Abstract In this study， noncumulative slope length （NCSL） calculation method and spatial analytical calculation （SAC） method were respectively applied to extract slope length and slope length factor from 10 sample areas， which are located in Ansai County， north Shaanxi Province. The comparison of computation precision between variable DEM resolutions showed that NCSL was superior to SAC entirely. And the results were best when the DEM resolutions were 5 and 10 m. Besides， the results of slope length factor were nearly the same under the two conditions. So DEM of 10 m resolution can be used to extract slope length.
　　 Key words DEM; Resolution; Slope length; Precision differentiation; Soil erosion model
　　
　　 Under the intervention of human unreasonable production activities in the context of rapid urbanization， the intensity of soil erosionis increasing. The severe soil erosion caused by this not only causes the decline of soil fertility and the loss of land productivity on a global scale， but also results in the elevation of the riverbedsof rivers and the reduction of the reservoir capacity and even the retirement， which further aggravate the occurrence of floods and other disasters. As a result， soil erosion has become one of the major environmental problems of human survival and sustainable development.
　　 Soil Erosion Model and Slope Length Extraction
　　 Progress in soil erosion model
　　In order to better grasp the laws and mechanisms of soil erosion and reduce the disasters and losses caused by severe soil erosion， scientists from all over the world have carried out a lot of research work in this field. Establishing a quantitative model is one of the important means of studying soil erosion. Wischmeier et al.[1]proposed the following general soil loss equation USLE （Universal Soil Loss Equation）：
　　A=R×K×LS×C×P （1）
　　The various factors in the equation represent rainfall erosivity， soil erodibility， slope length， slope， coverage and management， and soil and water conservation measures， respectively. After the equation was published， scholars from all over the world continually revised and improved various factors in above equation. These improvements were collectively referred to as RUSLE （Revised Universal Soil Loss Equation）[2]. Based on the above ideas， many countries in the world have successively established soil erosion models that are more suitable for their own country. For example， Chinese scholar Liu et al.[3]proposed the soil loss prediction equation in China， and changed the coverage and management factor and soil and water conservation measures in the aboveequation into such three factor as soil and water conservation biological measure （B）， engineering measure （E） and farming measure （T）： 　　A=R×K×LS×B×E×T （2）
　　RUSLE is currently the most widely used experience model for water erosion prediction. In recent years， there have been new progresses in this field. It is particularly worth mentioning that the emergence of physical models based on soil erosion mechanism process （water erosion and wind erosion） has opened up a new era of quantitative research on soil erosion.
　　The slope length is defined as the distance from the origin of the surface runoff to the point where the slope decreases until there is deposition， or the horizontal distance to a distinct channel， which is about the definition of a complete slope[1-3]. However， in most cases， the actual slope morphology is very complicated， so the definition of the above complete slope length is no longer applicable. To this end， Foster et al.[1，3]proposed a method for segmenting irregular slopes. This method considers the length of each section as the accumulation of the slope lengths of the upstream sections. According to this， Chinese scholar， Tang Guoan， defines the slope length as the projection of the maximum distance alongthe water flow direction from a point on the ground to the source point on the horizontal plane[4-6]. This definition is based on the theoretical basis for the automatic extraction of DEM slope length. In the abovementioned general soil loss equation USLE and the improved general soil loss equation RUSLE， the slope length appears as a slope length factor to describe the relationship between slope length and soil erosion. In RUSLE， the slope length factor is calculated as：
　　L=（λ/22.13）m （3）
　　Wherein λ is the horizontal projection distance of the actual slope; 22.13 is the horizontal projection length of the normalized slope length， the unit of which is m; and m is the variable slope length factor index， the value of which is related to the ratio of rill erosion and interrill erosion. For irregular slopes and segmented slopes， people do segmentation and treat the terrain features in each segment as uniform. The slope length factor corresponding to the ithsegment from the top of the slope is calculated as：
　　Li= λm+1i-λm+1i-1（λi-λi-1）22.13m （4）
　　λi in the equation is the length from the bottom of the ithsegment to the upstream boundary. This method takes the complexity of the slope into account and provides a basis for the quantitation of the slope length factor. If the bidimensionality of the water flow is taken into account， the runoff and soil erosion at a point on the slope are not completely dependent on the slope length value， but on the unit catchment area of the catchment at that point. This view is more representative of the dispersion and collection of runoff， and its hydrological significance is more obvious. Next， Desmet et al.[1，7-9]proposed that the slope length factor in the soil erosionmodel should be converted from the slope length to the unit catchment area in twodimensional conditions， such as the upstream catchment area of the unit contour length. Based on the aboveideas， Krikly et al.[1，3]improved the calculation method of terrain factor in USLE and obtained a new algorithm based on DEM automatic extraction. The algorithm replaces the slope length λ with the unit catchment area A： 　　In practical applications， people use GRASSGIS to calculate the slope length factor. Some people in China have used the algorithm to find the slope length factor， which was then substituted into formula 1 to invert the actual slope length. The results show that this method can only reflect the slope length differentiation law in the whole area， and the extraction results of slope length has low precision[10-13].
