Accepted Manuscript Intrinsic versus extrinsic variability of analogue sand-box experiments-insights from statistical analysis of repeated accretionary sand wedge experiments Tasca Santimano, Dr. Matthias Rosenau, Prof. Dr. Onno Oncken PII:
S0191-8141(15)00073-5
DOI:
10.1016/j.jsg.2015.03.008
Reference:
SG 3205
To appear in:
Journal of Structural Geology
Received Date: 5 November 2014 Revised Date:
5 March 2015
Accepted Date: 13 March 2015
Please cite this article as: Santimano, T., Rosenau, M., Oncken, O., Intrinsic versus extrinsic variability of analogue sand-box experiments-insights from statistical analysis of repeated accretionary sand wedge experiments, Journal of Structural Geology (2015), doi: 10.1016/j.jsg.2015.03.008. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Intrinsic versus extrinsic variability of analogue sand-box experiments- insights from statistical analysis of repeated accretionary sand wedge experiments. Tasca SANTIMANOa*, Matthias ROSENAUa, Onno ONCKENa a
German Research Center for Geosciences,
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Section 3.1 Lithosphere Dynamics Telegrafenberg C 14473, Potsdam
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*Author of Correspondence- Tasca Santimano
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Germany
Section 3.1 Lithosphere Dynamics Telegrafenberg C, Potsdam 14473, Germany
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Telephone- +49 331 288 1315
Email-
[email protected]
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Dr. Matthias Rosenau
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Email-
[email protected]
Prof. Dr. Onno Oncken Email-
[email protected]
Keywords- analogue modelling, variability, accretionary sand wedge, statistical analysis, Chi square test, ANOVA test
ACCEPTED MANUSCRIPT Abstract
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Analogue models are not perfectly reproducible even under controlled boundary conditions
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which make their interpretation and application not always straight forward. As any scientific
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experiment they include some random component which can be influenced both by intrinsic
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(inherent processes) and extrinsic (boundary conditions, material properties) sources. In order
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to help in the assessment of analogue model results, we discriminate and quantify the intrinsic
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versus extrinsic variability of results from “sandbox” models of accretionary wedges that
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were repeated in a controlled environment. The extrinsic source of variability, i.e. the
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parameter varied is the nature of the décollement (material, friction and thickness).
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Experiment observables include geometric properties of the faults (lifetime, spacing, dip) as
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well as wedge geometry (height, slope, length).
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For each variable we calculated the coefficient of variance (CV) and quantified the variability
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as a symmetric distribution (Normal, Laplacian) or asymmetric distribution (Gamma) using a
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Chi squared test (χ2). Observables like fault dip/ back thrust dip (CV = 0.6-0.7/ 0.2 – 0.6) are
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less variable and decrease in magnitude with decreasing basal friction. Variables that are time
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dependent like fault lifetime (CV= 0.19- 0.56) and fault spacing (CV-0.12 – 0.36) have a
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higher CV consequently affecting the variability of wedge slope (CV= 0.12-0.33). These
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observables also increase in magnitude with increasing basal friction. As the mechanical
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complexity of the evolving wedge increases over time so does the CV and asymmetry of the
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distribution. In addition, we confirm the repeatability of experiments using an ANOVA test.
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Through the statistical analysis of results from repeated experiments we present a tool to
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quantify variability and an alternative method to gaining better insights into the dynamic
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mechanics of deformation in analogue sand wedges.
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Introduction
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Results from experiments can vary because of extrinsic and intrinsic controls. Generally,
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extrinsic controls are varied systematically in order to test their influence on a system in so
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called parameter studies. This approach is well suited to identify control factors which have a
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strong influence on experimental outcomes. However, if the influence is smaller it becomes
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difficult to verify that the resulting variability is attributable to extrinsic controls and not
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intrinsic variability. Experiments including parameter studies with so called analogue models,
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i.e. physical lab scale models made of weak rock analogue material (e.g. loose sand, clay,
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wax) have been created since the early nineteenth century pioneering the work of Sir Hall
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(Hall, 1815, see Graveleau et al., 2012 for a comprehensive historical review) to understand
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geological observations (e.g. the structure of a fold-and-thrust belt) by means of forward
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simulation of the underlying geodynamic processes. In contrast to numerical or analytical
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models which will always generate the same result with insignificant variability from a given
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starting configuration, experiments with analogue models never give exactly the same result
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as a consequence of inherent variability. This might be viewed as a handicap of the analogue
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modelling approach but also vice versa as a chance to simulate process variability in nature.
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The fact that variability is a matter of interest in scientific studies stems from the rising
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awareness for our limited view on geodynamic processes with single observations being only
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part of a spectrum of possible observational outcomes. If the source of variability is similar in
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the model as in nature, we argue that any physically simulated variability is preferred over
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statistically implemented noise from which variability typically arises in numerical models
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(e.g. Rosenau et al., 2010).
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Several sources of variability exist in nature (and consequently in the analogue model as well
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as in the natural prototype) both extrinsic, i.e. in the boundary conditions or material
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properties, and intrinsic due to the inherently changing physical properties of the system as it
ACCEPTED MANUSCRIPT evolves. In most parameter studies extrinsic variability is the focus, i.e. the change in
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experimental outcomes as a consequence of changes in the controlled boundary conditions or
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material properties. However, if intrinsic variability is unknown in such experiments,
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differences in the outcomes are potentially misattributed to large parts of the extrinsic
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controls. Only if intrinsic variability is significantly smaller than extrinsic variability,
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meaningful inferences from analogue models can be drawn. This indicates the importance of
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quantifying intrinsic vs. extrinsic variability in order to get an idea for the precision of
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analogue model results. However, only few studies have researched this issue so far (Cubas et
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al, 2010).
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Benchmark study by Schreurs et al. (2006) is an example of studying extrinsic variability
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related to boundary conditions. In this benchmark a simple experiment of creating a
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compressional sand wedge, was performed in six laboratories around the world without much
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prescription of the setup. Benchmark experiments showed consistently an overall cyclic
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deformation pattern of the wedge but the detailed observables and in particular the final
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geometry varied significantly. This difference was attributed to the different setup preparation
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protocols and physical differences (grain size distribution, density and frictional properties) in
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materials used. In this case, it was understood that experiments needed to have a strict setup
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protocol if they were to be compared. In a follow-up study, materials used were identical, the
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setup was constrained more heavily and consequently the variability minimized to the true
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intrinsic variability (Schreurs, pers. Comm.).
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Here we build on these findings and study intrinsic and extrinsic variability in a series of
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classical sand wedges with systematically varying basal coefficient of friction (µ basal).
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Boundary conditions and material properties were fixed and stable to a degree that is typical
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for analogue models, i.e. we used always the same experimental device and preparation
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moisture. A similar approach has been undertaken by Cubas et al. (2010) studying along-
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strike variation in cross-sections of early thrusts in repeated sand wedge experiments. Here
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the cross-sections were considered independent observations. In contrast to their approach, we
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analyze variation along a single profile across many thrusts evolving during wedge evolution
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in repeated experiments. By inducing higher strain compared to Cubas et al. (2010) and
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adding variation of extrinsic parameters (basal friction) we therefore go beyond their work in
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analyzing variability related to intrinsic and extrinsic sources as well as to wedge evolution.
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Our findings show that the variability in the results are attributed to (1) variability in material
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properties and (2) the complexity of the evolving mechanics of the wedge. In our study fault
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dip is related only to the coefficient of friction of the granular material and therefore has a low
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CV. On the other hand, fault spacing is related to fault lifetime and the criticality of the wedge
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which may change over time, thus fault lifetime and spacing changes with time and can have
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a larger CV.
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2) Methodology
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2.1) Experimental setup and analysis procedure
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The analogue experiments were kept simple and consist of creating convergent accretionary
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wedges. The experimental set-up (Fig.1a) is a box with dimensions 300 cm long, 20 cm wide
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and has a horizontal base that consists of a rubber conveyor belt (coefficient of friction (µ)
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~0.68 against sand, Fig. A2) and glass sidewalls (µ ~ 0.15 against sand, Fig. A2). In the box, a
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total thickness of 3 cm layer of granular material (sand, or sand and glass beads- for frictional
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coefficient see Table 1 and Fig A2) is sieved on the conveyor belt for the entire length of the
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box. Sieving of the granular material produces a homogenous layer (Lohrmann et al., 2003;
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Maillot, 2013) which is void of any internal preexisting structures or zones of weakness that
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can be activated during the experiment. We took care that the final surface was horizontal and
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ACCEPTED MANUSCRIPT planar within the thickness of the beam of a laser level (ca. 1 mm). To create the wedge the
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layer of granular material is pushed against a stationary rigid back wall by moving the
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conveyor belt at a constant velocity of 1mm/s towards and under the back wall. The growth of
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the wedge is recorded from the side by a digital camera (CCD, 11 MPx, 16 bit) at 1 Hz. Image
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data are calibrated and cross-correlated for deriving incremental displacement (= velocity)
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fields by means of Particle Image Velocimetry (PIV) (Adam et al., 2005) (Fig.1b, 2). Further
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processing of the images and vector fields produces binary images (Fig. 1c, 2, 3a) and maps
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of the evolving geometry and displacement of the wedge (Fig. 1d, 3b,c) thus monitoring the
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formation and kinematics of faults (Fig. 3c). To visualize the evolution of the faults in time as
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the wedge grows, a horizontal profile (Fig. 3b, c) of the velocity field is created in the middle
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of the granular layer at a height of 1.5 cm from the base. These velocity profiles are then
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assembled against each other in time and displayed as a deformation map (Fig. 3d).
