Earth and Planetary Science Letters 246 (2006) 125 – 137
www.elsevier.com/locate/epsl
Flexing is not stretching: An analogue study of flexure-induced
fault populations
S. Supak a,b,⁎, D.R. Bohnenstiehl b , W.R. Buck b
b
a
Department of Civil Engineering, Columbia University, New York, NY 10027, USA
Lamont-Doherty Earth Observatory of Columbia University, Palisades, NY 10964, USA
Received 19 October 2005; accepted 20 March 2006
Available online 11 May 2006
Editor: R.D. van der Hilst
Abstract
Flexure-induced fractures are predicted to form along the axis of maximum tensile stress within a bending brittle plate. The
mechanics of this process differ from extensional fault growth in response to lithosphere stretching, where a distributed set of
simultaneously growing fractures evolves through elastic interaction. To simulate extensional fault growth during lithospheric
flexure, partially solidified plaster layers resting on a foam rubber substrate were depressed by a linear load and fractured in
analogue models. The length- and spacing-frequency distributions of the resulting crack populations were analyzed for a series of
nine thin (5 mm) and ten thick (15 mm) layer experiments. Previous analogue stretching models predict power-law lengthfrequency distributions and clustered spacings (Cv N 1) at low strains (b ~ 10%), evolving toward an exponential distribution and
more regular spacings (Cv b 1, often termed anticlusted) at larger stains. Crack populations formed at low strains during these
bending experiments, however, exhibit length-frequency distributions that are not well described by either a power-law or
exponential distribution model, being somewhat better fit by the exponential model in the thin layer experiments and somewhat
better fit by the power-law model in the thick layer experiments. One-dimensional spacing-frequency distributions are well
described by an exponential distribution model, and crack spacing can be characterized as anticlustered within both the thin and
thick layer experiments. Although similar spacing patterns may develop when fracture growth is limited by mechanical layer
thickness, the characteristic spacing does not scale with the layer thickness in these flexural experiments. Alternatively, the
development of power-law (fractal) populations may be inhibited by the growth history of flexure-induced faults, whereby
nucleation is localized spatially due to the distribution of stresses within bending plate. These analogue experiments may be
relevant to the outer-rise regions of subduction zones, where the oceanic plate is flexed downward, and the abyssal flanks adjacent
to fast-spreading mid-ocean ridge crests, where recent models for axial high development suggest that the plate is unbent as it rafts
away from the axis.
© 2006 Elsevier B.V. All rights reserved.
Keywords: extensional faulting; mid-ocean ridges; subduction; analogue models; lithosphere flexure; outer-rise
1. Introduction
⁎ Corresponding author. Present address: Department of Geological
Sciences, UC Santa Barbara, Building 526, Santa Barbara, CA 931069630, USA. Tel.: +1 805 893 2853.
E-mail address: [email protected] (S. Supak).
0012-821X/$ - see front matter © 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.epsl.2006.03.028
Extensional fault systems formed in association with
the bending of an oceanic plate during subduction (e.g.
[1]) or the unbending of the newly accreted lithosphere
126
S. Supak et al. / Earth and Planetary Science Letters 246 (2006) 125–137
along a fast-spreading mid-ocean ridge crest (e.g. [2–5])
provide critical pathways for fluid circulation and mantle
serpentinization. These processes alter the composition
and rheological properties of oceanic lithosphere and
ultimately impact many geochemical and mechanical
aspects of the subduction and mid-ocean ridge systems
[6]. Their influence may be most pronounced in the
former environment, where the distribution of intermediate depth earthquakes (e.g. [7,8]), the strength of the
bending plate (e.g. [9,10]) and the character of arc
volcanism (e.g. [11]) are linked to metamorphism within
the down going slab. Moreover, flexure-related outerrise earthquakes may play an important role in
transferring stress to the subduction interface [12] or in
triggering devastating tsunami events [13]. The geometry and mechanics of flexure-induced fault populations
therefore are topics of broad interest in the earth-science
community and may have implications for earth-hazards
research in subduction zone settings.
During the last decade, great effort has been expended
to statistically characterize populations of extensional
faults exposed in regions subjected to lithospheric stretching. In these settings, power-law distributions of fault
size s (length, throw, or spacing) versus frequency are
observed commonly [14,15], with the total number of
faults N(s) having size ≥ s expressed as:
N ðsÞ ¼ as−D
ð1Þ
where D is known as the power-law exponent and the
constant a reflects the total number of faults. Analogue
and numerical experiments of extensional fault growth
have confirmed the development of a power-law fault
population at low strains (e.g. [16–18]). Such scaling
implies a spatial correlation between faults, with each
fault interacting elastically with its neighbors [19]. Since
power-law distributions exhibit a self-similar geometry
with no characteristic length scale, they allow for prediction at scales smaller than those observed—a potentially
powerful tool in fluid flow modeling and other applications (e.g. [20,21]).
