Relationship between volume-based and number-based fractal dimensions of soil particle size distributions

A volumetric soil particle size distribution (PSD) can be used to determine volume- (DV) and number (DN)-based fractal dimensions simultaneously. Direct comparisons between DV and DN were not available before by theoretical and experimental methods. To present the theoretical relationship between DV and DN and analyze the sensitivity of parameters, twelve soil samples representing different soil textures were taken at different locations in China and analyzed by laser diffractometry. The theoretical relationship between DV and DN was developed with 3 parameters including Rupper (upper limit of particle radius), Ri (the ith radius) and Rlower (lower limit). The relationship can be used to explain why DN can be greater than 3. The estimated DV values (2.18 to 2.69) were smaller than the DN values (2.38 and 3.19) within unadjusted power-law ranges (PLR) which ranged from 38.6 to 85.8 μm for DV and 53.2 to 358 μm for DN. The adjusted PLR for estimating DV and DN were narrower than the unadjusted PLRs. Within the adjusted PLRs, the estimated DV values were from 2.11 to 2.56 and the DN values were from 2.28 to 3.02. The difference between DV and DN values estimated from adjusted and unadjusted PLRs implied that Rupper, Rlower, Ri and Rlower/Rupper can affect the DV and DN relationship, and sensitivity analysis of parameters is essential. Sensitivity analysis can distinguish the relative variations of the parameters in DV-DN relationship using coefficients of sensitivity (C), through which the most important parameter that affects DV-DN relationship can be determined. Based on the C values at different circumstances, Rupper was found to be the most important factor that influenced the estimation of DN values. A relatively small Rupper may result in a less accurate estimation of DN. Larger PLR can result in more accurate estimation of DN. The relatively small absolute values of relative errors (<12%) between calculated DN and observed DV implied the correctness of relationship between DV and DN. The research can provide a reference for estimating number-based fractal dimension based on volume-based fractal dimension calculated from soil particle size distributions.