Acta Geophysica
vol. 61, no. 6, Dec. 2013, pp. 1642-1658
DOI: 10.2478/s11600-013-0151-z
Self-affinities of Landforms and Folds
in the Northeast Honshu Arc, Japan
Kazuhei KIKUCHI1, Kazutoshi ABIKO2, Hiroyuki NAGAHAMA1,
Hiroshi KITAZATO3 and Jun MUTO1
1
Department of Earth Science, Graduate School of Science,
Tohoku University, Sendai, Japan; e-mail: [email protected]
2
Forestry and Fisheries Department, Yamagata Prefectural Government,
Yamagata, Japan
3
Institute of Biogeosciences, Japan Agency for Marine-Earth Science and Technology,
Yokosuka, Japan
Abstract
A method to analyze self-affinities is introduced and applied to the
large scale fold geometries of Quaternary and Tertiary sediments or geographical topographies in the inner belt of the Northeast Honshu Arc, Japan. Based on this analysis, their geometries are self-affine and can be
differently scaled in different directions. We recognize a crossover from
local to global altitude (vertical) variation of the geometries of folds and
topographies. The characteristic length for the crossover of topographies
(landforms) is about 25 km and is related to the half wavelength of the
crustal buckling folds or possible maximum magnitude of inland earthquakes in the Northeast Honshu Arc. Moreover, self-affinity of the folds
and topographies can be connected with the b-value in Gutenberg–
Richter’s law. We obtain two average Hurst exponents obtained from the
self-affinities of folds in the Northeast Honshu Arc. This indicates that
there are two possible seismic modes for the smaller and larger ranges in
the focal regions in the Northeast Honshu Arc.
Key words: landforms, folds, Northeast Honshu Arc, self-affinities,
Gutenberg–Richter’s law.
________________________________________________
© 2013 Institute of Geophysics, Polish Academy of Sciences
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1.
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INTRODUCTION
Many folds or geographical topographies (landforms) are apparently curved
or jagged on a wide range of scales, so that their geometries look the same
when viewed at different magnifications or reductions. Such a fundamental
and pervasive scale invariance of folds or geographical topographies has
been studied within a framework of fractal geometry (Mandelbrot 1977,
1982).
Kulik and Chernovsky (1996) analyzed fold geometries in the low-grade
Paleoproterozoic strata-bound banded iron ores of the Krivoy Rog basin in
the Ukraine through 5-8 different scales, using a polygonal approximation
method and a grid cell counting method (Mandelbrot 1982). They pointed
out that these fold geometries are self-similar. Moreover, Wilson and Dominic (1998) analyzed geographical topography and structure in two areas of
the North American central Appalachian Mountains: one in the intensely deformed Valley and Ridge province, the other in the Appalachian Plateau
province. They pointed out that variations in the fractal characteristics of topography are related to near-surface structure relief.
Kitazato (1971) reported power spectrum analyses of the large scale fold
geometries of the Quaternary and Tertiary (Quaternary and Tertiary sediments) or geographical topographies in the inner belt of the Northeast Honshu Arc, Japan. He found that the spectrum of the amplitude A(λ) of the fold
or topographical geometries is related to the wavelength λ as A(λ) ∝ λp
where p is a positive constant (Nagumo 1969a, b). This indicates that the geometries of folds or geographical topographies are scale-invariant. The result
reported by Kitazato (1971) is in accordance with the assumption that the
Fourier spectrum W(k) of crustal deformation is related to the wave number
k as
W (k ) ∝ k − p ,
(1)
Mizoue (1980) showed that the spectrum of the vertical crustal deformation
of the Japan inland is inversely proportional to the wave number of the
crustal deformation. This result is the special case where p = 1.
A variety of tectonic processes under anisotropic stress fields and continuing erosion produce the various geographical topographies (landforms)
of the Earth. Matsushita and Ouchi (1989a, b) proposed a method to analyze
the self-affinity of various curves differently scaled in different directions.
They applied this method to a few transect profiles of mountain topographies
and showed that the profiles are self-affine. They also obtained the Hurst exponent H which is a measure of the continuity of a given curve (Feder 1988,
Peitgen and Saupe 1988).