The experiments described in this paper are designed
as potential analogues for lithospheric flexure induced
by vertical line loads. Although stretching- and flexinginduced normal faults may exhibit many similar traits,
flexure differs fundamentally from stretching in that
fault nucleation is concentrated along lines of maximum bending stress aligned parallel to and at a characteristic distance from the applied load (Fig. 1). We
hypothesize that this condition may suppress the development of power-law fault size distributions,
creating a fault population that exhibits a fundamentally
different geometry than commonly observed in extensional stretching regimes. We are motivated largely by
the need to understand fault development in two difficult to study submarine environments, the unbending
abyssal flank regions at fast-spreading ridges (e.g. [2])
and the flexing outer-rise regions of subduction zones
(e.g. [1]).
Abyssal hill faults on the flanks of fast-spreading midocean ridges have long been thought to be the product of
tectonic stretching of brittle lithosphere [22,23]. A more
recent view, however, is that faults flanking axial highs
are formed during the unbending of lithosphere as it
Fig. 1. Cartoon illustrating extensional fault growth in regions undergoing: a) Lithospheric stretching. The extensional component of stress is constant
with depth. Faults nucleate over a broad region, and elastic interactions give rise to a power-law fault distribution. b) Lithospheric flexure. The
extensional component of stress is depth dependent, with a maximum in tension at the surface and compression at greater depth. Faults preferentially
nucleate and localize along lines of maximum bending stress—fundamentally different from lithospheric stretching.
S. Supak et al. / Earth and Planetary Science Letters 246 (2006) 125–137
moves away from the rise axis [2–4]. Faulting studies,
which utilize multi-resolution sonar and bathymetric
data, consistently report small b0.04–0.08 brittle strains
and non-power-law scaling relationships in this environment (e.g. [22,24]). Cowie et al. [25,26] has suggested that size-frequency data from the flanks of the fastspreading East Pacific Rise (EPR) can be described by a
negative exponential scaling relationship of the form:
N ðsÞ ¼ be−sk
ð2Þ
where N(s) is the number faults having size ≥ s, λ is the
reciprocal of the mean fault size and b is the total number
of faults. Other examples of exponential distributions
include the height distribution of seamounts [27] and the
duration-amplitude scaling of volcanic tremor [28].
Unlike the power-law model, which is self similar (Eq.
(1)), an exponential distribution exhibits a characteristic
length scale (1/λ) and may not afford any predictive
capabilities at scales lower than that observed. In analyzing our model results, we will test the relative fit and
overall goodness-of-fit for both the exponential and
power-law distributions.
Flexure-related faulting has long been recognized in
the subduction environment, where a pronounced toughparallel horst-and-graben topography is commonly
observed between the outer-rise and the trench (e.g.
[1]). Presently, we are not aware of any published sizefrequency data for these regions. Kobayashi et al. [29],
however, measured the fault displacement to length (d–
L) ratio for sections of the Japan and Kurile trenches,
where it can be shown that the outer-rise faults form
independent of the pre-existing abyssal fabric. They
report a similar range of the d–L ratios for those outer
trench walls as are measured for the EPR fault populations [26,23]. Bohnenstiehl and Kleinrock [30] show
that the d–L ratio of faults along the EPR is significantly
less than the d–L ratio of faults within the slowerspreading Mid-Atlantic Ridge and continental settings,
where a stretching origin for faults is not disputed. This
commonality in the d–L ratio of fast-spreading abyssal
hill and outer-rise faults could be interpreted as consistent
with a common origin; if, for example, bending promotes
rapid linkage along the line of maximum tension.
2. Mechanical differences between flexing and
stretching
Two-dimensional numerical models for fault development can be used to illustrate and investigate differences between faults formed in a brittle layer that is
stretched versus one that is bent. The numerical approach
127
we use has been applied to the problem of normal fault
formation in several studies of simulated lithospheric
stretching [31–33] and the details are described in Lavier
et al. [32].
An essential feature of these numerical models is that
the fault locations are not prescribed, but model faults
develop as a consequence of assumed weakening of the
areas that strain in a brittle manner. Up to a prescribed
Mohr–Coulomb yield stress the material deforms elastically, but when a region reaches the yield stress it
deforms plastically. The local yield stress is reduced as a
function of the plastic strain. For all cases shown here
the elastic behavior is specified by a Young's Modulus
of 5 × 1010 Pa and a Poisson's Ratio of 0.25. Mohr–
Coulomb yielding is set by a friction coefficient of 0.6
and an initial cohesion of 11 MPa. The cohesion is
reduced linearly with strain to a value of 2 MPa at a
plastic strain of 0.1%. The numerical grid size is 500 m.
The brittle layer is 10 km thick, 75 km wide and the
acceleration of gravity is 10 m/s2. The top of the layer is
stress free, and the base floats on an inviscid substrate (a
Winkler foundation) with the same density as the layer
(3000 kg/m3).