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From these deformation maps, fault spacing and lifetime for each accretionary cycle is
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measured. Specifically, fault lifetime is defined as the time between the formation of two
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consecutive faults. In this definition, fault life time only includes the time when a fault is the
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main active structure and does not include the time it is reactivated at a later stage. Fault
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spacing is the horizontal distance between the last fault and a new fault at the time of
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localization of the new fault (Fig. 3b, d). Kinematically, fault lifetime and fault spacing are
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related. Fault dip and back thrust dip are measured at the initiation of a new fault. Fault dip
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has been measured from the PIV images by visually fitting a straight line along the isolines of
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the velocity field where the gradient is highest (Fig. 3b, c).
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Binarized images of the wedge are used to derive the height, length and slope of the wedge
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for every time step automatically. The height and length are measured from the crest (highest
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point in the wedge) to the base, and from the toe of the wedge to the crest respectively (Fig.
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3a, 4). This part of the complete wedge is most representative of an active wedge (Lohrmann
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et al., 2003). Wedge slope is measured as the arc tan (height of the wedge / length of the
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wedge).
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In summary, the observables measured for each experimental setup consist of fault
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characteristics i.e. fault lifetime, fault spacing (Fig. 5), dip of the fault and its respective back
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thrust as well as wedge geometry i.e. height, length, wedge slope. Estimation errors attached
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to these observables are 4mm for all lengths, 2 seconds for periods and 2o for angles.
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In this study, we tested the effect of different basal décollement layers on the intrinsic
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variability of the wedge evolution. From a sand only (no detachment layer) reference
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experiment (static/kinetic µbasal ~0.68/0.55 ) basal properties were varied by introducing a
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weak décollement layer of different thickness (0.2cm vs. 0.5 cm) and made of different sizes
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of glass beads (100-200 µm, 200-300 µm, 300-400 µm with static/kinetic µbasal increasing
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with size ~0.47-0.53/0.40-0.43, Appendix 1). The five different experimental setups that were
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realized are as follows:
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Setup 0) 3cm layer of sand (Number of experiments (N) = 7, Fig. 2a)
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Setup 3) 2.5cm layer of sand on 0.5 cm 300-400 µm glass beads (N =5, Fig. 1),
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Setup 2) 2.5cm layer of sand on 0.5 cm 200-300 glass beads (N = 5, Fig. 2b),
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Setup 1a) 2.5cm layer of sand on 0.5 cm 100-200 µm glass beads (Setup a) (N = 5, Fig. 2c)
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and
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Setup 1b) 2.8cm layer of sand on 0.2 cm 100-200 µm glass beads (Setup b) (N = 5, Fig. 2d).
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Each experiment is repeated up to 7 times for a total of 27 experiments. The internal
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coefficient of friction (µinternal) for all granular material used in this study is measured in the
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Schulze Ring shear tester (Schulze, 1994). Table 1 displays the static friction for the first or
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primary failure or fault, reactivation static friction for the reactivation of the fault as well as
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the kinetic friction.
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ACCEPTED MANUSCRIPT 2.2) Statistical analysis
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The reason to conduct a statistical analysis on a large data base from repeated experiments is
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threefold: First, to quantify intrinsic variability in the experiments, second, to detect trends
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which would be statistically irrelevant and therefore go unnoticed in a small number database
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or because they lay close to/within the error bars and thirdly, to quantify the repeatability of
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the measurements in experiments within identical setups. We grouped all the data for each
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observable (fault lifetime, fault spacing, fault dip, back thrust dip and wedge slope) of each
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setup into one population. Hence we created 5 populations for each setup and a total of 25
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populations (Table 2). For each population the basic statistical values i.e. mean, standard
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deviation and coefficient of variation (CV) were calculated (Table 2) before more advanced
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statistical treatment followed.
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Chi-square test (χ2)
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As a way to compare data populations beyond the mean and standard variation we choose to
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calculate the probability density function (pdf) for each population and compare it to
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theoretical distributions namely the Normal, Gamma and Laplacian distributions (Fig 6). To
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quantitatively determine which of the three distributions fit the best to each population, a Chi
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square test (χ2) is implemented (Table 2). The χ2 is calculated using the difference between
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the observed values from experiments and the theoretical value based on the distribution type.
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We calculated the χ2 for the observables fault lifetime, fault spacing, fault dip and wedge
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slope for all setups according to Cubas et al., 2010- Appendix A1. More information about the
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theory of the χ2 test and the probability distributions used in this paper are available in
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Appendix 2.
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In general, normal distributions characterize natural data symmetrically distributed around a
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mean. Laplacian distributions characterize symmetrically distributed data with a very narrow
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away from the median value at the tails (Cubas et al., 2010). In contrast the Gamma
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distribution characterizes asymmetric distributions of larger variability. Importantly,
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symmetric vs. asymmetric distributions are interpreted here in terms of observables that don’t
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vary much vs. observable that are variable especially over time. For example, a periodic
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process over time characterized by a single frequency and size plus some scatter or noise
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generates a symmetric distribution while a more complex process characterized by self-
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similar growing frequency and size generates an asymmetric distribution.
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ANOVA test
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In addition, to detect subtle differences amongst similar populations in our large database, we
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employed the analysis of variance test (ANOVA) for each population. The ANOVA test
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primarily analyses the statistical variance within a population. Specifically, since a population
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is made up of measurements from different experiments of the same setup, the test analyzes
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variance between the different experiments and within the experiments. In this manner the test
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determines whether all the measurements from the different experiments can be classified as
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originating from undistinguishable experiments or source. The test allows therefore, to detect
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statistically relevant differences between experiments that would otherwise (e.g. by the
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classical standard deviation criterion) be unnoticed. This particular test is chosen by
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examining the population in particular 1) the extent of the intrinsic variability within the
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population 2) if the distribution of the population is close to normal and 3) the independence
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of the data in a population (McKillup & Dyar, 2010). In this study the ANOVA test is used
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for each observable of each set-up 1) to analyze the variation between experiments in each
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setup and determine repeatability of the data and 2) analyze the variation between different
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setups to determine if any difference due to basal friction exists. For this test the significance
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level is set at 5% (Fisher, 1954). Therefore, with a p value more than 5%, the null hypothesis
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which states that there is no difference between two or more experiments is accepted. We
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used the one – way ANOVA test since we have one parameter µbasal that is changing. This
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statistical test can be easily computed using statistical softwares or in Matlab or Excel using
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predefined functions. More details on the ANOVA are included in Appendix 2.
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As part of the ANOVA test we calculated the coefficient of determination (R2) value. R2 is a
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quantitative value of how much of the variance in a population is explained due to repeated
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experiments in contrast to intrinsic variability within experiments. A large R2 value indicates
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that the variation is due to the experimental setup or extrinsic sources in contrast to a small R2
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indicating that the variation is within the system (Zar, 2010).
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The Fisher (F) value is also calculated and compared to the critical F (Fcrit) value to determine
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if the null hypothesis should be rejected. F values lower than the Fcrit value indicates that null
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hypothesis is accepted. Fcrit can be found on a universal table based on the degrees of freedom
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and the significance level.
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In summary the p value indicates the difference in the variance between experiments and
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whether the null hypothesis should be accepted or rejected, the F value confirms this
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conclusion and the R2 value suggests whether the variance is intrinsic or extrinsic. The
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disadvantage of the ANOVA test is that if the variance of almost all measurements in a
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population is consistent but there is one measurement that is an outlier and has a differing
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variance from the rest of the population, then the p value and F value reject the null
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hypothesis. On the other hand, since the ANOVA test compares the magnitude of the
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variance, it only accepts the null hypothesis when the difference between variances of all
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measurements in a group is not significant.
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ACCEPTED MANUSCRIPT For the two statistical analyses – χ2 and ANOVA test, the first 3 faults from all experiments
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were omitted. The reason for omitting these faults is simply because faults at this stage are
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small, short lived and not representative of critical wedge evolution (Gutscher et al., 1996).
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They rather represent the initiation stage of wedge formation characterized by the dominance
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of uplift over lateral growth. This stage ends when the wedge reaches the critical taper after
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which wedge growth is quasi-stable. These faults are noticeable in the bottom right corner of
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the deformation map (Fig. 3d) and are clearly different in their size and temporal lifetime.
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Therefore when analyzing data from analogue experiments it is important to distinguish the
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parts of the experiment that are representative of the natural process studied.
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3)
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All analogue accretionary wedges regardless of µbasal evolve in a cyclic fashion. The cyclic
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evolution of a wedge involves the growth of the wedge until it reaches the critical taper it can
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sustain, at which time the formation of a new fault begins the next cycle (Chapple, 1978;
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Davis et al., 1983; Platt, 1988; Dahlen, 1990; Hoth et al., 2007)
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Experimental observations
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In this respect, wedge geometry alternates synchronizing from increasing length of the wedge
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i.e. horizontal growth by new thrust fault formation and thereby lowering the slope of the
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wedge, to increasing height of the wedge i.e. vertical growth by underthrusting thus allowing
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the slope to increase to the critical taper angle. Consequently, fault lifetime, fault spacing and
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wedge slope are interdependent. According to Dahlen (1990) (equation 99), wedge slope is
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related to the internal angle of friction of the granular material and basal friction which
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governs the fault dip angle, therefore wedge slope and fault dip are also related. However, in
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our study, since fault dip is measured only at the initiation of the fault it is dependent on the
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internal angle of friction of the granular material and independent of the other observables.
