Use of a landscape simulator in the validation of the SIBERIA catchment evolution model: Declining equilibrium landforms

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1 WATER RESOURCES RESEARCH, VOL. 37, NO. 7, PAGES , JULY 2001 Use of a landscape simulator in the validation of the SIBERIA catchment evolution model: Declining equilibrium landforms Greg Hancock and Garry Willgoose Department of Civil, Surveying, and Environmental Engineering, University of Newcastle Callaghan, New South Wales, Australia Abstract. This paper presents a study to test the ability of the current generation of landscape evolution models to correctly predict landscape form from measured erosion processes. Landscapes generated by the SIBERIA landscape evolution model were compared to experimental model landscapes in declining equilibrium. The model simulations were compared using geomorphologically and hydrologically significant parameters. Comparisons of the simulated landscapes with the experimentalandscapes were carried out using the hypsometri curve, width function, cumulative area distribution, and area-slope relationship. These comparisons demonstrate that SIBERIA can correctly simulate the experimental model landscape at declining equilibrium. The simulation showed sensitivity to the spatial distribution of the rainfall, particularly with respect to hypsometry. Using the correct measured distribution of rainfall was necessary rather than using a spatially uniform rainfall distribution. The results also highlighted the importance of digital terrain map (DTM) error in deriving geomorphic statistics. The observed landform had a consistently larger width function than the simulation. Only when the simulations were corrupted with elevation errors with statistical properties of the errors in the experimentalandscape DTM, as measured by photogrammetry, did the observed and simulated width functions match. 1. Introduction SIBERIA is a physically based model for the geomorphic evolution of landforms subject to erosion and mass transport [Willgooset al., 1989, 1991a, 1991b, 1991c, 1991d]. This paper discusses the comparison of SIBERIA against controlled laboratory landscape development. For this purpose a rainfallerosion landscape simulator was developed in which experimental model landscapes were allowed to evolve under controlled conditions of rainfall and material erodibility. Visual comparison of the experimental and simulated landscapes is not sufficient to ensure that SIBERIA is able to accurately simulate declining equilibrium landscapes and any variable quantitatively dependent on geomorphology [Hancock, 1997]. Quantitative geomorphology began with the development of the laws of stream numbers by Horton [1945] and Strahler [1952]. Subsequently, almost all catchment studied have been examined for compliance with the Horton/Strahler laws. It is now generally accepted that all catchments approximately fit these stream number laws. Shreve [1966] postulated that in the absence of environmental controls, a natural population of channel networks will be topologically random and that the stream number laws are, on average, a consequence. The use of these statistics is questionable because of their universal applicability [Kirchner, 1993]. Consequently, their use as a tool for testing or evaluating landscape evolution models is limited. If the statistics of all landscapes and landscape 1Now at School of Geosciences, University of Newcastle, Callaghan, New South Wales, Australia. Copyright 2001 by the American Geophysical Union. Paper number 2001WR /01/2001 WR models fit these universal laws, then the statistics are not able to falsify a poor model and are not useful in model testing. Previous work has shown that SIBERIA can produce simu- lated landscapes that closely match the Horton/Strahler statistics and drainage density [Willgooset al., 1989]. As a result, stream statistics are not examined in this study. Consequently, researchers have sought new measures for mathematically describing or categorizing landforms. Four separate geomorphological descriptors have been shown to be important measures of catchment geomorphology and hydrology [Hancock, 1997; Perera and Willgoose, 1998; Ibbitt et al., 1999] and have been used here. These are the width function, cumulative area distribution, area-slope relationship, and the hypsometric curve. Other descriptors, such as multifractal measures and optimal channel network energy, have not been examined as part of this study. A second paper (in preparation) will discuss the comparison of SIBERIA with transient or developing landscapes [Hancock, 1997] Width Function The width function, originally developed by Surkan [1968], is a plot of the number of channels at a given distance from the basin outlet, measured along the network [Naden, 1992] (Fig- ure 1). A slightly more general interpretation is adopted here, which is easier to apply for digital terrain maps. The width function used here is the number of drainage paths (whether they be channel or hillslope) at a given distance from the outlet as it is difficult to determine what is channel and what is hillslope on a digital terrain map. It has long been recognized that the width function is a good measure of hydrologic response since it can be strongly correlated with the instanta- neous unit hydrograph. If it is assumed that rainfall excess is routed with a constant velocity, then the width function can be linearly transformed into the instantaneous unit hydrograph.

