rock fragment characteristics, patterns and processes on

Rock Fragment Characteristics, Patterns and Processes on

Natural and Artificial Mesa Slopes

Zhengyao Nie

Thesis submitted for the Master of Science Degree

The University of Western Australia

School of Earth and Environment Faculty of Natural and Agricultural Sciences

In the Discipline of Soil Science and Geomorphology

2011

Supervisors: Professor Christoph Hinz and Dr. Gavan McGrath

ii

Declarations

My supervisors Prof. Christoph Hinz and Dr. Gavan McGrath contributed to the

conceptualisation and method development of the experimental and modelling study

and constructive reviews of the thesis. UWA Honour students Tia Byrd and Erin

Poultney contributed to the sampling of rock fragments and the rainfall simulation

experiment in Chapter 2 and Chapter 4. Rowan Jenner contributed to the methods of

statistical analysis in Chapter 2. Contributions to the thesis are acknowledged in each

chapter. Besides, the work presented is entirely my own unless stated otherwise.

Student: Zhengyao Nie Coordinating Supervisor: Christoph Hinz

(on behalf of all supervisors)

iii

Abstract

Rock fragments on hillslopes interact with fine soil and vegetation by affecting infiltration,

runoff, erosion and evaporation, and therefore have an important function in arid

ecohydrological systems. Previous studies have described rock fragments and their spatial

patterns using simple measures such as mean or median size. This study presents research

conducted on three natural mesa slopes and a post-mining waste rock dump, in the Great Sandy

Desert, Western Australia, to characterize the statistical properties of rock fragments and

interrelationships between particle shape and size. Digital images of surface rocks were

collected along transects placed on each hillslope. A total of 112,142 rock fragments from 263

locations were recorded. From these images perimeter, area, Feret’s diameter and circularity of

rock fragments were determined.

On natural mesas, mean Feret’s diameter, and similarly area and perimeter decreased while

mean circularity increased downslope. The results indicated that larger and more angular rock

fragments occurred on the top of these hills. From a suite of probability distributions tested,

lognormal distribution was found to describe the Feret’s diameter best. Furthermore, both the

location and scale parameters of the lognormal distribution decreased approximately linearly

with distance down each transect. None of the probability distribution functions tested

sufficiently characterised the distributions of circularity.

Transport process such as overland flow has been the predominant explanation for the observed

particle sorting on rock armoured slopes. It has been suggested as a mechanism that selectively

washes fine material away, leaving coarser particles on steeper part of hillslopes. In order to

evaluate whether a weathering phenomenon could instead cause rock size sorting, a dynamic

model of particle fragmentation was used to reproduce the changes in rock fragment size

distributions downslope. With the initial condition for the model taken as the particle size

distribution at the top of the hill, the two parameter fragmentation model reproduced observed

trends in particle size distributions and did even better than the linear regressions. Preferential

fragmentation of larger sized particles was observed. The sorting phenomenon and preferential

fragmentation were found to be analogous with abrasion phenomena in rivers. The similarities

suggest the potential for a more general principle underpinning physical weathering of rock

particles.

Following the analysis on rock fragments on natural mesa slopes, rock characteristics were also

assessed on a waste rock dump at the rehabilitated mine site that was designed to mimic the

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shape and surface rock cover of natural mesa slopes, for a comparison between the two

contrasting environment. With similar surface cover, rock fragments on the artificial slope were

generally smaller and more angular. Lognormal distribution was found to describe rock size.

Unlike rock shape on natural mesas, the circularity of mined rock fragments was well described

by beta distributions. No distinct spatial organisation of size and shape were observed on the

artificial slope; lognormal and beta distribution parameters were randomly distributed in space.

In addition to measurements of rock fragments, rill erosions were measured on the artificial

slope. A rainfall simulation experiment was conducted in laboratory for assessing the initial

development of the surface rock armour. Surface rock armour was likely to develop quickly as

fines were washed out, and converge in surface cover through a self-organizing pattern,

preventing soils against. The results potentially indicated a similar self-organization pattern of

surface rock armour through different processes on natural and artificial slope, and how to

achieve long term stability on artificial mine waste dumps.

v

Table of Contents

Declarations ...................................................................................................................... ii

Abstract ............................................................................................................................ iii

List of Figures ................................................................................................................. vii

List of Tables .................................................................................................................... ix

Acknowledgments ............................................................................................................. x

Chapter 1: Overview ......................................................................................................... 1

Chapter 2: Spatial patterns of rock fragments along mesa hillslopes in the Great Sandy

Desert, Australia ................................................................................................................ 8

2.1 Introduction .......................................................................................................... 8

2.2 Materials and Methods ....................................................................................... 11

2.2.1 Study area ................................................................................................. 11

2.2.2 Field sampling .......................................................................................... 13

2.2.3 Image analysis .......................................................................................... 13

2.2.4 Statistical analysis .................................................................................... 14

2.3 Results and discussion ....................................................................................... 16

2.3.1 Spatial patterns in rock fragment coverage .............................................. 16

2.3.2 Spatial patterns in rock fragment size and shape ..................................... 18

2.3.3 Relationship between rock fragment size and shape................................ 21

2.3.4 Spatial trends in probability distributions ................................................ 22

2.3.5 Implications for processes contributing to the formation of rock armour 24

2.4 Conclusion ......................................................................................................... 25

Chapter 3: Does fragmentation weathering explain rock particle sorting on arid hills? . 31

3.1 Introduction ........................................................................................................ 31

3.2 Methods .............................................................................................................. 33

3.2.1 Site description and sampling .................................................................. 33

3.2.2 Modelling fragmentation .......................................................................... 35

3.3 Results ................................................................................................................ 37

3.4 Discussion .......................................................................................................... 39

3.5 Conclusion ......................................................................................................... 41

Chapter 4: Self-organisation of rock fragment cover on engineered and natural mesa

slopes ............................................................................................................................... 46

vi

4.1 Introduction ........................................................................................................ 47

4.2 Method ............................................................................................................... 48

4.2.1 Study site .................................................................................................. 48

4.2.2 Rock fragments ........................................................................................ 49

4.2.3 Field evidence of erosion ......................................................................... 51

4.2.4 Rainfall simulation ................................................................................... 51

4.3 Result and discussion ......................................................................................... 52

4.3.1Descriptive data of rock fragments ........................................................... 52

4.3.2 Probability distributions ........................................................................... 53

4.3.3 Spatial patterns of rock fragments ........................................................... 54

4.3.4 Rill erosion in the field ............................................................................. 56

4.3.5 Rainfall simulation ................................................................................... 57

4.4 Conclusion ......................................................................................................... 58

Chapter 5: Summary ....................................................................................................... 62

Appendices ...................................................................................................................... 66

Appendix A. The R script: Assessment of gamma, Weibull and lognormal

distributions for particle size .................................................................................... 66

Appendix B. Results of the probability distribution assessments ............................ 69

Appendix C. The R script: Searching for best model parameters (Mesa 1 as an

example) ................................................................................................................... 72

Appendix D. Further results of rock fragment analysis on the artificial waste dump

slope as a comparison to its natural analogues ........................................................ 76

Appendix E. Field evidence of erosion .................................................................... 80

vii

List of Figures

Figure 2.1. Photograph of Mesa 1 with the hard cap formed by secondary crust formation

and rock fragments with patchy vegetation covering the slope. ................................. 12

Figure 2.2. Examples of image processing procedure: original image (left), rectified image

using Turbo-Reg within ImageJ, binary image with hand-traced rock fragments and an

example of fragment characteristics as determined by the particle size analyser in

ImageJ (after [Byrd, 2008]). ......................................................................................... 14

Figure 2.3. Example profile from transect M1T1 showing the original elevation data

obtained from the GPS records, together with the fitted power regression line. ......... 14

Figure 2.4. Trends in rock fragment coverage (%) with respect to distance from capping on

(a) Mesa 1, (b) Mesa 2 and (c) Mesa 3. ....................................................................... 17

Figure 2.5. Trends in rock fragment coverage (%) with respect to gradient on (a) Mesa 1

and (b) Mesa 2. ............................................................................................................ 17

Figure 3.1. Results of best fit parameters (α = -87, β = 0.31) on Mesa 2 as an example of

systematic examination of model parameters by minimizing root mean square (RMS)

error between modeled distributions and observed ones. ............................................ 36

Figure 3.2. Changes in the size distribution of rock fragments with position on each hill,

with the frequency histogram of observed data (bars), truncated lognormal probability

density functions (pdf) fitted to the data (line), and modeled initial (solid circle) and

predicted (open circles) size distributions. The top is 0 m, the middle 30 m (Mesa 1

and 2), 27 m (Mesa 3) and the bottom 60 m (Mesa 1 and 2) and 51 m (Mesa 3) from

the duricrust cap. .......................................................................................................... 37

Figure 3.3. Changes in particle size lognormal distribution location μ and scale σ

parameters (Eq. 3.5) as a function of distance from duricrust cap. Shown are the

empirical fits to observed data (crosses), fitted linear regressions and 95% confidence

intervals (lines) and predicted distribution parameters by the fragmentation model

(circles) given the initial particle size distribution, emphasized by the large point. .... 38

Figure 3.4. Changes in cumulative mass fractions of (a) empirical mass data converted

from observed particle size distribution; (b) best fit from model prediction (α < 0); (c)

model prediction while α = 0; and (d) model prediction while α > 0. ......................... 40

Figure 4.1. The digial elevation model (DEM) of designed waste rock dump and sample

points on five selected transects. .................................................................................. 49

Figure 4.2. Relationship between lognormal distribution parameter μ and σ from all sites on

(a) natural mesa slopes and (b) the waste rock dump. The dark grey, grey and light

grey solid circles are the combination of distribution parameters on the top, middle and

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bottom of a typical transect on each system – natural mesas and the waste rock dump,

respectively as shown in Figure 4.4. ............................................................................ 53

Figure 4.3. Spatial changes of rock fragment surface coverage on (a) Mesa 1; (b) Mesa 2;

(c) Mesa 3 and (d) the waste rock dump. ..................................................................... 54

Figure 4.4. Lognormal distribution parameters of particle size change with respect of

distance are shown as changes of (a) μ and (b) σ down a typical mesa transect and (f) μ

and (g) σ down a typical waste rock dump transect; probability density of Feret’s

diameter with fitted lognormal distribution line changes on the mesa transect from (c)

the top at 0 m from cap, (d) middle at 30 m, to (e) bottom at 60 m, and on the waste

rock dump from (h) the top at 0m, (i) middle at 40m, to (j) bottom at 80m. ............... 55

Figure 6.1. Graphical residual analysis for the fitted regression lines for the area data from

Mesa 1. ......................................................................................................................... 71

Figure 6.2. Boxplots of all Feret’s diameter data on three natural mesa slopes and the waste

rock dump. .................................................................................................................... 76

Figure 6.3. Boxplots of all circularity data on three natural mesa slopes and the waste rock

dump. ............................................................................................................................ 76

Figure 6.4. (a) relationship between beta distribution parameters β1 and β2, with (b), (c) and

(d) density circularity histograms corresponding to solid circles of different

combination of β1 and β2, and the fitted beta distribution line. .................................... 77

Figure 6.5. Inter-relationship between (a) mean circularity and mean Feret’s diameter; (b)

rock surface coverage and mean Feret’s diameter and (c) rock surface coverage and

mean circularity on five transects on the waste rock dump. ......................................... 78

Figure 6.6. Spatial changes of (a) surface rock cover; (b) mean Feret’s diameter and (c)

mean circularity along distance on five transects on the waste rock dump. ................. 79

Figure 6.7. Changes of Feret’s diameter orienting downslope on top, middle and bottom of

a typical mesa transect and a typical waste rock dump transect. .................................. 79

Figure 6.8. Photographs looking downslope from the top of (a) Transect B (in Treatment 2)

and (b) Transect D (in Treatment 1). ............................................................................ 80

ix

List of Tables

Table 2.1. A summary of previous studies of spatial distribution of rock fragments in arid

and semi-arid regions. .................................................................................................. 10

Table 2.2. General information of three mesas. ................................................................... 13

Table 2.3. Descriptive statistics of rock fragments for all samples points on all transects

from all three Mesas. ................................................................................................... 16

Table 3.1. Linear regression coefficients; best fit model parameters and root mean square

(RMS) errors. For the linear regressions x denotes distance in meters from the top of

each transect. ................................................................................................................ 38

Table 4.1. Descriptive statistics of rock fragment characteristics on the waste rock dump

and natural mesa slopes. .............................................................................................. 52

Table 4.2. Results of rill erosion and the corresponding surface cover of rock fragments on

each transect on the waste rock dump. ........................................................................ 56

Table 4.3. Changes in rock fragment surface coverage for each replicate of each volume

ratio during rainfall simulation. ................................................................................... 58

Table 6.1. Hartigan's Diptest Results. Please note that the distance here in the “Transect and

Location” column is the distance from the bottom of the transect. ............................. 69

Table 6.2. Results of fitted distributions (lognormal, gamma and weibull) to Feret’s

diameter of rock fragments on three mesas. ................................................................ 70

x

Acknowledgments: This research was supported under the Australian Research Council

Linkage Projects (project no. LP0774881 in conjunction with Newcrest Mining Ltd. and

Minerals and Energy Research Institute of Western Australia) funding schemes.

I thank my supervisors Prof. Dr. Christoph Hinz and Dr. Gavan McGrath for their incredible

support and sharing ideas and feedback;

Newcrest Mining Ltd. for giving me the opportunity doing this research;

Tia Byrd and Erin Poultney for doing an excellent job in data collection;

Rowan Jenner for the committed support in statistics and surviving together from the ‘not so fun’

data analysis;

Martha Orozco, Basu Dev Regmi, Deborah Lin, Daniel Dempster for the happy times during

work;

Ziwen Zhang for the loving care and sharing in this journey;

And all the people whose supports and comments made it a better thesis.

1

Chapter 1: Overview

Rock armoured hillslopes are common in many arid and semiarid environments around the

world. Cooke et al. [1993] provided an elaborate review of stone pavements and their evolution

in deserts. The presence of surface rock armour results mainly from the preferential removal of

fine materials by deflation or overland flow, leaving the coarse materials behind [Cooke et al.,

1993]. These surface rock fragments have drawn researchers’ interests for several reasons.

Importantly, rock fragments play a role in controlling erosion, especially in arid land areas with

sparse vegetation [Simanton et al., 1984; Cooke et al., 1993]. Secondly, downslope fining of

surface armoured rock fragments has been observed extensively on arid and semiarid hillslopes,

and this sorting phenomenon could have eco-hydrological implications [Dury, 1966; Parsons,

1988; Abrahams et al., 1990; Cooke et al., 1993]. Therefore, a number of studies have

investigated spatial patterns of rock fragment characteristics, such as size and cover, in relation

to geomorphic features of the landscape such as slope type, gradient, curvature and aspect

[Abrahams et al., 1984; Abrahams et al., 1985; Abrahams et al., 1990; Simanton et al., 1994;

Poesen et al., 1998; Canfield et al., 2001; Li, 2007; Zhu and Shao, 2008].

Rock fragments are defined as strongly cemented particles with a diameter greater than sand and

smaller than a pedon. In some studies, rock fragments are defined as being either greater than 2

mm [Poesen and Lavee, 1994] or greater than 5 mm [Li et al., 2007; Poesen et al., 1998]. These

studies commonly focus on the soil surface, in particular in semi-arid and arid environments,

but also on the rock fragment distribution within soil profiles [Bunte and Poesen, 1993;

Brakensiek and Rawls, 1994; Poesen and Lavee, 1994]. In soil science literature the fine earth

fraction (< 2mm) have received considerable attention, however much less effort has been

invested in understanding the coarse fraction such as stones and rock fragments. This has partly

been due to the higher surface area to volume ratio of the fine fraction which controls to large

extent physical, chemical as well as biological properties of the material. However rock

fragments in particular in arid and semi-arid environments exert significant effects on

hydrological processes by influencing the soil water balance which in turn affects erosion

[Poesen and Lavee, 1994]. On the one hand surface rock fragments stabilize soil by shielding

the soil beneath the surface from raindrop impact and runoff detachment, and on the other hand

rock fragments entrap splashed sediment. In addition, the presence of rock fragment on the soil

surface increases roughness which largely determines infiltration rates, surface runoff and

erosion [van Wesemael et al., 1996a]. A high cover of surface rock fragments promotes

infiltration by inhibiting surface sealing, but the effects also depend on rock size, shape and

position [Cerdà, 2001; Poesen and Ingelmo-Sanchez, 1992; Poesen and Lavee, 1994; Valentin,

1994]. In conclusion, rock fragments in soil, especially on soil surfaces, protect against erosion

2

and physical degradation in general [Poesen and Lavee, 1994].

Weathering of rock fragments is extensive in desert by thermal fracture, salt crystallization and

biochemical weathering processes [Cooke et al., 1993]. The combination of weathering,

hydrological processes such as runoff and erosion will then affect the spatial distribution of rock

fragments in the soil and on the soil surface. Spatial patterns of rock fragment characteristics

including coverage, mean and/or median size along arid hillslopes have been studied by a

number of researchers. Among these studies, Simanton et al. [1994] reported a logarithmic

relationship between surface rock fragment cover and slope gradient in Nevada and Arizona,

and later Simanton and Toy [1994] developed a hyperbolic equation to predict surface rock

fragment cover in relation to a soil-slope factor (SSF), which combines effects of slope gradient

and soil profile rock content. Abrahams et al. [1985] found particle size positively related to

slope gradient, independent of distance from the divide in the Mojave Desert. They calculated

regression coefficients of the size-slope relations and the changes in coefficient values seemed

to relate to different slope platform shapes (plan-concave and plan-convex). Poesen et al. [1998]

observed an increase in rock fragment size and coverage with gradient in cultivated areas in

semiarid southeast Spain. They also found the best fit regression of this increase as a function of

slope gradient and the cover percentage of rock fragments are controlled by lithology and the

aspect of the slope. All the above studies proposed that spatial patterns of rock fragments on

hillslopes were largely controlled by surface runoff, which selectively transports smaller rocks

faster downslope in comparison to large rocks. However, spatial trends in size sorting were also

observed for large rocks, too large to be expected to be transported as bedload [Abrahams et al.,

1990]. Runoff creep, which is a slow creep process of large blocks resulting from selective

removal of fines that supports rocks by runoff has been hypothesised, but has yet to be

confirmed as a sorting mechanism [de Ploey and Moeyersons, 1975; Abrahams et al., 1990]. In

some undisturbed desert pavements on the Eastern Libyan Plateau, Egypt however, Adelsberger

and Smith [2009] found no spatial relationship between pavement characteristics and local

geomorphic features. Desert pavement surfaces in this study region could have developed

without significant influence from transport mechanisms such as overland flow, but rather was

more influenced by in-situ mechanical breakdown and pedogenesis.