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This independence of the fault dip from other observables is also evident in a cross-
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ACCEPTED MANUSCRIPT correlation of observables from this study (Fig. A2.1) and in the study by Cubas et al. (2010).
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On a smaller time scale, the geometric change shows systematic cyclicity between
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accretionary cycles. At the formation of a new fault, the pop-up of the fault at the toe of the
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wedge provides a linear increase in length of the wedge. At this point the height of the wedge
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is stationary. The height increases at the end of an accretionary cycle thus steepening the
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slope of the wedge. On a larger time scale, over multiple accretionary cycles, wedge growth
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according to critical wedge theory with a roughly constant slope results in an increase in
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height and width over time according to power law functions (Dahlen, 1990).
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In a self-similar like wedge, which grows ideally by “regular” accretion (i.e. adding material
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in a fashion optimized to maintaining the critical taper throughout wedge evolution) fault
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lifetime and fault spacing are related and they evolve systematically throughout the evolution
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of the wedge: For high µbasal wedges fault lifetime and spacing increases over time. As a result
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of the underthrusting mode both are limited by the length of the wedge (Gutscher et al.,
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1996). For low µbasal wedges which grow rather by frontal than basal accretion fault lifetime
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and spacing stabilizes towards a value which is consistent in maintaining the critical taper
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(Fig. 5). In terms of intrinsic variability, fluctuations around these consistent trends become
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important. Noticeably fluctuations are minimized in wedges with a thin or absent décollement
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layer (setup 1b and 0). In wedges with a thick décollement layer (setup 3, 2, 1a), distinct
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faults can be extra-long lived (i.e. they are apparently longer active than necessary to maintain
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the critical taper) and thus increase the spacing between two adjacent faults (Fig. 5). Such
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extra-long lived faults occur when the fault that underthrusts the wedge is occasionally
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weakened due to the entrainment of low friction material along the fault plane (glass beads
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from the thick décollement layer). On the other hand, lack of such a weakening mechanism
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can cause older faults to be reactivated in order to heighten the wedge to the taper angle (Platt,
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1988).
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The reactivation of internal pre-existing faults can be classified as second order deformation
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and formation of new faults at the tip of the wedge as first order deformation. In both styles
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of deformation the height of the wedge is increased to conform to the critical taper. The two
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varying styles of deformation are observed in setup 1a and 1b which demonstrate
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underthrusting style of deformation (Fig. 2) and fault reactivation (distributed strain)
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respectively. Difference in the two setups is observed in the fault lifetime and spacing (Fig.
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5). Underthrusting produces long-lived faults followed by short lived faults, whereas wedges
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with uniformed spaced first order faults may consist of fault reactivation.
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4)
Statistical results
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Histograms of the observables for each setup are represented by one of the three distribution
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types, Normal, Gamma and Laplacian. The best fitting type of distribution for each population
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is calculated by the Chi square test (χ2) (Table 2). The main goal of representing the
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populations with distribution curves is to describe the symmetry and variability of the
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populations. For many of the populations, the difference between χ2 calculated for each
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distribution type is very small (Table 2), therefore, all three distribution curves have been
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displayed in Fig. 6. The majority of the observables are best represented by a Gamma or
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Laplacian distribution (Table 2). Histograms of the fault dip for the setup 3 and 1b represent a
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Normal distribution. Fault spacing for setup 0 is best represented by a Normal distribution but
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can also be described as a Gamma or Laplacian distribution since the χ2 test for all three
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distributions is very similar. Fault dip for many setups could be represented as a bimodal
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distribution, however the difference in dip between each peak and mean value is
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approximately 2o which is within the error bars of the estimation. Observable back thrust dip
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between different setups is indistinguishable within the standard deviation (approximately 3°)
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and is not further considered in this analysis. The maximum CV for all setups is for the
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ACCEPTED MANUSCRIPT observable fault lifetime followed by the related observable fault spacing that has the second
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highest CV. The minimum CV for all setup is for the populations related to dip (Table 2,
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Fig.7a). Fault dips also have mean values that decreases with increasing µbasal (Fig. 6). Wedge
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slope has a lower variability (CV ranges from 0.12-0.33 for all setups) compared to fault
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lifetime (CV- 0.19-0.56) and spacing (CV- 0.12-0.36), however the mean value for all three
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observables are linked and show a decreasing trend with decreasing µbasal in this probabilistic
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approach (Table 2, Fig. 6). Overall the observables from the reference setup (setup 0)
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experiments are more symmetrical than of lower µbasal where many of the distribution curves
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of the observables are right skewed. The distribution for fault lifetime-setup 3 and 2 can also
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be considered bi-modal. However, the second mode is not well developed and not present for
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the related observable fault spacing of the same setups. Aside from setup 0 and the
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experiment with a thinner décollement layer (setup 1b), the trend of the CV for the
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observables fault lifetime and spacing increase as µbasal decreases for the other three setups
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(setup 3, 2, 1a) (Fig. 7a).
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The ANOVA test is used primarily to quantify the variation in the results within a population
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and determines whether results in the population come from the same system. Fault spacing
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and lifetime are two observables that have the highest CV especially for setups with glass
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beads (except setup 1b) due to the co-existence of large and short lived faults. Despite this
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observation the ANOVA test concludes that all results for the population fault spacing and
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lifetime come from alike experiments. Therefore the ANOVA test also recognizes that the
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magnitude of the variation in the observables is also repeatable between the experiments. For
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the observable fault lifetime in all the setups, the p- value supports the hypothesis that there is
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no difference between experiments within the setups. On the contrary, results show that there
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is a difference (p-value below 5%) between experiments for the population fault dip for the
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setup 1a , fault spacing for the setup 0 and 1b and back thrust dip for the setup 2 (Table 3). In
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ACCEPTED MANUSCRIPT these populations, an outlier in any one of the experiments causes a different magnitude of
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variability than the other experiments and thus the results of the ANOVA test reject the null
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hypothesis. If there realistically is a difference between experiments in a population then the
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ANOVA test would show a rejection of the null hypothesis for all the observables in that
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setup, which is not the case.
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F values clearly support the hypothesis that there is a difference between experiments of
343
different µbasal. From the ANOVA, the three experimental setups with the same décollement
344
thickness - setup 3, 2, 1a show an overall decreasing trend in the p value for the observables
345
fault lifetime and spacing, indicating that the difference in the variance between experiments
346
increases as µ basal decreases (Fig. 7b). However the p values are not below the 5%
347
significance level and the null hypothesis that all measurements in the populations come from
348
similar sources is accepted. In contrast the R2 value for the above mentioned observables and
349
setups increases with decreasing µ basal indicating that the statistical variance due to variations
350
between experiments in a population (Fig. 7c) also increases. Compared to the other setups
351
the same observables - fault lifetime and spacing - of setup 0 and 1b have lower p values (the
352
null hypothesis for fault spacing of setup 0 and 1a are rejected) and higher R2 values. Also
353
indicating that the difference between the variances of different experiments is high but this
354
variation is not within the experiments but due to extrinsic sources. However the difference
355
between the variances for these two setups is not higher than that for setup 3, 2, 1a since the
356
CV for fault lifetime and spacing for setup 0 and 1a is still lower than that for setup 3, 2, 1a
357
(Table 2, Fig. 7a).
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Although majority of the p values (Table 3) for all observables and all setups support the
360
hypothesis that there is no difference in measurements between experiments, there are some
361
exceptions. These exceptions can be explained by outliers in the data. An outlier or value
ACCEPTED MANUSCRIPT 362
significantly larger or smaller than the mean value in an experiment causes a different
363
variance for that experiment. When this experiment is compared to other experiments of the
364
same setup but with a significantly different variance, the calculated p value appears to be less
365
than the 5% significance level.
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5)
368
Here we discuss the range of variability between different fault observables (i.e. fault lifetime,
369
fault spacing, fault dip) and wedge observables (i.e. wedge slope) and the difference in the
370
variability between different setups. We demonstrate that the mechanics of deformation
371
controls the intrinsic variability in observables and that regardless of the varying intrinsic
372
variability in the populations, the mechanical relationship between fault, wedge observables
373
and µ basal is apparent. Since our experiments consist of large amounts of shortening and
374
formation of many faults we also examine the time evolution of the wedge and it’s reflection
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on the statistical data.
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Discussion
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5.1) Intrinsic variability in fault observables
378
We observe that fault observables of the same setup have different magnitudes of variability.
379
For example fault lifetime and spacing have a higher variability and asymmetry than fault dip.
380
Since faults are first order structures that change the geometry of the wedge, they also
381
influence any variability in the wedge slope. As a feedback fault lifetime is determined by
382
wedge criticality. Any changes in the physical properties of the wedge changes the frequency
383
of fault formation therefore making the observables fault lifetime, fault spacing and wedge
384
slope dependent on each other. Since wedge slope is also dependent on the µ basal and µ internal
385
(Davis, et al., 1983; Dahlen, 1990) which determines the fault dip, all observables are related
386
to each other. However, fault lifetime and fault spacing show a significantly higher CV than
387
fault dip and back thrust dip (Table 2). A comparison of the CV for all the fault observables
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ACCEPTED MANUSCRIPT reveals that the CV for fault lifetime and fault spacing range from 0.12-0.56. The CV for fault
389
dip is significantly lower and ranges from 0.06-0.07. This observation for fault dip is also
390
evident in the study by Cubas et al. (2010). The main difference between the observables is
391
that fault lifetime and spacing are time dependent. As the wedge grows the mechanical
392
process related to fault frequency may change with the evolution of the wedge (Platt, 1988).
