Analysis of phosphorus soil sorption data: improved results from global least-squares fitting

Phosphate sorption data are often analyzed by least-squares fitting to the two- or three-parameter Freundlich model. The standard methods are flawed by (1) treating the measured pseudo-equilibrium concentration C as the independent (hence error-free) variable and (2) neglecting the weighting that should accommodate the varying precision of the data. Here, we address both of these shortfalls and use a global fit model to achieve optimal precision in fitting data for five acidic Australian soil types. Each individual dataset consists of measured C values for up to nine phosphate spiking levels C₀. For each soil type, there are three–five such datasets from varying levels of phosphate fertilizer pre-exposure (P_f) two years earlier. These datasets are fitted simultaneously by expressing the Freundlich capacity factor a and exponent b as theoretically predicted functions of the assay amounts of Fe, Al, and P measured for each P_f. The analysis allows for uncertainty in both C and C₀, with inverse-variance weighting from variance functions estimated by residuals analysis. The estimated presorbed P amounts Q depend linearly on P_f, with positive intercepts at P_f = 0, indicating residual phosphate in the soils prior to the laboratory phosphate treatments. The key takeaway points are as follows: (1) global analysis yields optimal estimates and improved precision for the fit parameters; (2) allowing for uncertainty in C is essential when the data include C values near 0; (3) varying data precision requires weighting to yield optimal parameter estimates and reliable uncertainties.