Relationship between fractal dimensions of stream and morphometric characteristics of basin for the soil conservation
Subject Areas : Farm water management with the aim of improving irrigation management indicators
sepideh mofidi
1
,
Abolfazl Moeini
2
,
Ali Torkashvand
3
,
Ebrahim Pazira
4
,
Hassan Ahmadi
5
1 - Ph.D. Student Department of Soil Science, Science and Research Branch, Islamic Azad University, Tehran, Iran.
2 - Assistant Professor, Department of Forests, Range and Watershed Management, Science and Research Branch, Islamic Azad University, Tehran, Iran.
3 - Associate Professor Department of Soil Science, Science and Research Branch, Islamic Azad University, Tehran, Iran.
4 - Professor Department of Soil Science, Science and Research Branch, Islamic Azad University, Tehran, Iran.
5 - Professor Department of Forests, Range and Watershed Management, Science and Research Branch, Islamic Azad University, Tehran, Iran.
Keywords: drainage network, bifurcation ratio, Fractal Geometry Strahler, form factor,
Abstract :
Background and Aim: River behavior is affected by two categories of natural, and human factors. Natural factors such as floods, soil erosion, mass movement and human factors such as facility construction, land-use change, and sand removal from the bed, play a key role in the behavior and intensification of river changes. The serious and irreparable dangers that may be posed by river movements and changes indicate the need to study its morphology in the study phase before any action is taken. The network of streams continuously changes its location based on time, environmental factors and human interventions. It is very important to study the stream changes to provide the should be omitted management solutions for the should be omitted soil conservation. One of the new methods in this regard is to use the should be omitted fractal geometry. The purpose of this study was to calculate the fractal dimensions of the stream and to investigate its relationship with the morphometric characteristics of the basinMethod: To do this, a should be added topographic map of the should be added basin in Firouzkouh, Tehran province was prepared. The map of the streams was prepared and the morphometric characteristics of the basin were determined by using the ARC GIS 10.3 software. Then, three fractal dimensions of the should be added drainage network (stream branch), drainage density, and basin area were calculated. Finally, the morphometric characteristics of the basin associated with the fractal dimensions were investigated by entering the data obtained from the calculations in the SPSS 18 and Curve Expert software Results: The results showed that the minimum and maximum fractal dimensions of the bifurcation ratio were 0.25 and 2.99, of should be omitted the drainage density was 0.19 and 2.34, and of should be omitted the area was 0.76 and 2.60. The fractal dimension of the should be added, the should be added dimension of drainage density, and the should be added fractal dimension of the should be added total area of the basin were 1.84, 0.71, and 1.46, respectively. The fractal dimension of the should be added ratio and the sub-basin area were inversely related with the determination coefficient of 0.90, and the fractal dimension of drainage density and the area and the fractal dimension of an should be added area with the sub-basin area was directly related with the determination coefficients of 0.88, and 0.87, respectively. The more elongated the basin and the lower the form, roundness, and elongation factors, the smaller the bifurcation dimension. the fractal dimension of the area is directly related to the compactness factor, elongation factor, form factor, bifurcation ratio, equivalent rectangle width, and equivalent rectangular length, and is inversely related to other variables. Based on the results, the more elongated the basin and the smaller the form, and elongation factors, the lower the area dimension. The fractal dimension of drainage density is directly associated with the circularity factor, compactness factor, elongation factor, form factor, area ratio, bifurcation ratio, equivalent rectangle width and equivalent rectangular length, and is inversely associated with other variables. Therefore, as the basin becomes more circular, the fractal dimension of the drainage density is increased. Therefore, the fractal dimension of drainage density is increased then as the basin becomes more roundedConclusion: According to the coefficient of explanation of the obtained models for the relationship between fractal dimensions, and morphometric properties, the studied fractal dimensions can be easily calculated and analyzed using morphometric properties. Due to the importance of stream characteristics in the management of watersheds in terms of flood, erosion, and soil conservation, the fractal models can be used to make quick and accurate decisions for the should be omitted management. Finally, considering that the use of fractal geometry is a new method in studying the characteristics of waterway networks, it is suggested that basins be analyzed fractally in different regions with different morphometric conditions.
