An adapted Weibull function for agricultural applications

Abstract

The Weibull function is applied extensively in the life sciences and engineering, but under-used in agriculture. The function was consequently adapted to include parameters and metrics that increase its utility for characterizing agricultural processes. The parameters included initial and final dependent variables (Y0 and YF, respectively), initial independent variable (x0), a scale constant (k), and a shape constant (c). The primary metrics included mode, integral average, domain, skewness and kurtosis. Nested within the Weibull function are the Mitscherlich and Rayleigh functions where c is fixed at 1 and 2, respectively. At least one of the three models provided an excellent fit to six example agricultural datasets, as evidenced by large adjusted coefficient of determination (RA2 ≥ 0.9266), small normalized mean bias error (MBEN ≤ 1.49 %), and small normalized standard error of regression (SERN ≤ 8.08 %). The Mitscherlich function provided the most probable (PX) representation of corn (Zea Mays L.) yield (PM = 87.2 %), Rayleigh was most probable for soil organic carbon depth profile (PR = 96.4 %), and Weibull was most probable for corn seedling emergence (PW = 100 %), nitrous oxide emissions (PW = 100 %), nitrogen mineralization (PW = 58.4 %), and soil water desorption (PW = 100 %). The Weibull fit to the desorption data was also equivalent to those of the well-established van Genuchten and Groenevelt-Grant desorption models. It was concluded that the adapted Weibull function has good potential for widespread and informative application to agricultural data and processes.

Beware of scaling artefacts and implicit model characteristics when fitting soil water release and moisture capacity data

“…Some versatile and widely used soil water release relationships include those of van Genuchten (1980), Assouline et al (1998), Grant (2004), andDexter et al (2008). An adapted Weibull relationship was also proposed recently by Reynolds et al (2021).…”

“…Generally speaking, larger R A 2 , smaller MPE N , and smaller SER N indicates better goodness of model-data fit, with "perfect fit" indicated by R A 2 = 1, MPE N = 0, and SER N = 0. As described in Reynolds et al (2021), 0 ≤ SER N < 10% indicates excellent model prediction of the data; 10% ≤ SER N ≤ 20% signifies good prediction; 20% < SER N ≤ 30% indicates fair prediction; and SER N > 30% signifies poor prediction. Further detail on the R A 2 , MBE N , and SER N calculations can be found in Reynolds et al (2021).…”

“…As described in Reynolds et al (2021), 0 ≤ SER N < 10% indicates excellent model prediction of the data; 10% ≤ SER N ≤ 20% signifies good prediction; 20% < SER N ≤ 30% indicates fair prediction; and SER N > 30% signifies poor prediction. Further detail on the R A 2 , MBE N , and SER N calculations can be found in Reynolds et al (2021).…”

“…Due to the extreme complexity of soil pore water systems, θ versus h functions are almost exclusively empirical, and many parametric forms have been proposed over the past 60+ years (e.g., Gardner 1956;Brooks and Corey 1964;Visser 1966;Campbell 1974;Haverkamp et al 1977;van Genuchten 1980;Fredlund and Xing 1994;Assouline et al 1998;Groenevelt and Grant 2004;Dexter et al 2008;Reynolds et al 2021). The ability of these empirical functions to fit and predict θ versus h data (and dθ /dh vs. h data) varies substantially, however, as they contain both obvious and implicit characteristics that may or may not be compatible with actual data behaviour.…”

Assessing performance of empirical models for forecasting crop responses to variable fertilizer rates using on-farm precision experimentation

“…The first model type was parametric and modified from a non-linear sigmoidal model called the beta function (Yin et al, 2003). Following the literature, a model that assumes an asymptotic nature of crop response curves to additional N fertilizer was tested here (Anselin et al, 2004;Reynolds et al, 2021;Yin et al, 2003). Initial tests compared the beta function to multiple forms of non-linear models including hyperbolic, Richards, Gompertz, Weibull and logistic forms.…”