393
Wedge slope in the experiments is also time dependent and changes both within an
394
accretionary cycle and as the wedge grows toward criticality over multiple cycles. Effectively
395
criticality in “real” wedges might vary over time because of the weakening of the wedge due
396
to incorporation of weak faults. The skew in fault lifetime and spacing populations (Fig. 6) or
397
possible development of a second mode in the fault lifetime distribution (setup 3 and 2) may
398
reflect evolving wedge dynamics. This explains why fault lifetime, fault spacing and wedge
399
slope have a higher CV than fault dip. Fault dip and back thrust dip in this study are not time
400
dependent since they are measured only at the instance when a new fault is formed and not
401
throughout the lifetime of the fault. Since these two observables only depend on the µ of the
402
granular material, there is little possibility for variation thus generating a smaller CV than the
403
other observables.
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5.1.1) Low variability of fault dip
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The low variability in fault dip is represented by a CV = 0.06-0.07, and mostly symmetrical
407
type of distribution. Since the dip is measured at the initial formation of the fault, the change
408
in dip i.e. rotation of the fault during the rest of the accretionary cycle is not recorded.
409
Therefore the measurement is not time dependent. Fault dip initiates at the velocity
410
discontinuity between the conveyor belt and the overlying granular layer. After initiation it
411
propagates into the overlying sand layer (Suppe and Medwedeff, 1990). The relationship
412
between fault dip and µ is defined as
413
δ = 45- -ψ
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(1)
ACCEPTED MANUSCRIPT where δ is the angle of the fault, ϕ is the angle of internal friction of the sediment (in this case
415
the sand) and ψ is the angle between the maximum compressive stress and the x axis
416
(Coulomb, 1773; Hafner, 1951; Lallemand et al, 1994). Therefore fault dip is related only to ϕ
417
and ψ at any point in time.
418
A range in the µ basal or a measuring error may explain the small variations in fault dip. In our
419
experiments we use granular material with a particular range in grain size to control the µbasal.
420
Since there is a range in the grain size, this also produces a range in the µbasal and may produce
421
the small range in fault dip measurements. On the other hand, the highest standard deviation
422
for the fault dip is ±1.92o (Table 2). The estimated measuring error is 2o and the highest
423
standard deviation for frictional coefficients measured in the Ring shear tester is ±0.047 for
424
sand on a conveyor belt (Fig A2-E). This would produce a deviation of ±2.67o (ϕ = tan-1 µ).
425
Therefore the variation in fault dip could come from the variation in the size of the granular
426
material or measuring error.
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5.1.2) Variability in fault lifetime and spacing
429
The µinternal and µbasal of the wedge controls the slope or critical taper of the wedge (Davis et
430
al, 1983; Dahlen, 1984; Dahlen, 1990; Suppe, 2007). The critical taper is directly related to
431
the fault strength, in particular to the properties of the décollement, and inversely related to
432
the wedge strength or cohesion and internal angle of friction of the wedge (Dahlen, 1990;
433
Suppe, 2007). Since faults are the first order structures that aid in changing the geometry of
434
the wedge and hence the taper, the lifetime and accordingly the spacing of the fault is such
435
that it is able to bring the wedge slope to the preferred taper. Therefore, the lifetime of a fault
436
is dependent on wedge slope (Chapple, 1978).
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In setup 0, the experiments have the highest µbasal and since there is a strong basal coupling,
439
the resulting wedge is steeper than the wedges from the other setups (Table 2, Fig. 6)
ACCEPTED MANUSCRIPT (Gutscher et al, 1996; Saha et al., 2013). In this setup thrust sheets continue to underthrust the
441
wedge to raise the height of the wedge. Compared to setup 1b where the µbasal is lowest, the
442
resulting wedge is shallower (Fig. 6) and the wedge deforms by frontal accretion. In both
443
these setups the fault lifetime and spacing have trends that are similar over time (Fig. 5).
444
However the slight skewness in the distributions (Fig. 6) is mainly because the two
445
observables are time dependent. At the beginning of the wedge formation, faults are short
446
lived and closely spaced. As the wedge is established and due to self-similar like growth of
447
the wedge the fault properties also grow creating a skew in the distribution (Saha et al., 2013).
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In a system (setup 3, 2, 1a) with altering internal properties due to the entrainment of glass
450
beads, the fault properties also change. Lowering of the fault plane friction can lengthen the
451
fault lifetime as a result, making deformation easier to continue along the fault for a longer
452
time (Platt, 1988; Cubas et al, 2008). To compensate for the long duration of these faults, long
453
lived faults are always followed by short lived faults. Perhaps, this is the wedges way to
454
return back to a state of criticality. In these setups, the long-lived faults produce a skew
455
towards larger faults or even the formation of a second peak in the histograms (Fig. 6). The
456
skewness or asymmetry in the histograms represents the evolution of the wedge over time in
457
particular the change in fault properties. As a result of the large difference in fault length, the
458
distributions for both fault lifetime and fault spacing are further skewed for these setups
459
compared to setup 0 and 1b.
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5.1.3) Influence of thickness on granular basal décollement
462
The evolution of the wedge is affected by the physical properties of the initial sand layer and
463
décollement layer. Our study demonstrates that the thickness of the granular décollement
464
layer also influences the style of deformation that the wedge employs to reach its critical
465
taper. The influence of the thickness of the décollement layer can be seen in the experiment
ACCEPTED MANUSCRIPT with setup 1a and 1b. The only difference between the two setups is the thickness of the basal
467
décollement layer. Setup 1a has a décollement layer of ~ 0.5 cm and setup 1b has décollement
468
layer of ~ 0.2cm. When a new fault is formed in the wedge with a thick décollement layer, the
469
glass beads from the décollement layer get entrained onto the fault plane lowering the
470
frictional coefficient along the fault plane. This allows the wedge to continue deforming along
471
the weak fault plane and to continue underthrusting the wedge allowing it to grow in height.
472
In this manner, a thick décollement layer weakens the fault plane by the entrainment of low
473
friction material changing the style of deformation from frontal accretion to underthrusting
474
(Platt, 1988). This interaction between materials of differing friction also affects the wedge
475
making it heterogeneous and changing the overall strength of the wedge. Since the critical
476
taper is dependent on a force balance that includes the cohesion and internal friction (Davis et
477
al., 1983), the taper changes and the wedge must adjust accordingly. The adjustment of the
478
wedge over time as a response to changing parameters and an evolving taper is represented by
479
an asymmetric distribution or possible second peak that represents the change in fault
480
behavior.
481
A thinner basal décollement layer allows the frictional properties of the fault and
482
consequently the physical properties of the wedge to remain more consistent throughout the
483
growth of the wedge. Therefore there is no change in the force balance or the preferred
484
critical taper angle. In comparison to the wedges with altering fault plane friction, the wedges
485
with thin décollement layer (setup 1b) or décollement layer with granular material that is the
486
same as the rest of the wedge (setup 0), have a very periodic and predictable fault lifetime and
487
spacing. This observation is strongly supported by the lower CV for observables fault lifetime
488
and spacing in setup 1b and 0 compared to the CV of the same observables in setup 1a (Table
489
2, Fig 7). Since our experiments with wedges of thin décollement layer (setup 1b) show little
490
or no evidence of underthrusting, deformation moves to older faults and reactivates these
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ACCEPTED MANUSCRIPT 491
older faults to raise the height of the hinterland part of the wedge in order to reach the
492
preferred taper angle (Platt, 1988).
493
5.2) Mechanical causes for the differences between setups
495
Fault dip, fault lifetime and spacing and consequently wedge slope are all related to the µ of
496
the entire wedge i.e. µ internal and µ basal of the wedge. In our study a difference as low as 0.03 in
497
the µbasal shows a systematic change in fault and wedge observables. The relationship shows
498
that as µbasal increases, fault lifetime and spacing also increase but fault dip decreases (Davis
499
et al., 1983, Dahlen, 1984; Konstantinovskaia and Malavieille, 2005). In addition, the
500
thickness of the décollement layer between setups causes a significant difference in the
501
mechanical evolution of the wedge for the different setups.
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5.2.1) Relationship between basal friction and fault lifetime and spacing
504
Fault lifetime and spacing decreases for decreasing µ basal primarily because of the frictional
505
properties of the décollement granular material and its relationship to the critical taper angle.
506
For low µ basal the critical taper angle is low therefore a deforming wedge of certain length
507
does not have to grow as much in height compared to a wedge with the same length but
508
higher µ basal. Thus, fault structures that are essential in changing the geometry of the wedge
509
are driven by the µ basal of the wedge. The lifetime of the fault is directly related to the height
510
of the critical wedge and how much the wedge must be underthrusted for the wedge to reach
511
the critical height (Davis et al., 1983; Cubas et al., 2008).
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512 513
In the wedges (setup 3, 2, 1a) with the thick (~0.5 cm) décollement layer the entrainment of
514
low friction material along the fault plane changes the style of deformation of the wedge.
515
These wedges with a thick décollement layer produce long faults followed by short lived
516
faults therefore producing a skewed distribution of fault lifetime and spacing (Fig. 6).
ACCEPTED MANUSCRIPT Accordingly, the CV for fault spacing and fault lifetime for setup 3, 2, 1a are higher than that
518
for setup 0 and 1b.The experiments with setup 0, 1a and 1b has the highest and lowest µbasal
519
respectively in this study. Setups 0 and 1b have little change to the physical properties of the
520
fault plane thus producing a more symmetric distribution of fault lifetime and spacing (Fig.