　　 Comparison of four slope length extraction algorithms
　　For GISbased terrain factor extraction， many factors such as slope and terrain are often studied， but its application in slope length extraction is not very mature. The most widely used GISbased slope length extraction algorithms are described below.
　　The first is the direct calculation method based on noncumulative flow （noncumulative slope length， NCSL）. The algorithm is based on the assumption that the direction of the water flow is consistent with the direction of the maximum slope. The main principle is the "Baliantong" rule. Accordingly， the water starts from the pixel of the source （such as the ridgeline， the mountain peak， etc.）， flows toward the pixel of the maximum slope direction， and finally gives the water flow path， from which the slope length is thereby obtained. The algorithm is characterized by simplealgorithm and high efficiency. Especially for the depressions and flat areas， it has a strong processing capacity. However， the algorithm considers that the direction of water flow is consistent with the maximum slope， while field experiments show that this is notcompletely consistent with the actual terrain and water flow[12].
　　The second is the specific contribution area （SCA） calculation method based on cumulative flow. The algorithm considers that the water flow distribution has a decentralized nature. The basic idea is to use the multiple flow direction algorithm to obtain the unit catchment area， to thereby replace the slope length factor in the equation. Since the catchment area is obtained by the flow accumulation method in the calculation， the algorithm is called the slopelength calculation method based on flow accumulation. The algorithm solves the problem of water flow in the first method， but the algorithm design is extremely complicated， and many specialcases must be considered. Therefore， although the algorithm can accurately calculate the catchment area， the program is complicated and has not high operational reliability and limited application[11]. 　　The third is the indirect calculation method based on stream power index. This algorithm is based on the slope length based on the flow index， rather than the slope length in the direct physical sense. This algorithm calculates the composite factor of the slope length and slope （LS factor in USLE）， and considers that the LS factor is a measure of surface runoff sediment transport capacity， thus interpreting the calculation of the LS factor representing the shape of the ground surface as the calculation of the index of the dimensionless sediment transport capacity having a nonlinear functional relationship with flow and slope. It is worth noting that the slope length obtained by the indirect calculation method based on the stream power index is not the physical slope length in terrain. The algorithm is less used in the field of terrain， but more in soil and water conservation and hydrology[13].