Fragmentation in this thesis refers to a physical weathering process whereby one particle breaks

down into two or more smaller particles while preserving the mass. Probability size

distributions of rock particles are proposed to be indicators of breakdown processes and have

been explored in various areas of research including fluvial geomorphology of river systems and

material processing in the mining industries [Krumbein and Tisdel, 1940; Friedman, 1962;

Grady and Kipp, 1985]. Turcotte’s [1986] fragmentation model has been applied to describe

3

power-law particle size distributions in soil [Bittelli et al., 1999; Perfect, 1997]. It assumes that

fragmentation occurs as an instantaneous cascade, with the probability of breakage independent

of size [Turcotte, 1986; 1992]. Instead of an instantaneous cascade, multiple breakage events

have been shown to lead to a lognormal particle size distribution [Kolmogorov, 1941 as cited by

Dacey and Krumbein, 1979]. However, the concept of using spatial changes of particle size

distributions to infer processes contributing to the formation of rock pavements have not been

used to date. Dunkerley [1995] pointed to the significance of understanding the rock particle

size distribution in an empirical study of surface stone size. Surface and subsurface water flow

as well as erosion could vary significantly with different types of rock size distributions, even

where the mean sizes are the same [Dunkerley, 1995].

There is a need to quantify the size distributions of rock fragments on debris mantled

slopes and to assess how they change spatially.

The research described in this thesis was conducted near Telfer, located in the south centre of

the Paterson Range on the edge of the Great Sandy Desert, Western Australia [Parker, 2006].

Newcrest Mining Ltd. mines gold and copper at Telfer [Parker, 2006]. Stability of waste rock

dumps is an issue faced by the mining industry as surface and tunnel erosion occurred

frequently on the traditional berm-and-bench waste rock dump design. As Hancock et al. [2008]

suggested, slope shape and soil properties (e.g., rock fragment content) are largely responsible

for the slope stability. As a result a waste rock dump was designed and constructed at mine site

to mimic the concave shape and rock armour cover of natural mesa hillslopes in the Telfer

region. As slope surface develop, a number of processes may take place during the landform

evolution, including water wash, vertical sorting of rock fragments, creep and surface

weathering [Cooke, 1970]. It is likely that wash off is the dominating mechanism shaping the

two year old engineered slope, while surface weathering is more important on the natural mesas.

However, experimental evidence of the performance of engineered mesa-shaped slopes in

improving rehabilitation is missing.

This thesis consists of a series of three papers, aiming (1) to investigate characteristics and

spatial patterns of both rock size and shape on arid mesa slopes in Telfer region, and further

quantify probability distribution which could provide us with a better understanding of rock

pavement evolution; (2) to assess the hypothesis that in-situ fragmentation is a possible

mechanism explaining the change in size distributions along the natural mesa hillslopes; and (3)

to investigate rock characteristics and spatial trends on the mesa shaped rock dump at

Newcrest’s Telfer gold mine, and compare these with those of the natural mesas. As the thesis is

presented as a series of three papers, some parts of the introductions and methodology sections

4

in various chapters will appear repeatedly.

5

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7

Africa, Catena, 23(1-2), 87-97.

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8

Chapter 2: Spatial patterns of rock fragments along mesa

hillslopes in the Great Sandy Desert, Australia

Zhengyao Nie1, Christoph Hinz1, Gavan S. McGrath1, Rowan Jenner1 and Tia Byrd1

1. School of Earth and Environment, The University of Western Australia, Western Australia,

Australia.

Abstract

Rock mantled hillslopes are common in arid and semiarid environments. Investigating the size

and shape of the mantled rock fragments and their spatial distributions has implications for

geomorphic processes. In this study, rock fragments on three mesa hillslopes were characterized

by size (area, Feret’s diameter, and perimeter) and shape (circularity) using image processing of

digital photographs taken of the slope surface. A total of 93415 individual rock fragment

measurements were obtained from 222 positions along the mesa slopes. Particle size and

circularity were negatively related. Spatial patterns of these rock fragments were assessed by

relating mean particle size and shape to geomorphic features including slope gradient and

surface distance from the hill top. Distance was found to be a better independent variable

describing the changes in rock size and shape along the slope then gradient. On two mesas,

mean particle size decreased nonlinearly with distance from the cap, while mean circularity

increases linearly. The other mesa exhibited a weaker spatial trend of rock particles. In addition

to simple descriptive statistics, a set of probability distributions were tested to describe the

distribution of size and shape of rock fragments along each hillslope. Truncated lognormal

distribution proved to describe the particle size distribution well, with both distribution

parameters decreasing linearly with distance downslope on all three mesas. None of the three

distributions beta, Weibull and logit-normal consistently described the shape parameter.

Keywords: rock fragments; mesa slope; distance; spatial patterns; lognormal distribution

2.1 Introduction

Rock fragments have various geomorphic and ecological functions in arid ecosystems. One of

them is stabilizing landforms by forming rock armours over potentially erodible materials

[Cooke et al., 1993]. The important role of rock fragments has been recognized in the soil water

balance by reducing bare soil evaporation and increasing soil water storage, and preventing

erosion and physical degradation [Cooke et al., 1993; Poesen and Lavee, 1994; Valentin, 1994;

Poesen et al., 1998; van Wesemael et al., 2000].

9

The distribution of rock fragments in space is a reflection of landform evolution processes under

geomorphic and hydrological controls. Investigations of spatial patterns of rock fragments often

relate their size and/or surface coverage to geomorphic features of the landscape [Abrahams et

al., 1985; Simanton et al., 1994; Poesen et al., 1998; Canfield et al., 2001; Li et al., 2007; Zhu

and Shao, 2008; Adelsberger and Smith, 2009]. For example, Simanton et al. [1994] reported a

logarithmic relationship between surface rock fragment cover and hillslope gradient in semiarid

Arizona and Nevada. A hyperbolic equation was developed for surface rock fragment cover

from a combination of slope gradient and rock content in soil [Simanton and Toy, 1994].

Overland flow was also suggested to be the dominant factor in transporting rocks by Abrahams

et al. [1985], in the study relating mean particle size to slope gradient on the hillslopes underlain

by weak to moderately resistant rocks in the Mojave Desert. The rate of sediment transport G by

overland flow was proposed empirically to relate to horizontal distance to the slope divide x,

slope gradient S, and mean particle size D in the form pnm DSxG / , where m, n and p are all

constants of the equation with values larger than 0 [Abrahams et al., 1985]. Poesen et al. [1998]

observed a positive relationship between hillslope gradient and rock fragment cover as well as

size along semiarid hillslopes in southeast Spain, and furthermore found that rock fragment

cover was largely controlled by lithology and hillslope aspect. Dury [1966] found that particle

size decreases in orderly fashion with both slope and distance downslope. However, in some

undisturbed desert pavements on the Eastern Libyan Plateau, Egypt, Adelsberger and Smith

[2009] found no spatial relationship between pavement characteristics such as clast size, density,

lithology and orientation and local geomorphic features being typically slope gradient, and

aspect. They suggested that instead of transport mechanisms such as runoff the pavement

surface had developed by mechanical breakdown of surface clasts and in-situ pedogenesis

[Adelsberger and Smith, 2009]. Generally though, rock fragment cover and size are found to be

related to local geomorphic and geological features. Some of the results of spatial distributions

of rock fragments in arid and semi-arid regions are summarized in Table 2.1.

10

Table 2.1. A summary of previous studies of spatial distribution of rock fragments in arid and semi-arid regions.

Geomorphic location Slope shape Sample method Surface

coverage (%)

Size range(mm)

Slope gradient

(%) Rock type Result Source

RF cover RF size Gradient

A small catchment in wind-water erosion crisscross

region, Loess Plateau, China

convex-straight-concave

Digital photographing Compass 0 - 7 2-60 0 - 275 - cbSaSAc 2 [Zhu and

Shao, 2008]

Dolines in a semiarid Mediterranean mountain-

range, south Spain.


Line-intercept method of Mueller-

Dombois and Ellenberg

Tape measure

DEM 0-100 Median

diameter:>50-32.5 Limestone

bSaAc )log(bSeaD

[Li et al., 2007]

A catchment on a highland, northern Ethiopia

convex and concave

Photography and mode-count method.

Clinometer and tape measure

57-85 Median

diameter:>56-42

Tertiary basalt and

Antalo limestone

bSaAc )log(

bSaAc

[Nyssen et al., 2002]

Semiarid hillslopes, southeast Spain.


Photography and point-count method

Clinometer and tape measure

0-80 Median

diameter: 5-80

4-66 Micaschist, Andesite,

Conglomerate

bSaAc )log(

bc SaA

bSc eaA

[Poesen et al., 1998]

Catenas in semiarid Arizona and Nevada, USA

5 uniform, 4 convex and 3 concave

Line-point measurement

(Bonham, 1989)-

Abney level

0-75 - 2-61 - bSaAc )log( [Simanton

et al., 1994]

Debris slopes in Mojave Desert, USA

3 convex and 2

concave -

Sampling particles in a grid

Abney level

- Mean

diameter: 0-100

0-65

Gneiss, Latitic

Porphyry, Fanglomerate

baSD [Abraham

s et al., 1985]

* Ac = rock fragment coverage; D = rock fragment size; S = slope gradient. Symbols a, b and c are relationship function coefficients.

11

Particle size distributions may provide us with another means to infer landforming processes

and have been studied in soil science, fluvial geomorphology and processing engineering

[Turcotte, 1986; Bittelli et al., 1999; Cohen et al., 2009]. As fragmentation is caused by various

processes we expect different fragment size distributions depending on the type of

fragmentation such as physical weathering by salt, frost or temperature, blasting or saltation

during transport [Friedman, 1962; Grady and Kipp, 1985; Martin et al., 2009]. Scientists have

attempted to model fragmentation with different assumptions and algorithms leading to different

particle size distributions [Epstein, 1948; Turcotte, 1986; Perfect, 1997]. Among these studies,

Turcotte’s [1986] equation is widely applied in fragmentation models, which is a power-law

distribution of particle size derived from an instantaneous scale-invariant cascade of

fragmentation. In contrast, Kolmogorov [1941] (as cited by Dacey and Krumbein, 1979) argues

in an earlier paper that lognormal size distribution is caused by temporal fragmentation and sub-

particles obtained from larger particles are independent of size. In studies of surface rocks on

desert hillslopes, Dury [1966] and Dunkerley [1995] observed normal distribution of log particle

size. However, characterization of statistical distribution of particle size has been surprisingly

absent.

In addition to particle size, rock fragment shape, is considered to be an indicator of abrasion and

breakage during transport processes [Krumbein, 1941a]. On hillslopes, gravels are commonly

rounded when they are fluvial in origin, yet particle rounding can also be accomplished by

surface weathering [Dixon, 1994]. With respect to landscape origin, Al-Farraj et al. [2000]

reported changes in roundness of clasts on differently developed desert pavement surfaces of

alluvial fan. Again, to our knowledge, few studies assessed particle shape and probability

distribution on rock armoured hillslopes.

This study will identify the characteristics of the size and shape of rock fragments and the

surface cover on three natural mesa hillslopes in the Great Sandy Desert, Western Australia. In

addition, probability distribution of particle size and shape will be measured. The objective of

this study is to shed light in understanding landform stability through spatial organizations of

rock fragments on arid mesa slopes.

2.2 Materials and Methods

2.2.1 Study area

The study area was located near Newcrest Mining Ltd’s Telfer Gold Mine, 21.71ºS, 122.23ºE in

the Great Sandy Desert, Australia. The climate is arid, with typically 250 – 450 mm annual

rainfall with the majority of rainfall associated with 20 to 30 mm convective storms which

typically occur each year, while the annual potential evaporation is 3200 mm. The average

12

maximum daily temperature at Telfer varies between 25°C – 42°C [Bureau of Meteorology,

2011].

The terrain ranges from flat desert surfaces to steep mesa slopes. Interspersed with occasional

linear dune, rock fragments dominate the surface cover of mesa slopes. A duricrust capping

which is the apparent source of surface rock fragments is found at the top of the mesa (Figure

2.1). Telfer Gold Mine and the nearby mesas are part of the Paterson Range in Western Australia,

specifically the Mesoproterozoic to Neoproterozoic Yeneena Basin [Maxlow, 2005]. The mesas

are located in the Upper Yenenna Sub-Group of the lower Throssell Range Group in the west

and centre of Telfer [Maxlow 2005]. The Throssell Range Group consists of shallow marine

sediments. Mottled zone and lateritic duricrust have been eroded to reveal the underlying

saprolite and saprock [Henderson, 1996]. Samples of rock fragments were collected on three

selected mesas – simply called Mesa1, Mesa 2, and Mesa 3 (Table 2.2).

Figure 2.1. Photograph of Mesa 1 with the hard cap formed by secondary crust formation and rock

fragments with patchy vegetation covering the slope.

13

Table 2.2. General information of three mesas.

Longitude/

Latitude

Transect

aspects

Slope type Rock material Vegetation

Mesa 1 21°45’46.31’’S/

122.9’56.56’’E

NW Concave silcrete,siltstone acacia shrubs and

tussock grass

Mesa 2 21°56’0.96’’S/

122°12’43.00’’E

S Concave silcrete

conglomerate

acacia shrubs and

tussock grass

Mesa 3 21°52’3.23’’S/

122°7’29.21’’E

E Rectilinear Siltstone tussock grass

2.2.2 Field sampling

Four transects were placed along each mesa, beginning from the lower edge of the hard cap

(hard cap not included) downhill in a straight line along the direction of the steepest descent.

For Mesa 1 and 2 transects were 60 m in length, while for Mesa 3, transect lengths varied

between 24 and 51m as the hill was shorter. Sample points were located at 3 m intervals.

Positions of sample points were recorded with a differential GPS (Magellan ProMarkTM 3). At

each point a 1 m2 metal square grid was placed on the ground and digital photographs were

taken of the surface. This was repeated twice on each side of the transect line, resulting in a total

of four images per interval. This number was reduced where obstructions, such as larger plants,

prevented photographs being taken. Digital photographs were taken in JPEG/TIFF format with

the following cameras: Canon IXUS750 (3x optical zoon, 7.1 megapixels), Canon Powershot

A610 AiAF (4x optical zoom, 5 megapixels), Olympus u 1720SW (3x optical zoom, 7.1

megapixels), Olympus u850SW (5x optical zoom, 8 megapixels).

2.2.3 Image analysis

The Java based program ImageJ was used to create a binary image [Rasband, 2008]. Rock

fragments were hand traced with an Intuos3 Wacom graphics tablet system. The Turbo-Reg

plug-in for ImageJ was then used to rectify and register the image via a bilinear transformation,

altering the image to produce a standard resolution in IrfanView such that 1 pixel corresponded

to 0.28 mm2 [Thévenaz, 2008; Skiljan, 2011] (Figure 2.2). Due to the limitations of this

methodology, rock fragments less than 10 mm were excluded from the analysis.

For each individual rock fragment, the area (A) (mm2), Feret’s diameter (F) (mm), perimeter (P)

(mm) and circularity (C) (-) were determined. Feret’s diameter is defined as the longest distance

between any two points on the perimeter [Rasband, 2008]. Circularity, a measure of the shape of

particles, is defined by: 24 PAC and ranges between a small number near 0 for an

elongated thin plate and 1 for a perfect circle. For each image, surface coverage of rock

fragments (%) was calculated as:

14

%100

VT

Rc AA

AA (2.1)

where Ac, AR, AV, and AT represent rock fragment coverage percentage, the total rock fragment

area, total vegetation area and total area within the grid, respectively.

Figure 2.2. Examples of image processing procedure: original image (left), rectified image using Turbo-

Reg within ImageJ, binary image with hand-traced rock fragments and an example of fragment

characteristics as determined by the particle size analyser in ImageJ (after [Byrd, 2008]).

2.2.4 Statistical analysis

Mesa slope smoothing

The elevation at each sample location was obtained from the information recorded by the

differential GPS. To obtain a smooth transect profile, non-linear regression models of the form

bxxay )( max were fitted to the relative elevation data for each transect, where y is the

estimated relative elevation from the regression model, xmax is the horizontal length of the

transect, and x is the horizontal distance from the duricrust cap (see Figure 2.3). This enabled

the gradient at each location to be approximated by the first of the fitted equation. The following

notation is used to identify transects measured: M1 to M3 stands for Mesa 1 to Mesa 3 and T1

to T4 denotes Transect 1 to Transect 4.

Figure 2.3. Example profile from transect M1T1 showing the original elevation data obtained from the

GPS records, together with the fitted power regression line.

