Impact of DEM Resolution and Spatial Scale : Analysis of Influence Factors and Parameters on Physically Based Distributed Model

Physically based distributed hydrological models were used to describe small-scale hydrological information in detail. However, the sensitivity of themodel to spatially varied parameters and inputs limits the accuracy for application. In this paper, relevant influence factors and sensitive parameters were analyzed to solve this problem. First, a set of digital elevation model (DEM) resolutions and channel thresholds were generated to extract the hydrological influence factors. Second, a numerical relationship between sensitive parameters and influence factors was established to define parameters reasonably. Next, the topographic index (TI) was computed to study the similarity. At last, simulation results were analyzed in two different ways: (1) to observe the change regularity of influence factors and sensitive parameters through the variation of DEM resolutions and channel thresholds and (2) to compare the simulation accuracy of the nested catchment, particularly in the subcatchments and interior grids. Increasing the grid size from 250m to 1000m, the TI increased from 9.08 to 11.16 and the Nash-Sutcliffe efficiency (NSE) decreased from 0.77 to 0.75. Utilizing the parameters calculated by the established relationship, the simulation results show the same NSE in the outlet and a better NSE in the simple subcatchment than the calculated interior grids.


Introduction
Digital Elevation Model (DEM) data contains abundant topography, geomorphology, and hydrology information, which accelerated the development of physically based distributed hydrological model.As mature software such as ArcGIS and WMS has been completed to extract the information regarding slope, flow direction, and concentration, the spatial relationship between dynamic mechanics of water cycle and surrounding units could be taken into account.Based on conservation of mass, momentum, and energy, as well as catchment runoff and concentration characteristics, these components were deduced by numerical analysis method which makes the physically based distributed hydrological model deemed as a kind of model of high precision, scientificity, and effectivity [1,2].
Based on the assumption of some hydrological conditions, hydrological model is established on a certain spatial and temporal scale.It is significant to derive and demonstrate the hydrological variables and change factors on different scales [3][4][5][6][7].Higher DEM resolution provides a more accurate representation of topographic features [8], which makes the data on the small scale more precise and representative.However, the representativeness of small-scale data is doubted when used in large scale [9].When largerscale data are converted to small-scale data, there are some critical problems to be considered, such as the heterogeneity and the nonlinear response [10].Therefore, a complete system of theory and methodology should be constructed [11].Researchers focused on the scales problem fell largely into two different perspectives.The first perspective, which is proposed by Beven [12], is that two different hydrological models are still needed to solve the problem.One is used to describe the hydrological processes on a small catchment scale in detail while the other one is used to make a hydrological forecast on a larger catchment scale.Beven believes that only one hydrological model could not solve the scales problem and the scale dependence of distributed hydrological model must be introduced [13].The second perspective, which is proposed by Blöschl [14], indicates that there is a gradual process of solving the scales problem and eventually there will be significant achievements in theoretic and practical hydrology.The controlling equations of the model are established based on a point scale, which is deemed effective for different catchment scales, without taking into consideration how the model parameters are related to various scales.In practice, current techniques on hydrological measurement have limited the accuracy of observed parameters, and the accuracy of the measured values is much lower than the required unit scale of the model.
The recently developed distributed hydrological models are all physically based and DEM-based, with distributed parameters, input information, process description and results outputs, and hydraulic calculation method [15], which leads to a complex process to calibrate parameters and requires a long time to calculate the flow.The simulation results are strongly influenced by diverse factors, such as topographic characteristics, surface delineation methods, and DEM interpolation/aggregation methods.Generally, lower DEM resolution (or larger grid size) is generated from original higher DEM resolution by using certain interpolation/aggregation methods [16].However, DEM processing could change the accuracy of input information [17,18], and parameter calibration could give rise to serious equifinality with different parameters, which leads to an unreliable simulation in interior grids.However, in order to use the model in the study area, DEM processing, parameter calibration, and validation [19] are highly recommended and extremely necessary.
This research was focused on sensitivity analysis [20,21] of influence factors and parameters, and a numerical relationship between influence factors and sensitive parameters has been established.The selected relationship was used to improve parameter calibration efficiency, avoid overparameterization, and generate more accurate model outputs [22][23][24].The aim of the study is to enlarge the applicable scope for physically based distributed hydrological model, improve the simulation accuracy for interior grids, decrease the error caused by low DEM resolution, and provide a basis for parameter transformation.The research has several major objectives as follows.First, we hope to analyze the variations of hydrological information depending on different DEM resolutions, channel thresholds, and catchment scales.Second, we want to develop a numerical relationship between influence factors and sensitive parameters when the influence factors of runoff and concentration parameters are considered comprehensively.Thirdly, we want to make sure that the simulation results with different DEM resolutions and channel thresholds can achieve the same accuracy when the parameters are given by the established relationship.The ultimate objective of this research is to apply the calibrated parameters to improve simulation accuracy in interior grids.

