CSIS Discussion Paper #56

Drainage density and slope angle in Japanese bare lands from high-resolution DEMs

Zhou LIN a,* and Takashi OGUCHI ba

Department of Earth and Planetary Science, Graduate School of Science, theUniversity of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-0033, Japanb

Center for Spatial Information Science, the University of Tokyo, 4-6-1, Komaba,
Meguro-ku, Tokyo 153-8904, Japan* Corresponding author. E-mail address: lin@csis.u-tokyo.ac.jp (Z. Lin)

Abstract
Relations between drainage density and slope angle for three bare lands in Japan were analyzed withspecial attention to channels at early erosion stages and channels in a badland-type terrain. The bare lands
were caused by either volcanic eruptions or landsliding. Raster digital elevation models (DEMs) with a
1-m resolution and ortho aerial photos were generated using digital photogrammetry to enable detailed
stream-net extraction and topographic analyses. Data for drainage density, slope angle and relative heightfor 88 watersheds were obtained from the DEMs and derived stream-nets. The relation between drainage
density and slope angle for each watershed can be divided into two types: downward sloping and convex
upward. Although previous studies suggested that drainage density positively correlates with slope angle
if overland flow is dominant, such an upward sloping correlation seldom occurs in the study areas. The
two types of drainage density - slope angle relations correspond to differing channelization stages thatreflect the extension and integration of existing channels as well as the formation of new low-orderstreams in response to base-level lowering. The location of watersheds within each study area seems toplay a major role in determining the stages of channel development and, in turn, the types of density slope angle relations.
Key words: Stream-nets, Channel development, Digital photogrammetry, Geomorphometry, GIS

Introduction
Drainage density, total stream length per unit area (Horton, 1932, 1945), represents the degree of fluvial
dissection. Drainage density is positively correlated with slope angle or relative relief in some regions in
the United States (Schumm, 1956; Smith, 1958; Montgomery and Dietrich, 1992), but negatively inJapanese mountains (Mino, 1942; Yatsu, 1950). Oguchi (1997) has attributed the negative correlation to1

the decline of channel sidewalls on steep slopes related to slope failure. Tailing and Sowter (1999) havesummarized previous studies and note drainage density correlates positively with slope angle if overland
flow is dominant, but negatively if shallow mass wasting is dominant. Such differences have also beeninferred from a three-dimensional landscape evolution model by Tucker and Bras (1998). Howard (1997)
has similarly indicated drainage density and slope angle correlate negatively in quickly eroding areas, but
positively in slowly eroding areas.
The applicability of these relatively simple summaries, however, needs to be examined carefully. It is
still uncertain how the development of drainage systems with time affects the relation between drainage
density and slope angle. To solve this problem, channels at different stages of stream-net growth should besurveyed. Although high-resolution topographic data are needed to investigate shallow and narrow
channels at early erosion stages, obtaining such information in the field is tedious, and existing
topographic maps are often useless because of limited resolution. Instead, analytical and digital
photogrammetry can efficiently provide high-resolution topographic data from stereo photographs (Laneet al., 1993; Chandler, 1999).
This study examines drainage density and slope angle for three bare lands in Japan. High-resolution
raster digital elevation models (DEMs) were acquired using digital aerial photogrammetry. These barelands, originally caused by volcanic eruptions or landsliding, include channels at various erosion stages
because their eruption/landslide ages as well as local topographic relief are different.

