Geomorphology (2012) Contents lists available at SciVerse ScienceDirect. Geomorphology

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1 Geomorphology (2012) 1 11 Contents lists available at SciVerse ScienceDirect Geomorphology journal homepage: 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 Department of Geography, National Changhua University of Education, No. 1, Jin-De Rd., Changhua, Changhua, 500, Taiwan, ROC b Institute of Geophysics, National Central University, No. 300, Jhongda Rd., Jung-Li, Taoyuan, 320, Taiwan, ROC c Institute of Geoinformatics and Disaster Reduction Technology, Ching Yun University, No. 229, Chien-Hsin Rd., Jung-Li, Taoyuan, 320, Taiwan, ROC article info abstract 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 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 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 () 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 of a higher Strahler order. The 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 of 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 are in a non-steady state. The basin hypsometry may therefore be useful to probe into the nature of steady-state topography 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 () represents the relative proportion of the basin area below (or above) a given height. By using dimensionless parameters, drainage 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 (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). 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.: ; fax: address: kc0729@cyu.edu.tw (Q-C. Sung). Chen et al., 2003; Walcott and Summerfield, 2008). Hypsometric analysis of drainage 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 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 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. 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 X/$ see front matter 2012 Elsevier B.V. All rights reserved. doi: /j.geomorph

2 2 K-Y. Cheng et al. / Geomorphology (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 in Japanese mountains which resulted from concurrent tectonics and denudation. He inferred that 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, while Ohmori (1993) paid attention to the scale of mountain ranges. Stolar et al. (2007) found that of larger (>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 of smaller (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 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 ; 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 along the axis of the island of Taiwan to determine trends in topographic evolution. We also sampled drainage 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

3 K-Y. Cheng et al. / Geomorphology (2012) 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; thefigure shows the scale for this distance system) where a constant width of 87±4 km, a cross-sectional area of 118±24 km 2, 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). Fig. 3. Distribution of selected main drainage in the Taiwan Mountain Range. The plot of 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 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 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 in different physiographic divisions of the Taiwan Mountain Range. He found that drainage in a topographic steady state have a typical S-shaped hypsometric curve, a value of approaching0.5,andanearlynormal distribution of basin elevations. He also inferred that the subsequent breakdown of the state may be revealed by lower values of 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 for drainage 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 4 K-Y. Cheng et al. / Geomorphology (2012) 1 11 Table 1 Statistics of hypsometric integral for drainage studied. Main basin Distance N(km) a b main 5th order 4th order 3rd order 2nd order 1st order Area area area area area

5 K-Y. Cheng et al. / Geomorphology (2012) Table 1 (continued) Main basin Distance N(km) a b main 5th order 4th order 3rd order 2nd order 1st order Area area area area area All a b Distance of the basin center from to the southern tip of Taiwan along the N S profile. for either the 5th or 4th order main basin. Fig. 4. Plots of mean vs. N S distance for with different orders. Distance of the basin center from the southern tip of Taiwan is plotted. Bar: one standard deviation.

6 6 K-Y. Cheng et al. / Geomorphology (2012) 1 11 Fig. 5. Plots of mean vs. E W distance for 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 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 according to Strahler orders. The value of a drainage basin can be estimated by (H mean H min )/(H max H min ), where H mean and (H max H min ) 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 of higher Strahler orders from the Central Range and the Western Foothills of the Taiwan Mountain Range. We selected the main 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 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, including 67 of order 5 and 14 of order 4, were selected for grid-based hypsometric analysis (Table 1). The average size of order-5 is 230 km 2 with size varying from 65 to 725 km 2, while the average size of order-4 is 57 km 2 with size varying from 20 to 200 km 2. Drainage were further divided into subordinate according to Strahler's rule. A

7 K-Y. Cheng et al. / Geomorphology (2012) below the level of 0.5 (circles in Fig. 4). The higher the basin order, the lower the value of. 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 from the pro- to retro-wedges of the Taiwan orogen (Fig. 5). An eastward increasing trend of is observed in all profiles (north, middle and south). 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 for main (either order-5 or order- 4 ) shows a region roughly delimited by main >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 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. (A) for main ( main ). The region with main >0.4 is delimited by a bold dashed line. (B) Map of interpolated values based on the kriging method, showing the correlation between the reset zone inferred from thermochronometry and >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 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 Relationship between basin hypsometry and topographic steady state total of 19,928 of order 1 were extracted with an average size around 0.5 km 2 (Table 1). Then was plotted against distance (i.e. the central location of each basin from the southern tip of Taiwan) for of different orders. 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 for of the same order is used to minimize the effects of such local factors. 4. Results Although for all shows a weak positive correlation with distance from the southern tip of Taiwan, the mean values of with the same order shows a marked positive correlation with distance, and it becomes clearer as the basin order increases (Fig. 4). The mean values of drainage 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 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, respectively; the slope of the regression line for mean vs. distance also increases with the basin order (Fig. 4). However, of drainage in the Western Foothills (profile B B in Fig. 3) behaves differently. With a slightly increasing northward trend, the regression lines of 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. values are approximately 0.5 in the steady state region of the Taiwan Mountain Range (Fig. 4), and of the main approaches 0.5 posterior to that of the subordinate. We also found that the region defined by main >0.4 corresponds to the reset zone inferred from the AFT and/or ZFT thermochronometers; whereas, the region with main b0.4 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 main >0.4 is in both topographic and exhumational steady states. Therefore, we suggest that a drainage basin attains a steady state when 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). gives a further constraint for defining steady state landscapes. The geographic distribution of 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 8 K-Y. Cheng et al. / Geomorphology (2012) 1 11 Fig. 7. Hypsometric curves and elevation frequency of main. (A) Hypsometric curves of all main. (B) Hypsometric curves of selected. Solid curves: #3 to #32. Dashed curves: #41 to #81. (C) Elevation frequency for the selected. zone has rates of ~2.3 mm year 1. Therefore, 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 values. We also notice that in northern Taiwan, the region with main >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 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 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. 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 ~0.5, S-shaped hypsometric curves, and normally distributed elevations. Because a large drainage basin is composed of subordinate 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 to discuss whether the whole system is in a steady state 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 to tectonic rock uplifting using space-for-time substitution without taking climatic perturbations into consideration. The critical distance of a drainage basin when approaches 0.5 (Fig. 4) provides the time required for 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, 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 for which a large amount of data are available Scale dependence of basin hypsometry The scale dependence of basin hypsometry is manifested in our data because of main decreases with an increasing basin order and/or basin size (Table 1). However, the basin size vs.

9 K-Y. Cheng et al. / Geomorphology (2012) Fig. 8. Scatter plot of vs. basin area for 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 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 (Fig. 9), and the deviation from 0.5 is more distinct for the higher order. We conclude that is scale free for 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 of the 4th or 5th order. We found that of with steady state topography approached 0.5 irrespective of basin order or size; whereas, of the non-steadystate is lower particularly for higher order. The drainage at a steady state possess S-shaped hypsometric curves and normally distributed elevations. For the other, varies with basin size, pointing to the scale dependence of. Such 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 M , M and M-

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