Local fractal and multifractal characteristics of soil number-based particle size distributions | 土壤颗粒数量分布的局部分形及多重分形特性

Китайский. 采用定量方法全面判断土壤颗粒分布（PSD）的内在结构，以提高利用土壤PSD函数估测土壤水动力参数的精度。利用激光粒度仪测定红胶土、娄土、黑垆土和砂黄土4种黄土体积PSD，计算得出土壤颗粒数量PSD，然后基于分形理论研究了粒度的幂定律适用范围及相应的分形维数；在符合幂定律的粒度范围内进行多重分形计算，获得了土壤数量PSD的多重分形参数。分形分析结果表明，红胶土、娄土、黑垆土和砂黄土4种土壤符合幂定律的颗粒半径的下限均接近0.5 μm，而上限依不同土壤差别很大，相应的分形维数为2.17~3.43；多重分形分析表明，红胶土广义分形维数Dq值在整个q区间上变化最明显，显示出比其他土壤更强的颗粒分布不均匀性；4种土壤的τ(q)-q曲线均明显异于直线，表明4种土壤均具有多重分形分布特征。红胶土、娄土、黑垆土和砂黄土的f（α）谱宽分别为4.30，1.96，1.74和1.25，谱的不对称系数分别为-0.924，0.516，0.141 和0.490，表明红胶土的多重分形结构较其他3种土壤更明显，其不对称程度最大。多重分形理论能更细致地刻画土壤颗粒的数量分布特征；将分形和多重分形理论相结合可以更加全面地了解土壤颗粒分布的内在结构。

Английский. The inner structures of soil particle size distribution (PSD) were judged thoroughly by quantitative methods to increase the accuracy of soil hydraulic parameters estimation from soil PSD functions. Volume-based PSD data of four loess soils including Hongjiao soil, Lou soil, Heilu soil and Sand loess were measured by laser diffractometry and used to determine soil number-based PSD. The power law domain of particles size and the corresponding fractal dimensions were defined. Multifractal calculations were conducted within the particle range that followed the power law and the multifractal parameters were obtained. The fractal analysis showed that the lower particle limits of the power law domain for the four soils were close to 0.5 μm, while the upper limits differed a lot for different soils. The calculated fractal dimensions ranged from 2.17 to 3.43.From the multifractal analysis, the generalized dimension, Dq value of Hongjiao soil had the most obvious variations along regions of q and indicated stronger heterogeneous distribution of number-based PSD. The curves ofτ(q)-q for all 4 soils were concave downwards and quite different from straight lines, indicating the number-based PSD of all 4 soils had multifractal distributions. Widths of f(α) spectra for Hongjiao soil, Lou soil, Heilu soil and Sand loess were 4.30, 1.96, 1.74 and 1.25, respectively. The asymmetric coefficients of the spectra were -0.924, 0.516, 0.141 and 0.490, respectively, indicating that the multifractal structures of Hongjiao soil were more complicated than the other soils, its asymmeric extent was also the largest. Multifractal theory can depict number-based soil particle size distributions more in detail. Combining fractal with multifractal theory can understand the inner structures of soil PSD more comprehensively.