

5.4. Signi cance of geostatistical indexes

The properties of the elementary models describing the variability

of I B ( x, y) can be referred to the spatial and morphological propertiesof rock fragments.

Given the initial denition of the block indicator variable, I B ( x, y),it follows that:

• the range a is correlated with the average size of fragments;

• the sill C is linked to the percentage of calcareous lithology on theimage, being the variance of the indicator: σ I

2= p(1− p);• the variogram geometrical anisotropy is correlated with the

anisotropy of the iso-oriented shape of fragments;

• the periodicity T is a function of the average local distance betweenthe fragments and, therefore, it is only evident when, at the scale of investigation, the inclusions are arranged according to a regularpattern of spatial distribution.

Due to these properties, the three elementary models of theindicator variable, I B ( x, y), can be interpreted as follows (Figure 11,

Table 1):

• the spherical model takes into account the shape and spatialvariability of the very small calcareous fragments (set 1); it

expresses more than one-quarter of the total variability (total sill);

Fig. 7. a) Example of a binary SLCC image which can be interpreted as the realisation of the indicator variable I B ( x, y) (black features: calcareous fragments, white background:

matrix); b) experimental variograms calculated along the directions of 0° and 90°.

Fig. 8. a) Binary image I B ( x, y); b) experimental and model indicator variograms withnested structures along the directions of 0° and 90°.


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• the 1st exponential model refers to the variability of smallinclusions, with an average size of about 1 cm (set 2); it takes intoaccount about one-quarter of the total variability;

• the 2nd exponential model expresses the variability of medium tolarge fragments, with an average size between 3 cm and 30 cm (set3); it represents almost one-half of the total variability.

The presence of geometrical anisotropy is only highlighted formedium–large fragments (2nd exponential model), while it is not

visible for the small ones. The isotropic behaviour of the small-scaleelementary models (spherical and 1st exponential), however, does

not necessarily imply an isotropic shape of the small elements, butrather that the resolution of the actual working scale doesn't allow forproperly taking into account the shape variability of tiny elements.

The periodical/hole model is seldom observed and it usually has a

low impact given that its sill contributes only 4% to the totalvariability. No evidences from eld surveys and geological literature(cfr Section 2) suggest that this periodicity might be somehow relatedto a recurrent systematic internal zonation of the SLCC formation. The

periodical model is more likely to be due to a local arrangement of thefragments at the scale of the 4 m2 sampling window with no lateralcontinuity. Nonetheless, further multi-scale studies are required in

order to investigate the possible presence of structures representing

internal zonation of the bimrock at the scale of the entire outcrop.

6. Spatial correlation between strength and image parameters

6.1. Strength data by in situ tests

BimTests (Xu et al., 2007; Coli et al., 2011) are non-conventional insitu shear tests performed by the authors at the SB mine. The testing

procedure (Figure 12), in fact,signi

cantly differs from the classical insitu direct shear test (ISRM, 2007), i.e. the shear surface is not forcedto develop along a pre-dened horizontal plane, but rather it is free togrow in a tortuous way along the rock-matrix contact boundaries. In

addition, no vertical load is applied to the specimen.The tested specimen has a volume of 0.3 m3 and a size of 0.8 m

wide by 0.8 m long by 0.5 m high.Five BimTests were performed at the SB mine in order to

investigate the SLCC strength properties and to correlate them withthe volumetric block content (VBC) inside each specimen. The frictionangle was found to have a good positive correlation with VBC, whilecohesion showed a slight negative correlation with a sudden drop

above a critical VBC threshold of 20–25% (Figure 13).Strength parameters also depend on the BimTest location on the

outcrop surface, that is they are ReV, c ( x, y), φ ( x, y), dened on asupport of 0.3 m3 (the volume of tested specimens).

6.2. The auxiliary variable: total sill

Given the costs and the technical dif culties in performing the insitu tests, it is of signicant interest attempting to correlate thestrength parameters with some parameters characterising each one

of the 4 m2 sampling windows.The previous geostatistical analysis identied the parameters (i.e.

a, C ) of the nested elementary structures of block indicatorvariograms, which quantitatively highlight the spatial variability of

rock fragments at the scale of 4 m2.The values of range a are constant all over the eld S , while other

parameters vary among the pictures differently located over the

Fig. 9. a) Binary image I B ( x, y); b) experimental and model indicator variograms with

nested structures along the directions of 0° and 90°.

