The effect of uncertainty in predictions of nutrient concentrations from soil spectra on variable-rate fertilizer applications

The concentrations of available phosphorus (P) and potassium (K) in soil can be estimated by soil spectroscopy, and with sufficient sampling can be mapped to guide farmers to apply fertilizer at variable rates. Mapping errors arise from both spatial variation and calibration of the spectra against chemically determined concentrations. We aimed to develop a loss-function framework to explore how sizes of sample sets and calibration sets affect the likely profitability of variable-rate applications of P and K fertilizers. We demonstrate the approach through simulation. Based on our previous observations of variation in P and K from four fields in Cambridgeshire, England, we generated 100 realizations of P and K in each field using geostatistical simulation. We did so with various combinations of sizes of total sample and calibration set. For each such sample we assigned various proportions for calibration on which notionally both soil spectroscopy and chemical concentrations were determined. Knowing the costs for labour in the field for sampling, the preparation of soil in the laboratory, the spectroscopy, chemical analysis and amortization of equipment, we estimated the costs of acquiring data. Set against these were the costs of error, i.e. of uncertainty, in the final predictions by kriging arising from calibration error and spatial variation. For each combination we computed the fertilizer required to minimize the expected loss associated with predictions, where the expected loss is the difference in profit between applying fertilizer for the estimated concentrations of P and K and their true concentrations. The size of the calibration set outweighed the effect of total sample size on the uncertainty associated with predictions. Equally, for the same size of calibration set, there were large differences in the kriging variances between total sample sizes. When the costs of acquiring data were disregarded, the expected loss for available P was strongly affected by the total sample size. For available K, the effect of the size of the calibration sample dominated the expected loss. The expected loss showed diminishing returns with increasing sample size. None of the sample sizes considered would result in a financial gain: spectroscopy needs to become cheaper for it to be cost-effective for variable-rate applications of P and K fertilizer.