The alternative extended model-free gradient

Because of the equation S² = S²_f⋅S²_s and the form of the extended spectral density function (15.63) a convolution of the model-free space occurs if the model-free parameters {S²_f, S²_s, τ_f, τ_s} are optimised rather than the parameters {S², S²_f, τ_f, τ_s}. This convolution increases the complexity of the gradient. For completeness the first partial derivatives are presented below.

$\mathfrak{G}_j$ partial derivative

The partial derivative of (15.63) with respect to the geometric parameter $\mathfrak{G}_j$ is

$$\begin{multline} \frac{\partial J(\omega)}{\partial \mathfrak{G}_j} = \frac{2}{... ...{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2} \Bigg) \Bigg). \end{multline}$$

$\mathfrak{O}_j$ partial derivative

The partial derivative of (15.63) with respect to the orientational parameter $\mathfrak{O}_j$ is

$${\frac{{\partial J(\omega)}}{{\partial \mathfrak{O}_j}}} {\frac{{2}}{{5}}} \sum_{{i=-k}}^{k} {\frac{{\partial c_i}}{{\partial \mathfrak{O}_j}}} \Bigg( {\frac{{S^2_f \cdot S^2_s}}{{1 + (\omega \tau_i)^2}}} {\frac{{(1 - S^2_f)(\tau_f + \tau_i)\tau_f}}{{(\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2}}} {\frac{{S^2_f(1 - S^2_s)(\tau_s + \tau_i)\tau_s}}{{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}}} \Bigg)$$ (15.91)

S²_f partial derivative

The partial derivative of (15.63) with respect to the order parameter S²_f is

$${\frac{{\partial J(\omega)}}{{\partial S^2_f}}} {\frac{{2}}{{5}}} \sum_{{i=-k}}^{k} \Bigg( {\frac{{S^2_s}}{{1 + (\omega \tau_i)^2}}} {\frac{{(\tau_f + \tau_i)\tau_f}}{{(\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2}}} {\frac{{(1 - S^2_s) (\tau_s + \tau_i)\tau_s}}{{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}}} \Bigg)$$ (15.92)

S²_s partial derivative

The partial derivative of (15.63) with respect to the order parameter S²_s is

$${\frac{{\partial J(\omega)}}{{\partial S^2_s}}} {\frac{{2}}{{5}}} \sum_{{i=-k}}^{k} \Bigg( {\frac{{1}}{{1 + (\omega \tau_i)^2}}} {\frac{{(\tau_s + \tau_i)\tau_s}}{{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}}} \Bigg)$$ (15.93)

τ_f partial derivative

The partial derivative of (15.63) with respect to the correlation time τ_f is

$${\frac{{\partial J(\omega)}}{{\partial \tau_f}}} {\frac{{2}}{{5}}} \sum_{{i=-k}}^{k} {\frac{{(\tau_f + \tau_i)^2 - (\omega \tau_f \tau_i)^2}}{{\left((\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2 \right)^2}}}$$ (15.94)

τ_s partial derivative

The partial derivative of (15.63) with respect to the correlation time τ_s is

$${\frac{{\partial J(\omega)}}{{\partial \tau_s}}} {\frac{{2}}{{5}}} \sum_{{i=-k}}^{k} {\frac{{(\tau_s + \tau_i)^2 - (\omega \tau_s \tau_i)^2}}{{\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^2}}}$$ (15.95)