15

Data analysis

All statistical analyses were carried out using the software environment R [Ihaka and

Gentleman, 1996]. At each sample location, descriptive statistics such as mean, median,

standard deviation were calculated for the rock fragment size and shape measurements. The

relationship between rock fragment coverage and gradient was then analysed, in addition to the

relationship between coverage and distance along the transect. Probability densities of rock

fragment size and shape were estimated with kernel density approximation, which is a density

estimation insensitive to bin size [Sheather and Jones, 1991], to show the distribution of

particles at each sample point along the transect. The changes in mean rock fragment size

expressed either as Feret’s diameter, area or perimeter with distance along the transects were

examined by three regression models in the form of linear (Eq. 2.2), exponential (Eq. 2.3) and

power-law (Eq. 2.4).

bXLaD )(~ (2.2)

)](exp[~ XLnmD (2.3)

cXLpD q )(~ (2.4)

where D is either mean Feret’s diameter (mm), mean area (mm2) or mean perimeter (mm); L

denotes the total length of the transect (m), and X = distance from capping along each transect

(m). Model coefficients a, b, m, n, p, q, c were determined by linear and nonlinear least squares

methods [Bates and Watts, 1988; Chambers, 1992]. Regressions are deemed to be significant

with a p-value less than 0.01. In this paper, if not otherwise specified, distance refers to the

distance along the transect surface from the duricrust cap.

Hartigan’s Diptest was applied to test for unimodality of rock fragment size distribution

[Hartigan, 1985]. At each sample location, gamma, Weibull and lognormal probability

distributions were then fitted to the Feret’s diameter, while beta, Weibull and logit-normal

probability distributions were fitted to circularity. The chi-square test was applied to assess

whether the data are from the fitted distributions, and a relative measure of goodness of fit was

calculated [Ricci, 2005] (Eq. 2.5; see Appendix A).

n

ii

n

iii

y

yy

1

2

1

2* )(

(2.5)

where is the relative measure of goodness of fit; yi is expected frequency from estimated

distribution parameters; yi* is the observed frequency; n is the total number of bins, and i

denotes the ith bin. Distribution parameters were then estimated using maximum likelihood

method [Venables and Ripley, 2002].

16

As the chi-square test is sensitive to bin size, the test was carried out twice; firstly using Sturges

formula to determine bin size [Sturges, 1926] and secondly with a set number (n = 20) of bins,

each of equal width. The fit of distribution was deemed satisfactory at a 5% significance level.

Furthermore, geomorphic dependencies of the parameters of the fitted probability distributions

were assessed. The spatial dependency of the estimated distribution parameters was quantified

by linear regression as a function of distance.

2.3 Results and discussion

A summary of descriptive statistics of Feret’s diameter, area, perimeter, circularity and

percentage coverage for all transects of all three mesas comprising of a total of 222 sample

points and a total of 93415 individual rock fragments is presented in Tale 2.3.

Table 2.3. Descriptive statistics of rock fragments for all samples points on all transects from all three

Mesas.

Min Median Mean Max Standard Deviation

Coverage (%) 10.73 36.83 37.95 67.17 12.67

Feret’s Diameter (mm) 10 51 65 1008 51

Area (mm2) 53 1075 2709 484091 7456

Perimeter (mm) 24 138 184 3732 159

Circularity 0.02 0.72 0.70 0.92 0.11

2.3.1 Spatial patterns in rock fragment coverage

On most desert hillslopes, size of rock fragments has been found to decrease downslope [Cooke

et al., 1993]. We determined a significant negative linear relationship between rock fragment

coverage and transect distance from all three mesas (Figure 2.4). Although there is significant

variation in surface rock coverage at different sample points, in general higher coverage of rock

armour exists on the top part of the hillslopes.

17

Figure 2.4. Trends in rock fragment coverage (%) with respect to distance from capping on (a) Mesa 1, (b)

Mesa 2 and (c) Mesa 3.

Following Simanton and Toy [1994], Figure 2.5 shows the logarithm relationship between rock

fragment coverage and hillslope gradient on Mesa 1 and 2. Mesa 3 is not included in this

analysis because all four transects on Mesa 3 are approximately rectilinear and have nearly

constant gradients. For some individual transects, log-linear regression is able to describe the

relation between rock cover and slope gradient well, however, for most transects a linear model

better describes the spatial changes of data (see Figure 2.5).

Figure 2.5. Trends in rock fragment coverage (%) with respect to gradient on (a) Mesa 1 and (b) Mesa 2.

Above results indicate surface coverage of rock fragments is greater on the steeper parts of the

hills, where is the upper parts of concave shaped Mesa 1 and 2. However, distance provides

significant relationships with surface cover on all mesas, including Mesa 3, a rectilinearly

shaped slope with a constant slope gradient. Distance also appears to be a dependency that can

describe the changes of mean particle size and shape, and probability distribution parameters as

presented in the following sections. Although Nyssen et al. [2002] point out that distance is

0 10 20 30 40 50 60

1030

5070

(a)

R2=0.59

0 10 20 30 40 50 60

1030

5070

(b)

Distance (m)

R2=0.50

0 10 20 30 40 50 60

1030

5070

(c)

R2=0.19

Co

vera

ge

(%

)

0 10 20 30 40 50 60 70

1020

3040

5060

70

(a)

LinearR2=0.57

Log-linearR2=0.54

0 10 20 30 40 50 60 70

1020

3040

5060

70

(b)

LinearR2=0.42

Log-linearR2=0.34

Gradient (%)

Co

vera

ge

(%

)

18

important in the spatial distribution of rock fragments, there are few direct evidence for surface

cover or mean particle size – distance relationship in the literatures [Dury, 1966], as selective

particle transport process has often been hypothesized and slope gradient is closely related to

particle transport. However, the spatial changes of rock coverage and mean particle size with

respect to distance in this study are consistent with previous work on arid concave slopes

[Abrahams et al., 1985; Simanton et al., 1994; Poesen et al., 1998].

2.3.2 Spatial patterns in rock fragment size and shape

There are clear spatial patterns in the changes of probability densities of rock fragment size and

shape from top of the slope to the bottom (see Figure 2.6). The mode of size shifts from larger

to smaller sizes as one progresses downslope. Similarly the size distribution narrows downslope.

Circularity on the other hand shows the opposite trend with the mode, shifting from low to high

circularity. These distributions clearly show larger and more angular rocks with greater

variability can be expected in the upper part of the mesas, while smaller, rounder and less

variable particles are observed toward the bottom of each hill.

Figure 2.6. Changes in probability density of rock (a) area, (b) Feret’s diameter, (c) Perimeter and (d)

Circularity from the top 0m, the middle 30 m (Mesa 1), 33m (Mesa 2), 27 m (Mesa 3) to the bottom 60 m

(Mesa 1 and 2) and 51 m (Mesa 3) from cap.

0 50 100 200

0.00

00.

020

Mesa 1

Den

sitym

m1

TopMiddle

Bottom

0 50 100 200

Feret's Diameter (mm)

Mesa 2

0 50 100 200

(a)

Mesa 3

0 2000 6000 10000

0e+

006e

-04

Den

sitym

m2

0 2000 6000 10000

Areamm2

0 2000 6000 10000

(b)

0 200 400 600 800

0.00

00.

006

Den

sitym

m1

0 200 400 600 800

Perimeter (mm)

0 200 400 600 800

(c)

0.0 0.2 0.4 0.6 0.8 1.0

04

8

Den

sity

0.0 0.2 0.4 0.6 0.8 1.0

Circularity

0.0 0.2 0.4 0.6 0.8 1.0

(d)

19

The spatial trends of mean particle size along distance are described by power-law and/or

exponential regression models on all transects on Mesa 1 and 2 (Figure 2.7a shows a typical

transect). In contrast, a negative linear trend is observed between mean circularity and transect

distance (Figure 2.7b). Increases of standard deviation of size and shape with distance are

described by power-law and linear regressions, respectively (Figure 2.7c and 2.7d). Rock

fragments on Mesa 3 tend to be distributed more randomly in space (Figure 2.6). We found

smaller rock fragments in general on Mesa 3 that can result from a different lithology as we

observed more outcrops along the mesa slope. As lithology and particle size are responsible for

the spatial patterns of rock fragments [Abrahams et al., 1985; Poesen et al., 1998; Nyssen et al.,

2002], the relatively weaker spatial trends of rock particles on Mesa 3 could be controlled by a

different lithology and perhaps by a different source of surface armoured rock fragments.

Figure 2.7. Changes in (a) mean Feret’s diameter ( F ), (b) mean circularity (C ), (c) standard deviation

of Feret’s diameter (SdF) and (d) standard deviation of circularity (SdC) with respect to distance along a

typical mesa transect (Mesa 1 Transect 4).

In the above figures (Figure 2.7), power-law regressions were fitted to the changes of mean and

standard deviation of Feret’s diameter as a function of distance. By comparing r2 of regression

models, power-law regressions provided slightly better fits than exponential regressions. Using

Shapiro-Wilk normality test [Royston, 1982], residuals of power-law regression can be rejected

0 10 20 30 40 50 60

60

80

100

12

0

(a)

Me

an

Fe

ret's

Dia

me

ter

(mm

)

F = 0.011L X2.136 43.892

R2 = 0.901

0 10 20 30 40 50 60

0.6

60

.70

0.7

4

(b)

Me

an

Cir

cula

rity

C = 0.001X 0.675

R2 = 0.791

0 10 20 30 40 50 60

20

406

08

01

00

(c)

Distance (m)

Std

De

v o

f Fe

ret's

D (

mm

)

SdF = 0.003L X2.56 18.754

R2 = 0.923

0 10 20 30 40 50 60

0.0

70

.09

0.1

1

(d)

Distance (m)

Std

De

v o

f Cir

cula

rity

SdC = 0.001X 0.124

R2 = 0.783

20

as normally distributed at a p 0.1 significance level on some of the transect (Figure 2.8a and

2.8b) while a null hypothesis of normal distribution of residuals from exponential distribution is

accepted (Figure 2.8c and 2.8d), indicating that exponential regressions may be the more

appropriate description of the spatial dependence of mean rock fragment size. In fact, negative

exponential functions between particle size and distance have been observed in many sediment

studies [Krumbein, 1937; Sternberg, 1875 as cited by Krumbein, 1941a; Wentworth, 1931 as

cited by Krumbein, 1941a]. In this study, we cannot rule out unequivocally that the power-law

regression is inappropriate to describe the relationships between particle size and distance.

However, the possible consistency of particle size-distance relations in fluvial systems and

hillslope armoured rock fragments suggests the exponential relation may be applied more

generally, regardless of different geomorphic processes.

Figure 2.8. Comparison of (a) exponential and (c) power-law regression models of mean Feret’s diameter

along distance; and normal Q-Q plots of residuals from (b) exponential and (d) power-law regression

models on M2T1 as an example. W is the test statistic of Shapiro-Wilk test, denoting the ratio of the best

estimator of the variance.

The shape of rock particles is considered to be a significant indicator for rock evolution

0 10 20 30 40 50 60

60

80

10

01

20

(a)

Distance(m)

Fe

ret's

Dia

me

ter

(mm

)

F=50.441e0.014LDist

R2=0.799

-2 -1 0 1 2

-20

-10

01

02

03

0(b)

Theoretical Quantiles

Sa

mp

le Q

ua

ntil

es

Shapiro-Wilk normality test

W=0.98p-value=0.95

0 10 20 30 40 50 60

60

80

10

01

20

(c)

Distance(m)

Fe

ret's

Dia

me

ter(

mm

)

F =0.031L Dist1.878 53.845

R2=0.818

-2 -1 0 1 2

-20

-10

01

02

03

0

(d)

Theoretical Quantiles

Sa

mp

le Q

ua

ntil

es

Shapiro-Wilk normality test

W=0.92p-value=0.09

21

processes in sedimentary environments, which is usually measured by sphericity, a ratio of rock

surface area to its volume, or roundness that is based on the measurements of curvatures [Wadell,

1932; Krumbein, 1941b; Barrett, 1980]. In sedimentology, it is well known that the reduction in

particle size with transport distance down a river is usually accompanied by increased roundness,

as irregularly shaped rock particles wear. In an abrasion experiment using a tumbling barrel,

Krumbein [1941a] confirmed that both sphericity and roundness of rock fragments increased

with transport distance downstream. The concept of circularity measured in this study is similar

to sphericity, only in a 2-Demension. The increase in circularity downslope suggests that

transport is potentially a significant process contributing the spatial organisation of rock

fragments on hillslopes (Figure 2.6d and 2.7b). However, as abrasion studies show that the

increase in circularity slows down and reach a limit as rocks get more spherical during transport

[Krumbein, 1941a], the observed linear relationship between circularity and distance does not

show any obvious acceleration or deceleration in the change of shape (Figure 2.7b).

2.3.3 Relationship between rock fragment size and shape

Three size measurements – Feret’s diameter, perimeter and area are highly correlated as

expected. Circularity is an independent measurement that negatively correlated with size

variables with a comparatively weaker correlation (Table 2.4).

Table 2.4. Correlation matrix between rock fragment size and shape parameters from all samples.

Feret's Diameter Perimeter Area

Feret's Diameter - - -

Perimeter 0.989 - -

Area 0.967 0.988 -

Circularity -0.414 -0.346 -0.229

Measures of size could sometimes be influenced by particle shape [Dacey and Krumbein, 1979].

For example, Feret’s diameter, the longest distance between any two points on a certain particle,

could be exactly the same while rocks are diverse in shape – one spherical and one rectangle.

Relations between particle size and shape have also been studied in sedimentology. For example,

Krumbein [1941a] suggested a power function relating roundness and size in river rocks.

However, Russel and Taylor [1937] found no obvious correlation between the size and shape of

sand particles. In this study, we found a negative linear relationship between mean circularity

and mean Feret’s diameter (Figure 2.9).

22

Figure 2.9. Relationship between mean circularity and mean Feret’s diameter on (a) Mesa 1, (b) Mesa 2

and (c) Mesa 3.

Mean particle size on Mesa 1 and 2 are similar, however the mean circularity varies. On Mesa 3

mean particle size is smaller, but still significantly correlated with the shape. These slight

differences in size-shape correlation could be attributed to different rock textures and original

evolution processes. Among these three mesas, the cap rock of Mesa 2 consists of a

conglomerate with well-rounded pebbles embedded in the matrix. During weathering some of

those pebbles have been released from the matrix and are now part of the rock armour which

contributes to the larger circularity values.

2.3.4 Spatial trends in probability distributions

Results from Hartigan’s Diptest show that less than 2% of the locations have provided evidence

against unimodality at the 5% significance level (see Appendix B). The lognormal distribution

performed much better than gamma and Weibull distributions for describing Feret’s diameter. A

total of 85.14% sample points passed chi-square test for lognormal distribution, with 73.42%

pass for gamma distribution and 52.70% for Weibull distribution, while only 1% shows

evidence of consistency with power-law distribution. However, none of the tested probability

distributions were found to describe circularity.

Probability density function may provide us with an indication on the formation of the rock

fragment soil cover. For example, the Weibull distribution of particle size was predicted by a

cascade fragmentation model [Turcotte, 1992]. For small sized particles, Weibull distribution

can be reduced to a power-law relation by Taylor series [Turcotte, 1992]. Instead of a Weibull or

power-law distribution of particle size resulting from a single instantaneous fragmentation event,

we found lognormal distribution in particle size. As a natural form of particle size distribution,

lognormal distribution is derived from diffusion breakdown considering fragmentation as

temporal processes [Kolmogorov, 1941 as cited by Dacey and Krumbein, 1979; Epstein, 1948].

The fitted lognormal distributions are characterized by distribution parameters μ and σ from:

40 60 80 100 140

0.60

0.65

0.70

0.75

(a)

R2=0.621

RSE=0.026

Mea

n C

ircu

lari

ty

40 60 80 100 140

0.60

0.65

0.70

0.75

(b)

Mean Feret's Diameter (mm)

R2=0.751

RSE=0.013

40 60 80 100 140

0.60

0.65

0.70

0.75

(c)

R2=0.313

RSE=0.022

23

0],2

)(lnexp[

2

1),;(

2

2

xx

xxf

(2.6)

where x is the variable; μ and σ denote parameters of lognormal distribution that are the mean

and standard deviation of the variable’s natural logarithm , respectively, also known as location

parameter and the scale parameter.

Positive linear trends with distance are found for both parameters μ and σ on all three mesas,

including Mesa 3 (Figure 2.10). The negative relationship between distribution parameters and

distance is relatively weaker on Mesa 3 with lower r squared, but they are still significant. This

relationship is even stronger on each individual transect. Although the trends in the distribution

parameters seem minor, the equivalent changes in rock fragment size are immediately apparent.

For example, the change in the estimated μ on Mesa 1 decreases from μ=4.483 to μ=2.977

along the hillslope (μ = -0.014X + 4.093). This represents a threefold difference in the mean

particle size from 128 mm to 44 mm.

Figure 2.10. Trends in the parameters of the fitted distributions for all Feret’s diameter data. They are (a)

μ and (d) σ on Mesa 1; (b) μ and (e) σ on Mesa 2; and (c) μ and (f) σ on Mesa 3.

While mean or median size is often chosen to represent the spatial patterns of rock fragments on

hillslopes, few studies quantified the probability distribution of particle size and shape.

Probability distribution measured in this study overcomes the limitation that distributions with

0 20 40 60

3.0

3.5

4.0

4.5

(a)

R2=0.512

=0.014X 4.093

0 20 40 60

3.0

3.5

4.0

4.5

(b)

R2=0.498

=0.011X 3.927

0 10 30 50

3.0

3.5

4.0

4.5

(c)

R2=0.129

=0.006X3.523

0 20 40 60

0.6

0.8

1.0

1.2

1.4

(d)

R2=0.421

=0.004X 1.033

0 20 40 60

0.6

0.8

1.0

1.2

1.4

(e)

Distance (m)

R2=0.548

=0.006X 1.103

0 10 30 50

0.6

0.8

1.0

1.2

1.4

(f)

R2=0.191

=0.004X0.852

24

identical values for the mean or the median may have very different variance and skewness.