Agaard, K. and G. Hartvigsen. 2014. Assessing spatial patterns of plant communities at varying stages of succession. Applied Mathematics. 5: 1842-1851.
Agus Nur, A., Syafri, I., Muslim, D., Hiranawan, F., Raditya, P.P., Sulastri, M. and Abdulah, F. 2016. Earth and Environmental Science. International Symposium on Geophysical Issues.
Ahmadi, A., Neyshabouri, M.R., Rouhipour, H., and Asadi, H. 2011. Fractal dimension of soil aggregates as an index of soil erodibility. Journal of Hydrology 400 (3-4): 305-311.
Alimoradi, M., Ekhtesasi, M.R., Tazeh, M. and Karimi, H. 2018. Calculation of Fractal Dimension of the Geological Formations and Their Relationship to the Formation Sensibility. Physical Georaphy Research Quartrly. 50 (2): 241-253. [in Persian]
Andronache I., Peptenatu D., Ciobotaru A.M., Gruia A.K., Gropoșila N.M. 2016. Using Fractal Analysis in Modeling Trends in the National Economy, Procedia Environmental Sciences 32: 344-351.
Andronache, I., Ahammer, H., Jelinek, H.F., Peptenatu, D., Ciobotaru, A.M., Drăghici, C.C., Pintilii, R.D., Simio,n A.G., Teodorescu, C. 2016. Fractal analysis for studying the evolution of forests. Chaos, Solitons & Fractals. 91: 310–318.
Asadzadeh, F., Jalalzadeh, S. and Samadi A. 2017. Comparison of the physical and chemical properties of the bed and suspended sediments of the Roze-Chay river. Journal of Water and Soil Conservation. 24(2): 273-288. [in Persian]
Bartolo, S.G., Veltri, M. and Primavera L., 2006, Estimated generalized dimensions of river networks. Journal of Hydrology. 322, 181–191.
Bi, L., He, H., Wei, Z., Shi, F., 2012, Fractal properties of landform in the Ordos Block and surrounding areas, China. Geomorphology. 175, 151–162.
Chorley R.J., Kennedy B.A. 1971. Physical geography: a systems approach. Prentice-Hall International, 370, London.
Cui. Y., Li, J., Chen, A., Wu, J., Luo, Q., Rafay, L., He, J., Liu, Y., Wang, D., Lin, Y. and Wu, Ch. 2019. Fractal dimensions of trapped sediment particle size distribution can reveal sediment retention ability of common plants in a dry-hot valley. Catena. 180: 252-262.
Diaconu, D., Drăghici, CC., Pintilii, R.D., Peptenatu, D., Grecu, A. 2016. Management of the Protection Forest Areas in Region Affected by Aridity in Oltenia Southwestern Development Region (Romania), 16th International Multidisciplinary Scientific GeoConference-SGEM, Vienna, Austria, 477-483.
Ding, W.F., Huang, C.H., 2017. Effects of soil surface roughness on interrill erosion processes and sediment particle size distribution. Geomorphology 295, 801–810. https://doi.org/10.1016/j.geomorph.2017.08.033.
Enquist, B. J., G. B. West, E. L. Charnov and J. H. Brown. 1999. Allometric scaling of production and life-history variation in vascular plants. Nature. 401(6756): 907-911.
Fattahi, M.H. and Talebzadeh Z. 2017. The Relationship Between Watershed Compactness Coefficient and the Fractal Characteristics. Iran-Water Resources Research. 13 (1): 191-203. [in Persian]
Frontier, S. 1990. Applications of Fractal Theory to Ecology, In P. Legendre and C. Legendre (Eds.), Developments in Numerical Ecology: NATO ASI Series, Springer, Berlin.
Gavrila I.G., Man T., Surdeanu V. 2011. Geomorphological heritage assessment using GIS analysis for geotourism development in Măcin Mountains, Dobrogea, Romania, GeoJournal of Tourism and Geosites, 2 (8): 198-205.
Ghahroudi Tali M, and Derafshi K. 2015. The study of chaos in the flood risk pattern of Tehran. Journal of Spatial Analysis Environmental Hazards. 2 (2):1-16
Hekmatzadeh, A.A., Torabi Haghighi, Hosseini, K. and Klove, B. 2018. Fractal analysis of river flow time series: a case study on Shapur river. Geophysical Research Abstracts. 20.
