Logistic sigmoid function for modeling Phenology detailed in Elmore et. al 2011 (1) & Gao et al. 2011 (2). The function combines spring and autumn seasons into a single equation to account for the “greendown” phenomena by allowing for a gradual reduction of VI values in mid-summer. Typically the model is fit to all VI observations within a single year using Maximum Likelihood methods.

\[ v(t, M) = m_1 + (m_2 - m_7t)\left(\frac{1}{1+e^{(m_3-t)/m_4}}-\frac{1}{1 + e^{(m_5-t)/m_6}}\right) \]

where

• \(v(t,m)\) is the modeled VI value at time \(t\) in terms of day of year (DOY)
  • \(M = [m_1, \dots , m_7]^T\) is a parameter vector controlling the shape of the double-logistic function.
    • \(m_1\) represents the mean VI value in the dormant period;
    • \((m_2 − m_7t)\) represents the difference between VI value at time \(t\) and the dormant mean;
    • \(m_3\) and \(m_5\) are the inflection points that are commonly used to represent start-of-season (SOS) and end-of-season (EOS) dates in terms of DOY in the spring and autumn;
    • \(m_4\) and \(m_6\) are the corresponding slopes of the VI trajectories in spring and autumn;
    • \(m_7\) is the “greendown” parameter that accounts for the VI greendown phenomenon in the mid-summer time.

References

[1] Elmore, Andrew J. and Guinn, Steven M. and Minsley, Burke J. and Richardson, Andrew D., Landscape controls on the timing of spring, autumn, and growing season length in mid-Atlantic forests, Wiley, 2011.

[2] Gao, Xiaojie and Gray, Josh M. and Reich, Brian J., Long-term, medium spatial resolution annual land surface phenology with a Bayesian hierarchical model, Elsevier BV, 2021.