2 1982 HANCOCK AND WILLGOOSE: USE OF A LANDSCAPE SIMULATOR 8 lo s _, region 1 ', region 2 ireglen distance from outlet (distance) 2 2 number of 3 I channels intersected I (width) OUTLET. loo lo... i0... 1'60... i i"' 4 area (pixels) Figure 2. The cumulative area distribution for the Middle Creek natural catchment in Pokolbin region of Australia [Willgoose, 1994a] distance Region two is the area that represents the catchment area dominated by channelized flow. This part of the cumulative area distribution is generally observed to be approximately log-log linear. Region three consists of that part of the catchment dominated by the large channels and the junction of large channels near the catchment outlet. Its form is a function of boundary effects. Large area contributions are made in this part of the catchment where the ordinate of the distribution function rapidly decreases as a result of increasing drainage Figure 1. The width function, which is the number of channels cut at successive distances from the channel outlet (Figure area. 1, top). It is usually displayed as a bar or line graph (Figure 1, 1.3. Area-Slope Relationship bottom). The area-slope relationship is the relationship between the area draining through a point versus the slope at the point. It A comparison of the width function derived from the experimental landscape with that of the SIBERIA simulated landscape is one tool that allows us to test if SIBERIA can simulate quantifies the local topographic gradient as a function of drainage area. A relationship of the form the experimental catchment drainage network. The definition of the width function used here has the advantage that it can be A as = const, (1) easily derived from maps and digital terrain maps and provides where A is the contributing area to the point of interest, S is a picture of the spatial pattern of the water courses within a catchment [Naden, 1992]. It also eliminates the need to define the slope of the point of interest, and the value of a ranges between 0.4 and 0.7, which has been widely reported by many channels or the possibility that channel definition methodology authors for natural catchments [Hack, 1957; Flint, 1974; Gupta may impact on the model comparisons below. and Waymire, 1989; Tarbeton et al., 1989; Willgooset al., 1991c; 1.2. Cumulative Area Distribution Willgoose, 1994a]. Two distinct regions of the relationship are typically ob- The cumulative area distribution (CAD) is a function defining the proportion of the catchment which has a drainage area greater than or equal to a specified drainage area. The CAD describes the spatial distribution of areas and drainage network aggregation properties within a catchment. The cumulaserved (Figure 3). At small areas, region A is dominated by rain splash, interrill erosion, soil creep, or other erosive processes that tend to round or smooth the landscape. As the catchment area becomes larger, a break in gradient of the curve occurs. Region B is where slope decreases as catchment tive area distribution has been used as a means of character- area increases. This region of the catchment is dominated by izing the flow aggregation structure of channel networks [Rodriguez et al., 1992; LaBarbera and Roth, 1994], and it has also been used in the calibration of geemorphological models [Moglen and Bras, 1994; Sun et al., 1994a, 1994b, 1994c; Inaoka and Takayasu, 1993]. It is an important component in determining what sections of a catchment are saturated in a catchment dominated by saturation excess runoff generation thus contributing runoff during rainfall and characterises the scal- 0.1 ing properties of drainage area [Perera, 1998; Willgoose and Perera, 2001]. The cumulative area distribution typically has a form that Region A: Region B consists of three regions (Figure 2). Region one represents those small areas of the catchment where rain splash or inter area (pixels) rill erosion is the dominant erosive mechanism, and it is also Figure 3. Area-slope relationship of the Tin Camp Creek the region of hillslope flow aggregation. It is typically S shaped. natural catchment, Northern Territory, Australia.

3 HANCOCK AND WILLGOOSE: USE OF A LANDSCAPE SIMULATOR 1983.g youth - ' -... mature, ' _. monadnock ß 0.6._ ' 0.4 o normalised area Figure 4. Idealized hypsometric curve displaying the three stages of landscape development described by Strahler [1952, 1964]. housin simulator 1700 camera perspex '' viewing box access doors control )oints fluvial erosive processes, i.e., those processes that tend to incise the landscape. It has been shown that the area-slope relationship is a fundamental geomorphic relationship [Montgomery and Dietrich, 1988, 1989; Tarboton et al., 1989; Willgoos et al., 1991c; Tarboton et al., 1992; Montgomery and Foufoula-Georgiou, 1993; Moglen and Bras, 1994; Montgomery and Dietrich, 1994; Willgoose, 1994a, 1994b] and is sensitive to changes in the parameters characterizing the diffusive and fluvial transport pro- cesses Hypsometric Curve ß.'.{;. '.'..',..'.',...',..'.',.:.,.',.z.'.'..'.',.:.'.',..'.',.'.... outlet - ;:',,....'