Overview of CASC2D Model.
The CASCade 2 Dimensional (CASC2D) originally begins with a two-dimensional overland flow routing algorithm which was developed and written in APL by Professor Julien at Colorado State University [25][26][27].CASC2D has been developed to determine the runoff hydrograph generated from any temporally spatially varied rainfall event.For a given rainfall event, CASC2D model ignores the evapotranspiration during the rain and relies on inverse square distance weighting method for each grid to describe a fully spatially varied rainfall input [28], and once the initial soil moisture deficit has been supplied by rainfall, water begins to infiltrate.This step requires the adoption of an infiltration scheme that can predict the portion of the rainfall that drains into the ground.The Green & Ampt [29,30] infiltration equations accommodate spatial and temporal variabilities due to changes in rainfall and soils properties and take into account the accumulated infiltration.Using a Hortonian overland flow process, when the precipitation rate exceeds the infiltration rate, the excess rainfall will accumulate as surface water and begin to flow.Overland flow is routed into the channels using a 2D diffusive wave equation (( 1)-( 4)).In channels, the water is routed using a 1D diffusive wave equation [31] (( 5)-( 6)).
where,   0 is friction-resistance gradient of slope concentration in the and -direction, respectively;  0 is the gradient of slope concentration in the and -direction, respectively; ℎ/(, ) is an additional gradient of slope concentration in the and -direction, respectively.
where  is the discharge per unit width; ℎ represents the depth of surface runoff;  and  depend on the flow state and the turbulent flow.
where   0 /|  0 | determines the flow direction and  0 represents the Manning roughness coefficient of slope concentration.
where  1 is the Manning roughness coefficient of the channel,  is the hydraulic radius,   1 is the friction-resistance gradient of the channel, and  is the cross-sectional area.
where  1 is the gradient of the channel and Δℎ/Δ is an additional gradient.CASC2D is built on finite difference and finite volume numerical schemes and thus operates upon a uniform grid.This gridded approach allows spatial variability in watershed characteristics to be distributed across an entire basin at user-selected grid sizes [32].Small grid sizes are used when the physically based parameters can be accurately observed and spatially interpolated.However, larger grid sizes may be preferred because the spatial variability of catchment characteristics is not explicit and computational efficiency is an important issue [33].Actually, it is unrealistic to use small grid sizes in very large catchment or large numbers of events.Within a given catchment, input information (e.g., catchment area, stream links, soil distribution, and vegetation distribution) is defined through ArcGIS processing of DEM data.These geometric characteristics are used as the hydrological model components of CASC2D.

Statistics of Influence Factors.
The input data processing would cause some uncertainty in the process of DEM resolution conversion and drainage extraction by ArcGIS.On a statistical-information basis, some hydrological factors were calculated to analyze the sensitivity.CASC2D model divides the concentration process into two sections; one is slope concentration, and another is channel concentration.So the traditional concentration factors should be considered more specifically.This section describes the change of statistic and calculated hydrological factors after DEM and ArcGIS processing.The related hydrological factors are (1) the precipitation station number in the study area and the controlled area for every station; (2) total grids in stream and the ratio of catchment area and total grids in stream; (3) average slope for the study catchment in different DEM resolutions; (4) the slope concentration route distance  0 and the average concentration slope   0 for every DEM resolution and channel threshold calculated by D8 (deterministic eight neighbors) algorithm; and (5) the permanent main channel length  1 and the permanent main channel gradient   1 for every DEM resolution and channel threshold.Based on the original data and processed data, the major influence factors for nested catchment under different DEM resolutions and channel thresholds are calculated and summarized (Table 2).