Fig. 2a shows three time steps in a particular model of
brittle layer stretching. The sides of the layer are pulled
apart at a constant rate and are shear stress free. The
plots of plastic strain across the stretching layer show
that zones of concentrated brittle-plastic deformation,
analogous to faults, exhibit two important characteristics. First, the faults are randomly distributed across
the entire model domain. Second, the faults form first at
the top of the layer, where the yield stress is lowest, but
eventually cut the entire layer. The randomness of the
position of the incipient model faults is likely due to
computer round-off errors producing small random
perturbations in the stress field.
Fig. 2b shows three times steps in the development
of faults in a brittle layer that is bent and not stretched.
The dimensions and properties of the model layer are
the same as for the stretching case described above. The
only difference is in the side boundary conditions. The
right side of the bending layer is fixed horizontally, but
is free to move vertically. The left side is pushed vertically down, but is free to move horizontally. A normal
stress equal to the initial lithostatic pressure is applied to
the left side.
The pattern of surface deflection due to bending is
similar to that predicted by thin-elastic plate bending
theory [34]. Thin plate theory predicts that the maximum
bending stress occurs at a given distance from the load
applied to the plate and Fig. 2b shows that the first faults
do form at one horizontal position, where the extensional
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S. Supak et al. / Earth and Planetary Science Letters 246 (2006) 125–137
Fig. 2. Plastic strain from numerical models of deformation of brittle (Mohr–Coulomb) layers where cohesion is reduced with strain. The strain
weakening results in zones of concentrated strain that we term model normal faults. a) Results for a layer that is stretched. The sides are pulled apart
by 48, 64 and 120 m for the three examples. Note that faults form all across the top of the brittle layer, even at the smallest amount of stretching. b)
Results for a layer with the same elastic and brittle properties (given in the text) that is bent and not stretched. The left side of the model is pushed
down by 190, 207 and 477 m in the three examples. The model faults first form at a set distance form the left side of the model and later faults form
around the first formed fault.
bending stresses are a maximum at the surface of the
plate. With more bending, additional fault breaks occur
on either side of the initial break. In contrast to the
stretching faults, the bending-related normal faults do
not cut the entire brittle layer, but die out in the middle to
lower layer. Bending puts the lower part of the layer into
compression so that normal faults will not form there.
Though not shown here, we ran cases with different
strain weakening parameters and found that greater strain
weakening produced more widely spaced bending-related normal faults. As discussed in [31], the rate of strain
weakening affected the character of the bending-related
normal faulting, with no distinct faults produced for slow
strain weakening. The initial pattern of stretching-related
faults is not as sensitive to these parameters.
The very different distribution and progression of
faults produced by bending and stretching in the twodimensional models lead us to believe that the threedimensional development of faults should be very different for bending and stretching. Three-dimensional fault
development would be very difficult to simulate at high
enough resolution to see potential fault pattern differences. Thus, we turned to scaled physical models, similar
to those used previously to study stretching-related fault
development [16]. The specific idea that we test with
three-dimensional physical models is that bending stress
S. Supak et al. / Earth and Planetary Science Letters 246 (2006) 125–137
129
tends to organize faults and produce a different
geometric distribution than for stretching-related faults.
3. Experimental setup
Previous analogue work on bending-related crack
growth has focused on joint formation and mechanical
controls on joint spacing [35,36]. These studies utilized
very thin (b 0.5 mm) brittle coatings or polystyrene plates
subjected to four point bending, where the material is
bent with a constant radius of curvature (cylindrical
folding). More recently, a number of analogue studies,
using wet clay [16,17] and plaster [37,38], have
examined the scaling properties of fault populations
grown in response to stretching. Collectively, these
studies have found power-law scaling behaviors at low
(b ∼ 0.10) brittle strains, in agreement with field
observations [14], with the population evolving toward
an exponential distribution at larger strains. Also,
evident from these experiments is the importance of
brittle layer thickness, with power-law behavior break-
Fig. 3. a) Schematic of the apparatus used to simulate flexure-induced
faulting. A foam layer beneath the plaster represents the Earth's
hydrostatic force. The plaster, used to simulate lithosphere, fills the
inner box. The box is removed and a downward linear load is applied
across the plaster by a depressor. A maximum depression of 4.75 cm
created a deflection with a wavelength of ∼ 16 cm. b) Side view of the
apparatus after full deflection is reached.
Fig. 4. Time series photographs of plaster surface during progressive
bending. A linear line load is applied along the top of each frame.
Lighting angle for this series was positioned to highlight the early
stages of fault growth.
ing down at lower strain within thinner mechanical
layers [17]. To examine the growth and scaling of faults
during lithospheric flexure, we have conducted a similar
set of experiments with plaster layers of two distinct
thicknesses. Here, the deflection of a brittle layer and
elastic substrate by a vertical line load does not result in
cylindrical bending, but rather creates an evolving radius
of curvature that may simulate previously mentioned
environments.
Wet clay and partially solidified plaster are viscoelastic-plastic materials that both shear and flow during
deformation. Both scale similarly to the Earth [39];
however, for these experiments, plaster was a more
desirable material because of its relatively low initial
viscosity. This allowed it to be poured onto a weak
substrate, where it flowed to produce a uniform layer
thickness and smooth surface able to preserve a record of
fine scale cracking. In addition, because plaster does not
shrink, the risk of unrelated cracking due to drying was
eliminated—a problem associated with clay models.