... i ß erodible material.,!,.,.,., , _ Figure 5. Experimental setup showing position of soil box containing the erodible material, rainfall simulator, and cameras. Dimensions are in millimeters. The hypsometricurve [Langbein, 1947] is a nondimensional area-elevation curve (Figure 4). The nondimensionalization of the curve allows ready comparison of catchments with different infiltration capacity and low cohesion. The medium-grade fly area and steepness. The area below the hypsometri curve is ash had a particle size distribution where 90% of the material the hypsometric integral. was <310/xm in diameter and 50% of material was <113 The hypsometricurve has been used as an indicator of the whereas the fine grade material had a distribution where 90% geomorphic maturity of catchments and landforms. More re- of the material was <62/xm and 50% of material was <16 cent work has demonstrated its linkage with erosion processes, Fly ash was therefore used for all reported experiments in this catchment geometry, and network form [Willgoose and Hanpaper. cock, 1998]. An advantage of this curve is that it is not unduly A permeable base of geotextile allowed free drainage of affected by random aspects of network form [Willgooset al., infiltrated water. An adjustable flume at the outlet allowed a 1989], while being sensitive to changes in the parameters of the nickpoint or uplift event to be created. The whole apparatus erosion physics. was housed in a perspex box with a roof, which allowed viewing Strahler [1952, 1964] divided landforms into youth, mature, while protecting the rainfall simulator from external air movements. and monadnock characteristic shapes reflecting increasing catchment age (Figure 4). Willgoose and Hancock [1998] dem- The rainfall simulator used microsprinklers arranged around onstrated that these characteristic shapes were also consistent a frame. Rainfall distribution (Figure 6) was determined using with different catchment erosion processes and catchment ge- 225 rain guages spaced on a 100 mm by 100 mm grid over the ometry. 2. Experimental Landscape Simulator The landscape simulator (Figure 5) consisted of a rainfall simulator suspended above a box containing the erodible material, fly ash, the burnt remains of coal from coal-fired electricity generating stations [Hancock, 1997; Hancock and Willgoose, 2001a]. The fly ash had low cohesive strength and an asymptotic steady state infiltration rate of 20 mm/hr. An erodible material with little or no cohesion between particles was needed so that the material could readily be eroded at low shear stresses and have an infiltration capacity considerably less than the rainfall rate so that runoff could occur. Fly ash was available commercially in several size grades, and testing indicated that a blend of medium (two thirds by weight) and fine grade (one third by weight) had low outlet[ Figure 6. Distribution of rainfall over landscape surface. Contours are in millimeters per hour.

4 1984 HANCOCK AND WILLGOOSE: USE OF A LANDSCAPE SIMULATOR %! oo 1400 ' o 4.00 Figure 7. Open book type catchment. Dimensions are in millimeters O 0 landscape surface. Rainfall was measured for five separate periods of an hour prior to an experiment and once postexperiment to determine if there was any drift in spatial or temporal distribution. The temporal variation in rainfall at any one point was found to be <5% for most of the domain. Drop size was determined using a Malvern laser particle sizer. It was found that 90% of the drops were <330/xm in diameter and 50% of the drops <195 /xm in diameter. Rain splash has been demonstrated to occur at approximately twice the former drop diameter [Stow and Hadfield, 1981]. The kinetic energy of the simulated rainfall was many times less than natural rain. This effectively eliminated diffusive erosion processes, producing a landscape dominated by one process, fluvial erosion dominated by channelized flow, thus simplifying the model comparison in this paper. There was no indication of supercritical flow (i.e., hydraulic jumps), so it is believed that all flows were subcritical. Two rainfall intensities were used, 48 and 120 mm h -j, respectively, to ensure that the erosive potential of the experiments were very different. To input rainfall data into SIBERIA simulations, the steady state infiltration value of 20 mm h- was subtracted from the rainfall data. value of unity, suggesting that the digital terrain map was of sufficient accuracy for the analysis that follows [Gyasi-Agyei et al., 1995]. Pits in the experimentalandscape digital terrain map were removed by the software of Tarboton et al. [1989]. The low and steep slope experiments were started by lowering the outlet flume by 100 mm, creating a 100 mm nickpoint, simulating a single tectonic uplift event. Experiments were continued until the drainage network was fully developed and the landscape no longer changed visually. This was confirmed after the experiment by comparing the hypsometricurve at 720 (low-slope experiment) and 2400 min (steep-slopexperiment) of rainfall with hypsometricurves for earlier times in landscape development. The analysis that follows concentrates on the steep-slope experiments because of the highest reliability of the data. Despite this, some artifacts resulting from photogrammetric error were found. In these cases the steep slope catchment is compared with those of the low-slope catchment where artifacts from photogrammetry are more pronounced. This paper looks at declining equilibrium landforms, so only the declining equilibrium results are discussed below. Declining equilibrium landforms are defined as where the hypsometric properties of the landscape do not change with time. The transient landforms before declining equilibrium has been reached will be discussed in a future paper. Two open book type catchments were used as initial conditions for the experiments. The first "low slope" experiment had a 1% longitudinal grade upslope from the outlet and a 2.5% grade laterally across the catchment (Figure 7). The second "steep slope" experiment had a 7.5% longitudinal grade and a 2.5% lateral grade. The low slope catchment was subjected to 120 mm/hr "high intensity" rainfall, whereas the steep slope 3. SIBERIA Simulations catchment was subjected to 48 mm/hr "low intensity" rainfall. SIBERIA simulations were run using the constructed initial To analyze