　　The fourth is the spatial extraction algorithm. In addition to the above algorithms that need to be programmed， the hydrological analysis function of GIS software has a direct slope length extraction algorithm， which we call the spatial analysis extraction method. The general principle is that if the slope approximately satisfies the condition that the slope flow direction is perpendicular to the ridgeline， then the horizontal distance of each point along the vertical direction to the ridgeline can be calculated as an approximation of the slope length of the point. Compared with the above three extraction methods， this algorithm is relatively simple. The first step is to use the negative terrain method to extract the ridgeline of the study area. After extracting the ridgeline， the vertical distance of the grid unit to the nearest neighbor ridgeline is obtained， which is approximately the slope length. Because the method is easy to operate and fast in calculation， it is also called a fast calculation method. The advantage of the fast calculation method is that the calculation is quick and the operation is simple. The disadvantage is that its scope of application is relatively narrow， only limited to the landforms where the contour lines are roughly parallel to the ridges （such as plain and ridge）， or the slopes which are relatively smooth and gentle， and the computation precision is relatively not high[5，12]. The above four algorithms are the automatic slope length extraction algorithms that are currently more commonly used and relatively mature， and are widely used in water conservancy， land， transportation， environmental protection and agriculture departments. 　　In summary， the slope， slope length and aspect are the three most basic topographical features of the slope. However， compared with other topographical factors such as slope， aspect， gully density， slope curvature， etc.， the definition of slope length is still controversial. The research on slope length and slope length factor is not deep enough. The existing different extraction algorithms also have their own advantages and disadvantages， but there is no clear limitation and detailed discussion on these advantages and disadvantages and the adaptability of methods. The generated slope length often has large error and the precision is not well controlled. In addition， for the GISbased extraction algorithm， there is no research on the difference in the precision of the extraction results due to the difference in DEM resolution under highresolution DEM （Digital Elevation Model） （<20 m）. With a topographic map of a large enough scale as the source data which are digitized， the powerful GIS software can easily extract enough accurate DEM. The ideal resolution of DEM extracted from a 1∶ 10 000topographic map is currently known to be 5 m[4-6]. In the case where the precision of the source data is guaranteed， the higher the resolution of the DEM， the higher the degree of coincidence with the real terrain， and the better the extraction effect of the corresponding terrain features. However， using the program to extract the slope length from a highresolution DEM means high time consumption and low efficiency， which are particularly evident when dealing with massive data. Under a highresolution DEM， does its height have a significant effect on the extraction of the slope length and the precision of the calculation of the slope length factor？ There are no studies on this issue that give definitive conclusions.
　　 Generation Situation of the Research Area and Research Methods
　　 Generation situation of the research area
　　The research area selected in this paper is located in Ansai County， Yanan City， Shaanxi Province （Fig. 1）. The county is located in the core area of the Loess Plateau in northern Shaanxi， the southern margin of the Ordos Basin， as well as the Yanhe River Basin of the first tributary of the Yellow River. The county has an area of 2 950 km2. The types of regional geomorphology are complex and diverse. The gully density is 4.7 km/km2， and the average elevation is 1 371.9 m. The whole area is covered by loess having a large thickness， which is basically Holocene loess. This area has a midtemperate continental semiarid monsoon climate， with an average annual temperature of 9.0 ℃ and an average annual precipitation of about 500 mm. The soil and water loss is severe， and longterm observation data show that the average annual erosion sediment transport modulus of this area is as high as 9 370 t/（km2·a）， which means that the natural conditions are bad and the ecological environment is deteriorating. 　　 Feng KONG. Comparison of Slope Length Factor Extraction in Hillslope Soil Erosion Model with Different DEM Resolutions
　　 Data source
　　The data in this paper mainly involved the selection of sample areas and the generation of DEM from a topographic map with a large scale of 1∶ 10 000. The quality of sample areas is directly related to the reliability of the research results， and the selection of sample areas is based on the scientific principle， representative principle and comprehensive information principle. Based on the above principles and considering the actual situation of this study， a total of 10 typical watershed sample areas were selected. The coverage area of these sample areas was between 0.5 and 1.0 km2. The vector topographic maps of 1∶ 10 000 scale of these sample areas were used as the source data （Table 1）. The ideal resolution of a DEM generated from a 1∶ 10 000 topographic map as source data is 5 m， which is the upper limit of resolution. Yang Qinke et al. （2009） extracted the slope lengths based on DEMs of 10 and 50 m resolutions in such three sample areas as Yulin， Yanan and Qinling， and the results showed that the extraction results under the low resolution were nearly four times higher than those under the high resolution. It can be seen that for the slope length extraction， especially the slope length extraction of small watersheds， the 50 m resolution is too low， and the extraction results are inevitably to have a large error. In order to explore the difference in calculation results caused by the difference in DEM resolution， this study used 5 m resolution as the upper limit and 20 m resolution as the lower limit， and selected a resolution value every 5 m， so four groups were selected in total. Four groups of DEMs with different resolutions from high to low were generated for each test sample area， and then the slope length was extracted by the algorithm to analyze the results.