In this study, a layer of plaster was poured upon a
sheet of thin plastic velum that rests on a 12.5 cm thick,
60 × 136 cm block of foam rubber (Fig. 3). The foam
pad's function was to simulate hydrostatic forces in the
Earth. The plaster layer had dimensions of ∼ 25 × 40 cm
and thicknesses of 5 and 15 mm. These thicknesses
bracket those used by in the stretching models of
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S. Supak et al. / Earth and Planetary Science Letters 246 (2006) 125–137
Spyropolous et al. [16] (8 mm), with the thicker layer
being slightly thinner than the 18 mm-thick clay layer
used by Ackermann et al. [17]. Bending approximating
flexing was produced by applying a linear load across the
plaster and foam (Fig. 3). Final depression was reached
when the foam layer was indented 4.75 cm, creating a
deflection wavelength measured to be ∼16 cm, as
controlled by the foam pad's stiffness (Fig. 3). Additionally, the rate of depression, 0.18 cm/s, was slow enough
that inertial and acceleration effects are negligible but
rapid enough that the plaster's rheology was essentially
uniform during deformation. The goal of these experiments was to generate a set of flexure-induced brittle
fractures rather than maintain a perfect dynamic similarity
to the Earth. These physical experiments do not exactly
simulate plate flexure due to the foam pad's elastic
properties; however, the models do share the salient
features of flexure in that bending stress is maximum at a
distance from the applied line load, as evidenced by the
localization of the initial cracks (Fig. 4 bottom).
The rheological properties of wet plaster vary as a
function of time. At some time after pouring, the plaster
hardened to a strength range that represents similarity to
the Earth and at that time it was deformed [39]. A minislump test, as described by Sales [39], was performed in
conjunction with visual observations in order to
determine the correct strength for the partially solidified
plaster. Drying times averaged around ∼ 10 min, but
varied depending on the humidity and temperature of
the room, which were not controlled. At the time of
deformation, the plaster had enough strength to yield
through fracture while the cracked areas retained some
fraction of their initial flexural rigidity. Differences in
environmental conditions and drying time for each
model run no doubt led to slight variability in the
mechanical properties of each plaster layer. As we are
interested in the nature of the distribution, rather than the
exact value of the scaling exponent (i.e., λ or D),
multiple model runs were conducted for both the thin
and thick layer cases. This yielded an ensemble of crack
populations formed within plaster layers spanning a
range of mechanical conditions.
As the model surface and foam substrate were flexed
downward, cracks formed parallel to the bending axis
and lengthened through a combination of lateral
propagation and along-strike linkages. A time sequence
of the model surface during deflection, as shown in Fig.
4, demonstrates that the sequential faulting behavior
observed in the numerical example (Fig. 2b) is reproduced by these analogue models. The progressive
Fig. 5. a) Original image of plaster surface after bending. b) Binary image of cracks selected for analysis. Boxes indicate enlarged sections. c,d)
Enlarged section of binary image overlaying the original plaster image.
S. Supak et al. / Earth and Planetary Science Letters 246 (2006) 125–137
131
Table 1
Thin model length statistics
Run
1
2
3
4
5
6
7
8
9
n
299
198
267
185
194
219
243
247
241
Strain
0.055
0.039
0.054
0.051
0.042
0.054
0.064
0.058
0.031
Exponential
χ2crit
Power law
λ±
R2
χ2
D±
R2
χ2
−0.506 ± 0.007
−0.339 ± 0.005
−0.423 ± 0.003
− 0.311 ± 0.004
−0.292 ± 0.004
−0.323 ± 0.004
−0.356 ± 0.003
−0.313 ± 0.003
−0.598 ± 0.007
0.95
0.96
0.99
0.98
0.96
0.97
0.98
0.97
0.97
750.9
327.9
83.8
271.3
583.9
659.8
560.3
680.6
462.5
− 1.433 ± 0.023
− 1.163 ± 0.028
− 1.219 ± 0.031
− 1.057 ± 0.027
− 1.112 ± 0.022
− 1.136 ± 0.021
− 1.195 ± 0.023
− 1.137 ± 0.022
− 1.521 ± 0.024
0.93
0.90
0.85
0.89
0.93
0.93
0.92
0.92
0.94
1509.2
786.3
2059.2
548.4
454.2
513.4
728.6
910.9
763.4
339.3
230.7
305.0
216.0
226.4
253.4
279.3
283.6
277.1
Bold indicates χ2 values less than the critical value, for which the proposed model distribution cannot be rejected at the 95% confidence level.
flexure of the model surface approximates the movement of oceanic lithosphere across the outer-rise or the
progressive unbending of the lithosphere as it is rafted
away form the fast-spreading ridge axis.
When final depression was reached, digital images
were taken for each of the experimental runs with the
model surface illuminated at an oblique angle (Fig. 5a).