the experimentalandscape, stereo digital photogrammetry was used during the experimentalandscapevoconditions with the 10 mm grid measured by the photogramlution [Hancock and Willgoose, 200lb]. Rainfall was stopped metry. To simulate the initial roughness of the fly ash surface for the capture of the stereo photographs, which were taken at resulting from construction, a uniformly distributed random regular intervals during the evolution of the landscape. Obser- elevation perturbation of between + 1 mm was added to each vation suggested that the stopping and consequent recom- elevation to match the measured roughness. The perturbation mencement of rainfall appeared to have no effect on overall was added because the roughness measured by digital photolandscape development. Elevations were determined on a grid size of 10 mm by 10 mm. The accuracy of the elevation data grammetry was exaggerated. Two types of rainfall conditions were used in the modeling: (1) spatially constant rainfall equal was checked by comparison with points with known elevations. to mean rainfall and (2) spatially variable rainfall as calibrated Elevation data for the steep slope experiment was found to have a vertical error range of +6 mm with a standard deviation from the simulator (Figure 6). The simulations used the values of m and n calibrated from a one-dimensional catchment of 2.0 mm for a catchment relief of mm. The ratio of (section 4). The boundary condition that was used for the average drop per pixel in the catchment (7.6 mm) to the range of vertical elevations errors (6 mm) was also greater than the simulations was the case of the instantaneous drop, as used in the experiments.

5 HANCOCK AND WILLGOOSE: USE OF A LANDSCAPE SIMULATOR Ir... experiment -I N. '' ml = 1.0, n =2.1 n , - ß m =1.3,.;, m, =2.1, n, = '.,..... m, =2.5, n t 0.2 o o normalised area Figure 8. SIBERIA-simulated hypsometric curve for the one-dimensional catchment using a range of values for rn with n held constant. q¾= lqmls nl, (3) where Q is the discharge per unit width (m3/s - 1 m- 1 width), S (mm- ) is slope in the steepest downslope direction, and/3, rn, and n are parameters of the model. The rate constant controls the rate of erosion and can be scaled to match the observed erosion rate. The diffusive term q d (e.g., rain splash) is composed of qsa = OS, (4) where D (m 3 s- m- width) is Fickian diffusivity where transport is proportional to slope. The low kinetic energy of our rainfall simulator ensured that diffusive transport can be ignored. For our experiment the simplicity of the rainfall simulator means we can relate the discharge Q (m 3) to the area so that Q =/33A, (5) where A is specific area (m 2 m- width) and/33 is the runoff rate constant./33 can be varied spatially for nonuniform runoff over a catchment One-Dimensional Catchment To calibrate m and n, a small one-dimensional experimental catchment was used. A one-dimensional catchment has flow in only one direction with an absence of stream branches. A one-dimensional catchment is the simplest case of landform development. Calibration was performed by comparing hypsometric curves of the one-dimensional catchment [Willgoose and Hancock, 1998]. The calibration catchment was 1000 mm long by 100 mm wide and 300 mm deep with a slope of 10% that was uplifted in one instantaneous event. The outlet of the catchment was the full catchment width. To allow free draining, holes were drilled in the base of the box, and a 200 mm layer of beach sand was added. A 100 mm layer of the erodible material, fly ash, was then placed over the sand, filling the box completely, and the entire surface was smoothed. To expose the catchment to known and controlled rainfall conditions, the catchment was positioned within the larger e Calibration of SIBERIA 1.5 m x 1.5 m box. High-intensity rainfall was used, its distribution known from the overall calibration of the larger box. The erosion equation of SIBERIA consists of two terms, Before the commencement of erosion, the catchment was wetted fully by the simulated rainfall. Rainfall was commenced qs = q¾ + qsa, (2) and continued until landscape development had visually come where q s is the volumetric sediment transport rate per unit to completion after -20 hours. During the experiments, flow width (m 3 s - m - width), qsf is the fluvial sediment transport width equalled catchment width. Landscape maturity was also term, and q d is the diffusive transport term (both m 3 s- m- indicated by a low and relatively constant sediment output width). The fluvial sediment transport term q f is based on the from the catchment. Sediment output during the experiment Einstein-Brown model [Henderson, 1966] so that was measured Calibration Results Elevations of the final landform were measured by electronic level and a digital terrain map and a hypsometricurve produced. The hypsometri curve was then compared to SI- BERIA hypsometric curves generated by SIBERIA for the same size and slope catchment but developed over a range of values of rn, and n. Simulations used the single uplift case of the experiments. The values of rn and n in the SIBERIA sedimentransport equation were then adjusted (Figures 8 and 9) until the experimental and SIBERIA-predicted stream profiles [Willgooset al., 1991c] matched. These simulations provided a sensitivity analysis for the various values of rn and n. In this fashion the fluvial erosion equation was calibrated. The hypsometric curve was more accurate than using sediment outflow data, as the hypsometri curve is a measure of total sediment flux integrated over time taken over the entire catchment and is less sensitive to the rapid fluctuations observed in sediment output. Comparisons between SIBERIA simulations and