Parameters that describe shape and position of the distribution may provide us with another

window into assessing the evolution of rock armour. The linear change of the two distribution

parameters in space indicates that a simple modelling approach may be used to describe the

complex processes of rock armouring.

2.3.5 Implications for processes contributing to the formation of rock armour

Hillslopes are controlled by various processes depending upon elevation, slope angle, and

natural weathering properties of the bedrocks [Melton, 1965]. Spatial patterns of rock fragments

on hillslopes have been used to postulate the formation of rock armour [Abrahams et al., 1985;

Nyssen et al., 2002]. The downslope fining patterns of rock particles has been explained by

selective transport of fine materials by runoff events accumulating rock fragments on the

surface on both undisturbed and management affected hillslopes [Abrahams et al., 1985; Poesen

et al., 1998]. Surface runoff is also hypothesized as the dominant process in the studies of rock

fragments focusing on erosion and rock-soil interaction [Simanton et al., 1994; Li et al., 2007;

Zhu and Shao, 2008]. A few other processes were assessed by Nyssen et al. [2002], including

tillage, trampling, vertical sorting and rock fall on tillaged land. A “runoff creep” process, which

consists of larger rocks tilting and moving as the supporting fines are removed, was

hypothesized to be responsible for the spatial trends of particles too large to be transported by

overland flow [de Ploey and Moeyersons, 1975; Abrahams et al., 1990]. Dry ravel, which is

another mechanism for the movement of rock fragments by rolling, bouncing and sliding under

gravity is likely to result in downslope coarsening [Gabet, 2003].

The land surface of the Telfer region in the Great Sandy Desert is very old, probably older than

any of the landforms in the studies mentioned above [Henderson, 1996]. Rock fragments on the

mesa surface have a long history, with little human impact. Tillage can be ruled out in this study.

Trampling by animals tends to move large particles more rapidly downslope and is therefore

considered an unlikely mechanism for the observed sorting. Similarly, rock fall or dry ravel

which leads to a reverse sorting pattern, with larger particles found at the bottom of the hills,

contradict our findings and are therefore unlikely to be the sorting mechanism. It is evident that

the surface rocks are sourced from the cap on Mesa 1 and 2 as the rock texture of surface rocks

and the ones in soil profiles are very different, and there are insufficient rock particles in soil

profile to replenish the surface cover. Therefore, vertical sorting is unlikely to be significant.

While the surface coverage of rock fragments is slightly smaller on our mesa slopes in

comparison to other studies [Simanton et al., 1994; Poesen et al., 1998; Nyssen et al., 2002], the

particle size is however larger, indicating surface runoff alone is unlikely to generate the spatial

organization of rock armour. At a late stage of the landform evolution, surface weathering is

likely to be important for the formation of rock armour [Cooke, 1970] and is proposed to be a

25

highly significant process at our study sites.

Despite the importance of physical weathering processes for the formation of rock armour, few

studies have considered this process to explain the spatial distribution of rock fragments during

landform evolution. In the context of land use changes and engineered landforms, the formation

of rock armour as part of the physical weathering process and sediment redistribution is relevant

for designing stable landforms. Sediment transport and physical weathering as part of the

formation of stable land surfaces has been incorporated into landform evolution models

[Willgoose and Sharmeen, 2006; Wells et al., 2008; Cohen et al., 2009], However, few of these

models were able to describe the spatial patterns of rock fragments on mesa slopes as found in

this study. For example, instead of sourcing rock fragments from the cap, Willgoose and

Sharmeen [2006] hypothesized that surface rocks are being replenished from underneath after

finer soil particles have being washed out. They did not assess particle size distribution as

means of validating their modelling results against data. Contrary to our observations, Wells et

al. [2008] studied rock fragmentation and obtained multi-modal probability distributions during

the early stage of weathering. However, it appeared that a negative skewed unimodal

distribution was obtained as weathering proceeded, indicating that the later stages of weathering

may lead to unimodality that is consistent with the probability distribution of our dataset [Wells

et al., 2008]. The mARM model described by Cohen et al. [2009] yielded a downslope

coarsening in rock particles due to the combined effect of erosion, deposition and physical

weathering. There is a need to assess different processes such as fragmentation and creep that

could be responsible for rock armour evolution, and search for a proper model to describe our

observations.

2.4 Conclusion

With the investigation of geomorphic dependencies of rock fragment characteristics, we found

consistency with previous studies that surface coverage and mean particle size of rock

fragments increased with slope gradient on two concave shaped mesas. However, distinct spatial

patterns of both size and shape were also observed when surface distance from capping was

taken as the explanatory variable, including Mesa 3 which was approximately rectilinear with

constant gradient. Mean particle size decreased in a power-law and/or exponential function of

distance along the transects, while mean circularity increased linearly. Particle size was found to

be lognormally distributed on all the mesas. The lognormal distribution changed systematically

in space, with distribution parameters μ and σ decreasing linearly with distance down all mesa

slopes.

While wash-out certainly contributes to surface evolution, based on the age of the landform and

26

the very large particles, fragmentation is very likely to be a significant process on the old

landform. Further studies are required for deriving principles to aid the design of engineered

landforms.

27

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31

Chapter 3: Does fragmentation weathering explain rock

particle sorting on arid hills?

Zhengyao Nie1, Gavan S. McGrath1, Christoph Hinz1 and Tia Byrd1


Australia.

Abstract

Transport processes are often suggested as the underlying mechanisms explaining the sorting of

rock particles on arid hillslopes, whereby mean particle sizes typically decrease in the

downslope direction. Here we show that fragmentation of particles can also reproduce similar

emergent patterns. A total of 93,415 rock fragments were digitized from 222 photographs on

three mesa hills in the Great Sandy Desert, Australia. Rock fragment size was found to be

distributed lognormally, with both the location and scale parameters decreasing approximately

linearly with distance down each transect. As particles were often much larger than those which

could be expected to be transported downslope by fluvial processes on these short hills, we

assessed whether a fragmentation process could instead reproduce the observed patterns. A

dynamic fragmentation model, with just two parameters and using the particle size distribution

at the top of the hill as the initial condition, predicted the remaining particle size distributions

downslope. The results have analogies with Sternberg's Law of abrasion in rivers, which

suggests the potential for a more general principal underpinning physical weathering of rock

particles.

Keywords: Rock fragments; particle sorting; fragmentation; alternative to fluvial transport;

mesa hillslope

3.1 Introduction

Rock fragments play a significant role in the regulation of ecological and geomorphic processes

such as infiltration, evaporation and erosion. For example, surface rock size, fractional cover

and their position in the soil all affect infiltration rates in nonlinear ways [Poesen and Lavee,

1991]. Rock fragment size and cover are also key factors affecting erosion rates [de Figueiredo

and Poesen, 1998]. Due to their impacts on these processes, observational studies have

attempted to relate rock particle size to geomorphic features of the hill, such as its slope [Poesen

et al., 1998]. Both within - and between - hillslope differences in mean particle size have been

proposed to relate to slope [Abrahams et al., 1985; Parsons et al., 2009]. A feature of these

32

studies is the use of characteristic measures of rock size such as the mean or median [Abrahams

et al., 1985; Poesen et al., 1998; Zhu et al., 2008] rather than consideration of the size

distribution of particles. Dunkerley [1995] acknowledged this limitation, pointing out that the

impact on runoff, infiltration and erosion may be very different where a narrow range the rock

size distribution exists as compared to the case of a broader distribution even the means of the

two are the same. To the best of our knowledge, and rather surprisingly, spatial patterns of

surface rock size distributions on arid hills have not yet been quantified.

Preferential erosion of fine soils often explains the emergence of rock armoured surfaces

[Poesen and Lavee, 1994]. Implicit in this has been the assumption that a similar selective

transport of small rock fragments is the mechanism explaining within hill sorting that rock

fragments on the slope surfaces are often organized and typically display a decrease in particle

size downslope [Abrahams et al., 1985; Cooke et al., 1993; Parsons et al., 2009]. However,

selective transport is often observable up to certain thresholds in particle size [Kirkby and

Kirkby, 1974; Abrahams et al., 1984]. In the case of rock armoured slopes mechanisms such as

runoff creep for a much wider ranged size have been hypothesised yet remain to be confirmed

as a particle sorting process [de Ploey and Moeyersons, 1975; Abrahams et al., 1990].

This sorting phenomenon by particle size has been observed in a variety of geomorphic systems

including desert dunes, rivers and beaches [Friedman, 1962; Jerolmack and Brzinski, 2010], yet

particularly common for rock fragments on debris mantled hillslopes in arid and semiarid

environments [Cooke, 1972; Poesen et al., 1998; Parsons et al., 2009]. In some systems, rivers

for example, physical weathering that includes abrasion and/or impact breakage, as well as

selective transport is considered necessary to explain features of downstream sorting [Sklar and

Deitrich, 2006; Jerolmack and Brzinski, 2010; Chatanantavet et al., 2010]. In a fragmentation

and abrasion model that predicted sediment fining during transportation, both mechanisms were

considered for changes in particle size distribution during an experimental study [Le Bouteiller

et al., 2011].

Untangling selective transport and physical weathering in barrel tumbling experiments found

that the mean particle size of rocks decreased exponentially with distance travelled downstream

[Sternberg, 1985 as cited by Krumbein, 1941]. This is known as Sternberg’s Law of abrasion.

Krumbein [1941] and references therein not only confirmed these results but also noted rapid

and preferential initial abrasion and breakage of the largest particles. The consideration that

physical weathering of particles may play a role in size sorting has largely been neglected in

hillslope studies, with a few exceptions studying artificial rock dump geomorphology [Wells et

al., 2008; Cohen et al., 2009].

33

In deserts, where weathering is pervasive, rock exposed to the elements at the surface can

fragment by thermal fracture, salt crystal fracture, and other physical, chemical and biological

weathering processes [Cooke et al., 1993; Eppes et al., 2010] In various contexts, fragmentation

has been modelled in a number of disciplines including soil science, engineering and geology

[Turcotte, 1986; Perfect, 1997; Bittelli et al., 1999; Bird et al., 2009; Cohen et al., 2009]. In

stochastic models of instantaneous fragmentation, such that occurs in an explosion, a cascade of

fragments from a single large particle has been shown to result in power-law size distributions

when a scale invariant cascade is assumed [Turcotte, 1986; Perfect, 1997; Bittelli et al., 1999].

In earlier studies, where fragmentation was instead considered as a temporal phenomenon,

asymptotic lognormal particle size distributions were predicted to occur [Kolmogorov, 1941 as

cited by Bird et al., 2009; Epstein, 1948; Bird et al., 2009]. More recently, Bird et al. [2009]

proposed a dynamic model combining temporal and instantaneous cascade fragmentation

processes.

Motivated by the debate about the relative importance of physical weathering and selective

transport to explain downstream fining in rivers [Sklar and Deitrich, 2006; Jerolmack and

Brzinski, 2010], here we hypothesize that observed sorting of rock particle size distributions on

hillslopes can arise from a fragmentation process. In this study, we adapt a dynamic

fragmentation model to simulate the evolution of coarse rock fragments size distributions on

three arid mesa slopes from the Great Sandy Desert, Australia. Many of these rocks are expected

to be too large to be moved by overland flow. We therefore model fragmentation without any

selective transport being considered explicitly. A rock covered surface is assumed implicitly to

be maintained by selective transport of fine soil, finer than the smallest rock fragments

considered. Unlike previous studies reporting only mean or median particle sizes, this study

characterizes, quantifies and then models changes in the statistical distributions of rock particles

on arid hillslopes. Such an approach allows physical interpretation of not only of rock

weathering but also of the geomorphology of the hills studied.

3.2 Methods

3.2.1 Site description and sampling

Three mesa slopes in the Great Sandy Desert, Australia were selected for particle size analysis,

simply named as Mesa 1, Mesa 2 and Mesa 3. They are located at 21°45’46’’S, 122.9’57’’E;

21°56’01’’S, 122°12’43’’E; and 21°52’3’’S, 122°7’29’’E. These mesa slopes have a surface

coverage of up to 67.2% siltstone or silcrete conglomerate rock armour sourced from a duricrust

cap which had undergone secondary silicification, giving rise to relatively hard rocks in

comparison to the underlying siltstone [Henderson, 1996]. The typical lithology revealed by

excavation on several hills showed that the surface rock armour overlay a highly weathered

34

zone between 10 – 200 cm thick, including a clear zone without any rock fragments, before a

siltstone bedrock was intercepted. Vegetation on these mesas are usually acacia shrubs and

tussock grass, but only the latter were found on Mesa 3. The climate is arid, with a mean annual

rainfall of around 300 mm, and an average daily temperature varying seasonally between 25°C

– 42°C [Bureau of Meterology, 2011]. These mesa hills are analogous to those described in

Ollier and Tuddenham [1962] in which a duricrust capped hills, covered in a thin armour layer

sourced from the cap material retreated laterally over time. In addition, retreat of the cap was

hypothesised to control the rate of hillslope retreat.

Four transects were placed along each mesa, beginning from the lower edge of the hard

duricrust cap downhill in a straight line in the direction of the steepest descent, where duricrust

cap was not included in sampling. Transects on the three mesas were facing different aspects

respectively as northwest, south and east. Sample points were located at three meter intervals

along each transect to a distance of 60 m on Mesa 1 and 2, whereas varying from 24 m to 51 m

on Mesa 3 as the hillslope intercepted aeolian sandy deposits at the base of the hill made it

shorter. Slopes on Mesa 1 and 2 are concave shaped, with slope gradient ranging from 1% to

68%, and 2% to 61%; yet Mesa 3 slopes are approximately rectilinear, with gradient of 36%.

At each of these points a one meter square reference frame was placed on the ground and digital

photographs were taken of the surface with the following cameras: Canon IXUS750 (3x optical

zoon, 7.1 megapixels), Olympus u 1720SW (3x optical zoom, 7.1 megapixels), and Olympus

u850SW (5x optical zoom, 8 megapixels). Four reduplicate photographs were taken at each

sample point, but this number was reduced if taking photographs was prevented by obstructions,

such as shrubs. All rock fragments included in those four photographs were combined as rock

samples at each sample point for further analysis. Rock fragments larger than 1 m2 surface area

thus are not able to be included. The software ImageJ was used to rectify and standardize

images to a resolution of 0.28 mm2 pixel-1 [Rasband, 2008]. Individual rock fragments were

hand-traced with a graphics tablet system (Intuos 3 Wacom) in order to create binary images.

Rock fragment size was quantified by the Feret’s diameter, the longest distance between any

two points on the perimeter of a particle. Due to the limitations of this methodology, rock

fragments less than 10 mm in diameter, were excluded from the analysis.

The chi-square test was applied to assess whether it is reasonable to assume that particles at

each location came from truncated lognormal distributions. A significance level of 5% was

adopted for this test. Parameters of upper truncated lognormal distributions were estimated by

maximum likelihood methods using the R software environment [Ihaka and Gentleman, 1996].

35

3.2.2 Modelling fragmentation

The model, adapted from Bird et al. [2009], describes the temporal evolution of the cumulative

mass fraction of particles equal to or less than a size [L]. This mass fraction is denoted iM [-],

where i denotes to the ith size class, in which the largest size class being i = 0 and the smallest i

= n. Fragmentation causes a proportion p of this fraction to break down into smaller size, i.e.

ii pMM 1 . With this definition, the fraction of mass at time t in a given class i is:

)()()( 1 tMtMtm iii (3.1)

A size dependent proportion iQ of this mass is assumed to break down to the next size class 1ir

at time t through a time-dependent fragmentation. In addition, at time t the mass fraction

passing via a cascade from larger particles to a number of smaller sizes equal to and less than ir

through instantaneous fragmentation is given by:

)1()()( tMtMtc iii (3.2)

A size dependent proportion iP of this mass fraction is smaller than ir , then the total mass

fraction balance equation is given by:

)1()()1()( 11 tmQtcPtMtM iiiiii (3.3)

Substituting Eq.3.3 with Eq.3.1 and Eq.3.2 results in the fragmentation model:

)]1()1([)]1()([)1()( 111 tMtMQtMtMPtMtM iiiiiiii (3.4)

In order to relate the mass fraction for model to the measured Feret’s diameter, we assume mass

m can be calculated from size r via barm where [M L-3] denotes the rock mass density,

and for spherical particles 6/a and 3b . As we will demonstrate later, observed rock size

distributions were consistent with lognormal distributions. Given the above mass-size

relationship then rock mass is also lognormally distributed, with distribution parameters * and

* , that can be derived from size distribution parameters and by the form 2* b

and * [Sklar and Dietrich, 2006].

The boundary conditions necessary to solve the time dependent fragmentation model are given

by 1)(0 tM and the initial cumulative mass fraction size distribution )0(0 FM i which we

obtained from parameterised regression estimates of the and at the top of the hillslope. As

the model specifies a largest particle class, we used a truncated lognormal distribution function

with a upper bound 0r , which gives the mass fraction of class i as:

*2

*)log(1

*2

*)log(1

0

rerf

rerf

M

i

i (3.5)

36

Bird et al. [2009] gave examples of model behaviour with various forms of iP and iQ . One

particular case they presented iii pQP where p < 1 and < 1 that produced similar

preferential weathering of large particles, however, was found to be unable to reproduce

observed changes in particle size distributions. This failure initialized a search for a different

type of model, in which the probability of fragmentation was a function of particle size

explicitly. The simplest and most predictive model found was the following:

)/exp( iii rQP (3.6)

where β [T-1 or equivalently in space m-1] is a size-invariant rate of fragmentation, controlling

how quickly the distribution shifts to smaller sizes and α [L-1] is a rate coefficient controlling the

size dependence of the fragmentation rate. Best fit parameters were identified using a systematic

examination of the parameter space using the root mean square (RMS) error between modeled

size distributions and distributions fitted to the data (see Appendix C; Figure 3.1 is an

example). In each case the error surface was smooth giving confidence that the estimated

parameters were close to the global optimum. We used 50 size classes, binned exponentially,

between 10 mm and 3000 mm for modeling. The largest size class r0 = 3000 mm was

determined from estimates of the thickness of the duricrust capping on the mesas.

