Appendix 1: Second-order derivation for sample terms
1.1 Sample term A
The ET is a function of current soil moisture conditions and can be represented as follows:
$${\text{ET}}_{t} = \left\{ {\begin{array}{*{20}l} {0} & {{\theta _{t} \leq \theta _{{pwp}} }} \\ {{\frac{{{\text{ET}}_{{{\rm p}_{t} }} (\theta _{t} - \theta _{{pwp}} )}}{{(1 - p)(\theta _{{{\text{FC}}}} - \theta _{{pwp}} )}}}} & {{\theta _{{pwp}} \leq \theta _{t} < (1 - p)(\theta _{{{\text{FC}}}} - \theta _{{pwp}} )}} \\ {{{\text{ET}}_{{{\rm p}_{t} }} }} & {{\theta _{t} \geq (1 - p)(\theta _{{{\text{FC}}}} - \theta _{{pwp}} )}} \\ \end{array} } \right..$$
Now considered the term
$$E(\theta _{t}1_{{(\theta _{{pwp}} \leq \theta _{t} \leq (1 - p)(\theta_{{\rm FC}} - \theta _{{pwp}}))}} (\theta _{t}))$$
from the second moment expression in actual ET equation. This term can be divided into following three individual parts:
$$\begin{aligned}\, & nZ_{t}E(\theta _{{t - 1}}1_{{(\theta _{{pwp}} \leq \theta _{t} \leq (1 - p)(\theta _{{\rm FC}} - \theta _{{pwp}}))}} (\theta _{t})) + E((\hbox{Ir}_{t} + \hbox{Ra}_{t} + n(z_{t} - z_{{t - 1}})\theta _{r} - \hbox{ET}_{t}) \\&\quad 1_{{(\theta _{{pwp}} \leq \theta _{t} \leq (1 - p)(\theta _{{\rm FC}} - \theta _{{pwp}}))}} (\theta _{t})) + E{\left({\eta _{t} 1_{{(\theta _{{pwp}} \leq \theta _{t} \leq (1 - p)(\theta _{{\rm FC}} - \theta _{{pwp}}))}} (\theta _{t})} \right)}. \\\end{aligned}$$
(20)
The second-order approximation of two first parts is calculated next and the third part is determined similarly but not shown here.
AI. Using the property of the indicator function, the first term of Eq. 20, AI, can be expressed as follows:
$$nz_{t} {\int\limits_{\theta ^{{\min}}_{{t - 1}}}^{\theta ^{{\max}}_{{t - 1}}} {\,{\int\limits_l^u {f_{{\eta _{t}}} (\eta _{t})}}}}\,\theta _{{t - 1}} f_{\theta_{{t - 1}}} (\theta _{{t - 1}})\,\hbox{d}_{{\eta _{t}}} \hbox{d}_{{\theta _{{t - 1}}}} $$
(21)
\(f_{{\eta _{t}}} (\eta _{t})\) is the probability density function of the random noise term, η
t
∼ N[0,Var(η
t
)], for the current work.
\(f_{\theta_{t - 1}} (\theta _{{t - 1}})\) is the density function of the soil moisture state variable in the previous period. u=nz
t
θ
max
t
− (Ir
t
+Ra
t
+n(z
t
− zt−1)θ
r
− ET
t
) − nzt−1 θt−1 and l=nz
t
θ
min
t
− (Ir
t
+Ra
t
+n(z
t
− zt−1)θ
r
− ET
t
) − nzt−1 θt−1 are the upper and lower limits, respectively. (θt−1) and η
t
are assumed independent; thus it can be written:
$$nz_{t} {\int\limits_{\theta ^{{\min}}_{{t - 1}}}^{\theta ^{{\max}}_{{t - 1}}} {{\left[ {{\int\limits_l^u {f_{{\eta _{t}}} (\eta _{t}) \hbox{d}_{{\eta _{t}}}}}} \right]}}}\,\theta _{{t - 1}} f_{\theta{t - 1}} (\theta _{{t - 1}})\,\hbox{d}_{{\theta {t - 1}}},$$
(22)
which is the expectation of a function of (θ t−1) and it can be shown as:
$$ = nz_{t} E{\left\{{\theta _{{t - 1}} {\left[ {{\int\limits_l^u {f_{{\eta _{t}}} (\eta _{t})\hbox{d}_{{\eta _{t}}}}}} \right]}} \right\}},$$
(23)
where
\(f_{{\eta _{t}}} (\eta _{t})\) is presented as follows:
$$f_{{\eta _{t}}} (\eta _{t}) = \frac{1}{{{\sqrt {2\pi \hbox{Var}(\eta _{t})}}}}\hbox{exp} {\left[ {- \frac{1}{2}{\left({\frac{{(\eta _{t})^{2}}}{{\hbox{Var}(\eta _{t})}}} \right)}} \right]}.$$
(24)