field-scale 1 experiment landscapes have found that values of rn = 1.8 and n = 2.1 approximat erosive conditions in field catchments very well. 0.8,, -- m 1.62, n1=1.2 n 1.5 [Willgoose, 1994a]. These values have also been determined, '..... m 1 = 1.62, n = 1.8 mathematically and assume the surface to be a flat plain with... m = 1.62, n = 2.1 sheet flow conditions. The values of Willgoose [1994a] and 0.4 x x' ' x ' -- m = 1.62, n = 2.4 Kirkby [1971] provide a guide as to the suitability of values for c 0.2 rn and n for the experimentalandscape. However, as this was a model landscape, with unusual soil 0 ' "' ' ' (i.e., fly ash), m and n were calibrated, as it could not be normalised area assumed that any apriori values were appropriate. Calibration Figure 9. SIBERIA simulated hypsometricu e for the of rn and n was carried out separate of the main experiment one-dimensional catchment using a range of values for n with using the hypsometricurve [Willgoose and Hancock, 1998]. m held constant.

6 ,,, 1986 HANCOCK AND WILLGOOSE: USE OF A LANDSCAPE SIMULATOR 1 -- experiment N SIBERIA (spatially variable rainfall) 0.8 ll) experiment 150 ;,?,", :, -... SIBERIA.x ß ' normalised area Figure 10. Hypsometri curves from the steep-slopexperiment and SIBERIA simulation using calibrated spatially variable rainfall and spatially constant rainfall. 0 ' distance (pixels) Figure 11. Width functions from the steep-slop experiment and SIBERIA simulation using calibrated spatially variable rainfall. Values of m x = 1.62 and n x were found to produce similar and are not shown. These results indicate the impora curve that closely fitted the experimental data. Willgoose et al. tance of the rainfall runoff regime on the final landscape form. [1991c] demonstrated that the actual values of mx and n are The differences are clearly the result of the runoff regimes not important, but the value of a, expressed as a - (m x - eroding the landscape at different rates in different areas, 1)/n, is the important relationship for the case of declining resulting in landscapes with different shapes. These results equilibrium examined in this study. Willgoose and Hancock demonstrate that SIBERIA matches the experiment hypsom- [1998] demonstrated that the hypsometri curve for a twoetry satisfactorily and that the nonuniform rainfall results in dimensional catchment is also dependent on n x and the planar nontrivial deviations in the catchment hypsometry. All comgeometry not just the ratio of (mx - 1)/nx. The value of parisons that follow use the spatially variable rainfall and the mx is not greatly different than the value ofmx steep-slope catchment. derived mathematically by Willgoos et al. [1989] for widechannel flow and is close to the value of m x = 1.6 found 5.2. Width Function experimentally by Moore and Burch [1986]. The fitted value of m = 1.62 is also well within the range of m x = 1.45 to 1.71 Figure 11 shows the width function comparison. Width functions for the experiments and SIBERIA simulations were defound by Willgoose [1994a] for the Pokolbin field catchment. termined with all catchments constrained to flow through the The constant n x is the exponent on slope S and has been found by Willgooset al. [1989] to vary only slightly between channel single-node outlet. The major difference is the length of the width functions. In comparison, SIBERIA produced a catchgeometries. The value of n x = 2.1 found to be applicable in ment with fewer and straighter channels than the experiment, this case is also the value previously determined mathematiso that the distance from some point in the catchment is cally by Willgooset al. [1989] for wide-channel flow. It is also shorter for SIBERIA than the experiment. The experiment very close to the value of n x = 2 suggested by Henderson was also more highly branched, with SIBERIA generating [1966]. The amount of sediment leaving the catchment was highly long, straight catchments with less branching. These results are variable and was similar to other experimental geomorphologiqualitatively similar to those presented by Moglen and Bras [1994]. cal studies [Parker, 1977; Weaver, 1984]. During each experi- To test whether these differences were simply due to less ment, sediment was observed to exit the catchment in pulses. tortuous flow paths in the simulations and not to more funda- These pulses were caused by nickpoints moving headward. Consequently, measured sediment output depended on mental problems, such as differences in network statistics, Figure 12 shows the normalized width function. Width and length whether the sediment was sampled when the surface was stable have been normalized by dividing each width and length value or when a nickpoint was present. The collected sediment data from the calibration catchment was consistent with the paramby the maximum width and length, respectively. When normalized, the differences are greatly reduced, indicating that the eters calculated from the hypsometric curve, but the highly variable flux with time made it difficult to accurately calibrate m, and n directly from sediment output data. 5. Quantitative Comparison Between the Experiments and SIBERIA The hypsometricurve, width function, cumulative area diagram, and area-slope relationship were used for the comparison between the experimental model landscape and SIBERIA simulations. Visual or qualitative similarities are also presented. The results of the comparisons are discussed below ' 13.6,_ i 13.4 o experiment SIBERIA,Ir,¾ Hypsometric Curve Figure 10 displays the steep-slop experiment and simulation results. Results from the low-slope experiment were very normalised distance Figure 12. Normalized width functions for the steep-slope experiment and SIBERIA simulation.