　　 Research methods
　　In this paper， two different slope length extraction algorithms were used to calculate the slope length and slope length factor of a corresponding sample area under a corresponding resolution with DEMs of different resolutions of several representative small watershed test areas as the source data， and then， statistical analysis methods were used to process the result data， to obtain corresponding indexes. Next， the relationship between DEM resolution and extraction precision was determined， and the causes of the error were analyzed. Considering the complexity of algorithm implementation and the limitation of thesis length， this paper used noncumulative slope length （NCSL） method and spatial analysis extraction method （fast calculation method） to extract slope length. If the resolution of DEM has no significant effect on the precision of the extraction result， the calculation efficiency can be improved by reducing the DEM resolution when ensuring the precision， which is meaningful when dealing with massive data. The statistical methods adopted in this paper are mainly conventional mathematical statistics analysis methods， and there are mainly four types. The first is absolute error and relative error， of which the absolute value is the difference between the extracted value and the standard slope length value， and the relative error is obtained by multiplying the ratio of the absolute error to the standard slope length value by 100%. The second is quantification rate. 30% was the upper limit of relative error in this paper， and when the error between the extracted slope length value and standard value was less than 30%， the result was qualified. Otherwise， it was unqualified. The third is correlation coefficient. The fourth is certainty coefficient. This coefficient was proposed by Nash and Sutcliffe in 1970 to analyze the correlation between the extracted value and the actual value. The coefficient reflects the closeness of the calculated value to the 1∶ 1 line of the measured value， and its value range is （-∞， 1）. When the certainty coefficient is greater than 0， it indicates that the extracted value is in good agreement with the standard value; when the certainty coefficient is closer to 1， the extracted value is more precise; and when the certainty coefficient is 0， the squared sum of the differences between the calculated results and the measured values is just equivalent to the squared sum of the differences between the measured values and their mean value， indicating that the effect of estimating with the average of the measured values is the same as the calculated value. In addition， oneway ANOVA was used to determine whether the DEM resolution had a significant effect on the calculation of slope length factor. 　　 Results and Analysis
　　 Comparison of two extraction methods
　　Since there is a great randomness in automatically selecting the center sample point of 100 m， there may be cases where sample points fall on the ridgeline or the mountaintop， and some points fall at the bottom of the valley. First， the sample points participating in the analysis should be screened， and those sample points that are not suitable for comparison with the true value of the slope length should be excluded. Finally， the number of sample points participating in the comparison was 218 （the original number of sample points was 292）. The statistical results （Table 2， Table 3） showed that the extraction precision of both algorithms decreased with the decrease of DEM resolution， indicating that the higher the DEM resolution， the better the slope length extraction effect. The slope length extraction effect of the direct calculation method based on noncumulative flow was much better than the spatial analysis extraction method.
　　The scatter diagrams of the values obtained by the two extraction methods and the true values showed that the extraction effects were quite different （Fig. 2， Fig. 3）. The scatter diagram of the direct calculation method based on noncumulative flow （Fig. 2） showed that when the DEM resolutions were 10 and 5 m， respectively， the quantification rates of the extraction results were 87.41%and 81.15%， respectively， and the extraction effects were good. The certainty coefficients were all greater than 0 and the correlation coefficients were close to 1， indicating that the extracted values were in good agreement with the true values. It can be seen that if a DEM is generated using a contour vector diagram with a scale of 1∶ 10 000， the resolution is generally set to be 10 m. The scatter diagram of the spatial analysis extraction method （Fig. 3） showed that when the resolution of the DEM was 5 m， the extraction precision was 61.47% and the certainty coefficient was greater than 0， so the extracted value had certain degree of coincidence with the true value; and when the DEM resolution was lower than 10 m， the calculation effect was not ideal， and with the decrease of DEM resolution， the slope length extraction had a tendency of being too large， and some extreme deviations can reach tens times of the true value. Therefore， in order to use this algorithm to extract the slope length， a DEM with a resolution of 5 mis required to ensure that the extraction result has practical significance and value. When the DEM resolution was 15 or 20 m， the extraction precision of both algorithms was significantly reduced， and especially when the resolution was 20 m， the quantification rate of the direct calculation method based on noncumulative flow was only 39.27% and the degree of coincidence with the standard slope length was also very low. It is known that the higher the DEM resolution， the better the extraction result. Therefore， when doing similar slope length extraction， if the DEM resolution is lower than 20 m， the error of the extraction result will be large. We should try to use largescale topographic map data as much as possible to get high resolution DEMs. Although the extraction precision of the spatial analysis algorithm is much lower than the extraction precision of the noncumulative flow algorithm under the same resolution condition， the algorithm based on the readymade GIS software hydrological analysis module is easy to operate， stable in operation and high in efficiency. Especially when the terrain of the research area meets the condition that the ridgeline is substantially parallel to the contour line， the slope length extracted by the algorithm is also quite accurate （Tang Guoan et al.