Matsushita and Ouchi (1989a, b) suggested that transect profiles of
mountain topographies are characterized by at least two regimes: one local
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and the other global. They stated that the local and the global structures are
brought about by small-scale erosion and plate tectonics, respectively. Similarly, for the across-strike sections of folded bed surfaces, the scale invariance of the fold might be affected by a variety of tectonic processes under
the anisotropic stress field. However, analyses of folds in previous studies
(Nagumo 1969a, Kitazato 1971, Wilson and Dominic 1998) have been based
on only one parameter such as the fractal dimension D or the spectral exponent p. So, it is necessary to determine whether the scale invariance of the
folds or the crustal deformations is isotropic (self-similar).
Here we briefly introduce a method of self-affinity analysis, and apply it
to the across-strike section of folded bed surfaces in Quaternary and Tertiary
sediments and the transect profiles of geographical topographies (landforms)
in the inner belt of the Northeast Honshu Arc, Japan. Then, we discuss the
self-affinity and a crossover from local to global vertical variation of the
transect profiles of folds and landforms in the Northeast Honshu Arc. Moreover, we point out that the characteristic length estimated by the crossover
from local to global altitude variation is related to the wavelength λ of the
crustal buckling folds in the Northeast Honshu Arc. Finally, we discuss
a new relationship between the self-affinity of folds and the scale invariances
proposed in the previous studies.
2.
A METHOD TO ANALYZE SELF-AFFINITY
Self-affinity means that a curve or pattern is differently scaled in different
directions. Many researchers have analyzed the self-affinity of various patterns in geosciences, e.g., fractured surfaces (Power et al. 1987, Nagahama
1991, 1994), fault surfaces (Scholz and Aviles 1986, Power et al. 1987, Nagahama 1991, 1994), geographical topographies (landforms) (Scholz and
Aviles 1986, Matsushita and Ouchi 1989a, b, Malinverno 1989, Mareschal
1989, Ouchi 1990, Turcotte 1992, Wilson 2000), and fluctuation curves
(e.g., Malinverno 1990, Turcotte 1992, Korvin 1992). In this section, we
briefly introduce a method developed by Matsushita and Ouchi (1989a, b)
for the analysis of the self-affinity.
Let us first define the smallest fixed length scale as a unit length scale α
and measure by this scale the curve length Nα between arbitrary points A and
B on the curve (Fig. 1). Then calculate the x- and y-variances, X2 and Y2, of
all measured points between the two points A and B
X2 =
Y2 =
1
N
1
N
N
∑ (x − x )
i
c
2
,
(2a)
i =1
N
∑(y − y )
i
c
2
,
i =1
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Fig. 1. Measurement of a curve length N between a pair of points A and B on a given curve. x and y are coordinates, (xi, yi) is the coordinate of the ith measured
point, Pi, on the curve.
with
xc =
1
N
N
∑ xi ,
i =1
yc =
1
N
N
∑y
i
,
(2b)
i =1
where (xi, yi) is the coordinate of the ith measured point Pi, on the curve. The
standard deviation of X and Y indicates the approximate width of that part of
the curve. Let us then repeat this measurement procedure for many pairs of
points on the curve and determine the distribution using log-log plots of X
and Y versus N whether they scale as
X ∝ N νx ,
Y ∝ N νy ,
(3)
where the exponents νx and νy are different in general. If so, they are related
to each other as
Y ∝ X H , H = νy / νx ,
(4)
where H is the Hurst exponent which is an index of the continuity of a given
curve (Feder 1988, Peitgen and Saupe 1988).
A curve satisfying Eq. (4) is called self-affine. For example, for the time
trace of a one-dimensional random walk (Peitgen and Saupe 1988), H = 0.5.
In particular, if νx = νy (= ν), the given curve is self-similar (fractal) with the
fractal dimension D = 1/ν. So, H = 1.0 for self-similar patterns. Using this
method, Matsushita and Ouchi (1989a, b) showed that νx ≈ 1.0 for a mountain topography (landform) near Mt. Yamizo in Japan. From this result, the
real mountain topography can be characterized by a single parameter H
(Matsushita and Ouchi 1989a, b).
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3.
K. KIKUCHI et al.
DATA OF THE TRANSECT PROFILES OF LANDFORMS
AND FOLDS
The Japanese Islands are located in the circum-Pacific seismic zone and suffer extreme seismicity. The Northeast Honshu Arc on the North American
plate is pushed and deformed by the subducting Pacific Plate (Fig. 2) (Seno
et al. 1993). Mountains and plains in this area are aligned in a north-south
direction. Many active faults exist at the boundary between mountains and
plains (The Research Group for Active Faults of Japan 1991). It is thought
that the active faults raise the present mountains. It is well known that the
half wavelength of crustal buckling folds is about 25 km (Otsuki 1995).
