Evaluation of fractal dimension of soft terrain surface Jiang Chunxia a , Lu Zhixiong a,⇑ , Zhou Jin a , Muhammad Sohail Memon a,b a College of Engineering, Nanjing Agricultural University, Nanjing 210031, China b Faculty of Agricultural Engineering, Sindh Agriculture University, Sindh 70050, Pakistan Received 8 January 2016; received in revised form 9 July 2016; accepted 10 January 2017 Abstract By using a laser profiler, the roughness of sowed and plowed surfaces was obtained. Through evaluation of the precision of fractal dimensions based on the Weierstrass–Mandelbort (W–M) function, we found the rescaled (R/S) analysis method and the structure spectral method were not suitable for the calculation of the fractal dimension on a soft terrain surface. Therefore, the fractal dimension, non-scale range and correlation coefficient for each kind of terrain were analyzed using the following fractal computational methods – (i) variate-difference method, (ii) power spectral density method, and (iii) root mean square method. The results showed that: fractal dimen- sion of plowed terrain was large with small fluctuations, while its internal structure was complex. The power spectral density method was not robust enough to compute the fractal dimension of a soft terrain surface. The fractal dimension computed using the root mean square method was found to be more accurate for the soft terrain surface. Moreover, the correlation coefficient of linear regression when using the root mean square method was good and the range of non-scale variation was small. Ó 2017 ISTVS. Published by Elsevier Ltd. All rights reserved. Keywords: Soft terrain surface; Root mean square method; Fractal theory; W–M function 1. Introduction In the 1970s, a French mathematician named Mandelbrot (1982) proposed the fractal theory which has been applied in the research area of nonlinear systems. He indicated that models witnessed at different scales could be connected to each other by a power function, and he called its index ‘fractal dimension’. It has been expanded upon by Feder (1988) and has been widely used in the study of river networks (Rosso, 1991; La Barbera and Rosso, 1989; Feng, 1997) and geometric shapes having rough and irregular surfaces. Donald (1997) reported that the applica- tions of fractal geometry in the earth and geological sciences have been summarized. Zhou et al. (2006) used fractal dimension to study the soil structure on the Ziwuling Mountains at various steps of plant succession. The fractal theory is widely used all over the world in the field of soil science (Cai et al., 2011; Zhou, 2009) as it provides a very good representation of the characteristics of soil which is irregular, unstable and has a highly complex structure. A good method of calculating fractal dimension could assist in effectively characterizing soil surface. At present, the major techniques of computing fractal dimension include variate-difference method, power spectral density method, structure function method, root mean square method and rescaled (R/S) analysis method. Each compu- tational method has a different application is suitable for a specific nonlinear system. If an inappropriate fractal dimension computation method is inconsistent with actual fractal sets, this will generate a huge error (Majumdar and Bhushan, 1990; Zhu and Ge, 2004). For example, if one method used to calculate the fractal dimension of a soil http://dx.doi.org/10.1016/j.jterra.2017.01.003 0022-4898/Ó 2017 ISTVS. Published by Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail addresses: chunxia_njau@163.com (C. Jiang), luzx@njau.edu. cn (Z. Lu), zhoujing306@163.com (J. Zhou), memon.sohail@outlook.com (M.S. Memon). www.elsevier.com/locate/jterra Available online at www.sciencedirect.com ScienceDirect Journal of Terramechanics 70 (2017) 27–34 Journal of Terramechanics