， 2002）. In addition， the spatial analysis extraction method has a lot of room for improvement. For example， the incomplete ridgeline identification is one of the important reasons for the low precision of the algorithm. If a method for accurately extracting the ridgeline can be found in the future， the calculation error caused by the ridgeline identificationproblem will be greatly reduced， and the computation precision of this algorithm can be greatly improved. 　　 Analysis of the effect of DEM resolution on slope length factor
　　The oneway analysis of variance was used to determine whether the DEM resolution had a significant effect on the calculation of slope length factor. When the DEM resolutions were 5 and 10 m， the precision of the slope length extracted by the direct calculation method based on noncumulative flow was ideal， and both were above 80%. Moreover， the precision under the two resolutions was not obvious. However， when the DEM resolutions were 15 and 20 m， the precision of the extraction results was not ideal. The slope length determines the slope length factor， and the computation precision of the slope length factor depends on the extraction precision of the slope length. Therefore， when the DEM resolution is greater than 10 m， the extraction precision of the slope length factor is inevitably not high. It is only necessary to determine whether there is a significant effect on the extraction of the slope length factor under the DEM resolutions of 5 and 10 m. Since the randomness and typicality are considered in selecting the sample area and the sample point， the slope length factor values of the 218 sample points participating in the statistical analysis can be regarded as samples from the normal population. Under the two levels of DEM resolution of 5 and 10 m， whether there was a significant difference in the mean value of the slope length factor at these two levels was tested at the significance level of 0.05， and then whether the resolution had a significant effect on the calculation of the slope length factor when the DEM resolution was over 10 m can be determined. The null hypothesis and alternative hypothesis for establishing the given problem were： H0∶ μ1=μ2， H1∶ μ1≠μ2 （μ1 and μ2were the expected values for the two groups of data）. The results showed F=0.02， which was smaller than the critical value of F=3.86）， and P=0.89， which was larger than α=0.05 （Table 4）， so H0 was accepted， which meant that there was no significant difference in the calculation result between the two groups of slope lengths. After the above analysis， it can be concluded that there was no significant difference between the slope length factors extracted from the DEM of 5 m resolution and the DEM of 10 m resolution， that is， the resolution had no significant effect on the calculation of the slope length factor when the DEM resolution was higher than 10 m.
　　
　　 Conclusions and Discussion 　　 Conclusions
　　There were three main conclusions. First， the effect of the slope length extracted by the direct calculation method based on noncumulative flow was much better than that of the spatial analysis method. The reason for this is that the noncumulative flow algorithm has a clear physical meaning， its design idea has a strict hydrogeomorphology basis， and the slope length calculated according to the calculation method of the segmented slope is close to the real terrain feature. In addition， when the DEM resolution was large enough （over 10 m）， the extraction precision of the direct calculation method based on noncumulative flow was already quite satisfactory. Therefore， the direct calculation method based on noncumulative flow is mature enough to be popularized in slope length extraction. Secondly， when the DEM resolutions were 5 and 10 m， respectively， there were no significant differences in the precision of the slope length and the slope length factor extracted by the direct calculation method based on noncumulative flow. This tells us that when extracting the slope length and the slope length factor， it is usually sufficient to set the DEM resolution to 10 m， and it is not necessary to set it to 5 m. Reducing the DEM resolution means increasing the computational efficiency of the program to a large extent. Thirdly， the overall precision of the spatial analysis calculation method was poor. The precondition for accurate spatial analysis is that the ridgeline is substantially parallel to the contour， which is obviously inconsistent with or even different from the actual terrain， and there is no recognized reliable standard for the threshold setting when extracting the ridgeline. Blindness and randomness can easily lead to misjudgment and missed ridgeline， which results in low computation precision. Especially when a lowresolution DEM is used as the source data， the calculation result has a very large error and no practical value. It shows that the application of the algorithm in the extraction of slope length is still not mature and needs to be improved.