521
6).
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Although the experiments have a systematic change in the µ basal, the statistical results i.e. CV,
524
p value and R2 value show that the experiments can be further divided into two categories
525
based on the thickness of the décollement layer and the resulting style of deformation.
526
Accordingly the CV is lower for the experiments with evenly spaced faults (setup 0 and 1b)
527
and higher for experiments with underthrusting due to weakened faults (setup 3, 2, 1a) (Fig.
528
7a). In addition for the latter experiments the CV increases as µ basal decreases. The p value
529
shows a decreasing trend and R2 an increasing trend (Fig. 7b, c), indicating that as µ basal
530
decreases the repeatability value of experiments decreases. This observation could imply that
531
wedges with lower µ basal become more susceptible to extrinsic changes if all other
532
components within the system are the same for each setup. It is known that experimental
533
artifacts become significant when basal friction is very low in comparison to lateral friction
534
(Vendeville, 2007), and this would also produce very low p values.
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5.2.2) Increase in fault dip due to decreasing basal friction.
537
For all the setups the fault dip for the fore thrusts ranges from 26o +/- 1.82 to 30o +/- 1.92, and
538
increases with decreasing µbasal. As µ basal decreases by an average of 0.07 fault dip increases
539
by a minimum of 0.6o. In our experiments the décollement layer is sand (setup 0) or glass
540
beads (setup 3, 2, 1a and b). Fault dip is controlled primarily by the mechanics of faulting, in
541
particular the internal angle of friction (Hafner, 1951). Since the fault propagates from the
542
velocity discontinuity between the conveyor belt and the décollement layer, fault dip is related
ACCEPTED MANUSCRIPT to the internal angle of friction for sand (setup 0) and glass beads (setup 3, 2, 1a, b).
544
Mechanically, the dip of a fault is also related to the orientation of the maximum component
545
of stress (σ1). In a completely Andersonian setting i.e. where shear stress at the base is zero, ψ
546
(equation 1) is also zero. In this case, the orientation of the maximum stress component is
547
parallel to the convergence direction in a uniform thickness of incoming granular material.
548
However, in our experimental settings, we have the formation of shear zone between the
549
conveyor belt and the granular material. Here the magnitude of the shear stress is proportional
550
to the µbasal. As shear stress at the base increases due to increasing µbasal so does the rotation of
551
the maximum stress component and increase in ψ. Therefore, according to equation 1, the
552
consistent decrease in fault dip with increasing µbasal is an effect of the rotation of the
553
maximum stress component from the horizontal. This is also consistent with the findings of
554
Davis and Engelder (1985) studying salt detachments. Statistically fault dip demonstrates a
555
consistently low CV for all the setups regardless of µ basal, style of deformation and in
556
comparison to the other observables (Fig. 7a). This supports the idea that in each setup fault
557
dip is dependent only on the internal angle of friction of the granular material and
558
independent of other observables. For the three setups that demonstrate underthrusting (setup
559
3, 2, 1a), p values for fault dip tend to decrease and R2 show a shallow but increasing trend,
560
also indicating an increase in intrinsic variability possibly due lower stability in the system
561
when µ basal is lowered. This observation is similar to the other observables of the same setups.
563 564
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5.3) Relating interpretations of the statistical analysis to geodynamic natural observations.
565
The statistical trends produced from this study are more beneficial if they can be related to
566
observations in nature. It is important to note the mechanical interaction of parameters (i.e.
567
coefficient of friction, basal friction) when interpreting data from nature or creating models.
568
In particular, for observables (fault lifetime, spacing or wedge slope) that may have a high
ACCEPTED MANUSCRIPT variability, a distinctive solution may not be apparent (for example, in our study, the change
570
in fault lifetime and spacing is due to the change in fault plane friction). Therefore to make
571
reliable interpretations a larger data set is preferred. In addition if the probability distribution
572
curve is known for that observable then this may also aid in the strength of the interpretations.
573
On the other hand, if the observable is highly variable, it presents a clue that there are other
574
controlling factors involved and that need to be investigated. Ideally, when creating a model
575
(analogue or numerical) of a natural system, quantitative values of the physical parameter
576
(µ internal, µ basal, fault plane friction) of the system are taken from nature. However, if the
577
parameter is known to vary especially over time, then it is also important to test a range of
578
values in the model.
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579
It is also important to know the evolutionary stage of deformation. For example one system
581
can have two styles of deformation over time - frontal accretion at earlier stages of the
582
evolution of the wedge and subsequent predominance of underthrusting. Statistically they
583
may originate from the same system. However measurement taken from one or the other stage
584
or at the transition between stages can result in varying interpretations about the initial setting.
585
Furthermore, a change in the fault behavior observed in the field may not necessary mean a
586
different system but only a change in the deformation style due to internal or external
587
influences on the system. Our study is an example of this case and shows that this change in
588
the style of deformation can be initiated by a small change in the thickness of the décollement
589
layer.
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6)
Conclusion
592
Observables related to faults show different magnitudes of variability although they are from
593
the same experimental setup. For example fault dip (CV- 0.06-0.07) shows a lower variability
594
than fault lifetime (CV- 0.19-56), spacing (CV- 0.12-0.36) and wedge slope (CV- 0.12-0.33).
ACCEPTED MANUSCRIPT The difference in variability is related to the dependency of the observable on the interaction
596
of different components of the wedge as it evolves. Fault dip is only dependent on the internal
597
angle of friction of the basal and overlying granular material and measured at the initial stage
598
of formation of the fault, therefore the CV for fault dip is low. In contrast fault lifetime,
599
spacing and subsequently wedge slope varies with changes in the growth of the wedge. Other
600
dynamic changes in the wedge that affect fault lifetime and spacing is the change in fault
601
plane friction due to entrainment of low friction material from a thick décollement layer.
602
A difference in µ basal between 0.03 and 0.15 among the varying setups is statistically
603
distinguishable for all observables. Fault lifetime, fault spacing and related wedge slope
604
decrease with decreasing µ basal. Fault dip increases with decreasing µ basal. These observations
605
are consistent with the mechanics of faulting and deformation of the wedge defined by the
606
critical taper theory (Davis et al., 1983). The setups can be further categorized based on the
607
style of deformation resulting from the difference in initial setup design. The statistical test
608
ANOVA confirms that data from experiments within each setup are repeatable to a high
609
degree (low R2 values – Table 3). Moreover, the statistical results (p value and R2 from the
610
ANOVA test) show that repeatability decreases in setups with high CV and decreasing µ basal.
611
Overall, results signify that the variability of the observables is related to its dependency on
612
the complexity of the system. Therefore, when applying a theoretical analysis, for example,
613
wedge mechanical analysis to natural systems one needs to take into consideration the
614
possible range of values for each observable from natural observations (Maillot et al., 2007;
615
Cubas et al., 2013).
616
Repeating experiments presents the opportunity to quantify the intrinsic variability that is due
617
to geodynamic processes in the experimental system. It allows one to identify the robustness
618
of the results. A single result from one experiment could be near or far from the precise mean
619
value and this uncertainty can be minimized by the repetition of experiments. Thus adding a
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ACCEPTED MANUSCRIPT higher degree of precision to the interpretations. This does not imply that data from previous
621
analogue modelling studies that lack repetition are incorrect. Our study also shows that
622
despite the higher variability in some observables, the relationship between µ basal and mean
623
values for fault and wedge observables is evident. Many observables from previous
624
experiments that are quantitatively measured like fault dips have a low variability, therefore
625
interpretations are credible. Repeating experiments also allows the experimenter to assess
626
whether the variability of the results is due to the experimental setup or due to the processes
627
involved in the system and accordingly improve the design. Thus, we advise experimenters to
628
repeat experiments especially those created to study complex tectonic processes, if the
629
technical possibility exist.
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630 631
Below is a summary of the statistical tools that can be used to analyze data from repeated
632
experiments for better interpretation of the results.
1) Represent the results or observables with distribution curves to qualitatively describe
634
the variability. Determine which distribution best fits the population by calculating the
635
χ2 for each distribution. We advise to use the statistical analysis by Cubas et al. (2010)
636
- Appendix 1.
638 639 640
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2) The type of distribution that represents a population indicates the variability and also
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the sensitivity of the observable. An observable that is more sensitive to changes in the system or dependent on other observables will have a population with a large statistical variation (fault lifetime, fault spacing and wedge slope). On the contrary,
641
populations with small statistical variations are from independent observables like
642
fault dip.
643
3) Use the variability between experiments to determine how reproducible data from
644
experiments are with the help of the ANOVA test. The p value and R2 value indicate
645
not only whether the experiments are repeatable but also to what extent. In our study it
ACCEPTED MANUSCRIPT is observed that experiments with low µ basal are less repeatable. This observation could
647
imply that wedges with lower µ basal become more susceptible to extrinsic changes if all
648
other components within the system are the same.
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ACCEPTED MANUSCRIPT Acknowledgements
650
We would like to thank Marina Morlock for her help in conducting the first analogue
651
experiments for this study and processing the data. TS would like to thank Nadaya Cubas for
652
initial discussions on the critical taper theory. Technical support from Fang-Lin Wang,
653
Thomas Ziegenhagen and Frank Neuman was greatly appreciated in the analogue lab. We
654
would also like to thank Nadaya Cubas and Bertrand Maillot for their constructive criticism
655
on this paper. Tasca Santimano is supported by the European Union FP7 Marie Curie ITN
656
project TOPOMOD (project # 264517).