For the current study, the integral in Eq. 23 is represented as follows:
$$\begin{aligned} \, & E\left({nZ_{{t - 1}} \theta _{{t - 1}} \hbox{Pr}\{ nz_{t} \theta ^{{\min }}_{t} - ({\text{Ir}}_{t} + {\text{Ra}}_{t} + n(z_{t} - z_{{t - 1}})\theta _{r} - {\text{ET}}_{t}) - nz_{{t - 1}} (\theta _{{t - 1}}) < \eta _{t}} \right. \\ & \quad \left. { \leq nz_{t} \theta ^{{\max }}_{t} - ({\text{Ir}}_{t} + {\text{Ra}}_{t} + n(z_{t} - z_{{t - 1}})\theta _{r} - {\text{ET}}_{t}) - nz_{{t - 1}} (\theta _{{t - 1}})\} } \right). \\ \end{aligned}$$
(25)
$$nZ_{{t - 1}} E\left({\theta _{{t - 1}} \hbox{Pr}\{ \eta } \right._{t} \left. { \leq - nz_{{t - 1}} (\theta _{{t - 1}})\} } \right)^{{nz_{t} \theta ^{{\max }}_{t} - ({\text{Ir}}_{t} + {\text{Ra}}_{t} + n(z_{t} - z_{{t - 1}})\theta _{r} - {\text{ET}}_{t} }}_{{nz_{t} \theta ^{{\min }}_{t} - ({\text{Ir}}_{t} + {\text{Ra}}_{t} + n(z_{t} - z_{{t - 1}})\theta _{r} - {\text{ET}}_{t} }}.$$
(26)
The second-order Taylor’s series approximation of the expectation of Eq. 23 is thus given as:
$$\begin{aligned}\,& \left\{ {nz_{{t - 1}} E(\theta _{{t - 1}})\left\{ {\frac{1}{2}\left({{\text{erf}}{\left({\frac{{nz_{t} \theta ^{{\max }}_{t} - ({\text{Ir}}_{t} + {\text{Ra}}_{t} + n(z_{t} - z_{{t - 1}})\theta _{r} - {\text{ET}}_{t}) - nz_{t} E(\theta _{{t - 1}})}}{{{\sqrt {2{\text{Var}}\,\eta _{t} } }}}} \right)}} \right.} \right.} \right. \\&\quad \left. {\left. { - {\text{erf}}{\left({\frac{{nz_{t} \theta ^{{\min }}_{t} - ({\text{Ir}}_{t} + {\text{Ra}}_{t} + n(z_{t} - z_{{t - 1}})\theta _{r} - {\text{ET}}_{t}) - nz_{t} E(\theta _{{t - 1}})}}{{{\sqrt {2{\text{Var}}\,\eta _{t} } }}}} \right)}} \right)} \right\} \\&\quad - \hbox{exp} {\left({ - \frac{1}{2}\frac{{(nz_{t} \theta ^{{\max }}_{t} - ({\text{Ir}}_{t} + {\text{Ra}}_{t} + n(z_{t} - z_{{t - 1}})\theta _{r} - {\text{ET}}_{t}) - nz_{{t - 1}} E(\theta _{{t - 1}}))^{2} }}{{{\text{Var}}\,\eta _{t} }}} \right)} \\&\quad \times {\left[ {\frac{{{\left({nz_{{t - 1}} } \right)}^{2} }}{{{\sqrt {2\pi {\text{Var}}\,\eta _{t} } }}}{\left({2 + \frac{{(nz_{t} \theta ^{{\max }}_{t} - ({\text{Ir}}_{t} + {\text{Ra}}_{t} + n(z_{t} - z_{{t - 1}})\theta _{r} - {\text{ET}}_{t}) - nz_{{t - 1}} E(\theta _{{t - 1}}))nz_{{t - 1}} E(\theta _{{t - 1}})}}{{{\text{Var}}\,\eta _{t} }}} \right)}} \right]} \\&\quad \times \frac{{{\text{Var}}(\theta _{{t - 1}})}}{2} + \hbox{exp} {\left({ - \frac{1}{2}\frac{{(nz_{t} \theta ^{{\min }}_{t} - ({\text{Ir}}_{t} + {\text{Ra}}_{t} + n(z_{t} - z_{{t - 1}})\theta _{r} - {\text{ET}}_{t}) - nz_{{t - 1}} E(\theta _{{t - 1}}))^{2} }}{{{\text{Var}}\,\eta _{t} }}} \right)} \\&\quad \times {\left[ {\frac{{{\left({nz_{{t - 1}} } \right)}^{2} }}{{{\sqrt {2\pi {\text{Var}}\,\eta _{t} } }}}{\left({2 + \frac{{(nz_{t} \theta ^{{\min }}_{t} - ({\text{Ir}}_{t} + {\text{Ra}}_{t} + n(z_{t} - z_{{t - 1}})\theta _{r} - {\text{ET}}_{t}) - nz_{{t - 1}} E(\theta _{{t - 1}}))nz_{{t - 1}} E(\theta _{{t - 1}})}}{{{\text{Var}}\,\eta _{t} }}} \right)}} \right]} \\&\quad \times \frac{{{\text{Var}}(\theta _{{t - 1}})}}{2}. \\ \end{aligned}$$
(27)
AII. Using the property of the indicator function and assuming (θt−1) and η
t
are independent, the second term of the Eq. 20, AII, can be expressed as follows:
$$nz_{t} {\int\limits_{\theta ^{{\min }}_{{t - 1}} }^{\theta ^{{\max }}_{{t - 1}} } {{\left[ {{\int\limits_l^u {\eta _{t} f_{{\eta _{t} }} (\eta _{t}){\text{d}}_{{\eta _{t} }} } }} \right]}} }f_{\theta_{t -1}} (\theta _{{t - 1}}){\text{d}}_{{\theta _{{t - 1}} }},$$
(28)
which is the expectation of a function of (θt−1) and it can be shown as:
$$ = nz_{t} E{\left\{ {{\left[ {{\int\limits_l^u {\eta _{t} f_{{\eta _{t} }} (\eta _{t}){\text{d}}_{{\eta _{t} }} } }} \right]}} \right\}},$$
(29)
where
\(f_{{\eta _{t}}} (\eta _{t})\) is presented as follows:
$$f_{{\eta _{t} }} (\eta _{t}) = \frac{1}{{{\sqrt {2\pi {\text{Var}}(\eta _{t})} }}}\hbox{exp} {\left[ { - \frac{1}{2}{\left({\frac{{(\eta _{t})^{2} }}{{{\text{Var}}(\eta _{t})}}} \right)}} \right]}.$$
(30)
The expectation of Eq. 29 is thus given as
$$E{\left\{ { - \frac{{{\left({nz_{{t - 1}} } \right)}{\sqrt {{\text{Var}}\,\eta _{t} } }}}{{{\sqrt {2\pi } }}}\hbox{exp} {\left({ - \frac{1}{2}\frac{{(k - nz_{{t - 1}} \theta _{{t - 1}})^{2} }}{{{\text{Var}}\,\eta _{t} }}} \right)}^{{nz_{t} \theta ^{{\max }}_{t} - ({\text{Ir}}_{t} + {\text{Ra}}_{t} + n(z_{t} - z_{{t - 1}})\theta _{r} - {\text{ET}}_{t})}}_{{nz_{t} \theta ^{{\min }}_{t} - ({\text{Ir}}_{t} + {\text{Ra}}_{t} + n(z_{t} - z_{{t - 1}})\theta _{r} - {\text{ET}}_{t})}} } \right\}}.$$
(31)
The second-order Taylor’s series approximation of the expectation of Eq. 31 is thus given as:
$$\begin{aligned} & \frac{{{\sqrt {{\text{Var}}\,\eta _{t} } }}}{{{\sqrt {2\pi } }}}\hbox{exp} {\left({ - \frac{1}{2}\frac{{(nz_{t} \theta ^{{\max }}_{t} - ({\text{Ir}}_{t} + {\text{Ra}}_{t} + n(z_{t} - z_{{t - 1}})\theta _{r} - {\text{ET}}_{t}) - nz_{t} E(\theta _{{t - 1}}))^{2} }}{{{\text{Var}}\,\eta _{t} }}} \right)} \\ &\quad \times {\left[ {1 + \frac{{{\text{Var}}(\theta _{{t - 1}})}}{2}{\left[ {\frac{{ - (nz_{{t - 1}})^{2} }}{{{\text{Var}}\,\eta _{t} }} + {\left({\frac{{nz_{t} \theta ^{{\max }}_{t} - ({\text{Ir}}_{t} + {\text{Ra}}_{t} + n(z_{t} - z_{{t - 1}})\theta _{r} - {\text{ET}}_{t}) - nz_{{t - 1}} E(\theta _{{t - 1}})}}{{{\text{Var}}\,\eta _{t} }}(nz_{{t - 1}})} \right)}^{2} } \right]}} \right]} \\ &\quad - \frac{{{\sqrt {{\text{Var}}\,\eta _{t} } }}}{{{\sqrt {2\pi } }}}\hbox{exp} {\left({ - \frac{1}{2}\frac{{(nz_{t} \theta ^{{\min }}_{t} - ({\text{Ir}}_{t} + {\text{Ra}}_{t} + n(z_{t} - z_{{t - 1}})\theta _{r} - {\text{ET}}_{t}) - nz_{t} E(\theta _{{t - 1}}))^{2} }}{{{\text{Var}}\,\eta _{t} }}} \right)} \\ &\quad \times {\left[ {1 + \frac{{{\text{Var}}(\theta _{{t - 1}})}}{2}{\left[ {\frac{{ - {\left({nz_{{t - 1}} } \right)}^{2} }}{{{\text{Var}}\,\eta _{t} }} + {\left({\frac{{nz_{t} \theta ^{t}_{{\min }} - ({\text{Ir}}_{t} + {\text{Ra}}_{t} + n(z_{t} - z_{{t - 1}})\theta _{r} - {\text{ET}}_{t}) - nz_{{t - 1}} E(\theta _{{t - 1}})}}{{{\text{Var}}\,\eta _{t} }}{\left({nz_{{t - 1}} } \right)}} \right)}^{2} } \right]}} \right]}. \\ \end{aligned}$$
(32)
1.2 Sample term B
Considered the term
$$E({\left({\theta _{{t - 1}}} \right)}^{2}1_{{(\theta _{{pwp}} \leq \theta _{t} \leq (1 - p)(\theta _{{\rm FC}} - \theta _{{pwp}}))}} (\theta _{t}))$$
(33)
from the second moment expression in actual ET equation (see sample term A).