Figure 3.1. Results of best fit parameters (α = -87, β = 0.31) on Mesa 2 as an example of systematic

examination of model parameters by minimizing root mean square (RMS) error between modeled

distributions and observed ones.

In summary we assume rocks armoring on slope surface are sourced from the duricrust cap at

the top of the hill, and weather in place as the hill retreats laterally the surface rock armor

weathers in place, relative to surrounding rock [Ollier and Tuddenham, 1962; Ollier, 1963]. One

model time step is assumed equivalent to one additional meter from the hard cap material, a

-90 -89 -88 -87 -86 -85 -84 -83 -82 -81

0.26

0.27

0.28

0.29

0.3

0.31

0.32

0.33

0.34

0.35

0.0

99

10

.10

18

0.1

04

50

.10

73

>=

0.1

1

RMS Error

37

time for space substitution. Selective transport of fine soil, finer than the smallest rock particle

considered, 10 mm, is assumed to maintain the surface rock armor but is not considered

explicitly in the modeling. Non-selective transport such as creep, moving the surface rock

irrespective of size, may also be occurring, but is also not considered explicitly.

3.3 Results

A total of 93415 rocks were digitized from the image. Size measurements of rock fragments in

85% of 222 photographs could not be rejected as coming from truncated lognormal distributions

(p < 0.05). There are obvious systematic changes of the distributions on Mesa 1 and 2, with

modes shifting from larger to smaller particle size as well as the distributions narrowing as one

progresses downslope (Figure 3.1). Although changes in distributions are not as obvious on

Mesa 3, all hills display a significant negative linear relationship between the fitted distribution

parameters (μ and σ) and slope surface distance from the cap (Figure 3.2). All slope coefficients

were found to be significantly different from zero (p < 0.01).

Figure 3.2. Changes in the size distribution of rock fragments with position on each hill, with the

frequency histogram of observed data (bars), truncated lognormal probability density functions (pdf)

fitted to the data (line), and modeled initial (solid circle) and predicted (open circles) size distributions.

The top is 0 m, the middle 30 m (Mesa 1 and 2), 27 m (Mesa 3) and the bottom 60 m (Mesa 1 and 2) and

51 m (Mesa 3) from the duricrust cap.

0.0

00

0.0

15

Fitted lognormal pdf

Model predictions

Initial condition

Top

Me

sa 1

Middle

Bottom

Pro

babi

lity

Den

sity

Me

sa 2

0 100 200 300 400

0.0

00

0.0

10

0.0

20

Me

sa 3

100 200 300 400

100 200 300 400Feret's Diameter (mm)

38

Table 3.1. Linear regression coefficients; best fit model parameters and root mean square (RMS) errors.

For the linear regressions x denotes distance in meters from the top of each transect.

Mesa 1 Mesa 2 Mesa 3

Best-fit Parameters α -81 -87 -75

β 0.31 0.31 0.21

Linear Regressions μ 4.385-0.001x 4.329-0.008x 4.131-0.007x

σ 0.753-0.005x 0.708-0.005x 0.595-0.003x

RMS Error Model vs. data 0.10 0.10 0.14

Regression vs. data 0.57 0.48 0.36

Values of model parameters α and β are summarized in Table 3.1. All the values are in a

narrow range and both α and β are smaller on Mesa 3. The modeled μ and σ were derived from a

nonlinear least squares fit of Eq. 3.5 to the modeled particle distribution using the conversion of

distribution parameters between mass fraction and particle size described earlier. The modelled

µ and σ both decrease with distance downslope, corresponding well with observed distribution

parameters (Figure 3.2). While σ decreases in an approximately linear way, μ appears a

nonlinearly decrease on Mesa 1 and 2. Comparing predictions of observed parameters with

linear regression and fragmentation model, root mean square errors in Table 3.1 are showing

advantages of the model.

Figure 3.3. Changes in particle size lognormal distribution location μ and scale σ parameters (Eq. 3.5) as

a function of distance from duricrust cap. Shown are the empirical fits to observed data (crosses), fitted

linear regressions and 95% confidence intervals (lines) and predicted distribution parameters by the

fragmentation model (circles) given the initial particle size distribution, emphasized by the large point.

3.8

4.0

4.2

4.4

Fitted lognormal

Modelled

Linear reg.

95% Cl

Mesa 1

Mesa 2 Mesa 3

0 10 20 30 40 50 60

0.3

0.5

0.7

0 10 20 30 40 50 60 0 10 20 30 40 50 60

Distance (m)

39

3.4 Discussion

In this study, the consideration of the whole particle size distribution has elucidated systematic

variation in two parameters which quantify the whole size distribution. The lognormal

distribution was found to provide reasonable fits to the data, and its two parameters were found

to decrease approximately linearly with distance in a surprisingly simply way. A lognormal

particle size distribution is consistent with the sizes emanating from a temporal fragmentation

process [Epstein, 1948; Bird et al., 2009] rather than a pure instantaneous cascade which tend to

produce power-law distributions [Turcotte, 1986; Perfect, 1997]. Moreover, beyond general

measures of rock characteristics, the lognormal distribution allows the fragmentation model

reproducing observation well without considering real physical fracture of individual rock

fragments, and give clues not only to the rock fracture but also to the geomorphology evolution.

For the first time, a fragmentation model other than a regression to observations has described

changes in surface rock particle size distributions along a hillslope. Consistent with the results

from investigations of relationship between a characteristic particle size (e.g., mean and/or

median size) and geomorphic features (e.g., slope gradient), we found mean rock size to be

larger where the slope is steeper, as two of the three hills were concave in shape. However an

approximately rectilinear hill, Mesa 3, also had rock particle sizes decreasing downslope with

respect to distance, while size and slope obviously obeyed on relationship. Few studies report

distance downslope but similar trends are implied from some studies based upon descriptions of

hillslope shape (e.g., Abrahams et al., 1985). It remains unclear how the distribution in rock

fragment size within a hillslope relate to its form. The relationship between rock size and slope,

in the case of the concave hills, may just be coincidental and not causal.

The apparent behaviour of these rock fragments has a number of similarities with observations

of abrasion studies. Changes in the size distribution downslope suggest a preferential reduction

in the numbers of larger particles. In the condition that α < 0 the fragmentation rates of larger

rocks are faster than smaller ones; when α > 0 smaller sized rock particles preferentially

fragment (Figure 3.4). This more frequently breaking down of larger rocks had also been

observed by Cooke [1970]. In the context of rivers, Krumbein [1941] noted preferential abrasion

and some breakage of the largest rocks in a tumbler experiment, particularly during the initial

phase of the experiment. Based on similar experiments Sternberg [1875] and Schoklitsch [1914]

(as cited by Krumbein, 1941) found mean rock size, r , decreases downstream exponentially as a

function of travelled distance, x, via )exp()0()( xrxr , known as Sternberg’s Law of

abrasion. For lognormally distributed rock sizes, mean particle size )2/exp()( 2 xr . In

this study as described above, 2 and is approximately a linear function of distance

40

cx , we obtain )exp()0()( xrxr for rock particle size on mesa hills. This suggests

the potential that Sternberg’s Law is potentially a more general result applicable to physical

weathering of rock fragments.

Figure 3.4. Changes in cumulative mass fractions of (a) empirical mass data converted from observed

particle size distribution; (b) best fit from model prediction (α < 0); (c) model prediction while α = 0; and

(d) model prediction while α > 0.

Selective transport of fine materials by runoff events is the dominant explanation for the sorting.

However, more than 30% of rock fragments in this study above the threshold previously

reported for transport by overland flow of 65 mm, and with sorting already evident at very large

rock sizes, we find this explanation unsatisfactory [Abrahams et al., 1984]. A mechanism

observed by de Ploey and Moeyersons [1975] which induces a process of slow creep as a result

of selective removal of supporting fines by runoff, has been hypothesised as a sorting process

[Abrahams et al., 1990]. However, there is currently no evidence for sorting of large rock

fragments by this process. Another mechanism for the movement of dry debris and sorting as a

result of rolling, bouncing and sliding is dry ravel [Gabet, 2003]. However, this results in

downslope coarsening, opposite to the observations in our field sites. Deflation processes

described by Cooke [1970] illuminated how coarse rock fragments remained and concentrated

on surface during removal of underlying fines, which corresponded with hillslope retreating

0.0

0.2

0.4

0.6

0.8

1.0

(a) (b)

10 50 200 500 2000

0.0

0.2

0.4

0.6

0.8

1.0

(c)

10 50 200 500 2000

(d)

Diameter (mm)

Cu

mu

lativ

e M

ass

Fra

ctio

n

41

process in this study. Yet this hypothesis failed to explain particle sorting still. Nevertheless,

irrespective the mechanism for the sorting of rock fragments, we consider that selective

transport of fines which maintains the rock armour layer, removing fine physically weathered

fragments is still an essential process on these hills.

Thermal expansion and contraction, which is due to extremes of diurnal variability, significant

seasonal variations and sometimes short term fluctuations in temperature, is believed to be a

likely mechanisms, leading to rock fragmentation in the form of parallel breakdown and

irregular crazing crack [Ollier, 1963; Smith, 1988; Cooke et al., 1993; Eppes et al., 2010]. With

large fluctuations in temperature occurring in the study site, those rock fragments partially

buried into fine matrix are said to be prone to insolation weathering, because of constraint

volume change and reduced heating and cooling in the buried part leading to higher stress

gradient for fracture [Ollier, 1963; Smith, 1988]. Yet, it is unlikely that pure temperature change

having sufficient energy to break down rocks. Salt crystallization, chemical and biological

activities, abrasion and collision within subsequent fluvial, and freeze-thaw could also

contribute to particle breakdown [Cooke, 1970; Smith, 1988; Cooke et al., 1993].

In a pediment survey on Broken Hill in New South Wales of Australia, Lanford-Smith and Dury

[1964] claimed that the standard slope elment are independent of the presence of caprock.

However, rock material of the fragments on mesas in this study is consistent with the one of the

caprock. Ollier and Tuddenham [1962] suggested that the rate of hillslope retreat is controlled

by the rate of retreat of the duricrust cap. Therefore, it is the weathering of mesa cap being the

rate-limiting factor of hillslope retreat and the supply of surface rock armour. Preferential

fragmentation of larger particles will remain a rock mantled surface, but in contrast, if smaller

particles more easily breakdown, the rate of rock supply from the cap would be unable to catch

up with the rate of rocks weathering further downslope, resulting in the failure of maintaining

surface armour along the entire slope. A definitive assessment of the hypothesized

fragmentation process as an explanation for particle sorting appears to require a study to date

rock fragment surfaces. A physical mechanism explaining the estimated probability of fracture

)/exp( ii rP , (where 0 ) remains to be determined.

3.5 Conclusion

We investigated the spatial distribution of rock fragments on three debris mantled mesa in the

Great Sandy Desert, Australia. This study demonstrated the size distributions were consistent

with lognormal distributions and that the distributions changes systematically as a function of

distance from the hard duricrust cap, thought to be the source of the rock armour. A dynamic

42

model of fragmentation reproduced the observed changes in rock size distributions along the

hills. In addition some similarities with another physical weather process, abrasion, were found

with similarly preferential weathering of large particles initially and an approximate change in

mean particle size that may be consistent with Sternberg’s Law of abrasion, indicating an

exponential decline in mean particle size with distance (time). We believe further studies to

assess whole particle size distributions will be valuable for interpreting the geomorphologic and

ecological processes altering arid hillslopes.

43

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Gabet, E. J. (2003), Sediment transport by dry ravel, J. Geophys. Res., 108(B1), 2049.

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Abrahams, Springer.

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46

Chapter 4: Self-organisation of rock fragment cover on

engineered and natural mesa slopes

Zhengyao Nie1, Christoph Hinz1, Gavan S. McGrath1 and Erin Poultney1


Australia.

Abstract:

Erosion control and slope stabilization is essential for mine waste rock dumps (WRD). Rock

fragment armour on hillslopes protects subsoils from erosion. With the interest of the initial

surface development and self-organisation on engineered slopes in post–mining landscapes, the

study compared rock fragment characteristics on a mine waste rock dump and natural mesas,

and further hypothesized a wash-out process responsible for early evolution of rock armour. The

waste rock dump was designed to mimic concave shaped natural mesa to prevent erosion and to

blend in with the natural surrounding; two treatments were applied on the slope surface in the

construction, one with topsoils only, and the other with soil and rock fragment mixtures. Rock

fragments were characterized by surface cover, size, shape, and their probability distributions.

Rock fragments on the artificial slope were generally smaller, and more angular. Particle size

followed lognormal distribution, consistent with its natural analogues; circularity were from

beta distribution that has not been found on natural mesas. However, rock fragments on the

artificial slope did not exhibit a spatial pattern as natural mesa did. The differences in artificial

and natural mesas could be attributed to different lithology and evolution process. Erosion rills

measured on the artificial slope suggested surface soils without the presence of rock fragments

were less resistant to erosion. Fines were more easily to be washed out with less rock content

until the formation of rock armour which converged to values of about 32% irrespective of

initial rock content which was slightly lower than natural mesa slopes but fairly close. Rainfall

simulation experiments using different mixtures of top soil and rock fragments experiment

confirmed the convergence to a constant surface coverage after only 50 minutes of applied

rainfall. This study indicates that the formation of surface rock armour is a self-organized

phenomenon caused by different processes acting at different time scales for engineered and

natural hillslopes.

Keywords: rock fragments; artificial slope; rill erosion; rainfall simulation; water wash process;

initial surface development; self-organized pattern

47

4.1 Introduction

Reconstructing disturbed lands requires a design of stable landforms that will allow us to re-

establish post-disturbance land use and prevents rapid erosion. Post-mining landscapes are the

typical example for which stable landforms need to be engineered. A common problem with

current practice of engineered post-mining landforms is severe erosion risk in particular shortly

after slopes have been established [Lin and Herbert Jr, 1997; Lefebvre et al., 2001; Walker and

Powell, 2001; Johnson and Hallberg, 2005; Hancock et al., 2008]. The placement of rock

material and disturbed soil on the surface of engineered slopes makes the erodibility much

higher on mine waste rock dump than natural surfaces [Riley, 1995]. As a result rill and gully

erosion are common on such post-mining lands in particular in arid and semi-arid climates

[Nearing et al., 1997, Poesen et al., 2003; Hancock et al., 2008].

Hancock et al. [2003] proposed to use natural landforms as an analogue for engineered slopes of

rock waste dumps. Based on a case study in the Pilbara, Western Australia, they carried out a

geomorphic analysis of the surrounding landscape to derive the shape of the slopes that

engineered mine waste rock dumps should be designed with and suggested that concave slopes

are more likely to prevent erosion. While such concave slopes are common in arid lands,

engineered rock dumps consist usually of linear slopes interrupted by berms that accumulate

surface runoff and are major cause of tunnel and gully erosion. Concave slopes along with rock

mulch on the surface are the more desirable design to stabilize engineered landforms, but no

experimental evidence was provided to support this [Hancock et al., 2003]. Waste rock dumps at

Telfer goldmine followed the traditional design of a series of terraced benches with linear slopes

of approximately 20° with reverse graded berms. Problems of surface and tunnel erosion as well

as weed invasion occurred on those linear slopes in contrast to the natural concave shaped

slopes. Therefore a field trial was established to assess how engineered concave slopes with

different covers improve the resilience of the rehabilitation effort. Those were designed based

on natural mesas, which are stable landforms with flat tops often capped with hard debris, and

concave slopes armoured by rock fragments. These rock fragments placed in topsoils act as

mulch and play an overall positive role in infiltration and erosion control, to stabilize hillslopes

against interrill, rill and gully erosion [Brakensiek and Rawls, 1994; Poesen and Lavee, 1994;

Valentin, 1994; van Wesemael et al., 1996; Cerdà, 2001]. Accordingly, this experimental study

will focus on evaluating the role of rock fragments on stabilizing an engineered rock dump

slope and assess the self-organizing processes leading to rock armour.

Cooke [1970] proposed a wide range of different processes responsible for surface armour

development. In the early age of a hill, wash by water dominates the formation of the stone

pavements, before surface weathering gains significance. Vertical sorting with large stones

48

tending to concentrate in the soil surface due to saturation and desiccation cycles becomes

subsequently more important. In the long term all processes slow down and surface weathering

becomes the most important process in stone pavement development [Cooke, 1970]. Based on

this we hypothesize that rock fragment accumulation on the young engineered slope is caused

by wash-out. In contrast, rock fragment distribution of the old mesa slopes is most likely caused

by surface weathering and fragmentation as outlined in Chapter 3.

Accordingly, this paper reports on the characterisation of rock fragment distribution of two

contrasting site both located in the Great Sandy Desert in Western Australia, one being a

concave engineered slope and the other being natural mesa slopes with established rock armour.

The engineered slope was covered by rock fragments extracted from an open cut mined mixed

with topsoils. As this field trial was carried out as part of the mine operation to support closure

planning, the initial state of the trial was not assessed and documented. We therefore treat the

site the same way as the natural mesa site: We observe one point in time during the evolution of

the land surface and compare both sites based on the same measurements. The objective was to

perform a comparative field assessment of the coverage and size and shape distribution of

surface rock fragments on both landforms and measure the erosion indicators with a clear view

to assess how effective engineered rock pavements protect the surface in comparison to

naturally evolved surfaces. Furthermore, we carried out controlled rainfall simulation with

various mixes of soil and rock fragment to determine if our wash-out hypothesis contributes to

the self-organisation of rock armour during very early stage of hillslope evolution

4.2 Method

4.2.1 Study site

Telfer goldmine is located on the edge of Great Sandy Desert, Western Australia, at 21.71ºS,

122.23ºE. Telfer experiences hot summers and warm winters, with daily average maximum

temperature varying from 25ºC to 42ºC. The study site is in an arid environment, and the annual

rainfall is 250 – 400 mm mostly delivered as convective rain [Bureau of Meteorology, 2011].

