## 1 Introduction  
The characteristics of soil particle distribution and hydraulic properties (hereafter \"soil properties\") are the basis for the study of soil moisture and solute transport [1]. However, due to soil heterogeneity, soil properties vary spatially. The determination of distribution characteristics of the soil properties at a certain scale is significant for reasonable sampling and parameter prediction. Formerly, classical statistics and geostatistics were the main methods used to study spatial variation. Geostatistics establishes the relationship between lag distance and variables through the semivariance function, which is the main means of quantitative analysis of spatial variability and the basis of Kriging interpolation. However, geostatistics is frequently limited by the sample capacity, distribution and anisotropy, which always leads to errors in practical application. With the advancement of fractal theory, the validity and potential of applying fractals to analyse soil spatial variability have been verified by early research over the past decades; they are thus an important tool with which to quantify the spatial variability and scale transformation of soil properties [2]. The combination of fractals and geostatistics can realise the quantitative characterisation of soil property spatial variation [3]; that is, the fractal dimension increases with the distribution uniformity. Researchers all over the world have calculated the fractal dimension values of various soil properties and environmental variables of different soil types. Among them, the representative studies include soil organic matter [4; 5], total nitrogen [6], pH value [7], iron manganese nodules [8], and so forth. Due to different soil types, research scales and sample capacity, the fractal dimension values are also varied, but they are all in the range of 1.7\(\sim\)1.96. A single fractal dimension is simple and convenient, but it can only represent the holistic spatial variability of soil properties [9]. With the development of multifractal theory, the multifractal spectrum is used to quantitatively evaluate the spatial variability of soil properties. Multifractal parameters can not only reflect the spatial variation of the soil properties, but also reveal the small-scale or partial characteristics embedded in the holistic space [10]. [PERSON] and [PERSON] [11], [PERSON] et al. [2], [PERSON] et al. [12] and [PERSON] et al. [13] have analysed the spatial variability of soil properties, such as electrical conductivity, organic matter content, soil pH value, soil nitrate, soil water and salt, according to multifractal theory and obtained good results. In recent years, the aforementioned methods have been used to study the spatial variation of soil properties such as soil particle distribution [14; 15; 16], bulk density [17; 18], saturated hydraulic conductivity [19; 20; 21; 22] and unsaturated hydraulic conductivity [23; 24], allowing researchers to make a series of achievements. However, multifractal research is mostly limited to the farmland scale [13; 25; 26; 27; 28]. The study of multifractals when the scale increases gradually (with a sampling interval of approximately several hundred metres) are rarely reported in the literature. In addition, there is a less joint application of multiple methods and a lack of comparative study [29; 30]. Yanlou township in Lankao county is in a former old channel of the Yellow River and the Yellow River flood area. Owing to the frequent sedimentary processes and human activities, the soil structure is complex and strongly changes spatially, which is typical of the alluvial plain area in the lower Yellow River area. Therefore, this study focuses on Yanlou township and comprehensively analyses the spatial variation characteristics of six soil properties in the research area by classical statistics, geostatistics and single fractal and multifractal methods. Meanwhile, the causes of the spatial distribution of soil properties were analysed according to the spatial variability.