The study areas
The study areas are located in Usu Volcano, Kusatsu-Shirane Volcano, and the Aka-Kuzure Landslidein northern and central Japan (Fig. 1). Usu, a stratovolcano in southwestern Hokkaido, consists mainly of
hypersthenes dacite (Kadomura et al., 1988). The studied watershed has an area of 0.18 km2 and altitudesof 450 to 725 m (Fig. 1, U). The watershed underwent thick deposition of volcanic ash during eruptions
between August 1977 and October 1978. The master gully in the central watershed first formed at the latestage of the 1977-1978 eruptions. Then the network of rills and gullies widely extended by the summer of1979 due to overland flow (Kadomura et al., 1983). Only sparse grass vegetation existed in the study area
until the early 1980s.
Kusatsu-Shirane is an active volcano in central Japan. Although it consists mainly of Tertiary andesite,
historical eruptions occurred in 1882, 1832, 1942, 1976 and 1982 (Uto et al., 1983). The area studied islocated on the eastern slope of the main volcanic cone with a crater lake (Fig.1, K). It has an area of 0.19
km2 and altitudes of 1,900 to 2,200 m, and was subjected to ash fall during the historical eruptions. Gully
erosion by overland flow has been dominant, and only grass vegetation occurs in the study area.
Aka-Kuzure is a large landslide in the Southern Japanese Alps, forming a steep watershed with an areaof 0.43 km2 and altitudes of 1,800 to 2,000 m (Fig.1, A). The watershed is underlain by sandstone, shale,
and their alternations (Chigira and Kiho, 1994). The formation age of the landslide is unknown, but it has
existed since at least one hundred years ago. The surface of the landslide is finely dissected and almost
devoid of vegetation, showing a badland-type landscape. Shallow slope failure and subsequent sedimentremoval by running water are major hillslope processes.

2

Fig. 1 Locations and orthophotos of three study areas.

3

Data
Stream-nets DEMs and orthophotos
Digital photogrammetry was applied to 1:8,000 vertical airphotos to produce 1-m DEMs for the threeareas. The periods of the airphotos are October 1979 for Usu, August 1985 for Kusatsu-Shirane, andOctober 1995 for Aka-Kuzure (Table 1). Previous studies (Powers et al., 1996; Derose et al., 1998; Brown
and Arbogast, 1999; Lane et al., 2000; Westaway et al., 2000) have suggested that DEMs with a gridinterval of 1 m can be produced from 1:8,000 airphotos, and the DEM grid interval should be at least 5 to
10 times as large as the pixel spacing of scanned images (Pyle et al., 1997; Butler et al., 1998; Lane, 2000).
Therefore, the positive films of the airphotos were scanned into raster images with a 20-µm resolution or apixel dimension of 0.162 m using Leica DSW500, a high-quality photogrammetric scanner. Then 1-m
DEMs for Usu and Kusatsu-Shirane were generated using the VirtuoZo DPW (Digital PhotogrammetricWorkstation) at Center for Spatial Information Science, the University Tokyo. The 1-m DEM for
Aka-Kuzure was also generated using the Leica-Helava DPW at Tamano Consultant Co. Ltd., Nagoya,Japan. First, interior, relative and absolute orientations were undertaken to restore the internal geometry ofcameras, establish the geometric relationship of stereo-pairs, and provide shifted photographic models.
Then 1-m DEMs and orthophotos (Fig. 1) were generated using the obtained models.

Table 1 Airphotos used for digital photogrammetry and RMSE values at GCPs.

Usu

Oct
1979

Airphotos (1:8,000)
Frame
Cameranumber
153.12 mm
3RC-10

KusatsuShirane

Aug
1985

152.56 mm
RC-10

4

15

Aka-Kuzure

Oct
1995

152.95 mm
RC-10

2

5

Area

Period

DEM Errors
GCP
number
10

Model

X

Y

Z

1
2
1
2
3

0.1522
1.21960.9506
0.96480.56571.00261.69091.11970.28961.03470.04510.3517
0.66110.68840.42391

1.9991

1.83870.7818The quality of the constructed DEMs looks sufficient for general topographic analysis. RMSE values
showing errors at ground control points (GCPs) are smaller than 2 m in X and Y directions and 0.8 m in a
Z direction (Table 1), which are comparable to those reported in other digital photogrammetric studies
using medium-scale airphotos (Derose et al., 1998; Brown and Arbogast, 1999). The shaded relief images
generated from the DEMs also clearly depict detailed topographic features such as distribution of small
channels and ridges. However, small unnatural straight cliffs occur in Usu and Kusatsu-Shirane where
DEMs derived from different stereo pairs were merged. To remove the artifact, a smoothing filter wasapplied to narrow zones along the cliffs.