A 1.5 year old waste rock dump in the study region was constructed to mimic local mesas, with

a plateau of large rock fragments on top, and decreasing steepness downhill: top slope is 37°,

middle 20° and bottom 15°. This mesa shaped design was to minimize erosion and to assimilate

the rock dump with the natural surroundings. The waste rocks from blasting and crushing are

siltstones, sand stones and quartzites. A 20 cm depth layer of sandstone or quartzite was placed

on top of the waste rock dump, and two treatments were applied during slope construction.

Treatment 1 was spreading topsoil to 5 cm depth on the slope surface; Treatment 2 was

spreading a mixture layer of 5 cm depth of topsoil and 20 cm depth of sandstone that was

49

premixed. Topsoils in the treatments were designed for the purpose of revegetation. They are

not from natural mesa slope, but from swales, which is essential as native plants will only return

with topsoil as the initial growth medium. Soil properties were very similar in both treatments.

Since the completion of the construction in 2006, the waste rock dump had experienced three

tropical cyclones – cyclone George, Jacob and Kara on March 2007, which brought 466 mm of

rainfall in one-month time, contributed 81.2% of annual total rainfall, and the highest rainfall

daily reached 93 mm [Bureau of Meteorology, 2011].

Five transects were placed along the waste rock dump slope from top to bottom for rock

fragment sampling, named as Transect A, B, M, C and D (Figure 1). Transect A and B were 70

m in length whereas the other three were longer at 90 m. Transect A and D intersected

Treatment 1, with bare soils on top; Transect B and C intersected Treatment 2, that surface soils

were mixed with rock fragments, forming a mixture layer on slope surface; Transect M was

located in between of the two treatments.

Figure 4.1. The digial elevation model (DEM) of designed waste rock dump and sample points on five

selected transects.

4.2.2 Rock fragments

Sample points were located at 10 m interval on each transect, resulting in a total of 41 locations

50

(Figure 4.1). A 1 m2 metal frame was placed on the surface at each sample point, where two

replicated photographs were taken for further image analysis, using a Canon Digital IXUS 70

(7.1 mega pixels) and an Olympus μ850SW (8.0 mega pixels) in JPEG format. Locations of the

sampling points were recorded by a Magellan ProMark3 RTK Differential Global Positioning

System (GPS).

All the images taken at sample points in field site and in rainfall simulation were rectified to

square and transformed to bilinear format using TurboReg plugin in Image J [Rasband, 2008].

Individual rock fragments in these bilinear images were then traced using a Wacom tablet and

Analyse Particles plug-in in ImageJ. Pixels in images were then converted to physical length

and area of rock fragments, where 1 pixel corresponded to 0.28 mm2pixel-1. Rock fragments

were characterized by size and shape, represented by Feret’s diameter and circularity

respectively, where Feret’s diameter (F) (mm) was defined as the longest distance between any

two points in the rock boundary; circularity (C) (-) was calculated based on area (A) (mm2) and

perimeter (P) (mm) via 2/4 PAC [Rasband, 2008]. Surface coverage (Ac) (%) was

calculated as:

%100T

Rc A

AA (1)

where AR denotes the sum of rock fragment area and AT is the total image size of a sample point,

which is 106 mm2.

Software environment R was used for statistical analysis of rock fragments [Ihaka and

Gentleman, 1996]. In addition to the descriptive statistics of rock fragments fragment such as

mean and median of size and shape, probability distributions were assessed. Four probability

distributions (lognormal, gamma Weibull distributions) were fitted to Feret’s diameter, and two

(beta and logit-normal distributions) to circularity. Null hypotheses of fitted distributions were

accepted at a significance level of 5% through the chi-square goodness of fit test [Ricci, 2005].

Parameters of the fitted lognormal (Eq. 4.2) and beta distribution (Eq. 4.3) were estimated by

maximum likelihood method [Venables and Ripley, 2002]. Spatial changes of descriptive

statistics and probability distributions of rock fragments were assessed, as well as inter-

relationships among surface cover, particle size and shape. These characteristics of rock

fragments were than compared to the ones on the natural analogue of this artificial slope.

0],2

)(lnexp[

2

1),;(

2

2

xx

xxf

(4.2)

where x is the variable; μ and σ denote parameters of lognormal distribution that are the mean

and standard deviation of the variable’s natural logarithm , respectively, also known as location

parameter and the scale parameter.

51

10,)1(),(

1),;( 11

2121

21 xxxB

xf

(4.3)

where x is the variable; β1 and β2 denote two positive shape parameters of beta distribution; B is

a normalization constant to ensure that the total probability integrates to unity.

4.2.3 Field evidence of erosion

In the field, erosion rills were measured in addition to rock fragment sampling. The position of

the rills were mapped using a 100 m measuring tape and a 3 m tape was used to measure rill

geometry [Nearing et al., 1997]. The data were interpolated to estimate the total length of rills,

and the rill volume. Rill shape was highly variable on the slope, however for the purpose of

estimating the rill volume, the shape was assumed to be rectangular.

4.2.4 Rainfall simulation

Rainfall simulations were conducted on the topsoil collected from the field and rock fragments

for the rock dump at Telfer, using an oscillating boom rainfall simulator [Loch et al., 2001].

Rainfall simulator SMI smart motor (Smart Motor SM2315D, Animatics Corporation, USA)

was used in this experiment to increase and decrease rainfall intensity by stepwise changing the

rate of sweeps across the plot.

Rainfall simulation trays were 75 cm x 75 cm, and were 20 cm deep with the bottom 15 cm

filled with coarse yellow sand. Shade cloth was placed on the top of yellow sand before rock

fragment and topsoil mixtures were applied, to allow movement of water while preventing

movement of soil. Rock fragments (> 2 mm) were separated from silt (< 2 mm) using sieves.

Rock fragments larger than 10 cm were not included in the simulations because of their large

size in relation to the total tray area influencing runoff [de Figueiredo and Poesen, 1998]. A

single slope angle at 15° was set up for all rainfall simulation experiments [Evans, 1980; Fox

and Bryan, 1999].

Four different compositions comprised of 50%, 60%, 70% and 80% of rock fragments by

volume with the remaining volume being top soil. Rainfall was applied at an intensity of 100

mm/hr for a short period of time until the soil surface was homogeneously wetted. Simulations

ran with rainfall intensities from lowest to highest (20, 40, 60, 80 and 100 mm/hr) for each tray,

and then from highest to lowest (100, 80, 60, 40 and 20 mm/hr). For every intensity in one run,

simulated rainfall lasted for 5 minutes, resulting in 25 minutes for one run and 50 minutes for

the complete simulation. Three trays with same rock fragment volume were used for each

simulation. Photos were taken before, between and after simulation runs to determine change in

surface rock fragment cover (Ac) using the image analysis method described in Section 4.2.2

52

4.3 Result and discussion

4.3.1Descriptive data of rock fragments

A comparison of rock fragments on artificial and natural mesas by descriptive statistics is

summarized in Table 4.1. Average surface rock cover is slightly higher on natural mesas, but

there is no major difference, consistent with our field observation that appearances of rock cover

are very similar on natural and artificial slopes. Rock fragments on natural mesas have larger

Feret’s diameter and smaller circularity meanwhile, indicating larger and rounder particles.

Coefficient of variation of both particle size and shape tends to be greater on the waste rock

dump, suggesting rock fragments on the artificial slope are more variable. The statistics of rocks

on the waste rock dump is closer to Mesa 3, an approximately rectilinear shaped mesa. As

described in Chapter 2, we did not find the same sorting of rock fragments on Mesa 3 as on

Mesa 1 and 2 when assessing mean particle size and surface cover, indicating rock fragments on

Mesa 3 were more randomly distributed in space. With outcrops consisting of quartzitic

siltstone found along the hillslope, the source for rock fragments on Mesa 3 consists of

materials from the cap as well as weathering resistant rock from within the slope. However,

lognormal distribution of particle size followed the same spatial patterns on Mesa 3 as on the

other two mesas.

Table 4.1. Descriptive statistics of rock fragment characteristics on the waste rock dump and natural mesa

slopes.

Mesa 1 Mesa 2 Mesa 3 WRD

No. of sample points 83 83 56 41

No. of rocks 37632 26693 29089 18606

Surf. Cover (%) 39.83 39.09 34.98 32.02

Feret's Diameter

Mean

(mm) 69 69 57 47

CV (%) 82 78 65 82

Min (mm) 11 14 10 11

Max (mm) 911 1008 710 762

Circularity

Mean 0.68 0.71 0.7 0.66

CV (%) 16.38 14.8 17.22 17.98

Min 0.1 0.11 0.02 0.2

Max 0.91 0.89 0.92 0.93

The differences between artificial and natural mesas are significant at p<0.05 level in the t-test,

Further assessments in probability distributions and spatial patterns of rock characteristics will

provide further insights into the characteristics of both systems.

53

4.3.2 Probability distributions

Among lognormal, gamma, Weibull and power-law probability density functions describing the

Feret’s diameter distribution of rock fragments at each sample point, lower-end truncated

lognormal distribution described the observations best. A total of 92.68% sample points passed

the chi-square test of lognormal distribution at 0.05 significance level, very close to the number

of 92.77% on Mesa 1. Maximum likelihood estimated parameters μ and σ are significantly

correlated with each other on natural mesa slopes, but not on the artificial one (Figure 4.2).

There is a marked difference in the distribution of the parameters for the mesas and the waste

rock dump. A significant positive relationship exists between μ and σ for the mesa sites, in

contrast to the artificial slope. This relationship suggests that a group of particles with smaller

mean size are less variable compared to a group of larger particles on natural mesas. This

relationship does not exist on the artificial slope. With most points having lower μ values than

the natural site, which is indicative of a smaller rock fragments, the variation in particle size on

the artificial slope is relatively greater. The μ-σ relation reflects the spatial pattern of probability

distributions on the natural site as both μ and σ decrease downslope, which does not occur on

the rock dump (see Figure 4.4). According to our fragmentation model described in Chapter 3,

μ will decrease in a weakly nonlinear pattern and σ decrease more linearly once a fragmentation

process occurs. Considering the distribution of the rock dump as the initial condition of the

model of Chapter 3, it is unlikely the probability distribution of particle size on the artificial

slope will evolve in a similar way as the natural site.

Figure 4.2. Relationship between lognormal distribution parameter ì and σ from all sites on (a) natural

mesa slopes and (b) the waste rock dump. The dark grey, grey and light grey solid circles are the

combination of distribution parameters on the top, middle and bottom of a typical transect on each system

– natural mesas and the waste rock dump, respectively as shown in Figure 4.4.

While most of the blasting studies use Weibull or power-law particle size distribution to

quantify rock fragment size distribution [Kuznetsov, 1973; Turcotte, 1986], we found that rock

0.6 0.7 0.8 0.9 1.0 1.1 1.2

2.5

3.0

3.5

4.0

4.5

(a)

0.6 0.7 0.8 0.9 1.0 1.1 1.2

(b)

54

fragments on the artificial slope are lognormally distributed, consistent with its natural

analogues. The lognormal distribution can be the result of further grinding and crushing of mine

rocks after blasting. Selective transport of fines and movement of rock fragments during

placement may have further affected the distribution. Fragmentation and creep mechanisms that

are considered to be important occurring on natural mesas for ages are less likely to dominate on

the young artificial slope.

Beta, logit-normal and Weibull distributions were assessed for the shape measurement of rock

fragments. Circularity from 70.73% locations on the artificial slope are consistent with beta

distribution. Two beta distribution parameters β1 and β2 are positively correlated and the

correlation is 0.71. Beta distribution of circularity was not found on the natural mesa slopes.

Lithology and preferred tectonic directions are likely to be the controlling factors in rock

fracture and lead to different shape distribution.

4.3.3 Spatial patterns of rock fragments

On natural Mesa 1 and 2, distinct spatial patterns of surface rock coverage, particle size and

shape were found along transect distance. The spatial trends on Mesa 3 are less obvious, but still

significant. However, these patterns hardly exist on the artificial slope. For example, although

the surface cover of rock fragments on the two contrasting systems are similar, the significantly

decreasing trends in surface cover with distance downslope are absent on the artificial slope

(Figure 4.3).

Figure 4.3. Spatial changes of rock fragment surface coverage on (a) Mesa 1; (b) Mesa 2; (c) Mesa 3 and

(d) the waste rock dump.

Furthermore, we observed linearly decreasing distribution parameters μ and σ on all three

natural mesas (Figure 4.4a and 4.4b), with distribution shape narrowing down and distribution

mode shifting to the smaller size down the mesa slopes (Figure 4.4c, 4.4d and 4.4e). No

obvious spatial trends in probability distributions are found on the waste rock dump (Figure

4.4f to 4.4j).

0 20 40 60

1030

5070

(a)

Su

rfa

ce C

ove

rag

e (

%)

0 20 40 60

(b)

0 20 40 60

(c)

0 20 40 60 80

(d)

Distance (m)

55

Figure 4.4. Lognormal distribution parameters of particle size change with respect of distance are shown

as changes of (a) μ and (b) σ down a typical mesa transect and (f) μ and (g) σ down a typical waste rock

dump transect; probability density of Feret’s diameter with fitted lognormal distribution line changes on

the mesa transect from (c) the top at 0 m from cap, (d) middle at 30 m, to (e) bottom at 60 m, and on the

waste rock dump from (h) the top at 0m, (i) middle at 40m, to (j) bottom at 80m.

Spatial trends of rock fragment characteristics (e.g., coverage, mean or median size) have been

found widely found on arid and semiarid hillslopes [Simanton et al., 1994; Abrahams et al.,

1985; Poesen et al., 1998]. For example, Poesen et al. [1998] found surface coverage of rock

fragments up to 80% and it increased with slope gradient; with the mean particle size up to 100

mm, Abrahams et al. [1985] found it increased with respect to slope gradient. As the particle

size on the natural mesas is in a similar range with Abrahams’ [1985] study, the surface cover is

much lower than Poesen’s [1998] observations. However, as concave shaped slopes were

2.8

3.4

4.0

Natural Mesa

(a)

0 20 40 60

0.7

0.9

1.1

(b)

Distance (m)

00.

03

(c)

00.

03

Pro

b. D

ensi

ty

(d)

0 100 300

00.

03


(e)

WRD

(f)

0 20 40 60 80

(g)

Distance (m)

(h)

(i)

0 100 300

(j)


56

investigated in these two studies, surface rock cover and mean particle size can be expected to

decrease downslope, which is consistent with our observations on natural mesas. Rock

fragments on the waste rock dump are more randomly distributed possibly due to the initial

placement of rock fragments and topsoil using earthmoving equipment (see Appendix D).

However, the unexpected similar surface coverage of rock fragments and the same probability

distribution found on the two contrasting systems indicate some degree of self-organized surface

development on the artificial slope. Based on Cooke’s [1970] surface evolution and the major

cyclone events accompanied with a significant amount of rainfall the engineered site

experienced, it is very likely that wash-out by surface runoff dominates the early stage of rock

armour development on the waste rock dump. The role of sediment transport by water is well

documented by the field observation of rill erosion on the waste rock dump.

4.3.4 Rill erosion in the field

In field observation, rills are affecting a much higher percentage of transect length where

Treatment 1 was applied (topsoil only); the total lengths of rills measured on Transect A and D

are 23.02 m and 60.84 m, much higher than the number on Transect B and C, which intersected

with Treatment 2 mixing topsoil with rock fragments (Table 4.2). Total rill volume is 20.33 m3

on two transects in Treatment 1, but only 3.2 m3 in Treatment 2. The longest rills with the

highest starting point are observed on Transect D with the bare soil treatment, which also has

the lowest mean rock fragment surface coverage (21.74%) (see Appendix E). On all three

natural mesas however, no rills were found.

Table 4.2. Results of rill erosion and the corresponding surface cover of rock fragments on each transect

on the waste rock dump.

Treatment 1

(bare soil)

Treatment 2

(soil and rock

fragment mixture)

Transect A D B C M

Total length of all rills (m) 23.02 60.84 4.07 6.79 -

Rill volume (m3) 7.02 13.31 1.85 1.35 -

Surface rock cover (%) 33.25 21.74 34.32 32.38 39.21

The presence of rills clearly proves the importance of sediment transport by water during the

initial stage of development. After 716.6 mm total rainfall from 69 rain days during the 18

months since the completion of construction, the treatments showed marked differences. Daily

rainfall was less than 1 mm in 33 days out of 69, 1 – 20 mm in 27 days, 20 – 50 mm in 6 days,

and 50 – 100 mm in 5 days. Daily rainfall more than 50 mm were all brought from the tropical

cyclones in one month, with the highest number of 93 mm per day. As the field trial was

designed and executed within an operational context of the mine site, we were able to collect

57

evidence only at one point in time, without a thorough characterisation of the initial conditions.

The increased stability on Transect B and C can be attributed to the presence of rock fragments

in surface soils, forming a layer of rock armour as fine sediments are removed, and protecting

the soils beneath from further erosion [Barrett, 1980; Poesen et al., 1994, Cerdà, 2001; Knapen

et al., 2007]. Accordingly greater volume of soil has been eroded from Treatment 1 without rock

fragments in surface soils on Transect A and D. Although Transect D has a lower surface cover

comparing to the convergent coverage on the other four transects, it is likely the rock cover will

increase as more soil will be eroded with less surface rock protection, and finally evolve to a

similar coverage. This convergence is also indicated by Poesen et al. [1998], that surface

coverage seemed to be in a similar range on different hillslopes when lithology was the same.