During some of the model runs, additional still images
were taken during the experiment (e.g., Fig. 4). The
crack populations, however, have not been analyzed as a
function of time, due to the fact that the number of cracks
remained quite small at most time steps and our experimental set up did not allow the lighting angle to be
adjusted as the model surface was increasingly deflected.
A directional filter, or first derivative edge enhancement filter, was used to highlight image features having
specific directional components (gradients) and to
remove the effect of long-wavelength shadowing
associated with the curvature of the model's flexed
plaster surface. For these models, the filter was used to
specifically enhance shadows associated with cracks
paralleling the flexing axis. The result of this process was
a gradient map where areas with uniform pixel values
were zeroed in the output image, while those that were
variable (cracks) were presented as bright edges. From
this gradient information, a binary image was created
with additional low gradient values zeroed using a
threshold corresponding to roughly the 92nd quantile of
pixel values. This was followed by a majority analysis,
with the long axis of the kernel aligned parallel to the
linear load. Objects were selected from the final binary
image using 2D connectivity criteria, whereby pixels
were associated with an object if either an edge or a
corner touches. As shown in Fig. 5, this processing
routine was very successful in selecting visually identifiable cracks within the plaster. Only identified objects
with major axis lengths greater than 0.3 cm were considered in our analysis. Inspection of the detection results
and model surface suggested that smaller cracks were not
consistently recognized by the detection algorithm.
4. Statistical analysis
For each crack, the straight-line distance between its
two tips was measured in order to derive the cumulative
length-frequency distribution of cracks within each
Table 2
Thin model spacing statistics
Run
1
2
3
4
5
6
7
8
9
n
110
68
109
53
63
71
83
78
89
Cv
0.72
0.73
0.76
0.83p
0.74
0.77
0.60
0.86p
0.67
Exponential
χ2crit
Power law
λ±
R
χ
D±
R
χ
− 0.997 ± 0.017
− 0.532 ± 0.012
− 0.728 ± 0.010
− 0.395 ± 0.009
− 0.562 ± 0.017
− 0.563 ± 0.024
− 0.909 ± 0.020
− 0.452 ± 0.012
− 0.764 ± 0.016
0.97
0.97
0.98
0.98
0.95
0.89
0.96
0.95
0.97
32.5
28.9
62.2
13.2
38.4
89.2
85.8
53.0
53.4
− 1.301 ± 0.055
− 0.935 ± 0.071
−1.209 ± 0.054
− 0.635 ± 0.064
− 0.896 ± 0.077
− 0.803 ± 0.074
− 1.175 ± 0.081
−1.117 ± 0.051
− 1.390 ± 0.068
0.84
0.72
0.83
0.66
0.69
0.63
0.72
0.86
0.83
760.1
444.0
977.2
250.6
412.2
589.2
629.9
338.9
781.1
2
2
2
2
134.4
87.1
133.3
69.8
81.4
90.5
104.1
98.5
110.9
Bold indicates χ2 values less than the critical value, for which the proposed model distribution cannot be rejected at the 95% confidence level. p — Cv
value not significantly anticlustered.
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S. Supak et al. / Earth and Planetary Science Letters 246 (2006) 125–137
Table 3
Thick model length statistics
Run
1
2
3
4
5
6
7
8
9
10
n
236
336
292
453
346
323
453
396
369
473
Strain
0.058
0.073
0.052
0.092
0.097
0.066
0.075
0.092
0.069
0.076
Exponential
χ2crit
Power law
λ±
R2
χ2
D±
R2
χ2
− 0.349 ± 0.007
− 0.535 ± 0.007
− 0.468 ± 0.008
− 0.415 ± 0.008
− 0.507 ± 0.006
− 0.502 ± 0.009
− 0.670 ± 0.009
− 0.452 ± 0.006
− 0.534 ± 0.010
− 0.907 ± 0.014
0.91
0.94
0.93
0.86
0.96
0.90
0.92
0.94
0.89
0.90
1708.09
2137.39
2296.74
5922.33
2627.90
3262.84
3727.75
3721.57
3599.83
3417.20
− 1.309 ± 0.015
− 1.530 ± 0.016
− 1.410 ± 0.015
− 1.457 ± 0.015
− 1.416 ± 0.018
− 1.539 ± 0.018
− 1.718 ± 0.015
− 1.393 ± 0.014
− 1.580 ± 0.017
− 2.020 ± 0.018
0.97
0.97
0.97
0.95
0.95
0.96
0.97
0.96
0.96
0.96
409.42
920.71
338.13
2830.22
1371.28
1617.35
1856.57
1124.08
1938.38
3098.28
271
378
332
502
389
365
502
442
413
523
Bold indicates χ2 values less than the critical value, for which the proposed model distribution cannot be rejected at the 95% confidence level.
model run. To examine the spacing distribution of
cracks, a series of 1-D scanlines were run across the
processed binary images in a direction normal to the
bending axis. The spacing between faults was recorded
along each scanline and then combined for each model
run. To minimize bias associated with sampling the same
pair of faults multiple times, scanline spacing was set
equal to the 90th quantile of crack length in each model.