Fig. 10. a) I B ( x, y) experimental and nested variograms and b) elementary models.


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outcrop surface. This is the case of the sills, which can be consideredas ReVs, C tot ( x, y). Moreover, since the total sill, i.e. the sum of sills of

elementary structures, is a function of the 2D fragment content, C tot( x, y) = p(1− p), a rst attempt was carried out to verify if theparameter C tot is correlated with the strength parameters.

For this reason, a new geostatistical study at the outcrop scale was

performed by calculating the spatial variability of a synthetic

parameter dened on each image: the total sill C tot ( x, y). Theworking scale of this second step of the geostatistical study is thewhole investigated domain of the outcrop surface S , and the support

of the new variable C tot ( x, y) is a surface of 4 m2.

A correlation study usually takes into account variables measuredon the same samples. This would have required pictures to be takenon the same spots where the BimTest had been performed.

Nevertheless, at the time the SLCC pictures were collected, theBimTests were already done and, therefore, pictures were only takenon nearby SLCC outcrops. As a consequence, the samples of the twovariables, strength data and image total sill, are non-isotopic

(samples are not taken at the same locations) and, therefore, anyattempt to study their possible correlation cannot be performed usingthe standard statistical and probabilistic techniques (i.e. covariance,

index of correlation, standard regression).

The spatial cross-covariance (C ⁎12), instead, is a geostatistical toolthat, in principle, makes it possible to study the correlation of twonon-isotopic variables. If we consider two ReVs, z 1( x) and z 2( x),as the

realisation of two StRF, Z 1( x) and Z 2( x), their spatial cross-covariance,

Table 1

Parameters of the elementary models.

Picture Spherical Exp I Exp II Periodical/

hole

C a

(mm)C a

(mm)C amax

(mm)Dir

anis.

(°)

amin(mm)

C T

(mm)

1 3000 8 1800 11 4200 142 88 125

2 1500 8 1850 11 700 40 30 100

3 3000 8 3800 11 3350 75 65 40

4 1500 8 2500 18 3900 90 90 80

5 4000 8 2400 11 3400 70 70 50

6 1000 8 2100 28 1200 35 24 1207 1500 8 2700 13 4800 180 70 100

8 3000 8 2800 11 5500 65 80 50

9 3000 8 1800 11 3700 30 85 25

10 2400 8 1600 11 1600 130 85 80

11 3000 8 4000 11 3300 100 85 70

12 300 8 1200 11 8600 70 80 50 2500 770

13 2600 8 3000 11 3400 160 80 120

14 1500 8 2400 11 3000 120 80 80

15 2500 8 1000 11 2500 90 85 75

16 2400 8 1100 11 3000 90 80 75

17 1900 8 1400 11 3000 110 80 70

18 2400 8 1600 11 1600 130 85 80

19 1400 8 2500 11 2900 80 85 55

20 2000 8 2500 11 4100 110 85 8021 2000 8 2200 11 4900 65 85 5522 2200 8 1200 11 4400 80 85 60

23 3000 8 4000 11 3300 100 85 70

24 2200 8 2000 11 2200 85 5 70

25 2200 8 1850 11 5300 130 85 90 100 180

26 1700 8 800 11 2100 55 85 45

27 1800 8 1000 11 2500 55 85 50

28 1000 8 1100 11 2100 45 85 40 200 520

29 1800 8 800 11 1850 45 85 35

30 1500 8 900 11 5000 125 80 70 2200 1300

31 2500 8 2000 11 3700 80 80 50 200 350

32 2100 8 2250 11 7300 120 80 85 200 350

33 2400 8 2200 11 3200 120 80 70 2700 350

34 2000 8 3100 12 6100 250 80 200

35 3000 8 2900 13 4950 100 80 9536 2100 8 2500 11 8000 220 80 150 6500 350

37 1800 8 2000 11 7800 85 80 80 7800 35038 1800 8 2000 11 2900 120 80 100

39 3100 8 2000 11 6000 140 80 120

40 3000 8 2400 12 4000 85 80 80

Fig. 11. Each of the elementary variograms in b) describes the spatial variability of a

specic dimensional range of the rock fragments in a).

Fig. 12. BimTest scheme. 1) Specimen, 2) frontal steel plate, 3) LVDT transducer, 4) load

cell, 5) hydraulic jack, 6) support plate and holder for the hydraulic jack, and 7) failure

surface.