Horton, R.E. 1932. Drainage Watershed characteristics. Am Geophys Union Trans. 13: 348-352.
Khan S. Ganguly A.R. and Saigal S. 2005. Detection and Predictive Modeling of Chaos In Finite Hydrologycal Time Series, Nonlinear Processes in Geophysics. 12: 41-53.
Khanbabaei, Z., Karam, A. and Rostamizad, G. 2013. Studying Relationships between the Fractal Dimension of the Drainage Basins and Some of Their Geomorphological Characteristics. International Journal of Geosciences. 4: 636-642.
Kršák B., Blistan P., Pauliková A., Puškárovác P., Kovanič L., Palková J., Zelizňaková V. 2016. Use of low-cost UAV photogrammetry to analyze the accuracy of a digital elevation model in a case study. Measurement, 91, 276–287.
Kusak, M., 2014, Methods of fractal geometry used in the study of complex geomorphic netwoks, AUC. Geographica. 49 (2): 99–110.
Kutlu T, Ersahin S and Yetgin B, 2008. Relations between solid fractal dimension and some physical properties of soils formed over alluvial and colluvial deposits. Journal of Food, Agriculture and Environment. 6: 445-449.
Lisi B., Honglin., H, Zhanyu, W. and Feng, S. 2012. Fractal Properties of Landforms in the Ordos Block and Surrounding Areas, China. Geomorphology. PP. 151-162.
Long, C. Y., Y. Zhao and H. Jafari. 2014. Mathematical models arising in the fractal forest Gap via local fractional calculus. Hindawi Publishing Corporation. Abstract and Applied Analysis. 6 pages.
Lyu, X., Yu, J., Zhou, M., Ma, B., Wang, C., Han, G., Guan, B., Wu, H., Li, Y., Wang, D., 2015. Changes of soil Patricle Size Distribution in Tidal Flats in the Yellow River Delta. J. Plos One. 10(3), e0121368.
Mandelbrot, B.B. 1982. The fractal geometry of nature. W.H. Freeman and Company. New York. 468 p.
Miller, V.C. 1953. A quantitative geomorphologic study of drainage watershed characteristics in the Clinch Mountain area. Virginia and Tennessee, Project Nr 389042, Tech Report 3. Columbia University Department of Geology, ONR Geography Branch, New York.
Mofidi, S., Eskandari, M., Pazira, E., Homaee, M., 2018. Using fractal models for quantifying soil structure and comparison with classical methods. water soil Resour. Conserv. 7, 89–101
Mohammadi, M., Shabanpour, M., Mohammadi, M.H. and Davatgar, N. 2019. Characterizing Spatial Variability of Soil Textural Fractions and Fractal Parameters Derived from Particle Size Distributions. Pedosphere. 29 (2): 224-234.
Morisawa, M.E. 1962. Quantitative Geomorphology of Some Watersheds in the Appalachian Plateau. Geological Society of America Bulletin, 73, 1025-1046.
Peterson, G. 2000. Scaling ecological dynamics: self-organization, hierarchical structure, and ecological resilience. Climatic Change. 44(3): 291-309.
Petrisor, A.I., Andronache, I., Petrișor, L.E., Ciobotaru, A.M., Peptenatu, D. 2016. Assessing the fragmentation of the green infrastructure in Romanian cities using fractal models and numerical taxonomy, Edited by: Ioja, I.C.; Comanescu, L.; Dumitrache, L.; Nedelea, A.; Nita, M.R., Ecosmart - Environment at Crossroads: Smart Approaches for a Sustainabl Development, Procedia Environmental Sciences, 32, 110-123.
Pintilii, R.D., Diaconu, D.C., Dobrea, R.C., Gruia, A.K. 2016. Dynamics of the deforested areas in regions affected by aridity in Romania, 16th International Multidisciplinary Scientific Geoconference-SGEM, Vienna, Austria, 401-407.