7 .:..... :. :: HANCOCK AND WILLGOOSE: USE OF A LANDSCAPE SIMULATOR 1987 :Ix...:. :..::.... t, lllm. t.....:. ;..;...;...:...:. :...:... :.:..... ß....: :,...:...: : Figure 13. (a) Drainage ne ork of steep-slope SIBERIA simulated catchment. Outlet is on the right-hand side in the middle of the wall. (b) Drainage ne ork of steep-slope SIBERIA simulated catchment with digital photogrammetric error added to elevations. width function appears to be controlled by basin shape, as 5.3. Cumulative Area Distribution suggested by Rinaldo and Rodriguez-lturbe [1996]. Figure 16 shows the cumulative area distribution compari- To examine whether the tortuosity difference was due to son. The log-log linear region 2 of the simulations has a slightly photogrammetric errors in the experiment data, random digital lower slope than the experiment. This is typical of a less intense photogrammetric errors as measured for the experimental convergence than the experiment and is consistent with the landscape elevation data (standard deviation 2 mm) were photogrammetric errors discussed above. The slope of approxadded to the SIBERIA simulations at final steady state elevaimately -0.5 is consistent with field observations by other tions. Figure 13 displays the drainage network for both the authors. simulated catchment with and without photogrammetric error, The SIBERIA simulations demonstrate that the cumulative while Figure 14 displays the experimental catchment drainage network. The resulting width function (no normalization on area distribution has the same behavior despite large variations length was done, Figure 15) fits the data extremely well, sugin the deterministic form of the surface topography (see secgesting that the differences in Figure 11 are due to photogramtion 5.5). As with the width function result, the cumulative area metric errors. The length and width are about the same, with distribution is not dependent on the detailed deterministic major peaks at 70 and 140 pixels. Addition of photogrammetric features of the mature catchment topographic form but is error did not significantly change the other statistics used in the directly related to the catchment size and catchment geometry. steep slope comparison reported here [Hancock, 1997]. In the The cumulative area distribution also appears to be dependent flatter areas of the low-slope catchment, there was significant on the value of a in the area-slope relationship and is sensitive bias in the area-slope relationship introduced owing to the pit-filling of photogrammetric error, but otherwise results were to changes in its value [Moglen and Bras, 1994; Hancock, 1997]. A sensitivity study for different values of a was conducted, similar to those for the steep-slope catchment. and no difference was found in the cumulative area distribu-

8 1988 HANCOCK AND WILLGOOSE: USE OF A LANDSCAPE SIMULATOR (b) Figure 13. (continued) tion for approximately the same a using different values of rn and n (a = 0.24 and 0.26 for rn i = 1.5, n = 2.1 and rn = 1.62, n = 2.4), while a large difference was found in the slope of regions 1 and 2 for different values of a (a = 0.14 for rn = 1.3, n = 2.1 and a = 0.62 formx = 1.62, n = 1.0) [Hancock, 1997]. The cumulative area distribution is not dependent on the individual values of m and n but is dependent on the relationship between m and n, i.e., a. This result reinforces the importance of the correct determination of a for reliable simulations Area-Slope Relationship The straight-line steep-slopexperiment area-slope relationship (Figure 17) is consistent with a catchment dominated by fluvial processes. The small drop size of the rainfall simulator has successfully eliminated rain splash transport, otherwise a break from log-log linearity would be observed [Willgooset al., 1991c]. The SIBERIA simulation a value of is a close match to the experimental value of a There appears to be more scatter in the experimental data than the SIBERIA simulation. That this scatter results from measurement error was confirmed by the addition of measured random photogrammetric error to the simulated landscapelevations, as was done for the width function [Hancock, 1997]. By comparison with the high-slope catchment, the low-slope catchment also showed signs of photogrammetric error for the large area-low slope region of the curve. In the higher-slope regions, where photogrammetry errors were relatively less important, on the left-hand side of the curve, a value of a = was fitted to the experimental data. In the lower-slope regions of the low-slope catchment, a dropped to near zero in areas affected by pit filling caused by photogrammetric error. As a result, as for the high-slope catchment, addition of measured photogrammetric error to the low slope SIBERIA simulation yielded results that matched the photogrammetry data Qualitative or Visual Comparison Despite these good quantitative matches, aspects of the SI- BERIA-simulated landscape (Figure 18) were visually different from the experimentalandscape (Figure 19). The SIBE- RIA landscapes had regular hills with constant slope, whereas the experimental landscape had less regularity and slopes much more akin to natural curvature. An alternative flow di- rection algorithm was tried, but this provided little insight into the problem [Hancock, 1997]. SIBERIA has been developed for physical processes that operate on a much larger scale than that occurring in the experimentalandscape and ignores momentum terms in the flow equations. Despite these constraints