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C 12(h), expressed by Eq. (5a), can be experimentally computed by

Eq. (5b),

C 12 hð Þ ¼ E Z 1 x þ hð Þ Z 2 xð Þ½ −m1m2 ð5aÞ

C 12 hð Þ ¼

Pnc hð Þi¼1

z 1 xi þ hð Þ z 2 xið Þð Þ

nc hð Þ −m

1m

2 ð5bÞ

where m1, m⁎1 and m2, m⁎2 are, respectively, the mean theoretical

and experimental values of the variables.The spatial cross-covariance for h→0 coincides with the classical

covariance between two random variables, C 12(0)=σ 12, which canbe roughly inferred by extrapolating C ⁎12 at the origin even in case of

non-isotopic variables:

C 12 h→0ð Þ≅C 12 0ð Þ ¼ E Z 1 xð Þ Z 2 xð Þ½ −m

1m

2 ¼ σ 12: ð6Þ

For the purpose of the present study, the BimTest strengthparameters (i.e. cohesion c and friction angle φ dened on a support

equal to the BimTest sample volume) represent the rst ReV, c ( x, y),φ ( x, y), while the variogram total sill of the block indicator of eachimage represents the second ReV, C tot ( x, y).

Due to the low cost and relative simplicity, the number of

collected images (40) was larger than the number of performedBimTests (5), thus allowing computing experimental spatial correla-tions even if a rigorous correlation study would have also required

more data of the rst variable.

A correct cross-covariance modelling requires the strict stationar-ity of both variables, therefore it is necessary to examine theirvariograms γ 1⁎(h) and γ 2⁎(h) at the working scale of the investigateddomain S .

The variogram γ ⁎2(h) of Fig. 14 shows the spatial autocorrelationof C tot at the scale of the investigated domain S . Beside a markednugget effect, the behaviour of variography calls for a spherical model

with a range of 250 m and a sill coherent with the experimental

variance. The stationarity is thus veri

ed at the multi-metric scale. Itis well known that the meaning of the nugget effect can be explainedby several causes, among which a spatial micro-variability, that is a

spatial autocorrelation at a scale smaller than the multi-metric one.Due to the lack of suf cient c and φ experimental data, the spatial

auto-correlation of the strength parameters, γ ⁎1(h), could not beinferred. However, the omnidirectional spatial cross-covariance C ⁎12(h)

was anyway calculatedthanks to thehigher number of image data (40).

6.3. Cross-correlograms

Cross-correlograms ρ12(h) were introduced in order to facilitatethe interpretation of the spatial cross-correlation. Cross-correlogramsare dened as the spatial cross-covariance C 12(h) of the reduced

variables Y ( x)= [ Z ( x)−(m/σ )]. They represent a dimensionless mea-sure of the spatial cross-correlation, variable in the interval [+1, −1],and, therefore, they are much easier to interpret than C 12(h).

Experimentally they can be calculated as:

ρ12 hð Þ ¼

Pnc hð Þi¼1

z 1 xiþhð Þ z 2 x ið Þð Þ

nc hð Þ −m1m

2

σ 1σ 2

: ð7Þ

Due to the limited number of BimTests, and to the elongatedshape

of the sampled area (Figure 4), the omnidirectional cross-covarianceswere computed, by a-priori assuming the isotropy of the spatial

structures; which requires the ρ12(h) to be an even function.Before calculating the cross-correlograms, it is of fundamental

importance to verify if there are some anomalous values in the inputvariables that might fake the true ρ⁎12 behaviour. One BimTestsample shows very different strength parameters compared to theother test results (Figure 15).

According to Coli et al. (2011) these values are in agreement withthe mechanical behaviour of the bimrock and, in principle, they shouldbe included. However, in order to verify the stability of the correlation,the experimental cross-correlograms ρ⁎12 were calculated by both

including and excluding these suspected anomalous strength values,which proved to have no inuence on the resulting ρ⁎12. Also, severallags h were tested in order to nd a good compromise between thenecessity to study the small distances (smaller lags) and to work on a

statistically meaningful number of z 1( x+h), z 2( x) pairs (larger lags): abase lag h of 30 m, with a tolerance of ±15 m, was found to be the bestcompromise.