A power-law length or spacing distribution should be
represented as a linear trend on a log–log plot of the
cumulative frequency data, and an exponential model
should be characterized by a linear trend on a log-linear
plot of the cumulative frequency data.
Two methods were used to establish a quantitative
goodness-of-fit for both the power-law and exponential
distributions of the data. First, correlation coefficients
(R2) were calculated from the linear regression of both the
cumulative length- and spacing-frequency data against
the proposed distributions. This is the same method used
to establish the exponential scaling of the EPR abyssal
hill fault populations [23,25,26]. The correlation coefficient provides a measure of variability about the modeled
distribution, with higher values indicating a better fit to
the model. The goodness-of-fit was further tested using
the Chi-squared (χ2) test for both model distributions.
The χ2 value was calculated using the observed Oi and
expected Ei cumulative frequency, and is defined as:
v2 ¼
n
X
ðOi −Ei Þ
Ei
i¼1
ð3Þ
where n is the number of observations. The null
hypothesis is that the data are drawn randomly from a
population having either a power-law or exponential
distribution (tested separately). This hypothesis can be
rejected if the χ2 value is greater than a critical value that
depends on the degrees of freedom (n − 1) and confidence
level at which the test is performed [40].
Coefficient of variation (Cv) analysis represents
another commonly used method for characterizing the
Table 4
Thick model spacing statistics
Run
1
2
3
4
5
6
7
8
9
10
n
90
150
111
203
146
124
224
180
174
235
Cv
0.58
0.66
0.69
0.67
0.75
0.70
0.63
0.68
0.65
0.65
Exponential
χ2crit
Power law
λ±
R2
χ2
D±
R2
χ2
−0.700 ± 0.011
−0.908 ± 0.007
−0.553 ± 0.010
−1.064 ± 0.012
−1.027 ± 0.025
−0.696 ± 0.008
−1.099 ± 0.008
−0.897 ± 0.011
−0.975 ± 0.008
−1.207 ± 0.010
0.98
0.99
0.97
0.98
0.92
0.98
0.99
0.98
0.99
0.99
76.31
81.14
73.62
220.55
163.67
101.36
208.49
168.10
157.24
228.47
− 1.291 ± 0.076
− 1.288 ± 0.054
− 1.096 ± 0.061
− 1.257 ± 0.050
− 1.305 ± 0.051
− 1.363 ± 0.049
− 1.370 ± 0.046
− 1.141 ± 0.049
− 1.222 ± 0.052
− 1.440 ± 0.043
0.77
0.79
0.75
0.76
0.82
0.86
0.80
0.75
0.76
0.82
11904.54
37736.15
46762.84
66792.52
89534.68
105400.97
86517.31
110519.81
119729.64
103193.38
112
178
135
236
174
149
258
211
204
271
Bold indicates χ2 values less than the critical value, for which the proposed model distribution cannot be rejected at the 95% confidence level. All Cv
values are significantly anticlustered.
S. Supak et al. / Earth and Planetary Science Letters 246 (2006) 125–137
133
Fig. 6. Thin model cumulative length-frequency distribution for nine experimental runs plotted on two different scales: a) Log–log scale, where a
power-law distribution should produce a linear trend. b) Log-linear scale, where an exponential distribution should produce a linear trend.
spatial distribution (or clustering properties) of a set of
objects [41]. For each experimental run, Cv is calculated
as the ratio of the standard deviation of the spacing to the
mean spacing, as derived from the scanline analysis. A
value of 1.0 indicates a perfectly random distribution, as
the mean and standard deviation of a Poisson distribution
are equal, and a value of 0.0 indicates a perfectly uniform
spacing. In stretching-induced fault populations that are
non stratabound, Cv typically takes a value of N 1 and
these populations are termed spatially clustered. Where
the spacing is controlled by the mechanical layer
thickness, as in joint or vein sets, or by some other
process, a more regular spacing is observed (anticlustering), with Cv b 1.0 [42].
5. Observations and results
During the experimental runs, cracks were observed
to nucleate preferentially along the flexural axis, where
maximum tension is predicted (Fig. 4). As the linear load
increasingly deflected the plaster layer, the spatial
position of the bending axis evolved slightly and regions
adjacent to the initial cracks were progressively brought
to failure (Fig. 4). This pattern differs from that observed
in stretching experiments, where the earliest stages of
crack nucleation occur at random locations distributed
throughout the stretched layer (Figs. 1 and 2). As in the
stretching models, cracks lengthen along-strike through
a combination of lateral propagation and linkage, with
overlapping fault segments and ramp-like structures
observed in map view (Fig. 5).
Following Spyropoulos et al. [16], brittle strain was
calculated by summing the pixel width of all faults and
normalizing by the total pre-faulted area of each image.