Rezaei Moghadam, M.H., Sarvati, M.R. and Asghari Sareskanrood S. 2010. Compared Investigation Meandering Pattern by Fractal Geometrical Analysis and Central Angles and Sinuosity Ratio Indices. Journal of Watershed Management Research. 2 (3). [in Persian]
Rodriguez, I., and Rinaldo, A. 1997. Fractal River Basins: Chance and Self-Organization. Cambridge Univ. Press, Cambridge. 547 p.
Schumm, S.A. 1956. Evolution of drainage systems and slopes in badlands at Perth Amboy, New Jersey. Geol Soc Am Bull. 67: 597-646.
Shaghaghian, M.R. and Taleb Bidokhti, N. 2008. Existence of turbulence in the river at various time scales. Quarterly journal on Water Engineering. 2: 1-8. [in Persian]
Shen, X.H., Zou, L.J., Zhang, G.F. Su, N. Wu, W.Y. and Yang., S.F. 2011. Fractal Characteristics of the Main Channel of Yellow River and Its Relation to Regional Tectonic Evolution, Geomorphology 127: 64-70
Singh, S. 1992. Quantitative geomorphology of the drainage basin. In: Chouhan TS, Joshi KN (eds) Readings on remote sensing applications. Scientific Publishers, Jodhpur.
Singh, S. 1995. Quantitative analysis of watershed geomorphology using remote sensing techniques. Ann Arid Zone. 34 (4), 243–251.
Sivakumar, B., and Berndtsson, R. 2010. Advances in Data-based Approaches for Hydrologic Modeling and Forecasting, World Scientific, Singapore. 441 pp.
Strahler, A.N. 1952 Hypsometric (Area-Altitude) analysis of erosional topography, Geological Society of America Bulletin, 63; 1117-1142.
Strahler, A.N. 1957. Quantitative analysis of watershed geomorphology, Eart&Space Science News, 38, 6; 913-920.
Strahler, A.N. 1958. Quantitative slope analysis, Geological Society of America Bulletin, 69, 3; 279-300.
Strahler, A.N. 1964. Quantitative geomorphology of drainage basins and channel networks. In Handbook of Applied Hydrology, edited by V. T. Chow, 4-39/4-76. McGraw-Hill, New York.
Tahmasebi, Z., Zal, F. and Ahmadi Khalaji, A. 2015. Morphology of Tourmaline in the Mashhad granites (g2) with using fractal analysis and Diffusion-Limited Aggregation. The Iranian Journal of Crystallography and Mineralogy. 23 (3):417-428. [in Persian]
Turcotte, D.L. 2007. Self-organized Complexity in Geomorphology: Observations and Models. Geomorphology. 91: 302-310.
Valjarević A., Srećković-Batoćanin D., Živković D., Perić M. 2015. GIS analysis of dissipation time of landscape in the Devil's city (Serbia), Acta Montanistica Slovaca, 20 (2), 148-155.
West, G. B., J. H. Brown and B. J. Enquist. 1997. A general model for the origin of allometric scaling laws in biology. Science. 276 (5309): 122-126.
Xia, D., Deng, Y.S., Wang, S.L., Ding, S.W., Cai, C.F., 2015. Fractal features of soil particle-size distribution of different weathering profiles of the collapsing gullies in the hilly granitic region, south China. Nat. Hazards 79(1), 455–478.
Yamani, M. and Sharafi, S. 2012. Geomorphology and effective factors on lateral erosion in Hor Rood River, Lorestan province. Geography and Environmental Planning Journal. 45 (1): 15-32. [in Persian]
Yang, S., Li, Y.H., Gao, Z.L., Niu, Y.B., Bai, H., Wang, K., 2017. Runoff and sediment reduction benefit of hedgerows and fractal characteristics of sediment particles on Loess Plateau slope of engineering accumulation. Trans. Chin. Soc. Agric. Mach. 48 (8), 270–278. https://doi.org/10.6041/j.issn.1000-1298.2017.08.031. (in Chinese with English abstract).
Yu, J.B., Lv, X.F., Bin, M., Wu, H.F., Du, S.Y., Zhou, M., Yang, Y.M., Han, G.X., 2015. Soil erosion processes and sediment sorting associated with. Sci. Rep. 5, 1–9.
