On the other hand, sellectchem RMSE deciles of the log-cubic method are all less than those of the log-linear method and less than or slightly greater than those of the log-spline method. As far as RMSE is concerned, the log-cubic method outperforms the log-linear and log-spline methods for 95% and 84% soil samples, respectively. Therefore, the log-cubic method is superior to the log-linear and log-spline methods for interpolating PSDs.Figure 2ME (a) and RMSE (b) deciles of the log-linear, log-spline, and log-cubic methods.Nemes et al. (1999) [15] used four procedures to estimate cumulative percentage at unmeasured particle size, including log-linear and spline interpolation methods, Gompertz curve regression method, and similarity method.

Their results of cross-validation indicate that the similarity method is the most effective, which yielded the lowest RMSE ranging from 2% to 11% for different distances between particle size limits. RMSE value of the present log-cubic method is 6.3%, which corresponds to lower or medium RMSE values of the similarity method at smaller distances between particle size limits [15]. Moreover, the log-cubic method does not require a large external reference data set and can be used to estimate cumulative particle percentages at any size in the PSD range. The computation of the log-cubic method is much simpler than that of the similarity method. Therefore, the log-cubic method is appropriate to be used in the estimation of cumulative percentage at unmeasured particle size.3.2.

Comparison of Generated PSD CurvesUsing the log-linear, log-spline, and log-cubic methods, cumulative particle percentages at unmeasured sizes can be estimated from measured data and used for the generation of PSD curve. Figure 3 illustrates two PSD curves for two soil samples generated from 7 measured data. All generated PSD curves pass through measured data, which is an essential requirement of interpolation methods. For the log-linear method, the generated PSD curves are monotone but not smooth. For the log-spline method, the generated PSD curves are smooth but not monotone since an interpolated cubic spline is monotone only in specified conditions of measured data [17]. Impractical fluctuations in these PSD curves show that the log-spline method itself is not appropriate to generate a PSD curve, unless it is modified in some way [8]. While for the log-cubic method, the generated PSD curves are both smooth and monotone, which meet the essential requirements of a PSD curve. PSD curves of other soils generated with the log-cubic method all follow these essential requirements. These characters are Dacomitinib the same as those of traditional freehand PSD curve, but the present result is objective and independent on persons involved.