## Appendix B Parametric Double S-Shaped Functions
### Sine Functions  
The Parametric Double Sine Function (PD_Sine_F) is composed of 2 sine functions, each characterized by its own set pf parameters:  
\[f(x) =p_{0}+f_{1}(x)+f_{2}(x), \tag{108}\] \[f_{1}(x) =\left\{\begin{array}{ll}0.0&\quad\text{if }x<p_{2}\\ p_{1}\,(1.0+\sin(-(\pi/2.0)+((x-p_{2})/(p_{3}-p_{2}))\,\pi))/2.0&\quad\text{if }p_{2}\leq x \leq p_{3}\\ p_{1}&\quad\text{if }p_{3}<x\end{array}\right.,\] (109) \[f_{2}(x) =\left\{\begin{array}{ll}0.0&\quad\text{if }x<p_{5}\\ -p_{4}\,(\sin((\pi/2.0)+((x-p_{5})/(p_{6}-p_{5})\,\pi))-1.0)/2.0&\quad\text{if }p_{5}\leq x \leq p_{6}\\ -p_{4}&\quad\text{if }p_{6}<x\end{array}\right., \tag{110}\]  
where  
\(\circ\quad f(x)\) is the value of the Parametric Double Sine Function at argument \(x\), returned by the IDL function pd_sine_f, \(\circ\quad f_{1}(x)\) is the value of the first Parametric Sine Function at argument \(x\), using parameters \(p_{1}\) to \(p_{3}\) and provided as the output positional parameter psf1, and \(\circ\quad f_{2}(x)\) is the value of the second Parametric Sine Function at argument \(x\), using parameters \(p_{4}\) to \(p_{6}\) and provided as the output positional parameter psf2. The 7 parameters have the following meanings: \(\circ\quad p_{0}=\) Base value of \(f(x)\), i.e., asymptotic value when \(x\to-\infty\). \(\circ\quad p_{1}=\) Amplitude of \(f_{1}(x)\) (positive for an increase, negative for a decrease), independently from the base value.  
\(p_{2}\) = Phase shift along the \(x\) axis for the start of \(f_{1}(x)\): a positive (negative) value shifts the function to the right (left) of \(x=0\).
* \(p_{3}\) = Phase shift along the \(x\) axis for the end of \(f_{1}(x)\): a positive (negative) value shifts the function to the right (left) of \(x=0\).
* \(p_{4}\) = Amplitude of \(f_{2}(x)\), independently from the base value.
* \(p_{5}\) = Phase shift along the \(x\) axis for the start of \(f_{2}(x)\): a positive (negative) value shifts the function to the right (left) of \(x=0\).
* \(p_{6}\) = Phase shift along the \(x\) axis for the end of \(f_{2}(x)\): a positive (negative) value shifts the function to the right (left) of \(x=0\).  
The partial derivatives of the PD_Sine_F model with respect to the 7 parameters are as follows:  
\[\partial f(x)/\partial p_{i} =\partial p_{0}/\partial p_{i}+\partial f_{1}(x)/\partial p_{i}+ \partial f_{2}(x)/\partial p_{i}, \tag{101}\] \[\partial p_{0}/\partial p_{i} =\left\{\begin{array}{ll}1.0&\text{if }i=0\\ 0.0&\text{if }i>0\end{array}\right.,\] (102) \[\partial f_{1}(x)/\partial p_{1} =\left[1-\cos\left(\frac{\pi\left(x-p_{2}\right)}{p_{3}-p_{2}} \right)\right]/2,\] (103) \[\partial f_{1}(x)/\partial p_{2} =\left[\pi p_{1}\left(x-p_{3}\right)\sin\left(\frac{\pi\left(x-p _{2}\right)}{p_{3}-p_{2}}\right)\right]/\left(2\left(p_{2}-p_{3}\right)^{2} \right),\] (104) \[\partial f_{1}(x)/\partial p_{3} =-\left[\pi p_{1}\left(x-p_{2}\right)\sin\left(\frac{\pi\left(x- p_{2}\right)}{p_{3}-p_{2}}\right)\right]/\left(2\left(p_{3}-p_{2}\right)^{2} \right),\] (105) \[\partial f_{1}(x)/\partial p_{4} =\left[1-\cos\left(\frac{\pi\left(x-p_{5}\right)}{p_{6}-p_{5}} \right)\right]/2,\] (106) \[\partial f_{1}(x)/\partial p_{5} =\left[\pi p_{4}\left(x-p_{6}\right)\sin\left(\frac{\pi\left(x- p_{5}\right)}{p_{6}-p_{5}}\right)\right]/\left(2\left(p_{5}-p_{6}\right)^{2} \right),\] (107) \[\partial f_{1}(x)/\partial p_{6} =-\left[\pi p_{4}\left(x-p_{5}\right)\sin\left(\frac{\pi\left(x- p_{5}\right)}{p_{6}-p_{5}}\right)\right]/\left(2\left(p_{6}-p_{5}\right)^{2} \right), \tag{108}\]  
and all other partial derivatives (e.g., \(\partial f_{1}(x)/\partial p_{0}\), \(\partial f_{1}(x)/\partial p_{5}\) or \(\partial f_{2}(x)/\partial p_{3}\)) are null.