_||_Agaard, K. and G. Hartvigsen. 2014. Assessing spatial patterns of plant communities at varying stages of succession. Applied Mathematics. 5: 1842-1851.
Agus Nur, A., Syafri, I., Muslim, D., Hiranawan, F., Raditya, P.P., Sulastri, M. and Abdulah, F. 2016. Earth and Environmental Science. International Symposium on Geophysical Issues.
Ahmadi, A., Neyshabouri, M.R., Rouhipour, H., and Asadi, H. 2011. Fractal dimension of soil aggregates as an index of soil erodibility. Journal of Hydrology 400 (3-4): 305-311.
Alimoradi, M., Ekhtesasi, M.R., Tazeh, M. and Karimi, H. 2018. Calculation of Fractal Dimension of the Geological Formations and Their Relationship to the Formation Sensibility. Physical Georaphy Research Quartrly. 50 (2): 241-253. [in Persian]
Andronache I., Peptenatu D., Ciobotaru A.M., Gruia A.K., Gropoșila N.M. 2016. Using Fractal Analysis in Modeling Trends in the National Economy, Procedia Environmental Sciences 32: 344-351.
Andronache, I., Ahammer, H., Jelinek, H.F., Peptenatu, D., Ciobotaru, A.M., Drăghici, C.C., Pintilii, R.D., Simio,n A.G., Teodorescu, C. 2016. Fractal analysis for studying the evolution of forests. Chaos, Solitons & Fractals. 91: 310–318.
Asadzadeh, F., Jalalzadeh, S. and Samadi A. 2017. Comparison of the physical and chemical properties of the bed and suspended sediments of the Roze-Chay river. Journal of Water and Soil Conservation. 24(2): 273-288. [in Persian]
Bartolo, S.G., Veltri, M. and Primavera L., 2006, Estimated generalized dimensions of river networks. Journal of Hydrology. 322, 181–191.
Bi, L., He, H., Wei, Z., Shi, F., 2012, Fractal properties of landform in the Ordos Block and surrounding areas, China. Geomorphology. 175, 151–162.
Chorley R.J., Kennedy B.A. 1971. Physical geography: a systems approach. Prentice-Hall International, 370, London.
Cui. Y., Li, J., Chen, A., Wu, J., Luo, Q., Rafay, L., He, J., Liu, Y., Wang, D., Lin, Y. and Wu, Ch. 2019. Fractal dimensions of trapped sediment particle size distribution can reveal sediment retention ability of common plants in a dry-hot valley. Catena. 180: 252-262.
Diaconu, D., Drăghici, CC., Pintilii, R.D., Peptenatu, D., Grecu, A. 2016. Management of the Protection Forest Areas in Region Affected by Aridity in Oltenia Southwestern Development Region (Romania), 16th International Multidisciplinary Scientific GeoConference-SGEM, Vienna, Austria, 477-483.
Ding, W.F., Huang, C.H., 2017. Effects of soil surface roughness on interrill erosion processes and sediment particle size distribution. Geomorphology 295, 801–810. https://doi.org/10.1016/j.geomorph.2017.08.033.
Enquist, B. J., G. B. West, E. L. Charnov and J. H. Brown. 1999. Allometric scaling of production and life-history variation in vascular plants. Nature. 401(6756): 907-911.
Fattahi, M.H. and Talebzadeh Z. 2017. The Relationship Between Watershed Compactness Coefficient and the Fractal Characteristics. Iran-Water Resources Research. 13 (1): 191-203. [in Persian]
Frontier, S. 1990. Applications of Fractal Theory to Ecology, In P. Legendre and C. Legendre (Eds.), Developments in Numerical Ecology: NATO ASI Series, Springer, Berlin.
Gavrila I.G., Man T., Surdeanu V. 2011. Geomorphological heritage assessment using GIS analysis for geotourism development in Măcin Mountains, Dobrogea, Romania, GeoJournal of Tourism and Geosites, 2 (8): 198-205.
Ghahroudi Tali M, and Derafshi K. 2015. The study of chaos in the flood risk pattern of Tehran. Journal of Spatial Analysis Environmental Hazards. 2 (2):1-16
Hekmatzadeh, A.A., Torabi Haghighi, Hosseini, K. and Klove, B. 2018. Fractal analysis of river flow time series: a case study on Shapur river. Geophysical Research Abstracts. 20.
