As highlighted in Section 2.1.1, crop simulation models can be a valuable tool to derive stochastic crop-water production functions by iteratively generating multiple realisations of the production function using multi-year weather time series (Equation 2.3). Despite the ability to account for interannual variability in the crop-water production function, the discussion in Section 2.1.1 highlights that the stochastic seasonal crop-water production function (Equation 2.3) may still be inconsistent with actual irrigation behaviour for a number of reasons. Here, a new stochas- tic intraseasonal formulation of the crop-water production is developed that introduces two key innovations over Equation 2.3. First, the irrigation decision is represented by the selection of an intraseasonal soil moisture management strategy as opposed to the choice of a total sea- sonal irrigation depth. Second, the crop-water production function varies in response to both interannual weather variability and differences in instantaneous application rates that can be related to available well yield and the choice of irrigated area. The mathematical formulation of the stochastic intraseasonal crop-water production function is detailed in Equation 2.4, and the
calculation framework and basis for each model innovation are described below. Yi= f (wi, Ωi) = f (M, Ωi) Wi = ni X d=1 wid= f (M, Ωi) subject to: 0 ≤ wid ≤ wmax: ∀w wmax= C A (2.4)
where wi and Ωi are vectors of daily irrigation application application depths and daily weather
conditions, respectively, in year i, ni is the number of days, d, in the growing season in year i, Wi
is the total depth of applied irrigation in year i, wmax is the maximum daily applied irrigation
depth, C is the well yield, and A is the irrigated area. M is the intraseasonal soil moisture target, equal to a specified proportion of the soil water holding capacity at which irrigation is initiated during the growing season where soil water holding capacity is defined as the water that can be held by the soil for crop use between field capacity and permanent wilting point. Given
M and wmax, the per-area applied irrigated depth on a given day, wid, is calculated during the
simulation of the stochastic intraseasonal crop-water production function using Equation 2.5.
wid = Zrz,id−1· Θf c− Θrz,id−1 , Θrz,id−1− Θwp Θf c− Θwp < M 0, Otherwise (2.5) subject to: wid ≤ wmax
where Θrz,id−1 and Zrz,id−1 are the simulated root zone soil water content and depth of the root
zone, respectively, at the end of day d − 1 of the growing season in year i, and Θf c and Θwp are
the soil water contents at field capacity and permanent wilting point, respectively, that define soil water holding capacity.
In the first step of the simulation framework, the Matlab-AquaCrop model is applied to pre- dict crop yield and total seasonal irrigation requirements in response to a range of intraseasonal soil moisture targets, given weather variables for a single year, using Equation 2.4. Nonpara- metric functions are fitted to the generated data points using a Piecewise Cubic Hermite Inter- polating Polynomial (PCHIP) function in MATLAB (Mathworks Inc., 2013). Soil moisture has been highlighted as being a key determinant of irrigation behaviour, reflecting the fact that crop
growth and crop yield are both fundamentally dependent on the maintenance of adequate soil moisture during the growing season (Steduto et al., 2012). Therefore, by describing the produc- tion decision in terms of intraseasonal management of soil moisture rather than the total depth of irrigation applied over the season (Equations 2.2 and 2.3), Equation 2.4 is able to provide a more realistic and flexible representation of the farmer’s actual irrigation decision making pro- cess. Specifically, Equation 2.4 conditions the production function on the realistic assumption that the daily decision on when to irrigate and how much water to apply are made solely on the basis of the current depletion of soil water holding capacity, without introducing any assumptions about future conditions later in the growing season.
The calculations of crop yield and irrigation requirements described in step one are condi- tioned on weather inputs for a single growing season and assume a biophysically unrestricted maximum daily irrigation rate, defined by specified upper bounds of well yield and irrigated area that allow the farmer to always meet full crop water requirements on the maximum available irrigated area. To account for production variability introduced by interannual weather vari- ability as in Equation 2.3, the second step of the procedure repeats the method described in step one using weather time series, Ω, for multiple different growing seasons at the site or region of interest. Multi-year weather time series may be obtained from historic weather records or, alternatively, may be synthetically generated using a numerical weather generator. The resulting output captures the potential distribution of crop yields (Figure 2.1) and total seasonal irriga- tion requirements (Figure 2.2) for each soil moisture target. Significantly, Figures 2.1 and 2.2 illustrate that intraseasonal irrigation decisions about soil moisture management will be based upon expectations not only about final crop yield (as is the case in Equations 2.2 and 2.3), but also about the total amount of irrigation water that will be required over the season to maintain root zone soil moisture above a threshold proportion of soil water holding capacity. Accounting for this uncertainty in expected irrigation requirements will be important in situations of limited water availability, where the inability to maintain adequate soil moisture levels in drier years may lead to changes in irrigation behaviour to mitigate potential production losses and risks.
Finally, the intraseasonal structure of Equation 2.4 provides a means of incorporating the effect of intraseasonal groundwater supply constraints on irrigation scheduling, and resultant crop yield and total irrigation requirements, within the generated crop-water production func- tion. Variations in well yield place an upper bound on the instantaneous rate at which irrigation can be applied to a specific area in a given time period. This in turn will affect the ability to maintain sufficient soil moisture in the root zone with resultant negative impacts on crop yield, net economic returns, and production risk (O’Brien et al., 2001; Peterson and Ding, 2005; Lamm
0 0.2 0.4 0.6 0.8 1 0 3 6 9 12 15
Soil moisture target
Crop yield (tonne ha
−1 )
Figure 2.1: Stochastic relationship, due to interannual weather variability, between the soil mois-
ture target and per-area corn yield (tonne ha-1). Each line represents the individual relationship
for a simulation using AquaCrop as described in Section 2.3.1.
0 0.2 0.4 0.6 0.8 1 0 200 400 600 800 1000 1200
Soil moisture target
Total seasonal irrigation (mm)
Figure 2.2: Stochastic relationship, due to interannual weather variability, between the soil moisture target and per-area irrigation use for corn (mm). Each line represents the individual relationship for a simulation using AquaCrop as described in Section 2.3.1.
et al., 2007; Colaizzi et al., 2009; Wines, 2013). However as previously noted, instantaneous irrigation application potential has been neglected as a constraint in existing simulation mod- elling approaches used to generate crop-water production functions. In the third step of the calculation framework the simulations in steps one and two therefore are repeated to generate stochastic intraseasonal production functions for a range of maximum instantaneous irrigation rates. Consequently, the complete stochastic intraseasonal crop-water production function de- scribes the relationship between soil moisture target and crop yield, and between soil moisture target and total seasonal irrigation use, accounting for the variability in these functions induced by both weather and instantaneous application rate constraints. Figures 2.3 and 2.4 illustrate examples of the crop-water production functions that are generated for four specific instanta- neous irrigation application limits, where the time period of the application constraint is defined as a calendar day to be consistent with the temporal resolution of both AquaCrop and real-world irrigation planning. It is clear that instantaneous irrigation supply constraints may have large effects on both crop yield and total irrigation water requirements. Given that the instantaneous application constraint is determined as a function of well yield and irrigated area (Equation 2.4 and Figure 2.5), this suggests that incorporating this additional variation in the crop-water pro- duction function may be an important innovation when seeking to evaluate optimal irrigation decision making under conditions of limited groundwater supply.