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Hoth, S., 2005. Deformation, erosion and natural resources in continental collision zones:
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insight from scaled sandbox simulations. Ph.D. Thesis, Freie University, Berlin.
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Hoth, S., Hoffmann-Rothe, A., Kukowski, N., 2007. Frontal accretion: An internal clock for
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bivergent wedge deformation and surface uplift. Journal of Geophysical Research 112,
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DOI: 10.1029/2006JB004357.
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Lallemand, S., Schnürle, P., Malavieille, J., 1994. Coulomb theory applied to accretionary and
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nonaccretionary wedges: Possible causes for tectonic erosion and/or frontal accretion. Journal
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of Geophysical Research 99(B6), 12,033-12,055.
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Lohrmann, J., Kukowski, N., Adam, J., Oncken, O., 2003. The impact of analogue material
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properties on the geometry, kinematics, and dynamics of convergent sand wedges. Journal of
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Structural Geology 25, 1691-1711.
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Maillot, B., 2013. A sedimentation device to produce uniform sand packs. Tectonophysics
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593, 85-94.
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McKillup, S., Dyar, M.D., 2010. Geostatistics Explained: an Introductory Guide for Earth
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Scientists. Cambridge University Press, Cambridge.
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Platt, J.P., 1988. The mechanics of frontal imbrication: a first-order analysis. Geologische
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Rundschau 77/2, 577-589.
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Rosenau, M., Nerlich, R., Brune, S., Onken, O., 2010. Experimental insights into the scaling
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and variability of local tsunamis triggered by giant subduction megathrust earthquakes.
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Journal of Geophysical Research 115, B09314, doi:10.1029/2009JB007100.
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Saha, P., Bose, S., Mandal, N., 2013. Varying frontal thrust spacing in mono-vergent wedges:
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An insight from analogue models. Journal of Earth System Science 122 (3), 699-714.
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Schreurs, G., Buiter S.J.H., Boutelier, D., Corti, G., Costa, E., Cruden, A. R., Daniel, J-M.,
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Hoth, S., Koyi, H.A., Kokowski, N., Lohrmann, J., Ravaglia, A., Schlische, R.W., Withjack,
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M.O., Yamada, Y., Cavozzi, C., Delventisette, C., Elder Brandy, J. A., Hoffmann-Rothe, A.,
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Mengus, J-M., Montanari, D., Nilforoushan, F., 2006. Analogue benchmarks of shortening
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and extension experiments. In: Buiter, S.J.H., & Schreurs, G. (Eds.), Analogue and Numerical
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Modelling of Crustal-Scale Processes. Geological Society of London, 1-27.
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Schulze, D.,1994. Entwicklung und Anwendung eines neuartigen Ringschergerätes.
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Aufbereitungstechnick 35(10), 524-535
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Suppe, J., and Medwedeff, D.A., 1990. Geometry and kinematics of fault-propagation
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folding. Eclogae Geol. Helv. 83, 409-454
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Suppe, J., 2007. Absolute fault and crustal strength from wedge tapers. Geology 35, 1127-
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Twiss, R.J., Moores, E.M., 1992. Structural Geology. W.H. Freeman and Company, New York
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Vendeville, B.C., 2007. The 3-D Nature of stress Fields in Physical Experiments and Its
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Impact on Models Overall Evolution. Solicited Oral Contribution. In: EGU General
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Assembly, Vienna, 15-20 April, 2007
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Zar, J.H., 2010. Biostatistical Analysis, Fifth Edition. Pearson Prentice-Hall, New Jersey.
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Figure 1 – Experimental box for all the experimental setups. Data and processed images
762
obtained from the PIV system include a) calibrated image of the wedge, b) binarized image of
763
the wedge and c) processed images of the wedge showing the horizontal velocity field (Vx)
764
along the wedge. The experiment shown in this figure is from setup 3- sand +0.5cm layer of
765
300-400µm glass beads.
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Figure 2- Examples of calibrated images, binarized images and processed images of the
768
wedge showing the horizontal velocity field (Vx) for a) sand (setup 0), b) sand +0.5cm layer
769
of 200-300µm glass beads (setup 2), c) sand +0.5cm layer of 100-200µm glass beads (setup
770
1a) and d) sand +0.2cm layer of 100-200µm glass beads (setup 1b). All images display 160
771
cm of shortening. The processed images show the difference in deformation style in particular
772
underthrusting in setup 0 and frontal accretion in setup 1b.
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Figure 3 – Method of how the observables are measured using the data collected by the PIV
775
cameras a) Binarized images of the wedge are used to calculate the height and length of the
776
wedge for each time step. The incoming layer is not included in the length or height of the
777
wedge. b) Fault spacing is measured as the horizontal distance, along the profile line, between
778
two consecutive faults, when the former fault is at the end of its activity as a first order fault
779
and a new fault is formed. The velocity along the profile line for each time step is plotted in d.
780
The fault spacing is accurately measured from this plot. c) Fault dip and back thrust dip are
781
measured after strain is completely localized and at the initiation of a new fault plane and
782
back thrust plane. d) Deformation map displaying the velocity for each profile line from c).
783
Therefore the x axis is the length of the wedge and the y axis is the time step. Pink area
784
represents the velocity of the incoming layer. The transition between pink to black in the x
785
axis is the location of the fault plane on the profile line. From this map the fault lifetime and
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fault spacing can be measured. Fault lifetime is the time (or number of time steps) the fault is
787
active as a first order structure.
788
Figure 4 – Wedge height and length for all experiments within a) sand (setup 0), b) sand
790
+0.5cm layer of 300-400µm glass beads (setup 3), c) sand +0.5cm layer of 200-300µm glass
791
beads (setup 2), d) sand +0.5cm layer of 100-200µm glass beads (setup 1a) and e) sand
792
+0.2cm layer of 100-200µm glass beads (setup 1b). Dashed black line represents the power
793
law trendline for both height and length of the wedge.
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Figure 5- Fault lifetime and fault spacing for all experiments within a) sand (setup 0), b) sand
796
+300-400µm glass beads (setup 3) , c) sand +200-300µm glass beads (setup 2), d) sand +100-
797
200µm glass beads (setup1a), e) sand +100-200µm glass beads (setup 1b). b) c) and d) show
798
the occurrence of long lived and widely spaced faults followed by short lived and closely
799
spaced faults, thus producing a zig-zag trend.
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Figure 6 – Histograms for each observables- fault lifetime, fault spacing, fault dip and wedge
802
slope for each experimental setup. Normal, Gamma and Laplacian probability distributions
803
are represented in green, red and blue curves respectively. Q value is for the best fit
804
distribution curve based on the Chi square test (Table 2) and N indicates the number of
805
samples.
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Figure 7- Plots showing the comparison of a) CV b) p value (ANOVA test) and c) R2
808
(ANOVA test) for each observable in the different setups. Linear trendlines are only shown
809
for setups 3, 2, 1a since they differ from the other setups (0 and 1b) in the style of
810
deformation and the thickness of the décollement layer.
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ACCEPTED MANUSCRIPT Table captions
813
Table 1 – Material physical properties for the grains used in the experiments include grain
814
size, density, static friction, static reactivation friction and kinetic friction. The coefficient of
815
frictions for all materials were measured in a Ring-shear tester and the data can be found in
816
Fig. A2
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Table 2 – Statistics for all the observables for each experimental setup. For each observable,
819
the number of samples (n), mean, standard deviation, coefficient of variation (CV), median
820
and median deviation are presented. The Chi-square test (χ2) is calculated between the
821
observed values and the expected values based on the Normal model, the Gamma model and
822
the Laplacian model. The best fit distribution according to the χ2 test is presented in bold. For
823
the Gamma model the shape parameter (a) and scale parameter (b) are also indicated.
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Table 3- ANOVA test results for each observable for each experimental setup. The calculated
826
p-value, R2, Fisher value and critical Fisher value are shown in the table.
827
indicate a rejection of the null hypothesis.
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Bold values
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Methods: Derivation of material properties using a ring-shear tester
830
Material properties were derived using a Schulze Ring shear tester specially designed to
831
measure friction coefficients in loose granular material accurately at low confining pressures
832
and shear velocities similar to sandbox experiments (Lohrmann et al., 2003; Schulze 1994). In
833
this tester, a sand layer is sheared either against a base or internally at constant normal load
834
and velocity while the shear stress is measured continuously. The resulting stress curve
835
(example shown in Appendix Fig. A1) typically rises from zero to a peak level (= static
836
friction) within the first few millimetres of shear before it drops and stabilizes after formation
837
of a shear zone in the material or at the interface (= kinetic friction). Stopping the shearing
838
and resetting the shear stress followed by a renewed shearing (re-shear) results in a second,
839
similar shear curve whose peak (= reactivation static friction) is somewhat lower than the first
840
peak. From these curves, the strengths at first and second peak as well as on the plateau are
841
picked manually and assigned static, reactivated static and kinetic frictional strength for the
842
applied normal load, respectively. Density of material is calculated based on the weight of the
843
material sieved into the shear cell and the volume which is monitored accurately throughout
844
the shear test.