4Stream-nets
Stream-nets for the three study areas were delineated using the 1-m DEMs and orthophotos. DEMs
permit the automatic extraction of stream-nets (e.g., Mark, 1984; Martz and Garbrecht, 1992), and themost common method, which is often implemented to major commercial GIS software, assumes aminimum contributing area to determine channel-head locations. However, minimum contributing areas
should vary even within a small watershed according to local factors such as topography and lithology(e.g., Tucker et al., 2001; Vogt et al., 2003). To solve this problem for Usu and Kusatsu-Shirane, we
determined the location of each channel head by visual interpretation of the orthophotos. Stream-nets
were also automatically extracted from the 1-m DEMs based on the threshold contributing-area method of
Jenson and Domingue (1988). The threshold area was set to be small to delineate both major and subtle
valley lines. Then the delineated streams above the visually determined channel heads were deleted. The
final stream nets (Fig. 2a, U and K) are consistent with both the DEMs and actual channel-head locations.
Although channel heads in Usu and Kusatsu-Shirane are topographically well-defined, those in
Aka-Kuzure are indistinct because of badland-type topography, requiring another definition of
stream-nets. We first assumed that a first order stream starts from a ridge, because channels in Aka-Kuzure
usually attain ridges due to widespread mass wasting. Then we allocated stream orders to all the DEM
cells following Strahler (1952), i.e., the stream order increases when streams of the same order meet (Fig.
3). We developed a macro program for ArcView for this stream-ordering. When cells equal to or higher
than the fourth order are regarded as streams and the other cells are as interfluves, mean drainage density
for Aka-Kuzure becomes closest to that for Usu and Kusatsu-Shirane. Thus, the fourth and higher-orderstreams were regarded as the elements of stream-nets in Aka-Kuzure (Fig 2a, A).
Sub-watersheds
To examine stream-net properties in relation to local topographic variations, sub-watersheds withineach study area were delineated using the DEMs. Both the area of each sub-watershed and the total
number of the watersheds should be adequately large for statistically meaningful analyses. Thus, wedelineated 20 second-order sub-watersheds in Usu (designated as U01 to U20), 14 second-order
sub-watersheds in Kusatsu-Shirane (K01 to K14), and 54 sixth-order sub-watersheds in Aka-Kuzure (A01
to A54) (Fig. 2b). The area of the sub-watersheds ranges from ca. 1,200 to 11,600 m2 for Usu, 4,900 to
11,800 m2 for Kusatsu-Shirane, and 2,200 to 5,600 m2 for Aka-Kuzure. Each sub-watershed was further
divided into 10 m * 10 m raster grid cells.
Morphometric parameters
The total length of channels in each sub-watershed was computed using our original C++ algorithm toderive drainage density. One of four channel lengths, 0 m, 1 m, 1.21 m, and 1.41 m, was allocated to eachgrid cell based on the occurrence of channels in neighboring cells (Fig. 4). The calculated drainage density
for each sub-watershed ranges from 0.02 to 0.11 m/m2 for Usu, 0.03 to 0.06 m/m2 for Kusatsu-Shirane,
and 0.04 to 0.14 m/m2 for Aka-Kuzure. Drainage density for each 10 m * 10 m cell within a sub-watershed
was also computed.

5

Fig. 2 (a) Stream-nets and (b) sub-watersheds in Usu (U), Kusatsu-Shirane (K) and Aka-Kuzure (A). White
watersheds: Type 1, gray watersheds: Type 2, hatched watersheds: non-classified.

6

Fig. 3 Part of stream-nets in Aka-Kuzure with badland-type topography without distinct channel heads.

Fig. 4 Stream lengths for grid cells. White squares: 1 m, gray squares: 1.21 m, black squares: 1.41 m,
non-squared cells: 0 m.