9 HANCOCK AND WILLGOOSE: USE OF A LANDSCAPE SIMULATOR 1989 Figure 14. Drainage network of the steep-slope experimental catchment. Outlet is on the right-hand side in the middle of the wall. for experimental-scale catchments we believe that the landscapes produced by SIBERIA are visually representative of what would occur for field-scale catchments. This is suggested by the ability of SIBERIA to correctly simulate both quantitatively and qualitatively (visually) field catchments at Tin Camp Creek, Northern Territory, Australia [Hancock and Willgoose, 1999; Hancock et al., 2001] and Scinto 6 postmining landscape [Hancock et al., 2000]. Discussion of the field assessments are outside the scope of this paper. 6. Conclusion This study has shown that SIBERIA is able to qualitatively and quantitatively match four key statistics of landscape geo [... [... [ experiment... SIBERIA [ experiment 104 IA 100 '" '", 5O ' - - loo distance (pixels) Figure 15. Width function from the steep-slope experiment and SIBERIA simulation with photogrammetric error added to the SIBERIA landscape surface. 25O 1,,,,,,I,,,,,,d,,...,I,, \ area (pixels) Figure 16. Cumulative area distributions from the steepslope experiment and SIBERIA simulation using calibrated spatially variable rainfall.

10 1990 HANCOCK AND WILLGOOSE: USE OF A LANDSCAPE SIMULATOR o experiment - SIBERIA -o :oo: I lo loo loo0 lo 4 area (pixels) Figure 17. Area-slope relationships from the steep-slope experiment and SIBERIA simulation using calibrated spatially variable rainfall. Circles and crosses are averaged values. The raw data were sorted from smallest to largest area, and then each 10 data points were averaged and their value was plotted. morphology: hypsometry, cumulative area distribution, width function, and the area-slope relationship. The results were independent predictions since the model was calibrated to runoff and erosion processes using independent data. Two of these statistics, the cumulative area diagram and hypsometric curve, are particularly strong validations, as they change significantly if the modeling of the physics is erroneous [Willgoose and Hancock, 1998; Perera and Willgoose, 1998]. They also exhibit only small random scatter, so that errors in the model can be distinguished from random fluctuations in these statis- tics. This study was conducted within the range of plausible values of m, n as suggested by previous work [Kirkby, 1971; Willgoose, 1994a] and, consequently, plausible values of a. We recognize that the individual values of m ] and n ] may not be unique but the correct value of a is essential for the correct modeling of the experimental model landscapes. Second, other researchers have found that the cumulative area distribution is sensitive to changes in a but not the values of m and n that produce the same a. This was demonstrated by Moglen and Bras [1994], who calibrated an erosion model on the cumulative area distribution. The cumulative area distribution is a particularly strong test of comparison as it is derived from catchment area only, and we have found that it is insensitive to digital terrain map elevation data quality at larger catchment areas [Walker and Willgoose, 1999]. At smaller catchment areas, the cumulative area distribution is sensitive to elevation data quality. Perera and Willgoose [1998] demonstrated that the slope of region 2 of the cumulative area distribution is linked to the catchment networking properties and the match here demonstrates that we are closely matching drainage aggregation characteristics. The variation in the hypsometricurve produced by spatially variable rainfall is an important finding for large-scale (thousands of square kilometers) and long-term landscape simulation (hundreds of thousands of years). This work demonstrates that the presence of a rain shadow or orographically derived rainfall in a large catchment is likely to have a large influence on landscape development. We speculate that changing mean rainfall patterns with time, such as might be generated by the greenhouseffect, could modify landscape shape. While the width function from SIBERIA approximated the experimental shape, it was significantly shorter in distance and taller in width, consistent with a shorter, more highly branched flow pattern than that of the experimental catchments. This is consistent with the findings of Moglen and Bras [1994]. It is likely that much of the detailed difference is due to errors in the digital terrain map of the experimentalandscape, in this case derived from digital photogrammetry. This is likely to be true of other digital terrain map (DTM) data derived from digital photogrammetry, such as Walker and Willgoose [1999]. We thus believe that the claim that the width function is not very sensitive to data quality [Naden, 1992] is only true for small errors associated with traditional maps derived from cartographic photogrammetry and not for the larger errors associated with digital photogrammetry. Gyasi-Agyei et al. [1995] demonstrated that high-quality data are needed for the reliable extraction of the width function from DTMs. The width function, when nondimensionalized by the maximum length and width, was well matched. It is likely that much of the detailed differences between the curves of the experi- 24O 2OO mm 8O mm 1800'% % oo Figure 18. SIBERIA simulation of steep-slop experimentalandscape after 2400 min of erosion. 4O 0