Based on shortening ratios of the folds in late Cenozoic strata in the Northeast Honshu Arc (Otuka 1933, Sato 1989, Uemura 1989), the deformation
belt in the Northeast Honshu Arc is roughly divided into a high deformation
belt (inner belt) as a seismic zone and a low deformation belt (outer belt) as
an aseismic zone.
In the next section, we apply the method of self-affinity analysis (Matsushita and Ouchi 1989a, b) to transect profiles of geographical topographies
(landforms) and folds along measurement lines numbered from 1 to 9 in the
inner belt of the Northeast Honshu Arc (Fig. 3). We analyze the sections
across the Northeast Honshu Arc based on a 1/25,000 scale geological map
(Kitamura 1986), and the transect profiles of the landforms based on a
1/25,000 scale topographic map. In this paper, the transects are labeled as
Landform 1 for geographical topography and Fold 1 for fold in transect pro-
Fig. 2. Tectonic regime near the Northeast Honshu Arc. The slant-lined region is the
area considered in this paper. Arrows represent directions of relative motion of the
plates (after Seno et al. 1993).
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Fig. 3. Map of the Northeast Honshu Arc. Numbered measurement lines are used for
the analyses of self-affinity after Kitamura (1986). Here we briefly mention the
change in stress field from late Cenozoic to Quaternary according to Sato (1994). An
extentional stress field started from 32 Ma and formed major normal faulting at 2520 Ma resulting from backarc lifting. These normal faults were trending nearly parallel to the arc. From 20 to 15 Ma, the couterclockwise rotation of the NE Japan developed normal faults with an oblique to the arc. The rapid clockwise rotation of
southwestern Japan arc from 16 Ma produced a transtensional stress regime oriented
to NW-SE direction in the NE Japan. After the end of the opening of the Japan Sea,
at about 14 Ma, a neutral stress field was developed. The increases in velocity of the
westward motion of the Pacific plate at around 4 Ma produced a strong compressional stress field across the arc.
file 1 with the number corresponding to the measurement line parallel to the
direction of the horizontal maximum compression in the Northeast Honshu
Arc. If there are multiple curves used for the analysis on a given measurement line, they are distinguished by adding a letter such as Fold a, b and so
on from west to east (e.g., Fig. 4).
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Fig. 4. Case of Fold 6 used for analysis. Multiple curves on the measurement line are
distinguished by adding a letter such as Fold 6a, 6b, and 6c from west to east. In this
figure, the brackets (a) and (b) embrace two curves. They adjoin into the one longer
lines, which form Fold 6a and Fold 6b, respectively. The upper lines and the lower
lines are west and east parts of the transect of Fold 6a and Fold 6b, respectively.
Moreover, the bracket (c) embraces two curves. They adjoint into the one longer
line, which forms Fold 6c. The upper line and the lower line are west part and east
part of the transect profile of Fold 6c, respectively.
4.
SELF-AFFINITY OF LANDFORMS AND FOLDS
We set the unit length α at 250 m and the x- and y-coordinates in the
horizontal and vertical directions, respectively. Representative examples for
the profiles (Landform 3 and Fold 1c) are shown in Figs. 5 and 6. From these
figures, it is found that log X-log N and log Y-log N can be approximated by
straight lines (dotted lines) with different slopes (νx ≠ νy). These figures show
that both profiles are differently scaled in different directions, that is they are
self-affine.
Moreover, it is also found that νx ≈ 1.0 in both figures. This means the
self-affine properties of Landform 3 and Fold 1c can be characterized by
a single parameter H (Hurst exponent: Feder 1988, Peitgen and Saupe 1988).
In Fig. 5, H = 0.55. This means the transect profile of Landform 3 has a
form close to a one-dimensional random walk with H = 0.5. On the other
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SELF-AFFINITIES OF LANDFORMS AND FOLDS
(a)
1649
(b)
Fig. 5. Trensect analysis of the self-affinity of Landform 3: (a) transect profile of
Landform 3; (b) log-log plots of horizontal and vertical standard deviations (X and
Y) and curve length N. Pairs (log X, log N) and (log Y, log N) are linearly approximated by the method of least squares. The slopes represent νx ≈ 0.98, νy ≈ 0.54, and
H ≈ 0.55.
(a)
(b)
Fig. 6. Trensect analysis of the the self-affinity of Fold 1c: (a) transect profile of
Fold 1c; (b) log-log plots of horizontal and vertical standard deviations (X and Y)
and curve length N. Pairs (log X, log N) and (log Y, log N) are lineally approximated by the method of least squares. The slopes represent νx ≈ 0.95, νy ≈ 0.65, and
H = 0.68.
hand, in Fig. 6, H ≈ νy = 0.68, that is, the transect profile of Fold 1c (Fig. 6)
has larger continuity than the trace of a one-dimensional random walk.
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(a)
(b)
Fig. 7. Trensect analysis of the the self-affinity of Landform 7: (a) transect profile of
Landform 7; (b) log-log plots of horizontal and vertical standard deviations (X and
Y) and curve length N. Pairs (log X, log N) and (log Y, log N) are linearly approximated by the method of least squares, and characterized by two Hurst exponents
H1 ≈ 0.52 (N < Nc) and H2 ≈ 0.12 (N > Nc). A standard deviation of the vertical variation Y exhibits a clear crossover around a scale of log Nc = 2.1.
(a)
(b)
Fig. 8. Trensect analysis of the the self-affinity of Fold 6a: (a) transect profile of
Fold 6a; (b) log-log plots of horizontal and vertical standard deviations (X and Y)
and curve length N. Pairs (log X, log N) and (log Y, log N) are linearly approximated by the method of least squares, and characterized by two Hurst exponents
H1 ≈ 0.87 (N < Nc) and H2 ≈ 0.50 (N > Nc). A standard deviation of the altitude variation Y exhibits a clear crossover around a scale of log Nc = 1.0.
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SELF-AFFINITIES OF LANDFORMS AND FOLDS
Different behaviour from the above examples are shown in Figs. 7 and 8
(Landform 7 and Fold 6a). In both figures, log X-log N is approximated by
a straight line (dotted line) with νx ≈ 1.0 as in the above examples. However, log Y-log N is approximated by two straight lines (dotted lines) crossing
at a point log Nc (the estimated characteristic length, L ≡ Ncα). In Figs. 7 and
8, two straight lines (dotted lines) cross around a scale of log Nc = 2.1
(L = 31 km) and log Nc = 1.0 (L = 2.5 km), respectively. Apart from the
crossover, the rest of the curve can be approximated by straight lines. Therefore, we can regard them as self-affine curves with an inflection point
(crossover point).
While the crossover from local to global altitude variations has already
been noted for some transect profiles of mountain topography (Matsushita
and Ouchi 1989a, b), it has not been reported for transect profiles of folds.
For a crossover from local to global vertical variation, two Hurst exponents
can be estimated from the two slopes of crossing straight lines (H1 and H2)
for the smaller and larger ranges, respectively. For Landform 7 (Fig. 7b),
H1 ≈ 0.52 and H2 ≈ 0.12, and for Fold 6a (Fig. 8b), H1 ≈ 0.87 and
H2 ≈ 0.50. These values for landforms and folds are summarized in Tables 1
and 2. For those without a cross-over, we identify the Hurst exponent H with
H1. The average values of H1, H2, and log Nc for transect profiles of landforms are 0.53, 0.17, and 2.0 (L = 25 km), and for transect profiles of folds
they are 0.81, 0.43, and 1.0 (L = 2.5 km), respectively.
T ab l e 1
Hurst exponents and crossover points for transect profiles of landforms
No. of transect profiles
of landforms
H1
H2
log Nc
1
2
3
4
5
6
7
8
9
0.59
0.60
0.55
0.47
0.40
0.48
0.52
0.56
0.63
0.06
0.18
–
0.15
–
0.24
0.12
0.25
–
2.3
1.7
–
2.0
–
2.1
2.0
1.9
–
Average
0.53
0.17
2.0
Note: A dash in this column represents no Nc value in the analyses of
the self-affinity.
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T ab l e 2
Hurst exponent and crossover point for transect profiles of folds
No. of transect
profiles of folds
1a
1b
1c
1d
1e
3a
3b
4a
4b
4c
4d
4e
5a
5b
5c
6a
6b
6c
7
8
9
H1
H2
log Nc
0.64
0.67
0.68
0.84
0.56
0.86
0.90
0.85
0.84
0.85
0.87
0.90
0.86
0.78
0.78
0.87
0.83
0.84
0.87
0.84
0.88
0.41
–
–
–
–
0.18
–
–
–
–
0.47
–
0.57
0.41
0.37
0.50
0.25
0.69
–
–
–
0.8
–
–
–
–
1.4
–
–
–
–
1.1
–
1.0
1.0
1.0
1.0
1.1
1.0
–
–
–
Average
0.81
0.43
1.0
Note: A dash in this column represents no Nc value in the analyses of
the self-affinity.
5.
DISCUSSION
Matsushita and Ouchi (1989a, b) pointed out that mountain topography is
characterized by two regimes: local and global. In this paper, we applied
self-affinity analysis to the transect profiles of geographical topographies
(landforms) in the Northeast Honshu Arc. Comparing the landform profiles
with geological fold profiles in the Northeast Honshu Arc, it is found that the
characteristic length (L = 25 km) estimated from the crossover from local to
global vertical variation corresponds to the half wavelength of crustal buckling folds in the Northeast Honshu Arc (Kitamura 1986, Otsuki 1995).
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Tsuboi (1956) provided an empirical relation between the magnitude M
of an earthquake and the diameter 2r of the spherical volume storing elastic
energy in the crust given by
log 2r = 0.5 M − 2.1 .
(5)
When the wavelength of a crustal buckling fold is regarded as the diameter of the spherical volume storing elastic energy in the crust (i.e., r =
25 km), we can estimate a possible maximum magnitude of earthquakes
M = 7.6 from Tuboi’s assumption (Eq. (5)). This estimated magnitude is
concordant with historical records of inland earthquakes in this area (Matsuda 1990). The characteristic length estimated from the crossover from local
to global altitude variations of landforms is also related to the possible maximum magnitude of earthquakes in the inland part of the Northeast Honshu
Arc. It is worth noting that the biggest event in the vicinity of Honshu was
the magnitude 9.0 plate boundary megaquake (the 2011 off the Pacific Coast
of Tohoku Earthquake). Because the megaquake was located offshore Honshu; it was beyond the scope of this work and it was not considered in our
study. Thus, the characteristic length estimated from the crossover from local to global altitude variations of landforms is also related to the possible
maximum magnitude of earthquakes in the Northeast Honshu Arc.
For the transect profiles of landforms and folds in the Northeast Honshu
Arc, we characterized self-affinity by two scaling parameters, νx and νy, but
these parameters are not necessarily equal. If the values of these two parameter are different, then the transect profiles of landforms and folds should be
differently scaled in different directions. However, analyses of folds in previous studies (Nagumo 1969a, Kitazato 1971, Kulik and Chernovsky 1996,
Wilson and Dominic 1998) have been based on only a single parameter, the
fractal dimension D or the spectral exponent p. Next, we discuss the relation
between these parameters and the Hurst exponents derived from the two
scaling parameters, νx and νy.
Generally, the Fourier spectral density S(k) of a scale-invariant curve is
related to wave number k by a power law equation which is similar to
Eq. (1). In this case, the spectral exponent β (= 2p) and the Hurst exponent
H (= νy /νx) are related as (Feder 1988, Peitgen and Saupe 1988)
β = 2H + 1 .
(6)
Kitazato (1971) reported the power spectrum of geometries of the large
scale folds of Quaternary and Tertiary sediments or geographical topographies in the inner belt of the Northeast Honshu Arc. He pointed out that the
spectrum of the amplitude A(λ) of the fold or landform geometries is related
to the wavelength λ as A(λ) ∝ λp with a positive constant p. As regards the
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spectrum of the crustal deformation, Mizoue (1980) found that the spectral
amplitude of the vertical deformation along the Japanese Island is inversely
proportional to the wave number of the crustal deformation. This result indicates β = 2.0 (p = 1.0). From these studies, we can point out that the profiles of fold or the deformation patterns are also self-affine.
Under many circumstances, a seismic statistical relation (Gutenberg–
Richter’s law; Gutenberg and Richter 1944) is expressed by
log N ( M ) = −bM + a ,
(7)
where N(M) is the number of earthquakes with a magnitude greater than M
and a and b are constants. The size distribution of earthquakes expressed by
Eq. (7) is equivalent to the fractal distribution of fault break area by the relationship between the magnitude of the earthquake and its fault break area,
and Gutenberg–Richther’s law is a scale invariant law in seismology (Nagahama 1991, Turcotte 1992). The b-value is widely used as a measure of
seismicity and represents the sharpness of the spectrum of deformation in the
focal region. A large b-value means a sharper spectrum. Mogi (1963) has
suggested that b represents the grade of heterogeneity of the medium, and
a larger value of b corresponds to a higher heterogeneity. Therefore, the bvalue of Gutenberg–Richter’s law is also related to the fractal dimension or
the uniformity of the crustal fragmentation (Mogi 1963, Nagahama 1991,
Turcotte 1992, Nagahama 2000).
Nagumo (1969a, b) proposed a relationship between the spectral exponent β and the b-value of Gutenberg–Richter’s law (Gutenberg and Richter
1944) given by
β = 6 − 4b ,
(8)
under the following assumptions:
(I) The density of earthquake occurrence is assumed to be proportional to
the curvature of the plastic bending deformation of the medium.
(II) The frequency distribution of earthquakes with respect to earthquake
size is assumed to be proportional to the spectrum distribution of structural
wave number of plastic deformation.
(III) The frequency distribution of earthquakes in a certain area is assumed to be proportional to the area of focal region.
Nagumo (1969a, b) pointed out that the b-value of Gutenberg–Richter’s
law is related to the sharpness of the plastic bending deformation of the medium. Under Nagumo’s assumptions and Eqs. (6) and (8), we can derive
a new relation between the b-value of Gutenberg–Richter’s law and the
Hurst exponent H for the crustal deformation as
H=
5 − 4b
.
2
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The Hurst exponent H for the crustal deformation can be related to the fractal dimension or the uniformity of the crustal fragmentation.
Based on the analyses of the transect profiles from local to global vertical variations of folds, we derive two Hurst exponents as H1 and H2. Average
values of H1 and H2 are 0.81 and 0.43, respectively. By Eq. (9), we can calculate two seismic parameters as b = 0.85 (H1 = 0.81) and b = 1.04 (H2 =
0.43). Moreover, the Hurst exponents take values in the range 0 ≤ H ≤ 1 so
that the b-values of Gutenberg–Richter’s law are limited in the range
0.75 ≤ b ≤ 1.25 (Nagahama and Teisseyre 2000). This range of b-values is
concordant with our results for the Hurst exponents of the fold profiles and
seismic data: the low values b = 0.61 ~ 0.8 on the Japan Sea side and the
high values b = 1.01 ~ 1.2 on the Pacific Ocean side (Miyamura 1962, Mogi
1964).
This paper has been concerned with two major problems (Nagahama
1998): (1) to establish the scale invariance of folds or geographical topographies (landforms), and (2) to understand the physical conditions under which
scale invariance was formed. Ord (1994) and Hunt et al. (1996) proposed
that naturally occurring fold systems may display spatial chaos in their geometries, in contradiction with previous linear theories for the formation of
fold systems, such as Biot (1961) which predicted periodic geometries. Hunt
et al. (1996) suggested that elasticity plays an important role in producing irregular (fractal) fold geometries, and indicated that localization of buckles is
an important (perhaps dominant) mode of deformation for an elastic layer
embedded within a viscoelastic medium. Therefore, fractal spatially chaotic
fold geometries might be produced by the non-linear dynamics of an elastic
layer embedded within a viscoelastic medium (Nagahama 1998).
6.
CONCLUSION
We used a new method to analyze the Quaternary and Tertiary sediments or
geographical topographies (landforms) in the inner belt of the Northeast
Honshu Arc, Japan. We recognized a self-affinity in the transect profiles of
landforms and folds and a crossover from local to global altitude (vertical)
variation. The characteristic length (25 km) estimated by the crossover for
landforms depends on the half wavelength of the crustal buckling folds in
the Northeast Honshu Arc, and is also related to the possible maximum
magnitude of inland earthquakes in this region, but this magnitude is lower
than the magnitude 9.0 of the plate boundary megaquake (the 2011 off the
Pacific Coast of Tohoku Earthquake). The self-affinity for the crustal deformation is related to the b-value in Gutenberg–Richter’s law, the fractal dimension and the uniformity of the crustal fragmentation. From the selfaffinity of the folds, two average Hurst exponents were obtained, indicating
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two possible seismic modes for the smaller and larger ranges in the focal regions in the Northeast Honshu Arc.
A c k n o w l e d g e m e n t s . The authors would like to thank Harry
A. Mavromatis, Gabor Korvin, Kazuhito Yamasaki and an anonymous referee for valuable comments. We thank Benjamin Cramer for checking the
English of our revised manuscript.
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Received 7 March 2013
Received in revised form 8 May 2013
Accepted 13 May 2013
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