7

The slope angle for each DEM cell was calculated from elevations at eight neighboring cells, based on
the method by Jenson and Domingue (1988). Then the mean slope angle for each sub-watershed and in a
10 m * 10 m cell was computed. The mean slope angle for each watershed is 15 to 24 degrees for Usu, 18
to 28 degrees for Kusatsu-Shirane, and 43 to 57 degrees for Aka-Kuzure.
Relative height (Hr) is defined to express the relative location of a 10 m * 10 m cell within a
sub-watershed:Hr =H − H minH max − H min

where H is the mean height of a 10 m * 10 m area, Hmin is the minimum height of a sub-watershed, andHmax is the maximum height of a sub-watershed. The relative height varies from 0 (bottom) to 1 (top).
Data analysis
Slope angle and drainage density
The relation between slope angle and drainage density was examined for each sub-watershed. Using
data for the 10 m * 10 m grid cells within a sub-watershed, mean drainage density for each 2-degree slope
bin was computed and plotted against slope angle. Then the quadratic equation was fitted to approximate
the relation between slope angle and drainage density for each watershed. Figure 5 shows six examples ofthe plots with the fitted trend lines. Their trends can be divided into two types: downward sloping showingthat drainage density tends to decrease with increasing slope angle (Type 1), and convex upward showingthat drainage density tends to increase with slope angle for smaller slope angles but decrease for larger
slope angles (Type 2). The relations for the other 82 watersheds can also be classified into the two types(Fig. 6) except two in Usu (U-07 and U-10) and eight in Aka-Kuzure (A-12, 20, 21, 24, 35, 39, 40, 44)with various other relations. The trend lines for Type 1 are mostly concave upward (Fig. 6).
Figure 7 is a plot of mean slope angle and mean drainage density for each sub-watershed classified
according to Types 1 and 2. The two types in Usu are clearly separated since Type 1 has lower drainage
density than Type 2. Conversely, some Type 1 watersheds in Aka-Kuzure tend to have higher drainage
density than Type 2, since most watersheds with drainage density higher than 0.08 m/m2 belong to Type 1.
Such differences between the two types are indistinct in Kusatsu-Shirane.
Distribution of watershed types
The Type 1 and Type 2 sub-watersheds tend to have different types of spatial distributions (Fig. 2b).
Most Type 1 watersheds in Usu are located in the peripheral area, whereas the Type 2 watersheds arelocated mainly in the central area. Most Type 1 watersheds in Kusatsu-Shirane occur in the northern part,but most Type 2 watersheds occur in the southern part. The Type 1 watersheds in Aka-Kuzure take placein various parts of the landslide including the central area, whereas the Type 2 watersheds are located onlyin the periphery.

8

Fig. 5 Examples of two types of drainage density - slope angle relations. Dashed lines show approximation
by quadratic equations. Type 1 is downward sloping (and concave). Type 2 is convex.

9Fig. 6 Trend lines of two types of drainage density - slope angle relations. U: Usu, K:
Kusatsu-Shirane, A: Aka-Kuzure.

10

Fig. 7 Plot of mean drainage density and mean slope angle for each watershed. U: Usu, K:
Kusatsu-Shirane, A: Aka-Kuzure. Triangles: Type 1, Circles: Type 2.

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Stream-net structure
Basic Horton’s parameters representing stream-net structure were calculated for Type 1 and Type 2 ineach study area, based on the total number and the mean length of channels belonging to each stream order(Table 2). In Usu and Kusatsu-Shirane, Type 2 has larger bifurcation ratios and stream-length ratios thanType 1. In Aka-Kuzure, by contrast, Type 2 has smaller bifurcation ratios and stream-length ratios thanType 1.

Table 2 Parameters of stream-net structure for three study areas. Note the method of stream ordering
for Aka-Kuzure differs from that for Usu and Kusatsu-Shirane.

Area

Type

Streamorder

Streamnumber

1
2
1
2
1
2
1
2
4
5
6
4
5
6

39
11
37
7
34
7
40
7
486
102
33
160
39
13

1
Usu
2
KusatsuShirane

1
2
1

Aka-Kuzure
2

Bifurcation
ratio
3.54
5.28
4.85
5.70
4.76
3.09
4.10
3.00

Mean
streamlength
38.8
50.5
63.4
110.7
50.8
130.5
57.1
194.3
14.0
33.3
84.0
17.7
31.4
77.4