Sensitive Factors of the Model.
CASC2D model is a physically based model with physical-equations-controlled runoff process and concentration process.The calibrated parameters are analyzed to improve relationships between model parameters and hydrological characteristics (i.e., precipitation, area, soils, vegetation, and topographic features).Researchers have found out that the average slope of the catchment is reduced along with the decrease of resolution [34][35][36][37].However, the average slope of catchments could not fully represent the concentration process.So the average concentration slope and concentration route distance were introduced for analyzing slope and channel concentration process.The concentration route distance, the shape and length of stream links, and the slope are determined by DEM resolution and channel threshold.
The relationship among the Manning roughness, the average concentration slope, and the concentration route distance can be established according to the physically based equations ( 1)-( 5).This section was used to study the numerical relationships between different DEM resolutions and catchment scales, so the specific ratio should be considered in the following section: (a)  ,  0 (unit area), and  1 (unit area) in nested catchment consisting of Dongwan, Tantou, and Luanchuan.The numerical relationship of  0 and  1 with different DEM resolutions, channel thresholds, and catchment scales can be established according to the calculated ratio (Table 3).
2.6.Topographic Index.Topographic Index (TI) [38] refers to the spatial distribution of soil moisture deficiency and runoff process (6).It describes the cumulative trend of runoff in every grid and the slope concentration trend under the effect of gravity [39].TI of the study catchment can be calculated (Figures 3-4) according to the multiple flow direction algorithm (MFD).As to nested catchment, Dongwan is 11.55, Tantou is 11.44, and Luanchuan is 11.16, which are similar in numerical value and distribution curve (Figure 3).However, they are 11.16, 10.02, and 9.08 for Luanchuan catchment with the DEM resolutions of 1000 m, 500 m, and 250 m, which are quite different in both numerical value and distribution curve (Figure 4).
where  represents the single wide catchment area, and tan  represents the local surface slope.

Model Parameter Estimation Experiment.
Model calibration and validation provide a nominal flow simulation for 13 typical events spanning the period of 1962 to 1998.Selection of events was based on the existence of a flow response at the outlet and obtaining a set of events with a range of peak flow values.In Luanchuan catchment, three resolutions of DEM, which are 250 m, 500 m, and 1,000 m respectively, and This research regards the nested catchment as highly similar catchments; DEM resolution of 1000 m and channel threshold of 35 are selected to extract catchment diagram.First 8 typical events are selected to calibrate the parameters and 10 events are selected for validation.To compare the

Impact of DEM Resolution and Channel Threshold.
According to statistical information of the Luanchuan catchment, the resolution has a significant influence on the grid number and channel threshold has an impact on the length and shape of the drainage line.A smaller channel threshold shows a complex drainage line; on the contrary, a larger channel threshold shows a simple one.The area calculation and spatial distribution of precipitation influenced by DEM resolution can be ignored.The average concentration slope has positive impacts on velocity; the Manning roughness has an adverse effect on velocity, and the route distance is positively related to velocity.−32%, −36%, and −38% (Figures 5-6).The simulation result with the same runoff parameters and numerical-relationshipbased concentration parameters shows the same accuracy.

Development of Nested Catchment.
For the sensitivity analysis of nested catchment, about 18 typical events were selected for flood simulation and comparison.The simulated runoff parameters and the calculated concentration parameters are shown in Table 6; the simulation and comparison  The model performs best in the Luanchuan and worst in the Dongwan in terms of NSE coefficient.Variation of NSE is greatly different in every flood event.The NES of  1 (from 0.44 to 0.92),  1 (from 0.52 to 0.91), and  1 (from 0.45 to 0.93) are much better than  2 (from 0.42 to 0.90),  2 (from 0.51 to 0.85), and  2 (from 0.42 to 0.90).The average NSE coefficient in  1 (0.70) is the same as  2 (0.70), and  1 (0.72) and  1 (0.75) are larger than  2 (0.70) and  2 (0.72).The variation