For the thin models, brittle strains ranged from 0.031–
0.064, with a mean of 0.050 and a standard deviation of
0.011 (Tables 1 and 2). For the thick models, brittle
strains ranged from 0.058–0.097, with a mean of 0.075
and a standard deviation of 0.015 (Tables 3 and 4). The
range of brittle strains measured for a given layer
Fig. 7. Thin model cumulative spacing distribution for nine experimental runs plotted on a) Log–log and b) Log-linear scales.
134
S. Supak et al. / Earth and Planetary Science Letters 246 (2006) 125–137
Fig. 8. Thick model cumulative length-frequency distribution for ten experimental runs plotted on a) Log–log and b) Log-linear scales.
thickness reflects variable amounts of brittle and plastic
deformation resulting from small differences in mechanical properties of each plaster run.
5.1. Thin model crack distributions
The cumulative length-frequency distribution of cracks
within each of the nine thin model runs is shown in Fig. 6. A
power-law length distribution should be represented as a
linear trend on a log–log plot of the cumulative frequency
data; however, the data show significant curvature when
presented in this form (Fig. 6a). The exponential model,
which is characterized by a linear trend on a log-linear plot
of the cumulative frequency data (Fig. 6b), appears to
provide a generally better fit to the data. The cumulative
spacing distribution of cracks is shown in Fig. 7. As with the
length distribution, the log–log cumulative frequency data
show considerable curvature for most model runs (Fig. 7a).
The more linear nature of the log-linear spacing data (Fig.
7b) is qualitatively consistent with a better fit to the
exponential model.
For the length-frequency data, R2 values range from
0.85–0.94 for the power-law model and from 0.95–0.99
for the exponential model, with higher relative R2 values
for the exponential model in each model run (Table 1).
For the spacing-frequency data, R2 values range from
0.63–0.86 for the power-law model and 0.95–0.98 for
the exponential model (Table 2). For comparison, R2
values of ∼ 0.92–0.99 were reported for the fit of an
exponential size-frequency model for faulting data on
the East Pacific Rise [23]. Arguably, the spacing data
could be fit reasonable well by the power-law model
over a more limited scale range, ∼0.4–1.0 cm (Fig. 7a).
Only one of the analogue runs, however, has a larger R2
value (0.99 vs. 0.98) for the power-law model, relative
to the exponential model, if both distributions are
compared within this scale range.
For the length-frequency data, the χ2 values are less
for the exponential model relative to the power-law
model for 7 of the 9 model runs (Table 1). This is
consistent with the better relative fit of the exponential
model, as inferred from the R2 analysis. However, the
Fig. 9. Thick model cumulative spacing distribution for nine experimental runs plotted on a) Log–log and b) Log-linear scales.
S. Supak et al. / Earth and Planetary Science Letters 246 (2006) 125–137
hypothesis that the observed distribution was drawn
randomly from an exponential population can be rejected
in all but one of the experimental runs with 95%
confidence. For the power-law model, the null hypothesis can be rejected at the 95% level in all model runs. In
analyzing the spacing-frequency distributions, lower χ2
values are obtained for the exponential model, relative to
the power-law model, for each experimental run (Table
2). With regard to the spacing distributions, for each
model run, we cannot reject the hypothesis that the data
were drawn from an exponential population, but we can
reject the hypothesis that the data were drawn from a
power-law population at the 95% level (Table 2).
For the thin-model crack populations Cv ranges
between 0.60 and 0.86, as listed in Table 2. The significance of these values, relative to a random distribution,
can be assessed by simulating a population derived from a
Poisson point-process (e.g. [43,44]). Accounting for
sample size, seven of the nine experimental runs generated
crack populations that can be considered significantly
anticlustered, indicating that the spacings are somewhat
more regular than random (Table 2). This is consistent
with the concept of sequential faulting, with the first crack
forming along the line of maximum tensile stress within
the flexing plate (Figs. 2 and 4). This initial break
‘shadows’ stress over some adjacent zone [45], but as
deflection continues additional cracks are formed to the
sides. These breaks occur where extensional stresses exceed the strength of the plaster, and therefore the characteristic spacing reflects both the flexural wavelength and
strength of the layer. This process differs from stretching,
where initial cracking is distributed throughout the layer
and the resulting cracks evolve and interact simultaneously. The possible role of layer thickness in controlling crack
spacing [e.g., 17] is addressed below through comparisons
with the thick model results.
5.2. Thick model crack distributions
The cumulative length-frequency distribution of
cracks within each of the ten thick model runs is
shown in Fig. 8. The cumulative spacing distribution of
cracks is shown in Fig. 9. For the length-frequency data,
R2 values range from 0.95–0.97 for the power-law
model and from 0.86–0.94 for the exponential model,
with higher relative R2 values for the power-law model in
each model run (Table 3). This is the opposite of the
behavior observed in the thin models. For the spacingfrequency data, R2 values range from 0.76–0.82 for the
power-law model and 0.92–0.99 for the exponential
model (Table 4). These spacing results are similar to
those observed for the thin layer models. The character-
135
istic spacing (1 / λ), however, does not scale with the
plaster thickness (Tables 2 and 4), as would be the case if
this parameter were solely modulating the spacing of
cracks [42].