To further investigate the accumulation of rock fragment on the soil surface as a self-organizing

process, we will evaluate rainfall simulation experiments.

4.3.5 Rainfall simulation

The changes in surface coverage of rock fragments suggest surface cover appears to converge to

a full coverage through the simulated rainfall events irrespective to the initial rock content

(Table 4.3). Surface cover increases rapidly in the first turn of rainfall, and the increase slows

down in the second turn. For the mixture with 80% of rock fragments initially, surface cover

reaches 100% during the first turn of simulated rainfall, and does not change anymore. The

surface with lowest rock fragment volume changed the most in total; the surface with the

highest rock fragment volume changed the least.

With rock fragments mixed in surface soils, rock armours develop quickly as fine materials are

washed away. Surface rock cover tends to converge to 95% coverage and more independent of

the initial rock volume ≥ 50% in the mixture. Results from rainfall simulation support our wash-

out hypothesis in addition to the field observation, suggesting selective transports of fine

materials by surface runoff will cause self-organization of surface rocks. The rock content in the

mixtures is higher than our observations in the field, but is in a similar range with the

investigations by Simanton et al. [1994] (up to 75%), Poesen et al. [1998] (up to 80%) and

Nyssen et al. [2002] (up to 85%). While the experimental conditions are different from field

situations in which rainfall is highly variable even during a single event and the spatial scale is

much larger, the mixing of topsoil and rock fragments in the trays reflect the disturbance of the

material during placement in the field. The advantage of using rainfall simulation is that the

material variability and the rainfall intensity and duration can be controlled and reproducibly

repeated, and our results should be viewed as a proof of concept rather than an experimental

mimicking of the field situation.

58

Table 4.3. Changes in rock fragment surface coverage for each replicate of each volume ratio during

rainfall simulation.

Rock

Fragment

Volume

(%)

Replicate

Surface cover (%) Difference (%)

Before Between After Before -

Between

Between -

After

Before -

After

1 66.26 90.43 94.14 24.17 3.71 27.88

50 2 66.49 92.78 95.54 26.29 2.76 29.05

3 68.25 94.92 96.89 26.67 1.97 28.63

1 73.43 93.58 94.44 20.15 0.86 21.01

60 2 74.08 92.78 100 18.7 7.22 25.92

3 72.25 92.61 95.54 20.36 2.93 23.29

1 73.63 89.71 93.53 16.08 3.82 19.9

70 2 71.74 90.06 94.23 18.32 4.17 22.49

3 74.22 91.53 96.82 17.31 5.29 22.61

1 97.35 100 100 2.65 0 2.65

80 2 95.33 100 100 4.67 0 4.67

3 94.87 100 100 5.13 0 5.13

In summary, sorting of rock fragments in size and surface cover that are widely observed on arid

and semiarid hillslopes including the natural mesas in our study site are however absent on the

artificial slope. The lack of spatial patterns can be attributed to the engineered construction

instead of natural hillslope evolution processes, as well as a different lithology. However, the

similar surface rock coverage on the two very contrasting systems initialized the hypothesis of

wash-out process leading to a potential self-organized surface development on the 18 months

old waste rock dump. Rills found on the artificial slope as field evidence of erosion suggest

surface soil with rock content are more resistant to erosion. However, all transects seem to

evolve to the same surface coverage irrespective to the initial rock content as a result of

selective transport as fines being washed away. The rainfall simulation in a controlled laboratory

environment has proved the convergence in rock cover with different rock content initially, and

provides evidence that early stage rock fragment accumulation at the soil surface will occur very

quickly [Brakensiek and Rawls, 1994; Poesen et al., 2003; Hancock et al., 2008].

4.4 Conclusion

A wash-out process by water was hypothesized to be responsible for the initial surface

development on the artificial slope. Evidence were observed as: (1) a lognormal distribution in

particle size of rock fragments on the artificial slope was found rather than Weibull or power-

law distribution usually resulting from engineered processes, indicating a likely transport

59

process on slope leading to a self-organization. (2) Rill erosion was heavier on Treatment 1 with

topsoil coverage without rock fragments; in contrast, rock armour developed as fines were

removed, leading to a convergence in surface rock cover. (3) A rainfall simulation with mixes of

top soil and rock fragments clearly showed that irrespective of the initial conditions, the surface

coverage converged generally to a value of ≥95%. The independent evidences support the

formation of self-organizing and self- stabilizing surface rock armour can occur very quickly on

the young engineered slope. It furthermore indicates that placement of rock fragments on

engineered slopes will indeed stabilize the surface and may become a leading practise in

rehabilitation of engineered landforms in particular in semi-arid and arid climates in which

vegetation alone is insufficient to prevent erosion. While wash-out is the most significant

process in the initial phase of rock armour formation, results from the natural site indicates that

other much slower processes such as fragmentation and creep will continue to replenish rock

armour and provide a means of long term stabilisation of arid hillslopes. Further research is

required to determine if there is an optimum of rock fragment content in relation to the type of

fine earth matrix providing the best protection against erosion.

60

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62

Chapter 5: Summary

Obvious sorting of rock fragments was found on three selected natural mesas in the Telfer

region. Distance along mesa slopes was found to be a better predictor of rock fragment patterns

than slope. Surface rock coverage and mean particle size were found to decrease linearly with

distance down each transect. Rock fragment shape, measured by circularity, was found to

increase (become rounder) linearly downslope. The spatial trends while significant, were

weaker on Mesa 3. Downslope decreases in cover and size have been widely observed [Cooke et

al., 1993] and have previously been associated with the change in slope (e.g., Abrahams et al.

[1985]; Poesen et al. [1998]). Rock lithology and hillslope morphology were suggested to be

both responsible for and associated with the spatial organisation of rock fragments [Abrahams et

al., 1985; Abrahams et al., 1990; Poesen et al., 1998]. As we observed minor outcropping along

the Mesa 3 slope, and slightly smaller rock fragments, it may be that the weaker spatial trends

on Mesa 3 resulted from a different lithology and perhaps a different source of surface rocks.

For example the rocks on Mesa 1 and 2 were more of a conglomerate of alluvial pebbles in a

siltstone matrix, whereas Mesa 3 was much more a homogeneous siltstone. The probability

distributions of particle size and shape were assessed in addition to descriptive statistics of rock

characteristics. A lognormal distribution of Feret’s diameter was found to fit the data well at

most of the sample points, including those on Mesa 3. The two distribution parameters μ and σ

decreased approximately linearly with distance downslope.

It is widely accepted that particle sorting on hillslopes results from selective transport processes,

as a result of overland flow, mass movement, rockfall and rolling, animal trampling etc.

[Abrahams et al., 1985; Abrahams et al., 1990; Simanton et al., 1994; Nyssen et al., 2002;

Poesen et al., 1998]. At our field sites rock fragments were much larger than could be expected

to be transported by overland flow, and yet downslope fining was observed, even near the top of

the hill [Kirkby and Kirkby, 1974; Abrahams et al., 1990]. Other than selective wash out, a wide

range of processes are suggested by Cooke [1970] that may take place during the evolution of

surface rock armour, including creep, vertical sorting of rock fragments, and surface weathering

that dominates in a later stage of the landforms. Due to the very old age of the natural mesa

slopes, surface weathering is likely to be more important than other processes. Therefore, we

hypothesized an in-situ fragmentation accompanied by hillslope retreat yielded the observed

sorting, while not ruling out the possibility that other non-sorting transport processes may also

be at work. With a fragmentation model based upon the probability distribution of rock size, we

assumed a space for time substitution, such that a 1 m displacement along the surface of the

slope was equivalent to one model time step. The modelled fragmentation was considered as a

diffusion process with a probability of fragmentation decreasing as particle size decreased. The

63

model described changes in the rock size distribution well, particularly the preferential loss of

larger particles from the size distribution. The observed changes in mean particle size and the

preferential loss of large particles have analogies with the phenomena of abrasion in rivers

[Krumbein, 1941]. These similarities indicate that the weathering of rocks, probably by salt or

thermal fracture, on hills and the physical weathering by abrasion in rivers share similar

emergent patterns.

Rock fragments were also studied on an artificial waste rock dump, designed to mimic the

natural mesas. The artificial mesa hill was concave shaped and covered by various mixtures of

topsoils and rock fragments, aiming to prevent erosion and achieve long term stability. With a

similar surface coverage, mean particle size and circularity on the artificial slope were slightly

smaller than natural mesas. However, rock fragments on the waste rock dump failed to show a

distinct spatial pattern. The lognormal distribution of Feret’s diameter was found again to

describe the artificial hills rock fragments, consistent with natural mesas. The Weibull or power-

law distributions predicted by rock blasting models however did not fit well [Kuznetsov, 1973;

Turcotte, 1986]. A wash-out process was hypothesized to dominate in the initial development of

rock armour on this engineered slope. Rills as field evidence of erosion supported out

hypothesis that fines were more easily to be eroded without the presence of rock fragments in

soil surface, and a convergence in surface cover is likely to be achieved eventually irrespective

of the initial rock content in soil. During a controlled rainfall simulation in laboratory conducted

on mixed soil and rocks, rock armour developed quickly as fines were washed away, and the

surface cover converged with different rock content initially. The results indicate that there is

likely a rapid initial phase of self-organizing of rock cover on the artificial hills. As rock armour

developed, they appeared to slow down the erosion and play a positive role in stabilization

[Brakensiek and Rawls, 1994; Poesen and Lavee, 1994; Cerdà, 2001]. From the field

observation and laboratory experiment, wash-out is likely to be the dominating mechanism on

the young artificial slope responsible for surface development rather than in-situ fragmentation

of rock fragments, which possibly occurs on natural mesas.

With a unique dataset including 112,142 rock fragments from 263 locations on natural and

artificial arid mesa slopes, size and shape of individual rocks were measured, and the particle

size and shape distributions were quantified for the first time. This opened a window to

understanding the self-organisation of rock spatial patterns in the context of geomorphic

processes and rock particle weathering. The comparison study of rock fragments on the

engineered waste dump also suggested an initial rapid self-organization of surface rock cover,

but through a wash-out process at the early stage of landform evolution. Therefore, the two

contrasting sites developed rock armoured surfaces contributing to stabilization by self-

organization, but through two very different processes.

64

Overall, the series of studies characterized, quantified and modelled surface rock fragments on

arid hillslopes, and tried to find implications to slope surface evolution of natural mesas and

artificial waste rock dumps. With these interests, the studies hopefully will contribute to a better

understanding in rock fragments and their geomorphologic impacts. Further studies of dating

the age of natural mesas will contribute to confirming rock surface processes leading to the

sorting phenomenon.

65

References

Abrahams, A. D., A. J. Parsons, R. U. Cooke, and R. W. Reeves (1984), Stone movement on

hillslopes in the Mojave Desert, California: A 16 year record, Earth Surf. Process.

Landforms, 9(4), 365-370.



Geol., 93(3), 347-357.



Brakensiek, D. L., and W. J. Rawls (1994), Soil containing rock fragments: effects on



Eur. J. Soil Sci., 52(1), 59-68.




Kirkby, A., and M. J. Kirkby (1974), Surface wash at the semi-arid break in slope, Zeitschrift

fur Geomorphologie Supplement Band(21), 151-176.


fragments, J. Geol., 49(5), 482-520.

Kuznetsov, V. (1973), The mean diameter of the fragments formed by blasting rock, J. Min. Sci.,

9(2), 144-148.

Nyssen, J., J. Poesen, J. Moeyersons, E. Lavrysen, M. Haile, and J. Deckers (2002), Spatial

distribution of rock fragments in cultivated soils in northern Ethiopia as affected by lateral

and vertical displacement processes, Geomorph., 43(1-2), 1-16.

Poesen, J., and H. Lavee (1994), Rock fragments in top soils significance and processes, Catena,

23(1-2), 1-28.

Poesen, J. W., B. van Wesemael, K. Bunte and A. S. Benet (1998), Variation of rock fragment


23(2-4), 323-335.Turcotte, D. L. (1986), Fractals and Fragmentation, J. Geophys. Res.,

91(B2), 1921-1926.

Turcotte, D. L. (1992), Fragmentation, in Fractals and Chaos in Geology and Geophysics, pp.

20-34, Cambridge University Press, New York.

66

Appendices

Appendix A. The R script: Assessment of gamma, Weibull and lognormal distributions for

particle size

fitting<-function(data){

f.g<-fitdistr(data,"gamma") # fit the gamma distribution

g.shape.est<-f.g[[1]][[1]] # extracting the parameter estimates

g.rate.est<-f.g[[1]][[2]]

f.w<-fitdistr(data,"weibull") # fit the weibull distribution

w.shape.est<-f.w[[1]][[1]]

w.scale.est<-f.w[[1]][[2]]

f.l<-fitdistr(data,"lognormal") # fit the lognormal distribution

meanlog.est<-f.l[[1]][[1]]

sdlog.est<-f.l[[1]][[2]]

return(data.frame(g.shape.est,g.rate.est,w.shape.est,w.scale.est,

meanlog.est,sdlog.est))}

library(MASS) # required for fitdistr () function

chi.fit.shift<-function(data){

data<-data-floor(min(data))

b<-seq(min(data),max(data),length=20)

data.bin<-cut(data,breaks=b)## binning data into 19 bins of equal length with breaks

defined by 'b' (see above line)

binned.data.table<-table(data.bin) ## put the data frequencies for each bin in a table

f<-fitting(data) ## fit the three different distributions to the data and estimate the

parameters

# FREQUENCY = PROBABILITY * TOTAL NUMBER

# GAMMA

f.ex.g<-vector()

for(i in 1:(length(b)-1)) f.ex.g[i]<-(pgamma(b[i+1],shape=f[[1]],rate=f[[2]])-

pgamma(b[i],shape=f[[1]],rate=f[[2]]))*length(data)

f.ex.g<-ceiling(f.ex.g) # expected frequencies vector

# WEIBULL

f.ex.w<-vector()

for(i in 1:(length(b)-1))

f.ex.w[i]<-(pweibull(b[i+1],shape=f[[3]],scale=f[[4]])-

pweibull(b[i],shape=f[[3]],scale=f[[4]]))*length(data)

f.ex.w<-ceiling(f.ex.w) ## expected frequencies vector

# LOGNOMRAL

f.ex.l<-vector()

67

for(i in 1:(length(b)-1))

f.ex.l[i]<-(plnorm(b[i+1],meanlog=f[[5]],sdlog=f[[6]])-

plnorm(b[i],meanlog=f[[5]],sdlog=f[[6]]))*length(data)

f.ex.l<-ceiling(f.ex.l) ## expected frequencies vector

# OBSERVED FREQUENCY

f.os<-vector()

for (i in 1:length(binned.data.table)) f.os[i]<-binned.data.table[[i]]

# Calculating the chi-sq statistics

X2.g<-sum(((f.os-f.ex.g)^2)/f.ex.g); X2.w<-sum(((f.os-f.ex.w)^2)/f.ex.w); X2.l<-

sum(((f.os-f.ex.l)^2)/f.ex.l)

# degrees of freedom is equal to (# bins - # parameters estimated (2 for each

distribution) - 1)

df<-(length(b)-1)-2-1

# Calculating the p-values

p.g<-1-pchisq(X2.g,df)

p.w<-1-pchisq(X2.w,df)

p.l<-1-pchisq(X2.l,df)

best<-vector()

if(p.l>=0.05) {best<-"Lognormal"} else {

if (p.g>=0.05) {best<-“Gamma”}} else{

if(p.w>=0.05) {best<-“Weibull”}}

n<-length(f.os)

##gamma distribution##

g.rgof.ind<-sum(abs(f.os-f.ex.g))/sum(f.os)

g.rgof.ind2<-(sum((f.os-f.ex.g)^2/n))^0.5/sum(f.os/n)

g.rgof.ind22<-(sum((f.os-f.ex.g)^2)/sum((f.ex.g)^2))^0.5

##weibull distribution##

w.rgof.ind<-sum(abs(f.os-f.ex.w))/sum(f.os)

w.rgof.ind2<-(sum((f.os-f.ex.w)^2/n))^0.5/sum(f.os/n)

w.rgof.ind22<-(sum((f.os-f.ex.w)^2)/sum((f.ex.w)^2))^0.5

##lognormal distribution##

l.rgof.ind<-sum(abs(f.os-f.ex.l))/sum(f.os)

l.rgof.ind2<-(sum((f.os-f.ex.l)^2/n))^0.5/sum(f.os/n)

l.rgof.ind22<-(sum((f.os-f.ex.l)^2)/sum((f.ex.l)^2))^0.5

p<-round(c(p.g,p.w,p.l),digit=3)

d22<-round(c(g.rgof.ind22,w.rgof.ind22,l.rgof.ind22),digit=3)

return(c(f,d22,p,best)) ## the output of the function consists of a vector of: the

estimated parameters, the p-values of the tests, the relative % goodness of fit measures,

the degrees of freedom and the best fitting distribution

}

colnames<-

c("GammaShape","GammaRate","WeibShape","WeibScale","LnormMean","LnormSd","d22Gamma","d22

68

Weib","d22Lnorm","pGamma","pWeib","pLnorm","BestFit")

# Apply to M1T1 as an example

DistrFit.F.m1t1<-t(sapply(F.m1t1,chi.fit.shift))

69

Appendix B. Results of the probability distribution assessments

Hartigan's Diptest

When Hartigan’s Diptest [Hartigan, 1985] was carried out on the data, the results were positive

with very few datasets providing evidence against unimodality at the 5% level. These are noted

in Table 6.1 below.

Table 6.1. Hartigan's Diptest Results. Please note that the distance here in the “Transect and Location”

column is the distance from the bottom of the transect.