11 _. HANCOCK AND WILLGOOSE: USE OF A LANDSCAPE SIMULATOR mm Figure 19. Digital terrain map of steep-slope experimental landscape after 2400 min of rainfall. ment and simulation is due to random noise. Allowing for the photogrammetric error improved the match between the experiments and the simulations. To date, we have been unable to provide a causal physicalinkage between the width function and changes in the physical processes in SIBERIA, so it is not possible at this stage to say whether it is capable of rejecting a poor model. Rinaldo and Rodriguez-Iturbe [1996] suggested that the width function was controlled by basin shape, and we are currently investigating this issue [Hancock and Willgoose, we reliably capture the experimental errors resulting from pho- 1999]. The area-slope curve was well matched. This is a weak test of the internal physics of the model because there is a direct togrammetry, variation in erodibility, etc.? Nevertheless, there is a need to develop methods for evaluation of error bounds that would allow hypotheses testing and thus a statistically link to the erosion processes [Willgooset al., 1991c; Willgoose, defensible methodology for rejecting an incorrect geomorphol- 1994a] and this link was used for calibration of the erosion physics from other independent and functionally different experiments. Thus the results signify, in part, at least, that the calibration of the runoff and erosion processes was, in fact, ogy model [Hancock and Willgoose, 1999]. The authors are also continuing this work using natural and postmining landscapes over a range of time scales [Hancock et al., 2000, 2001; Ibbitt et al., 1999]. performed correctly. In some experiments, characteristics consistent with photogrammetric errors were observed, and allowance for this improved the match in these statistics, without Acknowledgments. This project was supported by the Land and degrading the match in the other statistics. Water Resources Research and Development Corporation The question may be asked, what does it matter if the landscapes do not look exactly the same (Figures 18 and 19), if (LWRRDC) grant UNC2, and the Research Management Committee Scholarship from University of Newcastle. The assistance of J. Fryer, geomorphologically and hydrologically, the simulated landscape is identical to the experimental landscape? What is a minimal set of essential descriptors of catchment geomorphology? A match is provided to the hypsometric curve, width function, cumulative area distribution, and area-slope relationship. The recent analytic solution for runoff generation provided by Perera [1998] and Willgoose and Perera [2001], dependent on area-slope and the cumulative area distribution suggests that the runoff generation hydrology will be well matched. The possible importance of other geomorphology/ hydrological statistics is an issue that invites further research. No doubt, different applications of the model will highlight different statistics discussed here. While the results demonstrated here are not for field-scale catchments, they provide an important first step in testing long-term evolution models in a controlled environment. The authors firmly believe that the quantitative methodology out- lined in this paper is an advance from qualitative visual comparisons and the limitations of using field data. Further evaluation of models needs to be continued using a combination of both field and experimental data. The difficulty with using experimental data is that we cannot get multiple realizations of the experimental catchment because of the difficulty of the experimental setup and surface measurement, so we are forced to use Monte Carlo simulations of the model. How, then, do B. King, and especially, E. Kneist in the manual and digital photogrammetric analysis is gratefully appreciated. References Flint, J. J., Stream gradient as a function of order, magnitude and discharge, Water Resour. Res., 10, , Gupta, V. K., and E. Waymire, On the formation of an analytic approach to hydrologic response and similarity at the basin scale, J. Hydrol., 65, , Gyasi-Agyei, Y., G. Willgoose, and F. P. De Troch, Effects of vertical resolution and map scale of digital elevation models on geomorphological parameters used in hydrology, Hydrol. Proc., 9, , Hack, J. T., Studies of longitudinal stream profiles in Virginia and Maryland, U.S. Geol. Surv. Prof. Pap., 292(B), 45-97, Hancock, G. R., Experimental testing of the SIBERIA landscape evolution model, Ph.D. dissertation, University of Newcastle, New South Wales, Australia, Hancock, G. R., and G. R. Willgoose, The sensitivity of geomorphic

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