　　 Discussion
　　First， due to the restriction from objective conditions such as time， the number of sample areas was relatively small. In order to obtain more reliable and convincing conclusions， according to the practice of previous studies， about 100 sample areas need to be selected only in Ansai County. In the case of a small number of sample areas， it is difficult to reflect the geographical differentiation of the region. Secondly， due to the limited level of this research， only two simple methods in the four commonly used algorithms were selected for the comparison of the extraction effect， and it was not possible to broaden the ideas and discover more accurate and more efficient algorithms for geographical slope length extraction. In addition， the error was not systematically analyzed， the reason for the calculation error was not given， and a feasible method for reducing and correcting the error was not proposed. These are all worthy of further research. At last， the precision evaluation of the algorithm is a fine and complicated process， and especially for slope length which is an originally controversial terrain element， there must be many unsolved problems in the precision evaluation. The difficulty involved in these problems is far beyond the actual level of this study. In addition， from the digitization of the topographic map to the measurement of the true value of the slope length in the whole study， there were inevitably errors， and the set evaluation criteria were also relatively simple. Whether the conclusions drawn under such conditions are truly reliable remains to be further studied. 　　 References
　　
　　[1] WEILL. Universal soil loss equation[M]//Encyclopedia of Soil Science. Springer Netherlands， 2008：806-806.
　　[2] ONORI F， BONIS PD， GRAUSO S. Soil erosion prediction at the basin scale using the revised universal soil loss equation （RUSLE） in a catchment of Sicily （southern Italy）[J]. Environmental Geology， 2006， 50（8）： 1129-1140.
　　[3] LIU BY， XIE Y， ZHANG KL， et al. Soil erosion prediction model[M]. Beijing： China Science and Technology press， 2001.
　　[4] TANG GA， CHEN ZH， ZHAO MD， et al. Arc View geographic information system spatial analysis method[M]. Beijing： Science Press， 2002.
　　[5] TANG GA， LIU XJ， FANG L， et al. A review on the scale issue in DEMs and digital terrain analysis[J]. Geomatics and Information Science of Wuhan University， 2006， 31（12）： 1059-1066.
　　[6] TANG GA， YANG X. ArcGIS geographical information system spatial analysisexperimental course[M]. Beijing： SciencePress， 2006.
　　[7] LI J. Automatic extraction of slope length and analyst based on DEM in the loess plateau[D]. Xian： Northwest University， 2007.
　　[8] HOYOS N. Spatial modeling of soil erosion potential in a tropical watershed of the Colombian Andes[J]. Catena， 2005， 63（1）： 85-108.
　　[9] VENTE J， POESEN J， VERSTRAETEN G， et al. Spatially distributed modeling of soil erosion and sediment yield at regional scales in Spain [J]. Global and Planetary Change， 2008， 60（3/4）： 393-415.
　　[10] ZHOU GY， LIU Y， WU L. Algorithms for extracting drainage network based on digital elevation model[J]. Geography and Territorial Research， 2000， 16（4）： 77-81.
　　[11] CAI GQ， LIU JG. Evolution of soil erosion models in China[J]. Progress in Geography， 2003， 22（3）： 242-250.
　　[12] CAO LX， FU SH. A comparison of methods for computating slope length based on DEM[J]. Bulletin of Soil and Water Conservation， 2007， 27（5）： 57-62.
　　[13] ZHANG QW， AN JZ， WANG X， et al. Research progress on soil erosionrelated models in China[J]. Soil and Water Conservation in China， 2014（1）： 43-46.