The ρ⁎12 relative to φ −C tot (Figure 16) shows a clear positive

correlation when h→0 ( ρ⁎12=0.3 for h=30), while for c −C tot(Figure 17) the correlation is negative ( ρ⁎12=−0.38 for h=30).These are very interesting results, because they are in agreement withwhat was expected given the relationships between the strength

parameters and the physical VBC (Coli et al., 2011). Due to theinsuf cient amount of data, the cross-covariance between strengthand image parameters could not be fully modelled on the basis of theexperimental ρ⁎12 only. Nevertheless, the likely existence of a

positive/negative correlation, probably at a scale b100 m, betweenthe total sill of the block indicator and the in-situ strength parameters

is quite evident.

Fig. 13. Correlation of friction angle and cohesion with VBC.

Fig.14. Experimental omnidirectional variogram and model variogram of C tot

calculated over the investigated domain S .


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7. Conclusions

In the present work a propaedeutic analysis was carried out aimedat improving the morphological and mechanical characterization of a

bimrock by exploiting the information made available by imagesampling of the outcrop surface.

The study was developed in two stages; a rst geostatisticalanalysis on binary images was performed to identify parameters

which are able to describe some properties of spatial and morpho-logical variability of rock fragments. A second geostatistical analysiswas then performed to study the possible correlation between theexperimental in-situ strength data and the index “total sill”

previously identied on the images.The following conclusions can be drafted:

• the geostatistical analysis of the images well characterises the

spatial distribution of bimrock rock fragments. The variogrammodels of the rock fragment indicator I B allowed identifying a

number of indexes (range a, sill C , period T ) that quantitatively andsynthetically express some essential properties of the morpholog-ical and spatial variability of fragments. Moreover, the proposed

approach appears to be of general interest for the 2D investigation

of bimrocks;• one of the geostatistical parameter identied on the images, i.e. the

total sill C tot of the block indicator, is eligible to be considered a ReV,

expressing rock fragment meaningful properties all over theinvestigated eld. As a consequence, a geostatistical correlationstudy with the strength values obtained by in-situ shear tests (e.g.BimTests) was carried out. A preliminary cross-covariance analysis,

possible also in case of non-isotopic variables (the strength

parameters and the C tot) highlighted the likely presence of ameaningful correlation at the multi-metric scale;

• this study and its preliminary results justify the pursuit of the

research aimed at improving the direct geotechnical characteriza-tion of bimrocks by auxiliary variables deduced by image analysis, acheaper and faster tool compared to an extensive campaign of in

situ mechanical testing. An immediate possible improvement forthe correlation study might be the integration of the cross-covariance analysis with a consistent number of new imagestaken in the proximity of the BimTest locations. The new data will

conrm the existence of a good correlation at the short scale bystudying their spatial cross-correlogram. This work is a rst steptowards the further investigation required to get a condent andoperational tool and procedure for improving the geotechnical

characterization of bimrocks by simple image sampling.

Acknowledgements

The authors would like to thank Dr. Edmund Medley for hissupport provided during the development of the present researchand the revision of the manuscript. The authors are also veryappreciative of the reviewer Dr. Bill Haneberg and of the second

anonymous reviewer for their contributions.

References

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Boccaletti, M., Bonini, M., Moratti, G., Sani, F., 1996. Tectono-sedimentary evolution of the post-nappe basin in the Northern Apennines. Proceedings of the InterimColloquium R.C.M.N.S. “Relations tectono-sédimentaries dans les basins Néogenèsde la Méditerranéè Occidentale”, Rabat, 7–8 April 1994. Notes et MémoriesServices Géologique Maroc 387, 177–193.

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Button, E., Riedmüller, G., Schubert, W., Klima, K., Medley, E., 2004. Tunnelling inTectonic Melanges: Accommodating the Impacts of Geomechanical Complexitiesand Anisotropic Rock Mass Fabrics. Bulletin of Engineering Geology and theEnvironment. March 2004.

Castellucci, P., Cornaggia, F., 1980. Gli olistostromi nel Macigno dei monti del chianti:analisi stratigraco-strutturale. Memorie Società Geologica Italiana 21, 171–180

(In Italian).

Fig. 15. Correlation between the c and φ strength parameters obtained through theBimTests.

Fig. 16. φ −C tot omnidirectional cross-correlogram, the base lag h is 30 m. For h→0

ρ⁎12 shows a clearly positive correlation ( ρ

⁎12≥0.3).

Fig. 17. c −C tot omnidirectional cross-correlogram, the base lag h is 30 m. For h→0

ρ⁎12 shows a clearly negative correlation ( ρ⁎12≤−0.38).


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