Stream length
ratio
1.30
1.74
2.56
3.41
2.38
2.52
1.77
2.46

Relative height and drainage density
The relation between relative height and drainage density for each sub-watershed was also examined.
Relative height was divided into 10 bins with a 0.1 interval. Using the 10 * 10 m grid cell data within awatershed, mean drainage density for each bin was computed and plotted against slope angle, and thequadratic equation was fitted to their relations. Figure 8 shows six examples of the plots, and Fig. 9 shows
the approximated trend lines for all the Type 1 and Type 2 watersheds. Types 1 and 2 in Usu andKusatsu-Shirane also differ in the relation between drainage density and relative height, in that the lines
for Type 1 are mostly downward sloping and concave upward, while those for Type 2 is convex upward(Fig. 9). In other words, drainage density - slope angle relations and drainage density - relative height
relations tend to have similar trends. Such correspondences also hold true for Aka-Kuzure, although theconcavity of Type 2 is less marked, and more exceptions can be found (Fig. 9).

12

Fig. 8 Examples of drainage density - relative height relations. Dashed lines show approximation by
quadratic equations. Type 1 and Type 2 are determined based on drainage density - slope angle
relations (see Figs. 5 and 6).

13

Fig. 9 Trend lines showing drainage density - relative height relations, classified according to types ofdrainage density - slope angle relations. U: Usu, K: Kusatsu-Shirane, A: Aka-Kuzure.

14

Discussion
Usu area
Geomorphometric differences between Types 1 and 2 in Usu indicate they are at the different stages ofdrainage evolution. Lower drainage density, a lower bifurcation ratio, and smaller mean stream lengths forType 1 (Fig. 7, Table 2) show stream-nets in the Type 1 watersheds are more immature than those in theType 2 watersheds. The biggest difference is observed in the middle height of each watershed: drainage
density at a relative height of 0.5 is around 0.05 for Type 1, but around 0.11 for Type 2 (Fig. 9). Drainage
density of Type 1 for medium slope angles (15 to 20 degrees) also tends to be lower than that of Type 2(Fig. 6). These observations support the inference that Type 1 corresponds to earlier channelization stages,
since channel extension usually starts at lower flat areas and subsequently proceeds to higher steeper areas.
The change in drainage density - slope angle relations from Type 1 to Type 2 can be accounted for by theenhanced channelization in the middle to upper watershed as erosion progresses. Drainage density for the
lower part of the Type 1 watersheds is slightly higher than that of Type 2, which may reflect channel
integration with time.
Such differences in the stage of stream-net growth seem to reflect watershed locations. Most Type 2
watersheds in Usu are close to the master gully, and thus have reached more advanced channelization
stages than the Type 1 watersheds in the periphery.
Kusatsu-Shirane area
The Type 1 and Type 2 watersheds in Kusatsu-Shirane have topographic differences similar to those in
Usu, in that Type 1 has a smaller bifurcation ratio, smaller mean stream lengths, and lower drainage
density at the middle height of a watershed (Table 2, Fig. 9), suggesting that Type 2 is at more advanced
channelization stages. Unlike Usu, mean drainage density does not differ according to the types (Fig. 7).
The drainage density for the lowest part of the Type 1 watersheds (relative height = 0.05) is higher than0.07, whereas that of Type 2 is lower than 0.06 (Fig. 9), which compensates lower drainage density for the
middle height of the Type 1 watersheds. Drainage density for low slope angles (< 20 degrees) of Type 1 isapparently higher than that of Type 2 (Fig. 6). Therefore, marked stream integration seems to haveoccurred at the lower and flatter areas of the Type 2 watersheds. Moreover, the Type 2 watersheds arelocated in the southern area where incision by master gullies is more evident than the northern area (Fig.
1). These observations confirm that Type 2 is at advanced channelization stages with integrated channels
in the lower gentler areas and extended channels in the higher and steeper areas. Consequently, the two
types of drainage density - slope angle relations for Kusatsu-Shirane are also ascribable to different stagesof stream-net growth.
Aka-Kuzure area
Unlike the other two areas, it seems inappropriate to assume that Type 2 in Aka-Kuzure corresponds tomore advanced channelization stages than Type 1. Although the trend lines of drainage density - relative
relief relations for Types 1 and 2 in Aka-Kuzure differ in the same manner as those in the other two areas

