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土壤的体积颗粒分布(PSD)可用于同时确定数量分形维数(DN)和体积分形维数(DV)。以往的试验和理论研究均未直接比较DV和DN的关系。为建立DV和DN的关系并分析其参数的敏感程度,在中国各地取了12种不同质地土壤,利用激光粒度仪进行了测定。提出了含3个参数Rupper(颗粒半径的上限),Ri(第i个半径)和Rlower(半径下限)的DV-DN理论关系,该关系可说明为什么DN可能大于3。在未调整的幂定律范围内(PLR),DV的幂定律半径为38.6～85.8μm,DN的幂定律半径为53.2～358μm,估算的DV值(2.18～2.69)比DN值(2.38～3.19)小。调整后的PLR比未调整的窄。在调整的PLRs范围内,估算的DV值变化范围为2.11～2.56,而DN变化范围为2.28～3.02。调整和未调整PLRs得出的DV和DN的差异说明,Rupper,Rlower,Ri和Rlower/Rupper均可能影响DV与DN关系,有必要进行参数的敏感性分析。进行敏感性分析可基于敏感系数(C)来鉴别DV-DN关系中各参数改变所引起的DV和DN相对变化,从而找出对该关系影响最大的参数。就不同情况下计算的C值而言,Rupper是影响DN值的最重要参数,小的Rupper值可能导致估算DN的准确度降低。PLR范围越大,DN的估算越准确。根据DV估算DN的相对误差的绝对值在12%范围内,说明得出的DV-DN理论关系是正确的。
Soil volume particle distribution (PSD) can be used to simultaneously determine the number of fractal dimensions (DN) and the volume fractal dimension (DV). Previous experimental and theoretical studies did not directly compare the relationship between DV and DN. In order to establish the relationship between DV and DN and analyze the sensitivity of its parameters, 12 different soil samples were taken from different parts of China, and measured by laser particle sizer. A DV-DN theoretical relationship with three parameters, Rupper (upper limit of particle radius), Ri (ith radius) and Rlower (lower limit of radius) is proposed, which shows why DN may be greater than three. In unadjusted power law (PLR), DV has a power law radius of 38.6-85.8μm, a power law of DN of 53.2-358μm, an estimated DV value of 2.18-2.69 and a DN value of 2.38-3.19, small. The adjusted PLR is narrower than the unadjusted one. Within the adjusted range of PLRs, the estimated DV values ranged from 2.11 to 2.56, while the DN ranged from 2.28 to 3.02. Differences between DV and DN derived from adjusted and unadjusted PLRs suggest that Rupper, Rlower, Ri, and Rlower / Rupper may affect the relationship between DV and DN, and sensitivity analysis of parameters is necessary. Sensitivity analysis can be based on the sensitivity coefficient (C) to identify the DV-DN relationship between DV and DN changes caused by the relative changes in order to find the most influential parameters of the relationship. For the C values calculated under different conditions, Rupper is the most important parameter that affects the DN value, and a small Rupper value may result in a decrease in the accuracy of the estimated DN. The larger the PLR range, the more accurate the DN estimate. According to DV, the absolute value of relative error of DN is within 12%, indicating that the theoretical relationship of DV-DN is correct.