$${\left({nz_{{t - 1}}} \right)}^{2} {\int\limits_{\theta ^{{\min}}_{{t - 1}}}^{\theta ^{{\max}}_{{t - 1}}} {\,{\left[ {\,{\int\limits_l^u {(\theta _{{t - 1}})^{2}\,f_{{\eta _{t}}} (\eta _{t}){\kern 1pt}\,f_{{\theta} _{{t - 1}}} \hbox{d}_{{\eta _{t}}}}} \hbox{d}_{{{\theta _{{t - 1}}}}}} \right]}}}.$$
(34)
Using the property of the indicator function, Eq. 34 can be expressed as
$${\left({nz_{{t - 1}}} \right)}^{2}{\int\limits_{\theta ^{{\min}}_{{t - 1}}}^{\theta ^{{\max}}_{{t - 1}}} {(\theta _{{t - 1}})^{2}\,{\left[ {\,{\int\limits_l^u {f_{{\eta_{t}}}(\eta_{t})\hbox{d}_{{\eta _{t}}}}}} \right]}}}f_{{\theta} _{{t - 1}}} \hbox{d}_{{{\theta _{{t - 1}}}}},$$
(35)
which is the expectation of a function of (θ t−1) and it can be shown as:
$$ = {\left({nz_{{t - 1}} } \right)}^{2} E{\left\{ {{\left({\theta _{{t - 1}} } \right)}^{2} {\left[ {{\int\limits_l^u {f_{{\eta _{t} }} (\eta _{t}){\text{d}}_{{\eta _{t} }} } }} \right]}} \right\}},$$
(36)
where
\(f_{{\eta _{t}}} (\eta _{t})\) is presented as follows:
$$f_{{\eta _{t}}} (\eta _{t}) = \frac{1}{{{\sqrt {2\pi \hbox{Var} (\eta _{t})}}}}\hbox{exp} {\left[ {- \frac{1}{2}{\left({\frac{{(\eta _{t})^{2}}}{{\hbox{Var}(\eta _{t})}}} \right)}} \right]}.$$
(37)
For the current study, the integral in Eq. 36 is represented as follows:
$$\begin{aligned}\,& E\left({\left({nZ_{t - 1} \theta _{t - 1} } \right)^2 \hbox{Pr}\{ nz_t \theta _t^{\min } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_{t - 1} (\theta _{t - 1}) < \eta_t } \right. \\&\quad \left. { \leq nz_t \theta _t^{\max } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_{t - 1} (\theta _{t - 1})\} } \right). \\\end{aligned}$$
(38)
$$\left({nZ_{t - 1} } \right)^2 E\left({\left({\theta _{t - 1} } \right)^2 \hbox{Pr}\{ \eta _t } \right.\left. { \leq k - nz_{t - 1} (\theta _{t - 1})\} } \right)_{nz_t \theta _t^{\min } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t }^{nz_t \theta _t^{\max } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t }.$$
(39)
The second-order Taylor’s series approximation of the expectation of Eq. 39 is thus given as:
$$\begin{aligned}\,& \left({nz_{t - 1} } \right)^2 \left({\left({E(\theta _{t - 1})^2 \left\{ {\frac{1}{2}\left({{\text{erf}}\left({\frac{{nz_t \theta _t^{\max } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_t E(\theta _{t - 1})}}{{\sqrt {2{\text{Var}}\,\eta _t } }}} \right)} \right.} \right.} \right.} \right. \\& \quad \left. {\left. {\left. { - {\text{erf}}\left({\frac{{nz_t \theta _t^{\min } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_t E(\theta _{t - 1})}}{{\sqrt {2{\text{Var}}\,\eta _t } }}} \right)} \right)} \right\}} \right) \\& \quad + \left({2\left\{ {\frac{1}{2}\left({{\text{erf}}\left({\frac{{nz_t \theta _t^{\max } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_t E(\theta _{t - 1})}}{{\sqrt {2{\text{Var}}\,\eta _t } }}} \right)} \right.} \right.} \right. \\& \quad \left. {\left. { - {\text{erf}}\left({\frac{{nz_t \theta _t^{\min } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_t E(\theta _{t - 1})}}{{\sqrt {2{\text{Var}}\,\eta _t } }}} \right)} \right)} \right\} \\& \quad + \frac{{ - 4\left({nz_{t - 1} } \right)E(\theta _{t - 1})}}{{\sqrt {2\pi {\text{Var}}\,\eta _t } }}\hbox{exp} \left({ - \frac{1}{2}\frac{{(nz_t \theta _t^{\max } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_{t - 1} E(\theta _{t - 1}))^2 }}{{{\text{Var}}\,\eta _t }}} \right) \\& \quad - \frac{{(nz_t \theta _t^{\max } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_{t - 1} E(\theta _{t - 1}))\left({nz_{t - 1} E(\theta _{t - 1})} \right)^2 }}{{{\text{Var}}\,\eta _t \sqrt {2\pi {\text{Var}}\,\eta _t } }} \\& \quad \times \hbox{exp} \left({ - \frac{1}{2}\frac{{(nz_t \theta _t^{\max } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_{t - 1} E(\theta _{t - 1}))^2 }}{{{\text{Var}}\,\eta _t }}} \right) \\& \quad - \frac{{ - 4\left({nz_{t - 1} } \right)E(\theta _{t - 1})}}{{\sqrt {2\pi {\text{Var}}\,\eta _t } }}\hbox{exp} \left({ - \frac{1}{2}\frac{{(nz_t \theta _t^{\min } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_{t - 1} E(\theta _{t - 1}))^2 }}{{{\text{Var}}\,\eta _t }}} \right) \\& \quad + \frac{{(nz_t \theta _t^{\min } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_{t - 1} E(\theta _{t - 1}))\left({nz_{t - 1} E(\theta _{t - 1})} \right)^2 }}{{{\text{Var}}\,\eta _t \sqrt {2\pi {\text{Var}}\,\eta _t } }} \\& \quad \times \hbox{exp} \left. {\left({ - \frac{1}{2}\frac{{(nz_t \theta _t^{\min } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_{t - 1} E(\theta _{t - 1}))^2 }}{{{\text{Var}}\,\eta _t }}} \right)} \right)\frac{{{\text{Var}}(\theta _{t - 1})}}{2}. \\ \end{aligned}$$
(40)
1.3 Sample term C
Considered the term
$$2nz_{{t - 1}} E(\eta _{t} {\left({\theta _{{t - 1}}} \right)} 1_{{(\theta _{{pwp}} \leq \theta _{t} \leq (1 - p)(\theta _{{\rm FC}} - \theta _{{pwp}}))}} (\theta _{t}))$$
(41)
from the second moment expression in actual ET equation (see sample term A). This term is expressed as:
$$2nz_{{t - 1}} {\int\limits_{\theta ^{{\min}}_{{t - 1}}}^{\theta^{{\max}}_{{t - 1}}} {{\left[ {{\int\limits_l^u {(\theta _{{t - 1}})\eta _{t} f_{{\eta _{t}}} (\eta _{t})f_{{\theta} _{{t - 1}}} \hbox{d}_{{\eta _{t}}}}} \hbox{d}_{{\theta _{{t - 1}}}}} \right]}}}.$$
(42)
Using the property of the indicator function, Eq. 42 can be expressed as
$$2nz_{{t - 1}} {\int\limits_{\theta ^{{\min}}_{{t - 1}}}^{\theta ^{{\max}}_{{t - 1}}}{(\theta _{{t - 1}}) {\left[ {{\int\limits_l^u {\eta _{t} f_{{\eta _{t}}}(\eta _{t})\hbox{d}_{{\eta _{t}}}}}} \right]}}}f_{{\theta} _{{t - 1}}} \hbox{d}_{{{\theta _{{t - 1}}}}},$$
(43)
where
\(f_{{\eta _{t}}} (\eta _{t})\) is presented as follows:
$$f_{{\eta _{t}}} (\eta _{t}) = \frac{1}{{{\sqrt {2\pi \hbox{Var} (\eta _{t})}}}}\hbox{exp} {\left[ {- \frac{1}{2}{\left({\frac{{(\eta _{t})^{2}}}{{\hbox{Var}(\eta _{t})}}} \right)}} \right]}.$$
(44)
The expectation of Eq. 41 is thus given as
$$2nz_{t - 1} E\left\{ { - \theta _{t - 1} \frac{{\left({nz_{t - 1} } \right)\sqrt {{\text{Var}}\,\eta _t } }}{{\sqrt {2\pi } }}\hbox{exp} \left({ - \frac{1}{2}\frac{{(k - nz_{t - 1} \theta _{t - 1})^2 }}{{{\text{Var}}\,\eta _t }}} \right)_{nz_t \theta _t^{\min } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t)}^{nz_t \theta _t^{\max } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t)} } \right\}.$$
(45)
The second-order Taylor’s series approximation of the expectation of Eq. 45 is thus given as:
$$\begin{aligned}\, & \left({\frac{{ - 2nz_{t - 1} E(\theta _{t - 1})\sqrt {{\text{Var}}\,\eta _t } }}{{\sqrt {2\pi } }}} \right.\hbox{exp} \left({ - \frac{{(nz_t \theta _t^{\max } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_{t - 1} E(\theta _{t - 1}))^2 }}{{2{\text{Var}}\,\eta _t }}} \right) \\&\quad + \frac{{{\text{Var}}(\theta _{t - 1})\sqrt {{\text{Var}}\,\eta _t } }}{{2\sqrt {2\pi } }}\hbox{exp} \left({ - \frac{{(nz_t \theta _t^{\max } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_{t - 1} E(\theta _{t - 1}))^2 }}{{2{\text{Var}}\,\eta _t }}} \right) \\&\quad \times \left\{ {2nz_{t - 1} (nz_t \theta _t^{\max } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_{t - 1} E(\theta _{t - 1})) - \frac{{(nz_{t - 1})^2 E(\theta _{t - 1})}}{{{\text{Var}}\,\eta _t }}} \right. \\&\quad \left. { + E(\theta _{t - 1})\left({nz_{t - 1} \frac{{nz_t \theta _t^{\max } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_{t - 1} E(\theta _{t - 1})}}{{{\text{Var}}\,\eta _t }}} \right)^2 } \right\} \\&\quad - \frac{{ - 2nz_{t - 1} E(\theta _{t - 1})\sqrt {{\text{Var}}\,\eta _t } }}{{\sqrt {2\pi } }}\hbox{exp} \left({ - \frac{{(nz_t \theta _t^{\min } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_{t - 1} E(\theta _{t - 1}))^2 }}{{2{\text{Var}}\,\eta _t }}} \right) \\&\quad - \frac{{{\text{Var}}(\theta _{t - 1})\sqrt {{\text{Var}}\,\eta _t } }}{{2\sqrt {2\pi } }}\hbox{exp} \left({ - \frac{{(nz_t \theta _t^{\min } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_{t - 1} E(\theta _{t - 1}))^2 }}{{2{\text{Var}}\,\eta _t }}} \right) \\&\quad \times \left\{ {2nz_{t - 1} (nz_t \theta _t^{\min } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_{t - 1} E(\theta _{t - 1})) - \frac{{(nz_{t - 1})^2 E(\theta _{t - 1})}}{{{\text{Var}}\,\eta _t }}} \right. \\&\quad + \left. {E(\theta _{t - 1})\left({nz_{t - 1} \frac{{nz_t \theta _t^{\min } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_{t - 1} E(\theta _{t - 1})}}{{{\text{Var}}\,\eta _t }}} \right)^2 } \right\} \\\end{aligned}$$