Horton, R.E. 1932. Drainage Watershed characteristics. Am Geophys Union Trans. 13: 348-352.
Khan S. Ganguly A.R. and Saigal S. 2005. Detection and Predictive Modeling of Chaos In Finite Hydrologycal Time Series, Nonlinear Processes in Geophysics. 12: 41-53.
Khanbabaei, Z., Karam, A. and Rostamizad, G. 2013. Studying Relationships between the Fractal Dimension of the Drainage Basins and Some of Their Geomorphological Characteristics. International Journal of Geosciences. 4: 636-642.
Kršák B., Blistan P., Pauliková A., Puškárovác P., Kovanič L., Palková J., Zelizňaková V. 2016. Use of low-cost UAV photogrammetry to analyze the accuracy of a digital elevation model in a case study. Measurement, 91, 276–287.
Kusak, M., 2014, Methods of fractal geometry used in the study of complex geomorphic netwoks, AUC. Geographica. 49 (2): 99–110.
Kutlu T, Ersahin S and Yetgin B, 2008. Relations between solid fractal dimension and some physical properties of soils formed over alluvial and colluvial deposits. Journal of Food, Agriculture and Environment. 6: 445-449.
Lisi B., Honglin., H, Zhanyu, W. and Feng, S. 2012. Fractal Properties of Landforms in the Ordos Block and Surrounding Areas, China. Geomorphology. PP. 151-162.
Long, C. Y., Y. Zhao and H. Jafari. 2014. Mathematical models arising in the fractal forest Gap via local fractional calculus. Hindawi Publishing Corporation. Abstract and Applied Analysis. 6 pages.
Lyu, X., Yu, J., Zhou, M., Ma, B., Wang, C., Han, G., Guan, B., Wu, H., Li, Y., Wang, D., 2015. Changes of soil Patricle Size Distribution in Tidal Flats in the Yellow River Delta. J. Plos One. 10(3), e0121368.
Mandelbrot, B.B. 1982. The fractal geometry of nature. W.H. Freeman and Company. New York. 468 p.
Miller, V.C. 1953. A quantitative geomorphologic study of drainage watershed characteristics in the Clinch Mountain area. Virginia and Tennessee, Project Nr 389042, Tech Report 3. Columbia University Department of Geology, ONR Geography Branch, New York.
Mofidi, S., Eskandari, M., Pazira, E., Homaee, M., 2018. Using fractal models for quantifying soil structure and comparison with classical methods. water soil Resour. Conserv. 7, 89–101
Mohammadi, M., Shabanpour, M., Mohammadi, M.H. and Davatgar, N. 2019. Characterizing Spatial Variability of Soil Textural Fractions and Fractal Parameters Derived from Particle Size Distributions. Pedosphere. 29 (2): 224-234.
Morisawa, M.E. 1962. Quantitative Geomorphology of Some Watersheds in the Appalachian Plateau. Geological Society of America Bulletin, 73, 1025-1046.
Peterson, G. 2000. Scaling ecological dynamics: self-organization, hierarchical structure, and ecological resilience. Climatic Change. 44(3): 291-309.
Petrisor, A.I., Andronache, I., Petrișor, L.E., Ciobotaru, A.M., Peptenatu, D. 2016. Assessing the fragmentation of the green infrastructure in Romanian cities using fractal models and numerical taxonomy, Edited by: Ioja, I.C.; Comanescu, L.; Dumitrache, L.; Nedelea, A.; Nita, M.R., Ecosmart - Environment at Crossroads: Smart Approaches for a Sustainabl Development, Procedia Environmental Sciences, 32, 110-123.
Pintilii, R.D., Diaconu, D.C., Dobrea, R.C., Gruia, A.K. 2016. Dynamics of the deforested areas in regions affected by aridity in Romania, 16th International Multidisciplinary Scientific Geoconference-SGEM, Vienna, Austria, 401-407.