Discussion
CASC2D model is a catchment-based hydrological model with distributed inputs and distributed parameters, which was used for accurate hydrological information simulation in small scale catchment with high DEM resolution [27,32].This study investigated the influence factors and sensitive parameters to improve the accuracy and efficiency of physically based distributed hydrological model.Parameter analysis shows that   , , and   varied slightly in different resolutions, channel thresholds, and catchment scales when they were used as runoff parameters. 0 ,  1 ,   0 , and   1 played a dominant role in controlling the  0 and  1 .
Input information (i.e., precipitation, soil, vegetation, and other underlying surface conditions) which is influenced by DEM resolution and channel threshold plays an important role in controlling the simulation process.The inverse square distance weighting is an excellent method to interpolate the rainfall in spatial scales [40,41].However, it is still difficult to describe the uneven distribution of rainfall in temporal scales, which makes the Green-Ampt equation and its parameters inefficient when an hourly interval rainfall data is used.There is a precipitation station for every 170 km 2 in Luanchuan, 307 km 2 in Tantou, and 321 km 2 in Dongwan, which makes the NES coefficient and relative error of peak discharge the best in Luanchuan and the worst in Dongwan.It highlights the significance of raw data and interpolation methods both in spatial and in time scales.
Lower DEM resolution would limit the attributes of each grid [42] and further restrict the simulation of hydrological attributes in interior grids.TI was greatly influenced by the resolution and information content of a DEM [43], and it was regarded as an important index to describe the rainfall-runoff process.Calculated TI shows a great difference in Luanchuan and the same in the nested catchment, which means the similarity should be established based on the same DEM resolution.On the one hand, with 1,000 m DEM resolution in Tantou and Luanchuan, the simulation results are superior to the calculated interior grids when parameters are reasonably calibrated.These simulation results indicate the uncertainty of interior grids in distributed hydrological model.The input information, parameters, and physical equations need to be more reasonable with the change of scale and position.On the other hand, Dongwan always served as an outlet, no matter when simulated in the nested catchment or as a simple catchment.The calibrated parameters are different and the simulation results are revealed to be the same, which reveals a phenomenon of equifinality for different parameters.
According to the statistical hydrological information, the concentration parameters could be given reasonably, which could offset the error caused by DEM processing and improve the simulation accuracy in interior grids.Sensitivity analysis and scale change regulation of parameters are necessary and helpful for parameter transplantation and accuracy improvement with parameters reasonably given.
The physically based model links the laboratory scales and field application scales, which provides a basis to calibrate parameters.However, it is still difficult to use the measured parameters in hydrological model.So parameter calibration and validation are necessary and urgently needed.The statistics of influence factors and sensitive parameters provide the variations of runoff process as well as concentration parameters.In order to establish a more accurate relationship in different scales, more data regarding rainfall, discharge, groundwater distribution, accurate DEM, soil structure, and vegetation are needed.

Figure 2 :
Figure 2: The catchment and drainage networks of Luanchuan, (a) resolution of 250 m, channel threshold of 60, (b) resolution of 500 m, channel threshold of 15, (c) resolution of 1000 m, channel threshold of 35, and (d) resolution of 1000 m, channel threshold of 7.5.

Figure 3 :Figure 4 :
Figure 3: TI of Luanchuan catchment in the DEM resolution of 250 m, 500 m, and 1000 m.

Figure 5 :1000Figure 6 :
Figure 5: Comparison of the results under the DEM resolution of 250 m, 500 m, and 1000 m.

Figure 8 :
Figure 8: Comparison of the results of Tantou.

Table 1 :
Information and statistics of the nested catchments.

Table 3 :
Relations among concentration parameters.

Table 4 :
Parameters and initial values of Luanchuan catchment.

Table 4 ,
and the concentration parameters of Tantou and Dongwan catchment should be given according to the numerical relationship shown in Table3; next, runoff parameters of Tantou and Dongwan are calibrated based on the observed stream flow; (b) Dongwan was considered as a simple catchment, while Tantou and Luanchuan were set as two interior grids.The model parameters of Dongwan were directly calibrated and the simulated flood process of the two interior grids was extracted.
In Luanchuan catchment, channel threshold has a significant influence on  0 (634.82km to 1289.79 km) and   0 (0.124 km to 0.153 km); resolution has a major influence on   0 (0.124 km to 0.185 km) and  1 (29.64 km to 39.78 km); with the decreasing of DEM resolution,   0 and  1 decrease;   1 is relatively stable when DEM resolution and channel threshold changed.As to the nested catchment, major influence factors are  0 (1289.79km to 11010.55 km) and  1 (27.75 km to 109.31 km).The unit area ratio of  0 for Dongwan to Luanchuan and Tantou to Luanchuan is near 1.3.2.Simulation Results.The slope manning roughness  0 and channel manning roughness  1 under different resolutions and channel thresholds were given according to the calculated numerical relationship.The precipitation interception coefficient, saturated hydraulic conductivity   , saturated head capillary , and initial soil deficit   were calibrated based on observed floods (Table4).With the decrease of DEM resolution the slope Manning roughness decreases and the channel Manning roughness increases.According to the application in Luanchuan catchment with DEM resolution of 250 m, 500 m, 1000 m, and channel threshold of 35 and 7.5 under the resolution of 1000 m; NSE coefficients are 0.77, 0.76, 0.75, and 0.75; relative errors of peak discharge are −33%,

Table 5 :
Comparison results under different DEM resolutions and channel thresholds.

Table 6 :
Parameters and initial values of nested catchment.Figure 7: Comparison of the results of Dongwan.results are shown in Figures 7-9.Table 7 lists NSE coefficient and relative error of peak discharge on two different conditions, where  1 ,  1 , and  1 are the simulation results of Dongwan, Tantou, and Luanchuan which are considered as nested catchment;  2 is the simulation results of Dongwan;  2 and  2 are calculated results of Tantou and Luanchuan which are considered as interior grids.