Appendix 2: Soil moisture moment estimation for a specific growing season
The weekly soil moisture continuity equation can be considered as follows for a specific plant in a weekly basis:
$$n(z_{t} \theta _{t} - z_{{t - 1}} \theta _{{t - 1}}) = (\hbox{Ir}_{{t - 1}} + \hbox{Ra}_{{t - 1}} + n(z_{t} - z_{{t - 1}}) \theta _{r}) - \hbox{ET}_{{t - 1}} - L_{{t - 1}} + \eta _{t},$$
(46)
where z
t
is root zone depth at time t, n is the soil porosity, θ
t
is volumetric soil moisture at the beginning of a weekly period t, θ t−1 is volumetric soil moisture at end of a weekly period t−1, θ
r
is remainder soil moisture at the beginning of a weekly period, Irt−1 is long-term (yearly) average of a weekly irrigation depth, Rat−1 is long-term (yearly) average of weekly effective rainfall depth, ETt−1 is long-term (yearly) average of weekly actual evapotranspiration for a specific plant, L
t−1 if long-term (yearly) average of weekly leaching fraction, and η
t
is the random noise term of soil moisture balance equation.
The soil moisture balance Eq. 46 can be represented using the indicator function 1[·](·) as in Eq. 47. For simplicity in demonstrating the methodology, the weekly rainfalls is assumed to be normally and independently distributed in time.
$$\begin{aligned} nz_t \theta _t = & nz_{t - 1} \theta _{t - 1} \\ & + ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t + \eta _t)1_{(\theta _t^{\min },\theta _t^{\max })} (\theta _t) \\ & + \{ nz_t \theta _t^{\min } - nz_{t - 1} \theta _{t - 1} - n(z_t - z_{t - 1})\theta _r \} 1_{(- \infty ,\theta _t^{\min })} (\theta _t) \\ & + \{ nz_t \theta _t^{\max } - nz_{t - 1} \theta _{t - 1} - n(z_t - z_{t - 1})\theta _r \} 1_{(\theta _t^{\max }, + \infty )} (\theta _t), \\\end{aligned}$$
(47)
where θ
min
t
is the minimum allowable soil moisture which is determined based on the specific level of irrigation deficit for a plant. θ
max
t
is the saturated soil moisture, n is the soil porosity and z
t
is the weekly root zone depth for an specific plant. The indicator function of the random variable θ
t
(for example
\(1_{{(\theta^{{\min}}_{t}, \theta ^{{\max}}_{t})}} (\theta_{t})\)) is itself a random variable such that:
$$1_{(\theta _t^{\min },\theta _t^{\max })} (\theta _t) = \left\{ {\begin{array}{*{20}l} 1 & {\theta _{\min }^t \leq \theta ^t \leq \theta _{\max }^t } \\ 0 & {{\text{otherwise}}} \\ \end{array} } \right.$$
(48)
The expected value of the soil moisture can be determined as follows:
$$\begin{aligned} E(nz_t \theta _t) = & E(nz_{t - 1} \theta _{t - 1}) \\ & + E\left\{ {({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t + \eta _t)1_{(\theta _t^{\min },\theta _t^{\max })} (\theta _t)} \right\} \\ & + E\left\{ {\{ nz_t \theta _t^{\min } - nz_{t - 1} \theta _{t - 1} - n(z_t - z_{t - 1})\theta _r \} 1_{(- \infty ,\theta _t^{\min })} (\theta _t)} \right\} \\ & + E\left\{ {\{ nz_t \theta _t^{\max } - nz_{t - 1} \theta _{t - 1} - n(z_t - z_{t - 1})\theta _r \} 1_{(\theta _t^{\max }, + \infty )} (\theta _t )} \right\}, \\ \end{aligned}$$
(49)
where the second part of the right hand side of Eq. 49 represents the expected soil moisture deficit for an allowable specific level of plant’s yield reduction. To derive the extended from of the Eq. 49, the expected value of the indicator function of soil moisture is determined. According to definition, the expected value of the indicator function of a random variable over any region is the probability of that random variable occurring within that same region. Therefore the expected value is as:
$$\begin{aligned} E\{ 1_{(\theta _t^{\min },\theta _t^{\max })} (\theta _t)\} = & \hbox{Pr}\{ nz_t \theta _t^{\min } \leq nz_t \theta _t \leq nz_t \theta _t^{\max } \} \\ = & \hbox{Pr}\{ nz_t \theta _t^{\min } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) \\ & \quad \leq nz_{t - 1} \theta _{t - 1} + \eta _t \\ & \quad \leq nz_t \theta _t^{\max } - ({\text{Ir}}_t + \hbox{Ra}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t)\}, \\ \end{aligned}$$
(50)
$$ E\{ 1_{(\theta _t^{\min },\theta _t^{\max })} (\theta _t)\} = \int\limits_{\theta_{t - 1}^{\min }}^{\theta _{t - 1}^{\max }} {\int\limits_{l}^{u} f_{\eta_{t}} (\eta_t) f_{\theta_{t - 1}} (\theta_{t -1}){\text{d}}_{\eta_{t}} {\text{d}}_{\theta_{t - 1}}},$$
(51)
where
\(f_{{\eta _{t}}}\) is the probability density function of the random noise term of the rainfall in the current period t and
\(f_{{\theta _{{t - 1}}}}(\theta _{{t - 1}})\) is the density function of the soil moisture state variable in the previous period (t−1). The integral at Eq. 51 can be derived analytically, in terms of θt−1, thus allowing for no loss of generality with respect to the assumed distribution of the zero-mean random variable η
t
. The outer integration in Eq. 51 is just the expectation definition with respect to the random variable of the soil moisture at the previous period of time and so it can be shown that:
$$\begin{aligned} \, & = E\left({\hbox{Pr}\left\{ {nz_t \theta _t^{\min } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_{t - 1} \theta _{t - 1} \leq \eta _t } \right.} \right. \\ & \quad \left. {\left. { \leq nz_t \theta _t^{\max } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_{t - 1} \theta _{t - 1} } \right\}} \right). \\ \end{aligned}$$
(52)
Using the Taylor series approximation about point E(θt−1) we generate approximation for expectation at Eq. 52 as below. The first order approximation gives:
$$\begin{aligned} E\{ 1_{(\theta _t^{\min },\theta _t^{\max })} (\theta _t)\} = & \hbox{Pr}\{ nz_t \theta _t^{\min } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_{t - 1} E(\theta _{t - 1}) \leq \eta _t \\ & \leq nz_t \theta _t^{\max } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_{t - 1} E(\theta _{t - 1})\}. \\ \end{aligned}$$
(53)
Assuming a normal distribution for the random error term (η
t
), Eq. 53 can be represented using the erf(·) as:
$$\begin{aligned} \hbox{Pr} & \left\{ {nz_t \theta _t^{\min } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_{t - 1} E(\theta _{t - 1}) \leq \eta _t } \right. \\ & \left. { \leq nz_t \theta _t^{\max } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_{t - 1} E(\theta _{t - 1})} \right\} \\ & \quad = \frac{1}{2}\left\{ {{\text{erf}}\left({\frac{{nz_t \theta _t^{\max } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_{t - 1} E(\theta _{t - 1})}}{{(2{\text{Var}}(\eta _t))^{0.5} }}} \right)} \right. \\ & \quad \quad \left.{- {\text{erf}}\left({\frac{{nz_t \theta _t^{\min } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_{t - 1} E(\theta _{t - 1})}}{{(2{\text{Var}}(\eta _t))^{0.5} }}} \right)} \right\}. \\ \end{aligned}$$
(54)
Therefore, using the same procedure for all terms of Eq. 49 and first order Taylor approximation, the expected value of soil moisture can be determined as follows:
$$\begin{aligned} E(nz_t \theta _t) = & E(nz_{t - 1} \theta _{t - 1}) + \frac{1}{2}\left\{ {{\text{erf}}\left({\frac{{nz_t \theta _t^{\max } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_{t - 1} E(\theta _{t - 1})}}{{(2{\text{Var}}(\eta _t))^{0.5} }}} \right)} \right. \\ & \left. { - {\text{erf}}\left({\frac{{nz_t \theta _t^{\min } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_{t - 1} E(\theta _{t - 1})}}{{(2{\text{Var}}(\eta _t))^{0.5} }}} \right)} \right\} \\ & \times ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) + \left({nz_t \theta _t^{\min } - nz_{t - 1} E(\theta _{t - 1}) - n(z_t - z_{t - 1})\theta _r } \right) \\ & \times \frac{1}{2}\left\{ {1 - {\text{erf}}\left({\frac{{nz_t \theta _t^{\min } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_{t - 1} E(\theta _{t - 1})}}{{(2{\text{Var}}(\eta _t))^{0.5} }}} \right)} \right\} \\ & + \left({nz_t \theta _t^{\max } - nz_{t - 1} E(\theta _{t - 1}) - n(z_t - z_{t - 1})\theta _r } \right) \\ & \times \frac{1}{2}\left\{ {1 + {\text{erf}}\left({\frac{{nz_t \theta _t^{\max } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_{t - 1} E(\theta _{t - 1})}}{{(2{\text{Var}}(\eta _t))^{0.5} }}} \right)} \right\}. \\ \end{aligned}$$
(55)
Same as the expected value, it can be held, the second moment of the soil moisture can be determined using the first order analysis as follows:
$$\begin{aligned} E(nz_t \theta _t)^2 = & E(nz_{t - 1} \theta _{t - 1})^2 + \left\{ {({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t)^2 + {\text{Var}}\,\eta _t } \right. \\ & + \left. {2E\{ (\hbox{Ir}_t + \hbox{Ra}_t + n(z_t - z_{t - 1})\theta _r - \hbox{ET}_t + \eta _t)nz_{t - 1} \theta _{t - 1} \} } \right\} \\ & \times \frac{1}{2}\left\{ {{\text{erf}}\left({\frac{{nz_t \theta _t^{\max } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_{t - 1} E(\theta _{t - 1})}}{{(2{\text{Var}}(\eta _t))^{0.5} }}} \right)} \right. \\ & \left. { - {\text{erf}}\left({\frac{{nz_t \theta _t^{\min } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_{t - 1} E(\theta _{t - 1})}}{{(2{\text{Var}}(\eta _t))^{0.5} }}} \right)} \right\} \\ & + \{ (nz_t \theta _t^{\min })^2 - (nz_{t - 1} E(\theta _{t - 1}))^2 - n^2 (z_t ^2 - z_{t - 1} ^2)\theta _r ^2)\} \\ & \times \frac{1}{2}\left\{ {1 - {\text{erf}}\left({\frac{{nz_t \theta _t^{\min } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_{t - 1} E(\theta _{t - 1})}}{{(2{\text{Var}}(\eta _t))^{0.5} }}} \right)} \right. \\ & + \{ (nz_t \theta _t^{\max })^2 - (nz_{t - 1} E(\theta _{t - 1}))^2 - n^2 (z_t ^2 - z_{t - 1} ^2)\theta _r ^2 \} \\ & \times \frac{1}{2}\left\{ {1 + {\text{erf}}\left({\frac{{nz_t \theta _t^{\max } - ({\text{Ir}}_t + {\text{Ra}}_t + n(z_t - z_{t - 1})\theta _r - {\text{ET}}_t) - nz_{t - 1} E(\theta _{t - 1})}}{{(2{\text{Var}}(\eta _t))^{0.5} }}} \right)} \right\}. \\ \end{aligned}$$
(56)
Appendix 3: First moment of actual ET
The ET is a function of current soil moisture conditions and can be represented as follows:
$${\text{ET}}_t = \left\{{\begin{array}{*{20}l} 0 & {\theta _t \leq \theta _{pwp} } \\ {\frac{{{\text{ET}}_{{\rm p}_t } (\theta _t - \theta _{pwp})}}{{(1 - p)(\theta _{{\text{FC}}} - \theta _{pwp})}}} & {\theta _{pwp} \leq \theta _t < (1 - p)(\theta _{{\text{FC}}} - \theta _{pwp})} \\ {{\text{ET}}_{{\rm p}_t } } & {\theta _t \geq (1 - p)(\theta _{{\text{FC}}} - \theta _{pwp})} \\ \end{array} } \right..$$
(57)
Equation (15a, b) can be represented using the indicator function as follows:
$${\text{ET}}_t = {\text{ET}}_{{\rm p}_t } 1_{(\theta _t \geq (1 - p)(\theta _{\rm FC} - \theta _{pwp}))} (\theta _t) + \frac{{{\text{ET}}_{{\rm p}_t } (\theta _t - \theta _{pwp})}}{{(1 - p)(\theta _{{\text{FC}}} - \theta _{pwp})}}1_{(\theta _{pwp} \leq \theta _t \leq (1 - p)(\theta _{{\text{FC}}} - \theta _{pwp}))} (\theta _t).$$
(58)
Therefore, the expected value of ET
t
can be considered as follows:
$$E({\text{ET}}_t) = E\left\{ {{\text{ET}}_{{\rm p}_t } 1_{(\theta _t \geq (1 - p)(\theta _{{\text{FC}}} - \theta _{pwp}))} (\theta _t)} \right\} + E\left\{ {\frac{{{\text{ET}}_{{\rm p}_t } (\theta _t - \theta _{pwp})}}{{(1 - p)(\theta _{{\text{FC}}} - \theta _{pwp})}}1_{(\theta _{pwp} \leq \theta _t \leq (1 - p)(\theta _{{\text{FC}}} - \theta _{pwp}))} (\theta _t)} \right\}.$$
(59)
Because the second order approximation of mean actual ET can be extended using a similar procedure for actual ET variance with the same sample terms (as represented in Appendix 1), those procedures are not repeated here.