Transect and Location Data displaying evidence against unimodality

M1T1_54m Circularity



M2T2_54m Perimeter and Feret’s Diameter

M3T3_27m Perimeter, Circularity and Feret’s Diameter

M3T3_36m Perimeter, Circularity and Feret’s Diameter

M3T3_39m Feret’s Diameter

M3T4_6m Perimeter

Fitting distributions

Lognormal, gamma and Weibull probability distributions were assessed for all the sample

locations, and results are tabulated below. In Table 6.2, the “percentage passed chi-square test”

refers to the proportion of locations for which the data can be assumed to come from the

corresponding fitted distribution, i.e. p-value ≥0.05. “Best fit” refers to the best fitting

distribution, i.e. the distribution with the highest proportion of locations passed, the highest

mean p-value or the lowest mean d22 value.

70

Table 6.2. Results of fitted distributions (lognormal, gamma and weibull) to Feret’s diameter of rock

fragments on three mesas.

Mesa 1 - 83 locations Lognormal Gamma Weibull Best Fit

Percentage passed chi-square test (>= 0.05) 92.77% 71.08% 54.22% Lognormal

Mean p-value 0.341 0.280 0.220 Lognormal

Mean d22 value 0.118 0.154 0.185 Lognormal

Mesa 2 - 83 locations

Percentage passed chi-square test (>= 0.05) 83.13% 85.54% 69.88% Gamma

Mean p-value 0.312 0.349 0.260 Gamma


Mesa 3 - 56 locations

Percentage passed chi-square test (>= 0.05) 76.79% 58.93% 25.00% Lognormal

Mean p-value 0.330 0.231 0.067 Lognormal


Model validation

Further model validation was carried out using graphical residual analysis (Figure 6.1 as an

example). There are no major concerns of the fit of linear models in general.

The plots in Figure 6.1 are as follows:

1. A plot of the data points (distance v. Parameter value at that point) together with the

fitted regression line;

2. A histogram of the residuals εi = yi – ẏi where yi are the observed parameter values and

ẏi are the estimated parameter values from the fitted regression line;

3. A plot of the fitted values against the residuals εi;

4. A QQ-plot of the residuals with 95% confidence envelope for the normal distribution. If

the points lie within the envelope, there is no reason to suppose the residuals are not

normally distributed.

71

Figure 6.1. Graphical residual analysis for the fitted regression lines for the area data from Mesa 1.

72

Appendix C. The R script: Searching for best model parameters (Mesa 1 as an example)

# loading field observation data

load("E:\\Chapter 1 Spatial Distribution of Rock Fragments\\R

outputs\\data_prepare_filtered.RData")

# required library

library(stats); library(MASS)

###########################################################

############ Prepare Mesa 1 #############

###########################################################

# combine all 4 transects on mesa 1 with distance from bottom

# NB: RF samples were taken from slope bottom to top. So 1 is the bottom and 21 is the

top

F.m1<-list();

F.m1[[1]]<-c(F.m1t1[[1]],F.m1t3[[1]],F.m1t4[[1]])

for (i in 2:21) {F.m1[[i]]<-c(F.m1t1[[i]],F.m1t2[[i-1]],F.m1t3[[i]],F.m1t4[[i]])}

##smallest and largest value of Feret's D in Mesa 1 for truncating distribution later

min<-sapply(F.m1,min); max<-sapply(F.m1,max) ##max and min value of each F in each

interval

F.min<-min(min); F.max<-max(max) ##max and min of F in whole trasects

# Truncated lognormal distribution with upper bound

trnclognormal<-function(x,a,b){

pdf<-vector();

pdf<- dlnorm(x ,meanlog=a,sdlog=b)/plnorm(F.max,meanlog=a,sdlog=b,

lower.tail = TRUE)

return(pdf)

}

fit<-function(x){

fit.l<-fitdistr(x,trnclognormal,start = list(a = mean(log(x)), b =

sd(log(x))),method = "SANN")

meanlog.est<-fit.l[[1]][[1]]

sdlog.est<-fit.l[[1]][[2]]

meanlog.sd<-fit.l[[2]][[1]] ##estimated standard error of meanlog

sdlog.sd<-fit.l[[2]][[2]] ##estimated standard error of sdlog

return(data.frame(meanlog.est,sdlog.est,meanlog.sd,sdlog.sd))

}

# fitting lognormal distribution to observed Feret's D

fit.data<-sapply(F.m1,fit)

meanlog.data<-vector(); sdlog.data<-vector(); meanlog.sd<-vector();sdlog.sd<-vector()

for (i in 1:21) {meanlog.data[i]<-fit.data[,i]$meanlog.est; sdlog.data[i]<-

fit.data[,i]$sdlog.est}

meanlog.sd<-vector();sdlog.sd<-vector()

for (i in 1:21) {meanlog.sd[i]<-fit.data[,i]$meanlog.sd; sdlog.sd[i]<-

fit.data[,i]$sdlog.sd}

# linear regression of parameters

meanlm<-lm(meanlog.data~rev(m1t1.d)); am<-meanlm$coef[[2]]; bm<-meanlm$coef[[1]]; rm<-

summary(meanlm)$r.squared

73

meanlog.reg<-am*m1t1.d+bm

sdlm<-lm(sdlog.data~rev(m1t1.d)); as<-sdlm$coef[[2]]; bs<-sdlm$coef[[1]]; rs<-

summary(sdlm)$r.squared

sdlog.reg<-as*m1t1.d+bs

# transfer rock class size into Diameter represented as "D"

n<-50 ##50 size class of rocks

# let D.min be the smallest value of Feret's.

# Because the sampling method is limited to get larger particle size than 1m square

which may be in the real case. So assume a larger D.max is 3000 mm

D.min<-F.min; D.max<-3000

b<-(D.max/D.min)^(1/(n-1)) ##base

D<-vector()

for (i in 1:n) {D[i]<-F.min*b^(n-i)}

# the initial particle size distribution can be generated from the regression data

initial.cdf.freq<-plnorm(D,meanlog=meanlog.reg[1],sdlog=sdlog.reg[1])/

plnorm(D.max,meanlog=meanlog.reg[1],sdlog=sdlog.reg[1],lower.tail = TRUE)

# which means the initial mass distribution is known

meanlog.mass<-vector(); sdlog.mass<-vector()

meanlog.mass[1]<-meanlog.reg[1]+3*sdlog.reg[1]^2; sdlog.mass<-sdlog.reg[1]

initial.cdf.mass<-plnorm(D,meanlog=meanlog.mass[1],sdlog=sdlog.mass[1])/

plnorm(D.max,meanlog=meanlog.mass[1],sdlog=sdlog.mass[1],lower.tail = TRUE)

# cdf.mass: restores cumulative mass in every 3 time step, to correspond cumulative

frequency in dataset

cdf.mass<-list(); cdf.mass[[1]]<-initial.cdf.mass

###########################################################

##### Systematic Search of the Best Fit ALPHA and BETA ####

###########################################################

# the determination of these two parameters is to minimize the sum of RMS (root mean

square) of the modelled results and the observation

# beta and alpha are two model parameters. Beta is in [0,1], alpha is the size sensitive

parameter, when alpha<0 preferential fragmentation of larger particles; when alpha=0

no size dependency; when alpha>0 preferential fragmentation of small particles

opt<-function(m){

beta<-df[m,]$beta; alpha<-df[m,]$alpha

##INITIALIZATION

cdf.mass.old <-initial.cdf.mass; cdf.mass.new <-vector(); cdf.mass.new[1]<-1;

time<-60; ##total time steps for fragmentation process

time.inter<-3; ##time interval

for (t in 1:time){

for (i in 1:(n-1))

{cdf.mass.new[i+1]<-cdf.mass.old[i+1]+beta*exp(alpha/D[i])*(cdf.mass.new[i]-

cdf.mass.old[i+1])}

cdf.mass.old<-cdf.mass.new;

# restore the CDF every 3 time step (= 3 m length on slope)

74

if(t%%time.inter==0) {

j<-t/time.inter+1;

cdf.mass[[j]]<-cdf.mass.old

d<-data.frame(D,cdf.mass.old)

fit.mod<-nls(cdf.mass.old~plnorm(D,meanlog=a,sdlog=b,lower.tail = TRUE, log.p =

FALSE)/

plnorm(D.max,meanlog=a,sdlog=b,lower.tail = TRUE, log.p = FALSE),

data=d,start=list(a=3,b=0.5),

lower=c(0.1,0.01),upper=c(10,1),algorithm = "port",

control = list(maxiter = 50, printEval = FALSE, warnOnly = TRUE))

meanlog.mass[j]<-coef(fit.mod)[1];

sdlog.mass[j]<-coef(fit.mod)[2];

} ##end of if

}##end of for

# convert back from mass distr to particle size distr

meanlog.p.mod<-vector(); sdlog.p.mod<-vector();

sdlog.p.mod<-sdlog.mass;

meanlog.p.mod<-meanlog.mass-3*sdlog.p.mod^2;

# difference between modelled one and observed one

rms.meanlog.data<-(sum((meanlog.p.mod-

rev(meanlog.data))^2)/length(meanlog.data))^0.5

rms.sdlog.data<-(sum((sdlog.p.mod-rev(sdlog.data))^2)/length(sdlog.data))^0.5

rms.meanlog.reg<-(sum((meanlog.p.mod-meanlog.reg)^2)/length(meanlog.data))^0.5

rms.sdlog.reg<-(sum((sdlog.p.mod-sdlog.reg)^2)/length(sdlog.data))^0.5

rms.sum<-rms.meanlog.data+rms.sdlog.data

return(data.frame(beta,alpha,rms.sum))

}

# 10 time 10 combination of beta and alpha. We aimed preferential fragmentation of

larger particles to reproduce field observation, so alpha<0

beta<-rep(seq(0.1,1,0.1),10)

alpha<-c(rep(-80,10),rep-(85,10),rep(-90,10),rep(-95,10),rep(-100,10),

rep(-105,10),rep(-110,10),rep(-115,10),rep(-120,10),rep(-125,10))

df<-data.frame(cbind(p=p,alpha=alpha))

a1<-t(sapply(c(1:10),opt)); a2<-t(sapply(c(11:20),opt)); a3<-t(sapply(c(21:30),opt));

a4<-t(sapply(c(31:40),opt)); a5<-t(sapply(c(41:50),opt)); a6<-t(sapply(c(51:60),opt));

a7<-t(sapply(c(61:70),opt)); a8<-t(sapply(c(71:80),opt)); a9<-t(sapply(c(81:90),opt))

a10<-t(sapply(c(91:100),opt))

par.best.fit<-rbind(a1,a2,a3,a4,a5,a6,a7,a8,a9,a10)

write.table(par.best.fit,file="par.best.fit.txt")

# from the result we narrow down the range beta in 0.25:0.35; alpha in -85:-76

p<-rep(seq(0.26,0.35,0.01),10)

alpha<-c(rep(-85,10),rep(-84,10),rep(-83,10),rep(-82,10),rep(-81,10), rep(-80,10),rep(-

79,10),rep(-78,10),rep(-77,10),rep(-76,10),)

75

df<-data.frame(cbind(p=p,alpha=alpha))

b1<-t(sapply(c(1:10),opt)); b2<-t(sapply(c(11:20),opt)); b3<-t(sapply(c(21:30),opt));



b10<-t(sapply(c(91:100),opt))

par.best.fit<-rbind(b1,b2,b3,b4,b5,b6,b7,b8,b9,b10)

write.table(par.best.fit,file="par.best.fit.txt")

# The best fit result is: beta=0.31 and alpha=-81

76

Appendix D. Further results of rock fragment analysis on the artificial waste dump slope

as a comparison to its natural analogues

Characteristics of rock fragment size and shape

Welch two sample t test performed on two groups of rock size from natural site and waste show

the two means are statistically different from each other. Boxplots in Figure 6.2, drawn with

width proportional to the square-roots of the number of observations in the group, is showing

that waste rock dump and natural mesas have rock fragments sized in the similar range, and

both of them have extreme values. Rock size on waste rock dump is slightly smaller.

Figure 6.2. Boxplots of all Feret’s diameter data on three natural mesa slopes and the waste rock dump.

With the result from natural mesas, smaller rock fragments usually accompany with larger

circularity. However, even the rock size is smaller on the waste rock dump circularity is still

lower, indicating more angular rock fragments on the waste rock dump. The plot whiskers

extended range is wider on the waste rock dump comparing natural mesas, however with fewer

extremes (Figure 6.3).

Figure 6.3. Boxplots of all circularity data on three natural mesa slopes and the waste rock dump.

1020

5010

020

050

0

Fer

et's

Dia

met

er (

mm

)

M1 M2 M3 WRD

0.0

0.2

0.4

0.6

0.8

1.0

Circ

ular

ity

M1 M2 M3 WRD

77

Probability distributions

Beta, logit-normal and Weibull distributions were assessed for the shape measurement of rock

fragments. Circularity from 70.73% locations on the artificial mesa are consistent with beta

distribution. Two beta distribution parameters β1 and β2 are positively correlated (Figure 4a). As

two beta distribution parameters increase, circularity concentrates at a higher peak and the value

goes larger (Figure 6.4b, 6.4c and 6.4d). On natural mesas however, null hypotheses of beta,

logit-normal and weibull distributions of circularity have all been rejected.

Figure 6.4. (a) relationship between beta distribution parameters β1 and β2, with (b), (c) and (d) density

circularity histograms corresponding to solid circles of different combination of β1 and β2, and the fitted

beta distribution line.

Interrelationships among between rock fragment cover, size and shape

Mean particle size of rock fragments is found to be significantly related to particle shape on

Transect A, B and C (Figure 6.5a). Although relationship is missing on the other transects,

particle size is overall negatively correlated to the shape with the correlation of -0.52,

suggesting that larger rock fragments are often more angular. The size and cover relation is even

stronger appearing on four of the transects (Figure 6.5b). However, there are no strong

relationship between circularity and surface cover (Figure 6.5c).

8 10 12 14 16

45

67

89

1

2

(a)

Circ.

Den

sity

0.2 0.4 0.6 0.80

12

34

5 (b)

Circ.

Den

sity

0.2 0.4 0.6 0.8

01

23

45 (c)

Circ.

Den

sity

0.2 0.4 0.6 0.8

01

23

45 (d)

78

Figure 6.5. Inter-relationship between (a) mean circularity and mean Feret’s diameter; (b) rock surface

coverage and mean Feret’s diameter and (c) rock surface coverage and mean circularity on five transects

on the waste rock dump.

The negative relationship between rock fragment size and shape is usually recognized as an

indicator of transport processes, during which particles wear, getting smaller in size and rounder

in shape [Krumbein, 1941; Barrett, 1980]. Surface cover is higher on sample locations where

rock fragments are generally larger. This cover-size relation can be resulted from the limitations

in methodology, as individual large rocks occupy relatively large areas in a limited sample

square (1 m2), resulting in over-estimated rock coverage. However, we observed a more

frequent emergence of “stone pavement” in the field where larger rock fragments were found. It

can also be an indicator of transport process and fine materials were more easily to be washed

away at these locations where high intensity runoff was experienced.

Spatial changes in rock fragment characteristics

On natural Mesa 1 and 2, distinct spatial patterns of surface rock cover, particle size and shape

were found along transect distance, as well as the probability distribution of particle size.

However, spatial patterns are barely found on the artificial slope. No relationship was found

between surface rock coverage and distance, while Feret’s diameter shows a decreasing trend

downslope only on Transect B; circularity was significantly related with distance on Transect B

and C, but randomly distributed on other transects (Figure 6.6).

30 50 70

0.60

0.64

0.68

A(a

)Cir

cula

rity

30 50 70

B

30 50 70

C

Feret's Diameter (mm)30 50 70

D

30 50 70

M

30 50 70

1030

50(b

)Sur

face

Cov

er(%

)

30 50 70 30 50 70Feret's Diameter (mm)

30 50 70 30 50 70

0.60 0.64 0.68

1030

50(c

)Sur

face

Cov

er(%

)

0.60 0.64 0.68 0.60 0.64 0.68Circularity

0.60 0.64 0.68 0.60 0.64 0.68

79

Figure 6.6. Spatial changes of (a) surface rock cover; (b) mean Feret’s diameter and (c) mean circularity

along distance on five transects on the waste rock dump.

Size characteristics on top, middle and bottom slopes tell the spatial patterns in a clear way

(Figure 6.7). Larger rock fragments locate on the top of mesa slopes, resulting less samples in

the same sample area. Feret’s diameter decreases downslope, with smaller ranges in size.

Similar trends are not shown on waste rock dump transect.

Figure 6.7. Changes of Feret’s diameter orienting downslope on top, middle and bottom of a typical mesa

transect and a typical waste rock dump transect.

1030

50

A

(a)S

urfa

ce C

over

(%) B C D M

3050

70(b

)Fer

et's

Dia

met

er(m

m)

0 20 60

0.60

0.64

0.68

(c)C

ircul

arity

0 20 60 0 20 60Distance (m)

0 20 60 0 20 60

1020

5020

050

0

Fer

et's

Dia

met

er (

mm

)

Top Middle Bottom

(a)

1020

5020

050

0

Top Middle Bottom

(b)

80

Appendix E. Field evidence of erosion

Rill erosions on the waste rock dump are visualized in the photographs taken from the top of the

slope looking down (Figure 6.8). No distinct rill erosion can be seen on Transect B (Figure

6.8a), while rill erosion was clearly evident on Transect D (Figure 6.8b).

Figure 6.8. Photographs looking downslope from the top of (a) Transect B (in Treatment 2) and (b)

Transect D (in Treatment 1).

81

References

Barrett, P. J. (1980), The shape of rock particles, a critical review, Sedim., 27(3), 291-303.


fragments, J. Geol., 49(5), 482-520.

Hartigan, P. M. (1985), Algorithm AS 217: computation of the dip statistic to test for

unimodality, J. Roy. Stat. Soc. C-App, 34(3), 320-325.

rock fragment characteristics, patterns and processes on

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