845
Two analysis methods are applied to derive friction coefficients and a realistic variability
846
from that data: First, linear regression of the normal load vs. shear strength data pairs, i.e. by
847
fitting a linear failure envelope to the data in the Mohr space. The slope of the line and the y-
848
axis intercept are accordingly the friction coefficient and cohesion, respectively. However,
849
because this method overestimates real precision and may be biased by a curved failure
850
envelope we performed a second analysis which relies on calculating all possible two point
851
slopes and intercepts for mutually combined data pairs. These data are then evaluated by
852
means of univariate statistics by means of calculating mean and standard deviation and
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found that the peaks of the experimental pdf are much narrower than a normal distribution pdf
855
suggesting that the calculated standard deviation is a conservative value which tends to
856
underestimates real precision. The standard deviations reported in Table 1 represent a trade-
857
off between the mean and standard deviations derived by the two methods described above.
858
Because inferred cohesion values are low (in the order of view tens of Pa) and show a large
859
scatter (tens to hundreds of Pa) we do not rely on the cohesion values but conclude cohesion
860
is neglectable in our material.
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Figure A1.1: Example of stress curve raw data as derived from Ring shear testing. See text
863
for explanation.
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Figure A1.2: Friction data set. In Each panel A-F the static, reactivated static and kinetic
866
friction data are plotted in Mohrspace (with a linear fit indicated) and as histograms of fiction
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coefficients and cohesion as derived from mutual two-point regression analysis. See text for
868
explanation.
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Method: Statistical tests used in this study
871
1) Chi square test (χ2 test)
872
The theoretical formula for this test is similar to that in Cubas et al. (2010) and calculated as
873
follows
874
= ∑
875
Where O= observed values, E = expected values based on the distribution and m= number of
876
intervals from the histograms. χ2 test as shown in Table 2 is calculated as follows
877
= 1 −
878
Therefore a high test value indicates the least difference between the theoretical values and
879
the dataset.
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The theoretical distribution functions for the different distribution types are as follows
882
Normal distribution
!
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(3)
= √
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where ̅ = mean, σ2 = variance
(4)
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Gamma distribution
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(2)
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=
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where a = shape factor, b = scale factor and Γ is the Gamma function
( $ )*
(5)
889 890
Laplacian distribution
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= ( +
892
where / = median, 2b2 = variance
-| | . '
(6)
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2) ANOVA test
895
For a population consisting of measurements from several experiments, N is the number of
896
experiments and 01 is the total number of measurements in all experiments. The sum, mean
897
(̅ , and variance (2 is calculated for each experiment in the population. The degree of
898
freedom (3 , Sum of squares (SS) and Mean of squares (MS) are calculated between
899
experiments (B) and also within each experiment (W) as follows
900
34 = 5 − 1
901
36 = 01 − 5
902
774 = ∑8 0 ̅ − ̅ 1
903
where 0 , ̅ is the number of measurements and mean respectively for the 9 :; experiment,
904
and ̅ 1 is the mean of all measurements.
905
776 = ∑8 <= − ̅ >
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(7) (8) (9)
(10)
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where = is the ? :; measurement in experiment 9
907
@7 = BC
908
771D:*E = 774 + 776
909
@71D:*E = @74 + @76
910
As part of the ANOVA test we calculated the coefficient of determination (R2) value. R2 is a
911
quantitative value of how much of the variance of an observable is explained due to variation
912
between the experiments in contrast to within experiments.
913
G = AA
914
A large R2 value indicates that the variation is due to the experimental setup or extrinsic
915
sources in contrast to a small R2 indicating that the variation is within the system (Zar, 2010).
916
(11) (12) (13)
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(14)
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The Fisher (F) value is calculated to determine if the null hypothesis can be rejected.
918
L=
MA H
MAN
=
O*P*QR& H
(15)
O*P*QR&N
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3) Cross-correlation of fault observables.
921
Figure A2.1- Graphs showing the correlation between the fault observables- fault lifetime,
922
fault spacing and fault dip for all setups. There is a clear correlation between fault lifetime and
923
fault spacing showing their interdependence. There is a very low correlation between fault dip
924
and fault lifetime or spacing indicating that fault dip is independent of the latter observables.
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Glass beads Glass beads Glass beads Sand Sand on rubber conveyor belt Sand on glass
Internal Internal Internal Internal Basal
Density (g/m3)
Grain size (µm)
Static friction
Standard deviation
1.59 1.60 1.64 1.72
100-200 200-300 300-400 > 630
0.47 0.50 0.53 0.68 0.62
0.03 0.04 0.04 0.04 0.05
0.15
0.09
0.12
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Stable reactivation friction 0.46 0.47 0.51 0.63 0.60
Standard deviation
Kinetic friction
Standard deviation
0.01 0.02 0.02 0.02 0.02
0.40 0.42 0.43 0.55 0.56
0.02 0.02 0.01 0.01 0.02
0.07
0.12
0.06
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Type
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Material
Table 1 – Santimano et al.
b 20 20 0.12 0.12 0.26
N.A. 0.04 0.22 N.A. 0.73
0.47 0.27 0.06 0.02 0.26
6.34 16 286 2883 14
24 14 0.09 0.02 0.85
0.39 0.46 0.20 N.A. 0.76
129 227 26 45 12
50 43 1.14 0.41 2.49
0.57 0.80 0.19 0.48 0.78
106 93 1.59 2.16 3.04
0.65 0.86 0.36 N.A. 0.77
0.56 0.36 0.06 0.05 0.19
3.47 8.35 327 395 28
54 30 0.09 0.11 0.55
0.81 0.91 0.36 N.A. 0.86
176 237 29 45 15
85 76 1.41 1.33 2.43
0.59 0.78 0.23 0.67 0.77
84 71 1.79 2.74 4.14
N.A. 0.11 0.21 N.A. N.A.
0.54 0.33 0.06 0.06 0.33
5.33 12 268 235 10
29 16 0.11 0.19 1.23
0.17 0.42 0.21 N.A. 0.69
129 193 29 45 11
55 48 1.13 1.78 3.34
0.40 0.38 0.50 0.08 0.41
20 16 1.92 0.80 2.54
0.91 0.94 0.30 N.A. N.A.
0.19 0.12 0.06 0.02 0.26
26 75 255 3019 17
4.04 1.94 0.12 0.01 0.56
0.92 0.94 0.28 N.A. N.A.
105 148 30 45 9.10
17 13 1.66 0.31 1.86
0.87 0.98 0.16 0.47 0.78
n 58 58 64 64 1297
Sand + 300-400 µm GB setup 3
Fault Lifetime Fault Spacing Fault dip Back thrust dip Wedge slope
48 48 53 52 853
153 241 26 44 12
71 65 1.59 0.82 3.23
Sand +200-300 µm GB setup 2
Fault Lifetime Fault Spacing Fault dip Back thrust dip Wedge slope
39 39 43 43 896
189 258 28 44 15
Sand +100-200 µm GB setup 1a
Fault Lifetime Fault Spacing Fault dip Back thrust dip Wedge slope
40 40 45 45 938
155 217 29 44 12
Sand +100-200 µm GB setup 1b
Fault Lifetime Fault Spacing Fault dip Back thrust dip Wedge slope
63 63 68 67 764
106 146 30 44 9.69
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Population Fault Lifetime Fault Spacing Fault dip Back thrust dip Wedge slope
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a 9.51 14 217 390 65
Setup Sand only setup 0
Gamma model
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CV 0.33 0.26 0.07 0.06 0.12
Laplacian model χ2 Median test Median deviation 0.76 188 48 0.95 298 62 0.63 26 1.32 N.A. 45 0.92 17 1.68 0.96
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χ2 test 0.89 0.94 0.67 N.A. 0.91
Table 2- Santimano et al.
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Setup
Fault spacing
Fault dip
Back thrust dip
between experiments of each setup p value R2 F value F crit
0.10 0.17 1.85 2.28
0.02 0.24 2.74 2.28
0.06 0.18 2.08 2.26
0.91 0.03 0.33 2.26
Sand +300-400 µm GB setup 3
p value R2 F value F crit
0.92 0.02 0.22 2.58
0.90 0.02 0.26 2.58
0.08 0.15 2.15 2.56
0.66 0.04 0.60 2.57
Sand +200-300 µm GB setup 2
p value R2 F value F crit
0.94 0.02 0.19 2.65
0.85 0.03 0.33 2.65
0.08 0.19 2.25 2.61
0.03 0.23 2.92 2.61
Sand +100-200 µm GB setup 1a
p value R2 F value F crit
0.53 0.08 0.79 2.64
0.16 0.16 1.71 2.64
0.03 0.22 2.85 2.60
0.46 0.08 0.92 2.60
Sand +100-200 µm GB setup 1b
p value R2 F value F crit
8.36E-07 0.43 11.20 2.53
0.16 0.09 1.69 2.51
0.07 0.12 2.24 2.52
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Sand setup 0
between different basal frictions or setups
1.42E-08 0.59 16.85 2.57
2.94E-14 0.77 39.12 2.57
4.54E-13 0.62 26.54 2.51
0.03 0.17 2.73 2.55
Table 3- Santimano et al.
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ACCEPTED MANUSCRIPT Intrinsic versus extrinsic variability of analogue sand-box experiments- insights from statistical analysis of repeated accretionary sand wedge experiments Tasca SANTIMANO, Matthias ROSENAU, Onno ONCKEN
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Highlights
We statistically quantify variability in repeated analogue experiments.
-
Extrinsic source of variability is due to varying basal friction.
-
A probability based comparison between the different observables is presented.
-
The ANOVA test is used to confirm the repeatability of results.
-
We show that the mechanics of deformation control the variability.