15

(Fig. 9), Type 1 has larger bifurcation ratios and stream length ratios than Type 2, which is reverse to theother two areas (Table 2). The order of mean drainage density for the two types is also reverse to that for
Usu (Fig. 7), suggesting that Type 1 represents more advanced channelization.
Type 1 is characterized by relatively short, abundant lower-order streams (Table 2) and widespreadchannelization dominates at the lower parts of watersheds (Fig. 9), which can be related to base-level
lowering at the bottom of the landslide. The topography of a small alluvial fan immediately below the
Aka-Kuzure landslide indicates that the stream flowing from the landslide have undergone rapid
downcutting in recent years, resulting in the degradation of major tributaries and subsequent slope failure
and the creation of new low-order streams because of hillslope instability. It can be assumed that suchprocesses have been more enhanced in the Type 1 watersheds than Type 2, especially in the lower andflatter parts of the watersheds adjacent to the major streams. This assumption conforms to the peripheral
distribution of the Type 2 watersheds and their smaller mean slope angles (Figs. 2 and 7), because
hillslope incision in response to base-level lowering should be limited in gentle terrains apart from themaster stream.
Factors affecting drainage density - slope angle relations
The detailed analysis of stream-net structure and watershed geomorphometry for the three areas has
provided fresh insights into drainage density - slope angle relations. Although previous studies suggestthat slope angle and drainage density correlate positively if overland flow promotes erosion, such positive
correlations seldom occur in Usu and Kusatsu-Shirane where overland flow is dominant. Previous studies
also suggest slope angle and drainage density correlate negatively if shallow mass-wasting is dominant,which conforms to the fact that Type 1 often occurs in Aka-Kuzure where slope failure is widespread.
However, the common occurrence of Type 2 in all the three areas indicates more attention should be paid
to such non-linear relations between drainage density and slope angle. Differences in Types 1 and 2 most
likely reflect differing stages of channel development, which often corresponds to the location of
sub-watersheds. In Usu and Kusatsu-Shirane, drainage density - slope angle relations shift from Type 1 to
Type 2 with the progress of channelization stages owing to channel extension in the middle parts of
sub-watersheds and channel integration in the lower parts. In Aka-Kuzure, however, formation of newchannels in response to base-level lowering has led to shift from Type 2 to Type 1. In summary, theadvance of channelization stages can lead to bi-directional changes between the two types of drainage
density - slope angle relations.
Conclusions
Digital aerial photogrammetry was applied to the Usu, Kusatsu-Shirane, and Aka-Kuzure areas inJapan for detailed topographic analyses of channels and surrounding hillslopes using 1-m DEMs. Special
attention was paid to relatively new channels at early erosional stages as well as channels in badland-type
topography. Stream-nets were obtained based on the two approaches: DEM-based automatic extraction
and visual identification of channel heads using ortho-airphotos. Drainage density was analyzed inrelation to slope angle and relative height within sub-watersheds.

16

Although previous studies suggested that the relation between slope angle and drainage density
corresponds to dominant erosion types, this study has indicated that they correspond more directly to thestages of channelization. Change in slope angle - drainage density relations with the progress ofchannelization depends on the current major channelization processes such as the extension andintegration of existing channels as well as the formation of new low-order streams in response tobase-level lowering.
This paper suggests simple summaries concerning drainage density - slope angle relations have only
limited applicability, and careful analyses using high-resolution quantitative data are necessary to discuss
the details of the issue. Although this study on the three locations in Japan has provided some new insights,
different conclusions may be reached elsewhere, since fluvial and hillslope processes are highly variableaccording to regional and local factors. Therefore, channelized watersheds in various parts of the world
should be investigated in future for better understanding of topographic effects on channel distribution.
Acknowledgements
We thank Mr. S. Sano, Tamano Consultant Co. Ltd., for his support with the construction of the DEM
for the Aka-Kuzure area, and Prof. R. Shibasaki and Prof. Z. Shi, University of Tokyo, for allowing us to
use the digital photogrammetric workstation in their laboratory. Thanks are also due to Prof. Thad
Wasklewicz, University of Memphis, for his constructive review of an early draft of this paper.

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