For the length-frequency data, the χ2 values are less
for the power-law model relative to the exponential
model for all ten model runs (Table 1). This is consistent
with the better relative fit of the power-law model, as
inferred from the R2 analysis. However, the hypothesis
that the observed distribution was drawn randomly from
either a power-law or exponential population can be
rejected at the 95% confidence level for all of the
experimental runs. In analyzing the spacing-frequency
distributions, lower χ2 values are obtained for the
exponential model, relative to the power-law model,
for each experimental run (Table 4). We cannot reject at
the 95% confidence level the hypothesis that the spacing
data were drawn from an exponential population, but we
can reject the hypothesis that the data were drawn from a
power-law population (Table 4).
For the thick-model crack populations, Cv ranges
between 0.58 and 0.70, as listed in Table 4. Accounting
for sample size, all ten thick model runs can be
considered significantly anticlustered, indicating that
the spacings are more regular than random (Table 4)—
consistent with the thin model results.
6. Summary and discussion
Through elastic interactions during simultaneous
growth, fractures formed in response to stretching may
evolve from an initially random distribution of nucleation points to form a self similar network of fractures
[15,19]. Consequently, power-law size-frequency distributions and clustered spacings have been observed
previously in extensional stretching environments (e.g.,
[14,15]) and models of extensional stretching at low
brittle strains (e.g., [16–18]). In flexural settings,
however, the pattern of faulting is influenced by a stress
distribution that differs fundamentally from the stretching regime, with fault nucleation and growth occurring
preferentially in association with stress maxima along
the flexural axis within the plate. Two-dimensional
numerical simulations illustrate that this stress distribution results in a more sequential history of fault formation, differing from the simultaneous growth history that
is characteristic of stretching environments.
To gain further insight into how these different stress
regimes may influence the geometric scaling of fault
populations, a series of the analogue experiments have
been preformed in order to simulate brittle extension
during lithospheric flexure. The experimental set up
136
S. Supak et al. / Earth and Planetary Science Letters 246 (2006) 125–137
involves the vertical deflection of a partially solidified
plaster layer overlying a foam substrate that represents
the Earth's hydrostatic restoring force. The upper
surface of nine thin (5 mm) and ten thick (15 mm)
layer models underwent ∼ 5% and 7.5% brittle extension, respectively, during these experiments. The
resulting crack populations were analyzed to examine
the goodness-of-fit for both power-law and exponential
length- and spacing-frequency models.
Correlation coefficient analysis (R2) indicates that
length-frequency distribution of flexure-induced cracks
is somewhat better described by an exponential model
for the thin layer experiments and somewhat better
described by a power-law model in the thicker layer
experiments. The Chi-squared goodness-of-fit tests,
however, indicate that in general these length-frequency
populations are not well described by either the distribution. In contrast, spacing-frequency data from both the
thin and thick models are well described by an exponential distribution, and coefficient of variation (Cv)
analysis indicates values b 1 (anticlustered).
As simultaneous elastic interactions form the mechanical basis for power-law (fractal) fault scaling relationships
[46,47], the sequential growth history observed in these
models suggests that such development may be inhibited
in flexural regimes. Layer thickness effects provide an
additional mechanism for suppressing elastic interactions
[16,17,45]. Similar layer thickness stretching models,
however, show power-law scaling and clustered spacing
within the range of brittle strains observed (b 10%)
[16,17], and unlike vertically-restricted fracture sets
[42], the characteristic spacing (1/ λ) does not scale with
layer thickness in these flexural models. Nonetheless, we
cannot completely rule-out the influence of layer
thickness. For example, the relative better fit of the
length-frequency data to power-law distribution in the
thick layer experiments may indicate that layer thickness
continues to influence fault development during flexure.
Admittedly, the existence of a neutral surface within the
bending plaster layer (Fig. 1) could reduce the effective
layer thicknesses in these models, further suppressing
fault interaction.
Recent models for mid-ocean ridge axial high
development suggest that the unbending of the newly
accreted lithosphere may play a significant role in abyssal
hill fault development in the fast-spreading environment
[2–5,48]. Our results indicate that the non-power-law
nature of these fault populations, particular as recorded by
crack spacing, could be consistent with a flexural origin;
however, they are clearly not a conclusive line of
evidence in support of this model. Our models also
predict a non-power-law distribution of faults within the
outer-rise regions of subduction zones, where a flexural
origin is not in dispute. Future high-resolution sonar
surveys of significant spatial extent will be needed to test
this prediction for outer-rise faulting.
Acknowledgements
We are thankful for the support of the LDEO summer
intern program, its coordinator D. Abbott, and the
LDEO Director's Office. Technical and scientific
discussions with C. Scholz, Z. Karcz and T. Koczynski
are gratefully acknowledged. C. Small kindly assisted
with developing the image analysis routines being
implemented. Detailed and constructive reviews by L.
Montesi were very helpful in improving the quality of
this manuscript. The project was supported by the NSF
grant OCE01-37293.
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