Rezaei Moghadam, M.H., Sarvati, M.R. and Asghari Sareskanrood S. 2010. Compared Investigation Meandering Pattern by Fractal Geometrical Analysis and Central Angles and Sinuosity Ratio Indices. Journal of Watershed Management Research. 2 (3). [in Persian]
Rodriguez, I., and Rinaldo, A. 1997. Fractal River Basins: Chance and Self-Organization. Cambridge Univ. Press, Cambridge. 547 p.
Schumm, S.A. 1956. Evolution of drainage systems and slopes in badlands at Perth Amboy, New Jersey. Geol Soc Am Bull. 67: 597-646.
Shaghaghian, M.R. and Taleb Bidokhti, N. 2008. Existence of turbulence in the river at various time scales. Quarterly journal on Water Engineering. 2: 1-8. [in Persian]
Shen, X.H., Zou, L.J., Zhang, G.F. Su, N. Wu, W.Y. and Yang., S.F. 2011. Fractal Characteristics of the Main Channel of Yellow River and Its Relation to Regional Tectonic Evolution, Geomorphology 127: 64-70
Singh, S. 1992. Quantitative geomorphology of the drainage basin. In: Chouhan TS, Joshi KN (eds) Readings on remote sensing applications. Scientific Publishers, Jodhpur.
Singh, S. 1995. Quantitative analysis of watershed geomorphology using remote sensing techniques. Ann Arid Zone. 34 (4), 243–251.
Sivakumar, B., and Berndtsson, R. 2010. Advances in Data-based Approaches for Hydrologic Modeling and Forecasting, World Scientific, Singapore. 441 pp.
Strahler, A.N. 1952 Hypsometric (Area-Altitude) analysis of erosional topography, Geological Society of America Bulletin, 63; 1117-1142.
Strahler, A.N. 1957. Quantitative analysis of watershed geomorphology, Eart&Space Science News, 38, 6; 913-920.
Strahler, A.N. 1958. Quantitative slope analysis, Geological Society of America Bulletin, 69, 3; 279-300.
Strahler, A.N. 1964. Quantitative geomorphology of drainage basins and channel networks. In Handbook of Applied Hydrology, edited by V. T. Chow, 4-39/4-76. McGraw-Hill, New York.
Tahmasebi, Z., Zal, F. and Ahmadi Khalaji, A. 2015. Morphology of Tourmaline in the Mashhad granites (g2) with using fractal analysis and Diffusion-Limited Aggregation. The Iranian Journal of Crystallography and Mineralogy. 23 (3):417-428. [in Persian]
Turcotte, D.L. 2007. Self-organized Complexity in Geomorphology: Observations and Models. Geomorphology. 91: 302-310.
Valjarević A., Srećković-Batoćanin D., Živković D., Perić M. 2015. GIS analysis of dissipation time of landscape in the Devil's city (Serbia), Acta Montanistica Slovaca, 20 (2), 148-155.
West, G. B., J. H. Brown and B. J. Enquist. 1997. A general model for the origin of allometric scaling laws in biology. Science. 276 (5309): 122-126.
Xia, D., Deng, Y.S., Wang, S.L., Ding, S.W., Cai, C.F., 2015. Fractal features of soil particle-size distribution of different weathering profiles of the collapsing gullies in the hilly granitic region, south China. Nat. Hazards 79(1), 455–478.
Yamani, M. and Sharafi, S. 2012. Geomorphology and effective factors on lateral erosion in Hor Rood River, Lorestan province. Geography and Environmental Planning Journal. 45 (1): 15-32. [in Persian]
Yang, S., Li, Y.H., Gao, Z.L., Niu, Y.B., Bai, H., Wang, K., 2017. Runoff and sediment reduction benefit of hedgerows and fractal characteristics of sediment particles on Loess Plateau slope of engineering accumulation. Trans. Chin. Soc. Agric. Mach. 48 (8), 270–278. https://doi.org/10.6041/j.issn.1000-1298.2017.08.031. (in Chinese with English abstract).
Yu, J.B., Lv, X.F., Bin, M., Wu, H.F., Du, S.Y., Zhou, M., Yang, Y.M., Han, G.X., 2015. Soil erosion processes and sediment sorting associated with. Sci. Rep. 5, 1–9.