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static friction peak
kinetic friction level
20
5
10
15 shear [mm]
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Sand (G23) internal 0.68529 (95%: 0.68183<>0.68875) (54 measurements)
0.62777 (95%: 0.62428<>0.63127) (54 measurements)
0 0.5
0.9
100 0 -1000
-500 0 500 cohesion (Pa)
0 0.55
1000
Glass beads (300-400) internal 0.53131 (95%: 0.52525<>0.53737) (54 measurements)
0.75
100
0 -200
0 200 cohesion (Pa)
2000 2000
8000
4000
4000 2000
0
6000 8000 10000 12000 14000 16000 normal load (Pa) 0.53039+/-0.03972 (1215 data) -24.3646+/-162.6242 Pa (1215 data) 200 400 300
counts
150 100 50 0
0.4 0.5 0.6 friction coefficient
0.7
200 100 0 -2000
-1000 0 cohesion (Pa)
1000
static reactivation friction
6000
counts
0
2000
4000
6000 8000 10000 12000 14000 16000 normal load (Pa) 0.5481+/-0.012659 (1215 data) 83.6748+/-41.1281 Pa (1215 data) 200 300
200
100 50
0.55 0.6 0.65 friction coefficient
0.7
100
0 -100
0
100 200 cohesion (Pa)
300
0.42893 (95%: 0.42743<>0.43043) (54 measurements)
0
2000
4000
kinetic friction
6000 4000 2000 0
6000 8000 10000 12000 14000 16000 normal load (Pa) 0.50879+/-0.0159 (1214 data) 20.7723+/-68.224 Pa (1214 data) 150 600
6000 8000 10000 12000 14000 16000 normal load (Pa) 0.43114+/-0.0082835 (1213 data) 4.281+/-37.3185 Pa (1213 data) 150 600
100
100
400
50
0 0.45
0.5 0.55 0.6 friction coefficient
0.65
counts
4000
0
8000
counts
6000
0
0 0.5
400
shear strength (Pa)
shear strength (Pa)
static friction
8000
2000
150
10000
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shear strength (Pa)
0.6 0.65 0.7 friction coefficient
4000
0.50314 (95%: 0.50073<>0.50555) (54 measurements)
10000
counts
50
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0.6 0.7 0.8 friction coefficient
200
200
6000
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50
4000
100
counts
100
shear strength (Pa)
6000 8000 10000 12000 14000 16000 normal load (Pa) 0.62698+/-0.017266 (1215 data) 109.2084+/-55.9641 Pa (1215 data) 150 300
300
counts
counts
150
2000
counts
6000 8000 10000 12000 14000 16000 normal load (Pa) 0.67555+/-0.035547 (1214 data) 28.5609+/-114.4291 Pa (1214 data) 200 400
kinetic friction
8000
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4000
counts
2000
5000
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10000
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10000
static reactivation friction
shear strength (Pa)
shear strength (Pa)
static friction 10000
0
0.55032 (95%: 0.54768<>0.55296) (54 measurements)
15000
counts
15000
200
0 -500
0 500 cohesion (Pa)
1000
0
2000
4000
400
counts
A
50
0 0.4
0.45 friction coefficient
0.5
200
0 -400
-200 0 200 cohesion (Pa)
400
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Glass beads (200-300) internal 0.50221 (95%: 0.49782<>0.5066) (72 measurements)
0.47007 (95%: 0.46696<>0.47319) (70 measurements)
6000 8000 10000 12000 14000 16000 normal load (Pa) 0.50011+/-0.035773 (2160 data) 23.13+/-127.9183 Pa (2160 data) 400 800
6000 8000 10000 12000 14000 16000 normal load (Pa) 0.47237+/-0.020338 (2038 data) 41.4431+/-91.2477 Pa (2038 data) 300 800
600
100 0
D
0.4 0.5 0.6 friction coefficient
0.7
400 200 0 -1000
-500 0 500 cohesion (Pa)
100
0
1000
0.46663 (95%: 0.46184<>0.47141) (54 measurements)
0.4 0.5 0.6 friction coefficient
0.7
200
0 -500
0 500 cohesion (Pa)
2000
4000
6000 8000 10000 12000 14000 16000 normal load (Pa) 0.46853+/-0.029369 (1214 data) -10.092+/-114.9539 Pa (1214 data) 200 400 300
counts
150 100 50 0
0.4 0.5 0.6 friction coefficient
0.7
counts
0
200 100 0 -1000
-500 0 500 cohesion (Pa)
6000 8000 10000 12000 14000 16000 normal load (Pa) 0.41931+/-0.019954 (2158 data) 18.8197+/-94.0515 Pa (2158 data) 300 800
1000
static reactivation friction
4000 2000
0
0
2000
4000
100
0.4 0.45 friction coefficient
0.5
400 200 0 -1000
-500 0 500 cohesion (Pa)
1000
0.40278 (95%: 0.40043<>0.40513) (54 measurements)
shear strength (Pa)
6000
2000
600
0 0.35
1000
4000
kinetic friction
6000 4000 2000 0
6000 8000 10000 12000 14000 16000 normal load (Pa) 0.46173+/-0.013418 (1214 data) 1.237+/-49.6772 Pa (1214 data) 300 600
6000 8000 10000 12000 14000 16000 normal load (Pa) 0.40274+/-0.018254 (1215 data) -8.1448+/-64.4635 Pa (1215 data) 300 400
200
200
400
100
0 0.35
0.4 0.45 0.5 friction coefficient
0.55
counts
2000
0
8000
counts
4000
EP
shear strength (Pa)
static friction
6000
0
200
8000
AC C
shear strength (Pa)
400
2000
0.45969 (95%: 0.45836<>0.46102) (54 measurements)
8000
counts
4000
600
Glass beads (100-200) internal
0
2000
4000
TE D
200
0
200
counts
counts
counts
300
0
kinetic friction
6000
counts
4000
counts
2000
2000
M AN U
0
shear strength (Pa)
2000
4000
counts
4000
static reactivation friction
6000
RI PT
6000
8000
SC
static friction
8000
0
0.41889 (95%: 0.41569<>0.4221) (72 measurements)
8000
shear strength (Pa)
shear strength (Pa)
10000
200
0 -500
0 cohesion (Pa)
500
0
2000
4000
300
counts
C
100
0
0.35 0.4 0.45 friction coefficient
0.5
200 100 0 -400
-200 0 200 cohesion (Pa)
400
ACCEPTED MANUSCRIPT
Sand (G23) basal on black rubber belt 0.62128 (95%: 0.61663<>0.62594) (72 measurements)
0.58446 (95%: 0.58225<>0.58667) (72 measurements)
4000 2000 0
4000
6000 8000 10000 12000 14000 16000 normal load (Pa) 0.62227+/-0.046799 (2159 data) -20.1912+/-147.8079 Pa (2159 data) 600 1000
0 -1000
0 1000 cohesion (Pa)
counts
100 0 0.5
2000
0.13938 (95%: 0.13259<>0.14617) (72 measurements)
3000
0 -200
0
200 400 cohesion (Pa)
0
6000 8000 10000 12000 14000 16000 normal load (Pa) 0.56223+/-0.022762 (2158 data) 36.5417+/-72.0611 Pa (2158 data) 300 400
100
0.8
100 0 -200
0
200 400 cohesion (Pa)
600
0.11847 (95%: 0.11142<>0.12552) (72 measurements)
static reactivation friction
1000
0
6000 8000 10000 12000 14000 16000 normal load (Pa) 0.12026+/-0.061432 (2095 data) 54.2483+/-211.629 Pa (2095 data) 600 800
400
400
400
0 -0.5
0 0.5 friction coefficient
1
0 -2000 -1000 0 1000 cohesion (Pa)
2000
200
0 -0.5
0 0.5 friction coefficient
1
200
0 -2000 -1000 0 1000 cohesion (Pa)
2000
0
2000
4000
600
counts
400
counts
counts
200
4000
kinetic friction 2000
6000 8000 10000 12000 14000 16000 normal load (Pa) 0.12125+/-0.070649 (2080 data) 57.6545+/-224.6019 Pa (2080 data) 600 600
200
2000
0.5 0.6 0.7 friction coefficient
200
6000 8000 10000 12000 14000 16000 normal load (Pa) 0.14826+/-0.08989 (2051 data) 74.4117+/-232.3463 Pa (2051 data) 600 600
400
0
4000
300
0 0.4
600
1000
0
2000
3000
counts
4000
0.9
2000
AC C
2000
EP
shear strength (Pa)
static friction 2000
counts
shear strength (Pa) counts
4000
0
0.6 0.7 0.8 friction coefficient
0
0.11615 (95%: 0.1084<>0.1239) (72 measurements)
3000
1000
100
M AN U
1
200
2000
200
TE D
0.5 friction coefficient
counts
counts
counts 0
500
200
Sand (G23) basal on glass
0
4000
300
200
F
2000
4000
6000 8000 10000 12000 14000 16000 normal load (Pa) 0.59118+/-0.02397 (2157 data) 52.4794+/-71.7629 Pa (2157 data) 400 300
400
0
0
6000
counts
2000
6000
kinetic friction
8000
shear strength (Pa)
0
static reactivation friction
8000
RI PT
5000
10000
counts
shear strength (Pa)
shear strength (Pa)
static friction 10000
0
0.55444 (95%: 0.55241<>0.55647) (72 measurements)
10000
shear strength (Pa)
15000
SC
E
200
0 -0.2
0 0.2 0.4 friction coefficient
0.6
400 200 0 -2000 -1000 0 1000 cohesion (Pa)
2000
AC C
EP
TE D
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT