Scale independence of basin hypsometry and steady state topography

Geomorphology 171–172 (2012) 1–11
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Geomorphology
journal homepage: www.elsevier.com/locate/geomorph
Scale independence of basin hypsometry and steady state topography
Kuang-Yu Cheng a, Jih-Hao Hung b, Hung-Cheng Chang b, Heng Tsai a, Quo-Cheng Sung c,⁎
a
b
c
Department of Geography, National Changhua University of Education, No. 1, Jin-De Rd., Changhua, Changhua, 500, Taiwan, ROC
Institute of Geophysics, National Central University, No. 300, Jhongda Rd., Jung-Li, Taoyuan, 320, Taiwan, ROC
Institute of Geoinformatics and Disaster Reduction Technology, Ching Yun University, No. 229, Chien-Hsin Rd., Jung-Li, Taoyuan, 320, Taiwan, ROC
a r t i c l e
i n f o
Article history:
Received 14 September 2010
Received in revised form 30 August 2011
Accepted 26 April 2012
Available online 4 May 2012
Keywords:
Geomorphic index
Bain hypsometry
Scale dependence
Steady state topography
a b s t r a c t
Basin hypsometry has long been used as an indicator of stages in landscape evolution. It has also served as a
tool for detecting tectonically active regions. Whether hypsometric curve and its integral are independent of
differences in basin size and relief has been discussed in many recent studies. The Taiwan Mountain Range,
the result of an oblique collision between the Phillipine Sea Plate and the Eurasian Plate, provides an
excellent opportunity to study landscape evolution in relation to steady state conditions. Taking advantage
of the space-for-time substitution concerning the building process of Taiwanese mountains, this study
sampled major drainage basins for hypsometric analysis, from the southern tip of the island to the northern
end. This study found that the area previously known as steady state topography is characterized by 1) the
hypsometric integral (HI) close to 0.5, 2) S-shaped hypsometric curves, and 3) normally distributed
elevations. The response time required for a drainage basin of various Strahler orders to evolve from
exposure of sub-aerial erosion to steady state topography ranges from 0.5 to 2.0 My, and is longer for basins
of a higher Strahler order. The HI value of 0.5 for a drainage basin seems critical in terms of landscape
evolution. The scale dependence of basin hypsometry is manifested mainly by the fact that HI of basins
increases as basin order and/or basin size decreases. This study also found that basin hypsometry is scale
independent where topography is in a steady state; scale dependence occurs only when drainage basins
are in a non-steady state. The basin hypsometry may therefore be useful to probe into the nature of
steady-state topography.
© 2012 Elsevier B.V. All rights reserved.
1. Introduction
Basin hypsometry is usually based on the percentage hypsometric
curve (area–altitude curve) which relates the horizontal crosssectional area of a drainage basin to its relative elevation above the
basin mouth (Strahler, 1952). The hypsometric integral (HI)
represents the relative proportion of the basin area below (or
above) a given height. By using dimensionless parameters, drainage
basins can be compared irrespective of true scale. In other words,
the hypsometric curve is independent of basin size and relief, as
long as topographic maps or digital elevation models (DEMs) being
used have sufficient resolutions to characterize the basins
(Rosenblatt and Pinet, 1994; Hurtrez et al., 1999). Nevertheless,
previous studies showed that basin hypsometry is related to the
size, shape, and relief of the basin as well as other factors such as
the dominant erosion process (Hancock and Willgoose, 2001; Azor
et al., 2002; Chen et al., 2003; Korup et al., 2005). HI has also been
found to be sensitive to both uplift rates and the erosional resistance
of lithological units (Lifton and Chase, 1992; Hurtrez et al., 1999;
⁎ Corresponding author. Tel.: + 886 3 4581196 5729; fax: + 886 3 2503017.
E-mail address: kc0729@cyu.edu.tw (Q-C. Sung).
0169-555X/$ – see front matter © 2012 Elsevier B.V. All rights reserved.
doi:10.1016/j.geomorph.2012.04.022
Chen et al., 2003; Walcott and Summerfield, 2008). Hypsometric
analysis of drainage basins has been applied to areas experiencing
rapid tectonic activities (e.g. Hurtrez et al., 1999; Keller and Pinter,
2002; Chen et al., 2003; Korup et al., 2005; Pedrera et al., 2009;
Barbero et al., 2010). Korup et al. (2005) attested to the catchmentsize dependence of HI in the study of regional relief and denudation
of the western Southern Alps, New Zealand, where the active oblique
continental convergence occurs between the Australian Plate and the
Pacific Plate. However, Walcott and Summerfield (2008) did not
observe a correspondence between HI and indices of basin dimension
such as basin area and basin relief, in the southeast margin of
southern Africa, a region where the long-term drainage development
is evident in the relatively stable continental margin. It is noteworthy
that the tectonic environment affects the scale dependence of HI.
Basin hypsometry was further applied to characterize the
topographic steady state in Taiwan (Stolar et al., 2007; Chen, 2008).
Research on the relationship between basin hypsometry and the
steady state or equilibrium of a landscape can be traced back to the
classic work of Strahler (1952) who proposed a stage of geomorphic
equilibrium to explain a distinctive series of hypsometric forms. He
related the hypsometric curve to the Davisian stages of an erosional
cycle (Fig. 1A) on the assumption that a mountain is rapidly uplifted
without serious denudation and then decreases its altitude and relief
2
K-Y. Cheng et al. / Geomorphology 171–172 (2012) 1–11
Fig. 1. Basin hypsometry and landscape evolution. (A) Change in the hypsometric curve according to Strahler (1952). (B) Changes in relief inferred from the Davisian scheme (Davis,
1899). (C) Change in the hypsometric curve according to Ohmori (1993). (D) Changes in relief inferred from Hack's (1976) scheme (Ohmori, 1993).
due to erosion. Ohmori (1993) simulated a hypsometric curve of
drainage basins in Japanese mountains which resulted from
concurrent tectonics and denudation. He inferred that HI converges
to a critical value when the drainage basin is close to a steady state,
and proposed that the hypsometric curve of a mountain changes
from a concave curve to an S-shaped curve with the progress of
geomorphic stages from development to culmination (Fig. 1C).
Ohmori's (1993) notion of geomorphic stages differs from that of
Strahler (1952) but is similar to Montgomery's (2001) life cycle of
an orogen in that the developing stage is in a tectonic steady state,
the culminating stage is in a topographic steady state, and the
declining stage is in an erosional steady state. Strahler (1952)
performed his hypsometric analysis on small drainage basins, while
Ohmori (1993) paid attention to the scale of mountain ranges.
Stolar et al. (2007) found that HI of larger basins (>100 km 2) in the
steady state topography of the Taiwan Mountain Range is between
0.39 and 0.5, while Chen (2008) found that HI of smaller basins
(b10 km 2) is always about 0.50 on average. Again, the size of each
drainage basin seems critical when geomorphological stages are
discussed.
The Taiwan Mountain Range resulted from an oblique collision
between the Phillipine Sea Plate and the Eurasian Plate. The collision
has propagated southward during the past 5 My, and the age of
collision and the duration of sub-aerial erosion have increased
progressively towards the north (Suppe, 1981). The duration of the
evolution of the sub-aerial landscape is assumed to be equivalent to
the distance from the southern tip of the island. This space-for-time
substitution can only be applied if (1) growth of the mountain
range has a relatively steady and progressive impact, and (2) the
orogen is a result of steady migration of point convergence. Using
the implied space-for-time substitution, Suppe (1981) interpreted
that the northward increase in mountain width and cross-sectional
area toward constant values represents evolution into a large-scale
topographic steady state. Indeed, the Taiwan Mountain Range is
considered to be in a topographic, thermal and exhumational steady
state (Willett and Brandon, 2002; Fuller et al., 2006). Although the
average cross-sectional form of topography can reach a steady
value, geomorphic processes are intrinsically unsteady, especially at
local spatial scales (Willett and Brandon, 2002). Stolar et al. (2007)
analyzed the orogen size, drainage basin hypsometry, and regional
and local topographic slopes in Taiwan, and suggested that variability
in the topography would most likely preclude the recognition of a
steady state from smaller-scale studies. Thus, even in a setting like
Taiwan, a topographic steady state is revealed only by broad spatial
transitions between pre-steady and steady-state landscapes. Chen
(2008), however, discovered that the evolutionary stage of Taiwan's
topography complies with Ohmori's (1993) culminating stage,
which is characterized by S-shaped hypsometric curves and normally
distributed elevations for drainage basins in the Central Range. These
previous studies suggest that the Taiwan Mountain Range is ideal for
studying basin hypsometry in relation to the scale dependence and
evolutionary stages of landscapes.
The above discussion points to two issues to be examined: (i)
whether basin hypsometry is inherently dependent on the scale or
tectonics of drainage basins; and (ii) whether basin hypsometry can
serve as a geomorphic index of a topographical steady state or
geomorphic equilibrium. In this paper, we follow the sampling
strategy of Stolar et al. (2007), who take advantage of the space-fortime substitution by measuring drainage basins along the axis of the
island of Taiwan to determine trends in topographic evolution. We
also sampled drainage basins in a cross-island swath from prowedge to retro-wedge considering the kinematic model for
convergent orogen (Willett et al., 2001). Our methods also draw on
the work of Walcott and Summerfield (2008); they used the basin
hypsometry of a given Strahler order to group catchments with
similar areas. The aim of this paper is to take topography in Taiwan
as an example for studying the scale-dependency of basin
hypsometry and the implications of this on the stages of landscape
evolution.
2. Overview of the topographic steady state of Taiwan
The arc–continent collision, developed in the last 5 My, resulted in
the accretion of sedimentary units with several hundred kilometers of
crustal shortening, and created the island of Taiwan (Fig. 2). Collision
began in the north of Taiwan approximately 4 Ma and is just
K-Y. Cheng et al. / Geomorphology 171–172 (2012) 1–11
Fig. 2. Tectonic framework of Taiwan. Collision occurs at the Taiwan Mountain Range
due to flipping of the subduction polarity: the Philippine Sea Plate subducts along the
Ryukyu Trench into the Eurasian Plate, and the South China Sea Plate subducts towards
the Philippine Sea Plate along the Manila Trench. The Peikang High plays a role of
indentation.
beginning in the south. Suppe (1981) has suggested that if we assume
that the collision rate has been reasonably constant, moving north in
Taiwan by 90 km is equivalent to moving back 1 My in time. Although
different collision rates were adopted in the relevant studies (Fuller et
al., 2006; Stolar et al., 2007), this space-for-time substitution is a
powerful tool to develop schematic models of landscape evolution.
The substitution indicates that mountain growth has been steady
until a point 120 km north of the southern tip of Taiwan (120 km N
in Fig. 2; the figure shows the scale for this distance system) where a
constant width of 87 ± 4 km, a cross-sectional area of 118 ± 24 km2,
and a mean elevation of 1350 m are attained, with this elevation
being maintained for another 170 km (Fig. 2). This region reflects not
only a topographic steady state in a broad scale (Stolar et al., 2007),
but also an exhumational steady state in terms of the apatite and zircon
fission track thermochronometers (AFT and ZFT; Fuller et al., 2006). The
region of the mountain belt between 290 km N and the northern tip of
Taiwan manifests a subdued topography with decreasing width and
height. This region is encompassed by an extensional environment
due to the flipping of subduction polarity (Teng et al., 2001) and
southwest extension of the Okinawa Trough; it is interpreted as a region
of collapsed mountainous topography. While the oblique collision
propagates southwards, the mountain range in southern Taiwan is
undergoing growth through subduction and is interpreted as a region
of pre-steady state topography (Stark and Hovius, 1998).
3
Fig. 3. Distribution of selected main drainage basins in the Taiwan Mountain Range.
The plot of HI vs. basin distance is projected onto the profiles along the Central
Range (A–A′), the Western Foothills (B–B′), and E–W traverses (C–C′, D–D′ and E–E′).
Stolar et al. (2007) indicated that the topographic steady state in
Taiwan is expressed by the existence and coincidence of transitions in
several characteristics of landscapes at approximately 100–125 km N.
They adopted a rate of 55 km My− 1 for the southward propagation of
the mountain building by considering the clockwise rotation of the island
of Taiwan and suggested that a topographical steady state is achieved
1.8–2.3 My after emergence from the sea. They also assumed that the
topographical steady state in Taiwan exists only for large-scale
structures. Chen (2008) studied hypsometric integrals, hypsometric
curves and elevation histograms of small drainage basins in different
physiographic divisions of the Taiwan Mountain Range. He found that
drainage basins in a topographic steady state have a typical S-shaped
hypsometric curve, a value of HI approaching 0.5, and a nearly normal
distribution of basin elevations. He also inferred that the subsequent
breakdown of the state may be revealed by lower values of HI as well
as more concave hypsometric curves and skewed elevation distributions.
However, he did not clearly show the spatial transition between presteady and steady states. Considering the work of Stolar et al. (2007)
and Chen (2008), we examine whether the value of HI for drainage
basins with different orders approaches a critical value, to discuss the
scale dependence of basin hypsometry in steady state topography.
3. Methods
In this research, we used a 40-m resolution DEM of Taiwan for
grid-based analysis. The DEM was produced by the Bureau of Forestry
in the 1980s with the Universal Transverse Mercator (zone 51)
4
K-Y. Cheng et al. / Geomorphology 171–172 (2012) 1–11
Table 1
Statistics of hypsometric integral for drainage basins studied.
Main Distance HImainb 5th order
basin N (km) a
No. of
Area
basins
(km²)
Mean No. of
HI
basins
Mean area
(km²)
Mean No. of
HI
basins
Mean area
(km²)
Mean No. of
HI
basins
Mean area
(km²)
Mean No. of
HI
basins
Mean area
(km²)
Mean
HI
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
0.272
–
–
–
0.368
0.362
–
0.371
0.360
–
0.276
0.300
0.345
0.369
0.448
0.459
–
–
0.448
0.543
–
0.403
0.518
0.441
–
0.263
0.453
0.496
0.452
0.475
0.267
0.524
0.232
0.505
–
0.470
0.362
0.373
0.514
–
0.519
0.479
0.545
0.538
0.353
0.216
0.168
–
0.425
0.290
–
–
0.425
0.356
0.382
0.580
0.531
0.521
0.447
0.310
0.434
0.407
0.252
0.397
0.456
0.342
0.368
0.414
0.211
0.416
0.412
0.204
0.390
46.71
124.18
102.74
54.17
32.91
32.29
87.44
45.85
60.66
150.21
59.28
29.63
37.45
27.05
44.05
38.24
58.52
62.11
85.05
51.39
58.66
23.49
45.93
37.91
129.19
27.35
30.82
36.49
74.30
25.58
31.31
44.68
60.49
32.24
78.98
47.03
40.95
55.82
31.50
97.03
41.16
31.79
52.20
62.38
37.11
27.87
54.82
199.84
61.51
27.00
98.35
149.75
93.47
36.23
41.20
59.65
45.34
50.88
78.41
42.57
54.63
59.87
49.90
48.16
75.40
22.43
55.47
91.98
58.70
37.37
91.33
86.03
56.26
0.262
0.310
0.380
0.315
0.434
0.389
0.369
0.451
0.417
0.317
0.334
0.436
0.382
0.475
0.458
0.568
0.332
0.266
0.479
0.571
0.476
0.470
0.525
0.496
0.355
0.363
0.516
0.516
0.499
0.525
0.344
0.542
0.358
0.532
0.537
0.546
0.488
0.426
0.507
0.441
0.559
0.450
0.562
0.640
0.376
0.242
0.272
0.455
0.446
0.366
0.492
0.488
0.453
0.373
0.464
0.597
0.529
0.522
0.448
0.406
0.460
0.461
0.283
0.412
0.446
0.447
0.490
0.301
0.252
0.499
0.481
0.275
0.477
9.20
13.81
8.87
6.88
8.06
8.92
10.40
6.04
10.89
10.60
10.44
9.51
7.67
9.38
8.97
13.13
10.39
12.11
13.94
6.32
9.98
9.70
12.52
12.02
6.99
8.30
9.25
11.53
10.14
8.13
9.88
9.86
9.41
8.84
6.63
12.84
10.77
8.74
10.15
10.26
9.27
6.63
15.80
12.55
6.95
9.01
8.80
17.54
12.66
6.78
12.62
13.11
8.98
7.12
13.17
8.95
8.86
6.32
9.78
10.90
8.98
9.42
10.90
11.53
10.30
7.25
10.17
8.93
10.04
10.09
10.15
8.80
9.26
0.347
0.449
0.425
0.434
0.500
0.449
0.470
0.507
0.484
0.490
0.420
0.477
0.481
0.465
0.492
0.505
0.414
0.329
0.493
0.528
0.542
0.511
0.566
0.561
0.404
0.427
0.495
0.518
0.503
0.494
0.405
0.540
0.389
0.542
0.536
0.521
0.487
0.489
0.522
0.514
0.530
0.498
0.502
0.583
0.378
0.311
0.364
0.510
0.467
0.411
0.502
0.524
0.501
0.442
0.474
0.536
0.523
0.469
0.506
0.436
0.471
0.473
0.408
0.477
0.513
0.464
0.514
0.405
0.323
0.502
0.516
0.418
0.526
2.09
2.55
1.89
2.45
1.83
2.07
1.94
1.83
2.87
2.34
1.82
1.95
1.99
2.22
2.03
2.13
1.89
1.93
0.25
1.41
2.72
2.33
2.12
1.93
1.80
2.15
2.18
1.87
2.48
2.30
2.00
2.90
2.32
2.26
2.14
2.38
2.09
2.07
2.14
2.26
2.16
2.37
2.60
2.13
1.82
1.57
2.16
2.20
2.49
1.73
1.64
1.93
2.06
1.70
2.14
2.32
2.36
2.07
2.51
2.24
2.16
2.06
1.94
1.90
2.24
1.61
2.60
2.33
2.25
2.13
2.31
1.91
2.34
0.406
0.445
0.467
0.486
0.508
0.514
0.520
0.518
0.505
0.531
0.505
0.517
0.517
0.511
0.501
0.518
0.456
0.406
0.525
0.534
0.550
0.526
0.556
0.554
0.405
0.446
0.521
0.528
0.514
0.510
0.454
0.519
0.415
0.531
0.545
0.534
0.488
0.483
0.528
0.500
0.532
0.537
0.538
0.539
0.399
0.380
0.420
0.519
0.506
0.450
0.509
0.532
0.511
0.476
0.517
0.525
0.519
0.524
0.516
0.438
0.520
0.503
0.444
0.517
0.521
0.500
0.536
0.490
0.418
0.510
0.521
0.420
0.538
0.51
0.51
0.49
0.44
0.48
0.44
0.51
0.52
0.55
0.50
0.51
0.49
0.53
0.50
0.43
0.50
0.46
0.41
0.51
0.43
0.47
0.47
0.54
0.47
0.42
0.41
0.45
0.47
0.47
0.44
0.41
0.62
0.41
0.54
0.46
0.47
0.47
0.43
0.49
0.48
0.47
0.47
0.48
0.43
0.41
0.47
0.43
0.43
0.44
0.41
0.43
0.44
0.47
0.39
0.47
0.56
0.49
0.47
0.50
0.43
0.46
0.46
0.43
0.42
0.53
0.42
2.60
0.44
0.42
0.43
0.57
0.41
0.48
0.438
0.477
0.493
0.523
0.532
0.544
0.543
0.547
0.522
0.530
0.526
0.516
0.524
0.529
0.529
0.527
0.492
0.457
0.535
0.524
0.561
0.535
0.554
0.549
0.459
0.482
0.534
0.541
0.526
0.520
0.475
0.548
0.441
0.554
0.541
0.527
0.507
0.500
0.543
0.512
0.544
0.533
0.536
0.540
0.445
0.425
0.459
0.519
0.519
0.487
0.525
0.532
0.521
0.493
0.523
0.534
0.527
0.550
0.524
0.495
0.528
0.517
0.477
0.533
0.535
0.502
0.544
0.518
0.447
0.519
0.534
0.457
0.544
8.20
19.94
28.28
35.87
37.22
44.68
47.18
54.65
54.71
62.07
65.60
70.07
78.52
79.35
87.87
88.98
94.13
98.82
105.75
105.99
117.32
117.52
125.97
126.79
135.53
137.46
138.74
138.95
146.55
148.15
148.27
156.46
164.46
165.31
169.32
174.16
181.36
181.77
184.36
186.48
194.12
195.42
206.32
206.89
209.51
216.15
216.96
218.56
219.56
227.45
227.62
233.03
234.42
236.98
239.66
240.93
251.25
254.23
263.70
264.26
266.51
268.33
270.20
270.65
272.75
277.36
283.66
286.18
286.23
292.50
293.39
294.65
299.62
0.272
0.310
0.380
0.315
0.368
0.362
0.369
0.371
0.360
0.317
0.276
0.300
0.345
0.369
0.448
0.459
0.332
0.266
0.448
0.543
0.476
0.403
0.518
0.441
0.355
0.263
0.453
0.496
0.452
0.475
0.267
0.524
0.232
0.505
0.537
0.470
0.362
0.373
0.514
0.441
0.519
0.479
0.545
0.538
0.353
0.216
0.168
0.455
0.425
0.290
0.492
0.488
0.425
0.356
0.382
0.580
0.531
0.521
0.447
0.310
0.434
0.407
0.252
0.397
0.456
0.342
0.368
0.414
0.211
0.416
0.412
0.204
0.390
1
0
0
0
1
1
0
1
1
0
1
1
1
1
1
1
0
0
1
1
0
1
1
1
0
1
1
1
1
1
1
1
1
1
0
1
1
1
1
0
1
1
1
1
1
1
1
0
1
1
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
101.78
–
–
–
125.01
108.45
–
117.84
134.50
–
141.33
211.28
103.39
176.12
120.23
300.16
–
–
492.08
108.09
–
146.52
299.43
133.79
–
619.14
108.16
253.89
625.47
173.87
724.55
109.24
243.98
65.47
–
288.24
445.92
419.96
415.92
–
255.15
126.95
167.80
231.97
95.81
135.86
361.43
–
223.42
197.20
–
–
223.47
77.89
433.18
119.59
249.13
129.22
183.59
86.63
125.51
631.06
110.18
158.38
191.46
103.90
275.22
308.75
117.92
238.16
310.73
322.45
683.01
4th order
2
1
1
1
3
3
1
2
2
1
2
3
2
3
2
3
1
1
4
2
1
3
3
2
1
5
2
4
5
3
9
2
3
2
1
3
6
5
4
1
3
2
2
2
2
3
5
1
3
5
1
1
2
2
3
2
4
2
2
2
2
7
2
3
2
3
4
2
2
4
3
3
8
3rd order
7
6
7
5
9
7
5
10
9
9
9
14
9
10
7
17
3
3
23
10
4
6
15
15
9
36
7
15
27
12
41
6
14
4
6
12
27
26
24
6
16
9
7
12
6
11
25
9
13
18
5
8
14
7
23
7
17
8
12
6
8
37
6
8
11
9
17
13
6
12
20
21
45
2nd order
27
29
31
15
39
30
26
38
34
40
41
63
33
49
28
82
14
18
117
38
12
32
76
78
41
169
27
71
149
45
204
27
60
14
22
70
113
115
103
25
66
32
34
59
27
46
107
58
54
66
26
41
61
20
112
33
68
31
46
22
32
173
29
46
49
34
63
80
30
63
84
91
179
1st order
136
158
140
77
167
144
118
148
160
177
201
250
138
210
126
343
65
84
575
116
60
116
381
308
166
788
145
309
627
188
906
117
305
75
86
299
480
530
488
96
301
129
174
249
112
138
456
232
227
235
103
171
238
107
451
150
309
126
226
100
125
691
141
166
230
140
329
239
149
296
378
406
858
K-Y. Cheng et al. / Geomorphology 171–172 (2012) 1–11
5
Table 1 (continued)
Main Distance HImainb 5th order
basin N (km) a
No. of
Area
basins
(km²)
Mean No. of
HI
basins
74
75
76
77
78
79
80
81
All
0.363
0.168
0.425
0.469
0.249
0.426
0.399
0.300
0.392
a
b
304.62
316.18
318.06
327.91
328.48
336.60
344.85
354.67
0.363
0.168
0.425
0.469
0.249
0.426
0.399
0.300
1
1
1
1
1
1
1
1
67
175.12
223.44
227.67
85.05
184.52
80.17
114.51
130.49
230.00
4th order
3
4
4
2
2
3
2
3
222
3rd order
Mean area
(km²)
41.20
32.54
32.16
37.34
91.98
20.06
28.90
32.56
57.23
2nd order
1st order
Mean No. of
HI
basins
Mean area
(km²)
Mean No. of
HI
basins
Mean area
(km²)
Mean No. of
HI
basins
Mean area
(km²)
Mean
HI
0.439
12
0.400
16
0.460
13
0.507
5
0.301
13
0.458
6
0.379
5
0.358
10
0.434 1017
7.89
8.32
11.62
9.93
8.93
7.41
5.18
10.11
9.78
0.496
52
0.403
70
0.502
62
0.533
24
0.405
50
0.463
20
0.464
32
0.394
33
0.472 4550
1.91
1.74
2.26
2.16
2.07
2.28
2.30
2.46
2.11
0.483
219
0.449
298
0.538
277
0.560
116
0.467
235
0.501
97
0.464
137
0.448
164
0.497 19928
0.43
0.40
0.48
0.53
0.42
0.59
0.55
0.58
0.50
0.500
0.456
0.548
0.546
0.486
0.531
0.501
0.484
0.514
Distance of the basin center from to the southern tip of Taiwan along the N–S profile.
HI for either the 5th or 4th order main basin.
Fig. 4. Plots of mean HI vs. N–S distance for basins with different orders. Distance of the basin center from the southern tip of Taiwan is plotted. Bar: one standard deviation.
6
K-Y. Cheng et al. / Geomorphology 171–172 (2012) 1–11
Fig. 5. Plots of mean HI vs. E–W distance for basins with different orders. Distance of the basin center from the western coastal line of Taiwan is plotted.
projection. The RiverTools software (version 3) was used to extract
the river network and basin limits from the DEM. A drainage basin
is defined as the set of cells related by their flow pathways to the
basin outlet. A single flow drainage direction, following the steepest
downslope, is calculated for each cell of the grid. Sinks are filled to
ensure the continuity of drainage. Giving a constant threshold
contribution area, the channel network and the drainage basin limit
can be determined. We used a value of 0.2176 km 2 (equivalent to
136 grid cells of 40-m resolution) for the threshold area which was
adopted from Chen et al. (2003). The software can also delimit the
subordinate basins according to Strahler orders. The HI value of a
drainage basin can be estimated by (Hmean − Hmin) / (Hmax − Hmin),
where Hmean and (Hmax − Hmin) are the mean elevation and the
elevation drop of the basin, respectively (Pike and Wilson, 1971).
The hypsometric curve and elevation histogram of the basin were
then plotted using the GIS extension CalHypso (Pérez-Peña et al.,
2009).
We sampled drainage basins of higher Strahler orders from the
Central Range and the Western Foothills of the Taiwan Mountain
Range. We selected the main basins from both sides of the Central
Range starting at the southern tip of the island and extending to a
point of 300 km N. Following this, we selected the main basins
primarily from the Western Foothills at 100 km N and within the
area extending to the northern end of the island (Fig. 3). A total of
81 main drainage basins, including 67 basins of order 5 and 14 basins
of order 4, were selected for grid-based hypsometric analysis
(Table 1). The average size of order-5 basins is 230 km 2 with size
varying from 65 to 725 km 2, while the average size of order-4 basins
is 57 km 2 with size varying from 20 to 200 km 2. Drainage basins were
further divided into subordinate basins according to Strahler's rule. A
K-Y. Cheng et al. / Geomorphology 171–172 (2012) 1–11
7
below the level of 0.5 (circles in Fig. 4). The higher the basin order,
the lower the value of HI. HI vs. distance along the three E–W profiles
perpendicular to the long axis of Taiwan (C–C′ to E–E′ in Fig. 3) is also
plotted to show the progressive change in the form of drainage basins
from the pro- to retro-wedges of the Taiwan orogen (Fig. 5). An
eastward increasing trend of HI is observed in all profiles (north,
middle and south). HI for each basin order tends to increase eastward
from the Western Foothills, and crosses over 0.5 at the Cental Range.
The slope of the regression line for the middle (D–D′) and south (E–E′)
profiles is greater than that of the north (C–C′) profile. Similarly, the
higher the basin order, the higher the slope of the regression line. The
geographic distribution of HI for main basins (either order-5 or order4 basins) shows a region roughly delimited by HImain > 0.4 (Fig. 6).
The region covers most parts of the Central Range between 110 and
310 km N, and a small part of the Hseshan Range between 280 and
330 km N. The region maintains a constant width of ~45 km from 110
to 220 km N and increases its width in the north covering a part of the
Hseshan Range. The hypsometric curves and elevation histograms of
some representative 5th order basins are shown in Fig. 7. From the
southern tip of Taiwan to its northern end, we can see a serial change
of hypsometric form from concave, through S-shaped, then to concave
again. Examining the elevation distribution in the same manner, we
also found a successive change of the histogram from right-skewed to
normal, and then back to right-skewed.
5. Discussion
Fig. 6. Distribution of HI. (A) HI for main basins (HImain). The region with HImain > 0.4
is delimited by a bold dashed line. (B) Map of interpolated HI values based on the
kriging method, showing the correlation between the reset zone inferred from
thermochronometry and HI > 0.4.
Based on the assumption of steady state topography in the Taiwan
Mountain Range (Suppe, 1981; Fuller et al., 2006; Stolar et al., 2007),
we discuss the characteristics of basin hypsometry in detail.
Assuming a space-for-time substitution for the distance along the
island's axis and the duration of sub-aerial erosion (Suppe, 1981),
we also discuss temporal changes in HI as well as the evolutionary
stages of landscapes. A propagation rate of 55 km My − 1 for mountain
building (Byrne and Liu, 2002) is adopted here.
5.1. Relationship between basin hypsometry and topographic steady
state
total of 19,928 basins of order 1 were extracted with an average size
around 0.5 km 2 (Table 1). Then HI was plotted against distance (i.e.
the central location of each basin from the southern tip of Taiwan)
for basins of different orders. HI may be affected by local factors
such as geologic structure, lithology, and erosion processes (Lifton
and Chase, 1992; Hurtrez et al., 1999; Chen et al., 2003; Stolar et al.,
2007; Walcott and Summerfield, 2008). The mean value of HI for
basins of the same order is used to minimize the effects of such
local factors.
4. Results
Although HI for all basins shows a weak positive correlation with
distance from the southern tip of Taiwan, the mean HI values of
basins with the same order shows a marked positive correlation
with distance, and it becomes clearer as the basin order increases
(Fig. 4). The mean HI values of drainage basins in the Central Range
(profile A–A′ in Fig. 3) gradually increase northwards, and then
crosses 0.5 at a certain distance and stays on 0.5 until it drops
below 0.5 at farther north (see triangles in Fig. 4). Here we define
the distance for the mean HI value to cross 0.5 as the critical distance.
The distance increases with an increasing basin order: it is roughly 30,
40, 60, 85, or 110 km for 1st to 5th-order basins, respectively; the
slope of the regression line for mean HI vs. distance also increases
with the basin order (Fig. 4). However, HI of drainage basins in the
Western Foothills (profile B–B′ in Fig. 3) behaves differently. With a
slightly increasing northward trend, the regression lines of HI stay
Although the Taiwan Mountain Range has been considered to
have a steady state topography, regional variability also exists
because of differences in geologic structure, rock strength and rock
uplift rate (Stolar et al., 2007). Willett and Brandon (2002) also
suggested that the topography of a convergent orogen may not
reach a steady state at small scales. Nevertheless, our study indicates
that drainage basin hypsometry is less variable. Mean HI values are
approximately 0.5 in the steady state region of the Taiwan Mountain
Range (Fig. 4), and HI of the main basins approaches 0.5 posterior to
that of the subordinate basins. We also found that the region defined
by HImain > 0.4 corresponds to the reset zone inferred from the AFT
and/or ZFT thermochronometers; whereas, the region with HImain b 0.4
corresponds to the unreset zone (Fig. 6b). The nested pattern of reset
and unreset zones in the Taiwan Mountain Range suggests that the
orogen has reached an exhumational steady state over the regions
with respect to the AFT and ZFT (Fuller et al., 2006), and that the region
with HImain > 0.4 is in both topographic and exhumational steady states.
Therefore, we suggest that a drainage basin attains a steady state when
HI approaches to a critical value (0.5). The topographic steady state is
usually defined based on the orogen size including the height and
cross-sectional area along the longest wavelength of topography
(Willett and Brandon, 2002). HI gives a further constraint for defining
steady state landscapes.
The geographic distribution of HI may be better interpreted in
relation to local factors such as geological structure, lithology and
erosion rate. Fuller et al. (2006) estimated that the reset zone has
current average erosion rates of ~ 3.3 mm year − 1, whereas the unreset
8
K-Y. Cheng et al. / Geomorphology 171–172 (2012) 1–11
Fig. 7. Hypsometric curves and elevation frequency of main basins. (A) Hypsometric curves of all main basins. (B) Hypsometric curves of selected basins. Solid curves: basins #3 to
#32. Dashed curves: basins #41 to #81. (C) Elevation frequency for the selected basins.
zone has rates of ~2.3 mm year − 1. Therefore, HI tends to be greater if
exhumation rates are higher. The lithology of the reset zone is
characterized by low- to medium-grade metamorphic rocks and
more resistant to erosion than the lithology of the unreset zone
characterized by sedimentary rocks and low-grade metamorphic
rocks. The more resistant rocks seem to yield higher HI values. We
also notice that in northern Taiwan, the region with HImain > 0.4
narrows at 220 km N, but becomes wider in the AFT reset zone
(Fig. 6a). Fuller et al. (2006) assumed an exhumational steady state
in this part of the Hseshan Range. We attribute the narrowing and
widening patterns to the indentation of the Peikang Basement High
and the extension of the Okinawa Trough, respectively.
The drainage basins with steady state topography are typified by
S-shaped hypsometric curves and normally distributed elevations
(Fig. 7). We found that Ohmori's (1993) cycle of the hypsometric
curve gives a reasonable relative chronology for the drainage basins
studied. Fig. 7 indicates that concave hypsometric curves at low
altitudes abruptly transform into S-shaped curves with an increase
in altitude, and then into the concave form again. The distances
corresponding to the two transitions between the concave and S
curves are 100 and 270 km N. The elevation histograms also vary
from the right-skewed to the normally distributed, along with
changes in the mean altitude of the sampled drainage basins. These
observations indicate that topography of the mountains between
100 and 270 km N has attained a steady state.
In summary, the steady state topography is characterized by
HI ~ 0.5, S-shaped hypsometric curves, and normally distributed
elevations. Because a large drainage basin is composed of subordinate
basins of lower orders, different elements within a basin or a
geomorphic system may not attain the steady state simultaneously.
The criteria noted above seem to be applicable to both main and
subordinate basins to discuss whether the whole system is in a steady
state.
5.2. Response time of landscapes inferred from basin hypsometry
Whipple (2001) discussed the response time of detachmentlimited fluvial bedrock channels to tectonic and climatic
perturbations in the Taiwan Central Range. He suggested that the
response times generally range from 0.25 to 2.5 My, assuming a
quasi-steady-state form for modern stream profiles. He also argued
that the Central Range has attained a quasi-steady state on the
drainage-basin scale despite Quaternary climatic fluctuations, as
actively uplifting and eroding landscapes have been adjusted to the
mean climatic condition. Our study also estimates the response time
of drainage basins to tectonic rock uplifting using space-for-time
substitution without taking climatic perturbations into consideration.
The critical distance of a drainage basin when HI approaches 0.5
(Fig. 4) provides the time required for basins to evolve from the
beginning of sub-aerial erosion to a steady state: about 0.55, 0.73,
1.10, 1.55, and 2.00 My for 1st to 5th order basins, respectively. This
inference is consistent with the result of Whipple (2001), although
the former is on a drainage-basin scale whereas the latter is on a
channel scale. In a statistical sense, our estimation may be valid
only for lower-order drainage basins for which a large amount of
data are available.
5.3. Scale dependence of basin hypsometry
The scale dependence of basin hypsometry is manifested in our
data because HI of main basins decreases with an increasing basin
order and/or basin size (Table 1). However, the basin size vs. HI
K-Y. Cheng et al. / Geomorphology 171–172 (2012) 1–11
9
Fig. 8. Scatter plot of HI vs. basin area for basins with different orders in the reset zone.
relationship for the same order indicates that the scale dependence
does not exist in the reset zone of the Taiwan Mountain Range,
because HI tends to be ca. 0.5 irrespective of basin size (Fig. 8). On
the other hand, in the unreset zone, higher the basin order, the
lower the value of HI (Fig. 9), and the deviation from 0.5 is more
distinct for the higher order basins. We conclude that HI is scale
free for basins at a steady state while it is scale-dependent at a nonsteady state.
6. Conclusions
Taiwan provides an excellent opportunity to study the
characteristics of topographic steady state. For hypsometric analysis
we selected the main drainage basins of the 4th or 5th order. We
found that HI of basins with steady state topography approached 0.5
irrespective of basin order or size; whereas, HI of the non-steady-
state basins is lower particularly for higher order basins. The drainage
basins at a steady state possess S-shaped hypsometric curves and
normally distributed elevations. For the other basins, HI varies with
basin size, pointing to the scale dependence of HI. Such basins
particularly large ones possess concave hypsometric curves and
skewed elevation histograms. The response time for a drainage
basin required to research a steady state from the beginning of subaerial erosion is estimated to be 0.5 to 2.0 My. This empirical study
indicates that basin hypsometry is useful to discuss the nature of
steady-state topography.
Acknowledgments
This research was supported by the Taiwan Earthquake Research
Center (TEC) and funded by the National Science Council (NSC): grant
numbers 96-2119-M-231-004, 97-2745-M-231-002 and 98-2116-M-
10
K-Y. Cheng et al. / Geomorphology 171–172 (2012) 1–11
Fig. 9. Scatter plot of HI vs. basin area for basins with different orders in the unreset zone.
231-001. The TEC contribution number for this article is 00072. Y.C.
Chen is thanked for his early work on the scale issue. T.K. Liu of National
Taiwan University is also thanked for his fruitful comments. This
manuscript was improved by constructive reviews by R.C. Wallcott,
K.W. Wegmann, and T. Oguchi.
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