### Table of Contents

## Disaggregation of yield and irrigation

Files:

%curdir%/capdis/disyield.gms %curdir%/capdis/disyield_sets.gms %curdir%/capdis/m_hpdYieldIrri_init.gms %curdir%/capdis/m_hpdYieldIrri.gms %datdir%/capdishsu/pesetagrid_fractionfsu.gdx %datdir%/capdishsu/irriShare2000fsu.gdx

CAPREG disaggregation: simulation model m_hpdYieldIrri.gms

Crop yield and irrigation shares are distributed synchronously under the assumption that realized yield on a certain spatial unit is increasing linearly from the simulated water-limited yield (PESETA) to the simulated potential yield.

\begin{equation} y_{h,c}=y_{h,c}^{wly}+f_{h,c}^{irri}\cdot (y_{h,c}^{py}-y_{h,c}^{wly}) \quad \forall h,c \end{equation}

\(y_{h,c}\) = Yield [variable, kg/ha] of crop *c* in spatial unit *h* under irrigation share \(f_{h,c}^{irri}\)

\(y_{h,c}^{wly}\) = Water limited yield [parameter, kg/ha] from PESETA simulations for PESETA, remapped to spatial unit *h* and crop *c*

\(f_{h,c}^{irri}\) = Irrigation share [variable, dimensionless] of crops *c* in spatial unit *h*.

\(y_{h,c}^{py}\) = Potential yield [parameter, kg/ha] from PESETA simulations for PESETA, remapped to spatial unit *h* and crop *c*

As the total production in the region must be recovered, but the water is limited and potential yields obtained from PESETA simulations are based on assumptions and parameters that are independent from – and therefore not necessarily consistent with – statistical yields available in CAPRI, a scaling factor is defined which links the observed average yield in the region with the average simulated water limited yield from PESETA. The water limited yield of a crop in each spatial unit is introduced as well relative to the average simulated water limited yield from PESETA.

\begin{equation} y_{h,c}^{wly}=Y_{r,c} \cdot s_{r,c}^Y \cdot r_{h,c}^{wly} \quad \forall h,c \end{equation}

\(y_{h,c}^{wly}\) = as defined above

\(Y_{r,c}\) = Observed yield [parameter, kg/ha] for crop *c* in region *r* from CAPRI simulations (ex post or ex ante)

\(s_{r,c}^Y\) = Scaling factor [variable, dimensionless] linking observed yield in region *r* to the average water limited yield \(Y_{r,c}^{wly} = \sum_h (a_{h,c}⋅y_{h,c}^{wly} )/ \sum_h(a_{h,c} )\) reflecting differences in the assumptions and underlying data between PESETA and CAPRI, and technological progress

\(r_{h,c}^{wly}\) = Relative water limited yield [parameter, dimensionless] of crop *c* in spatial unit *h*.

In practice, the product of the observed regional yield and the scaling factor \(Y_{r,c} \cdot s_{r,c}^Y\) represents the regional average water limited yield that is consistent with observed (or simulated) irrigation shares and regional observed yields.

With the above, we can formulate the constraints of recovering total regional production (Equation TOTYIELD_) and reformulate the equation for the crop yield per spatial unit MCACTYIELD_

#### Equation 4 TOTYIELD_

\begin{equation} Y_{r,c} \cdot A_{r,c}=\sum_h \{y_{h,c} \cdot a_{h,c} \} \quad \forall r,c \end{equation}

\(y_{h,c}, Y_{r,c}\) = as defined above

\(a_{h,c}\) = Cultivated area [parameter, 1000 ha] of crop *c* in spatial unit* h*

\(A_{r,c}\) = Cultivated area [parameter, 1000 ha] of crop *c* in region *r*, with \(A_{r,c}=\sum_h\{a_{h,c}\}\)

#### Equation 5 MCACTYIELD_

\begin{equation} y_{h,c}=Y_{r,c}\cdot s_{r,c}^Y \cdot r_{h,c}^{wly}\cdot \left( 1 + f_{h,c}^{irri}\cdot (y_{h,c}^{relpy}-1)\right) \quad \forall h,c \end{equation}

\(y_{h,c}, Y_{r,c}, s_{r,c}^Y,r_{h,c}^{wly}, f_{h,c}^{irri}\) = as defined above

\(y_{h,c}^{relpy}\) = Relative yield increase [parameter, dimensionless] for crop c in spatial unit h if removing water limitation by full irrigation as from PESETA simulations. \(y_{h,c}^{relpy}=\frac{y_{h,c}^{py}}{y_{h,c}^{wly }\), and \(y_{h,c}^{relpy}-1=\frac{y_{h,c}^{py}-y_{h,c}^{wly}}{y_{h,c}^{wly}}\)

In principle, there are endless possibilities to adjust the irrigation shares to satisfy the equations TOTYIELD_ an MCACTYIELD_ so constraints in the irrigation shares itself are required. Those are found in statistical data on the irrigation shares by crop from the Survey on Agricultural Production Methods (SAPM, 2010) and the FAO irrigation map.

The objective function seeks to minimize the deviation of the irrigation shares by crop as given in SAPM (2010) base at regional level, and the total irrigation share as given in the Irrigation map (FAO), while trying to minimize the deviation of the yield in the spatial units from the average yield in the region. The first two terms in the objective function shall ensure that deviations from observed irrigation shares per unit and per crop are kept small, while the third term shall ensure that irrigation shares are higher where water limited yield is low.

#### Equation 6 YIELDHPD_

\begin{align} \begin{split} hdp_{irri} \cdot \left( \sum_c A_{r,c} \right)^2 = \left\{ \sum_h \left( \sum_c (a_{h,c}) \cdot \frac{f_h^{irri} -\hat{f_h^{irri}}} {s_{f_h^{irri}}} \right)^2 + \sum_{h,c} \left( A_{r,c} \cdot \frac{r_{r,c}^{irri}-\hat{r_{r,c}^{irri}}} {s_{r_{r,c}^{irri}}} \right)^2 \\ + \sum_{h,c} \left( a_{h,c} \cdot \frac {y_{h,c}-\hat{Y_{r,c}}}{s_{y_{h,c}}} \right)^2 \right\} \end{split} \end{align}

\(a_{h,c}, y_{h,c}, Y_{r,c}\) = as defined above

\(f_h^{irri}\) = Irrigation share [variable, dimensionless] over crops in spatial unit *h*. This is calculated as weighted average of irrigation shares of individual crops, using the cultivated area as weights

#### Equation 7 UAARIRRI_

\begin{equation} f_h^{irri} \cdot \sum_c \{a_{h,c} \} =\sum_c \{a_{h,c} \cdot f_{h,c}^{irri} \} \quad \forall h \end{equation}

\(r_{r,c}^{irri}\) = Relative irrigation share [variable, dimensionless] of crop c in region r. The relative irrigation share \(r_{r,c}^{irri}\) is calculated from average irrigation share of crop c over all spatial units in a region \(\sum_h \{a_{h,c} \cdot f_{h,c}^{irri} \} ⁄ \sum_h \{a_{h,c} \}\), divided by the average irrigation share of all crops \(c^\prime\) over all spatial units in a region \(\sum_{h,c^\prime}\{a_{h,c^\prime} \cdot f_{h,c^\prime}^{irri} \} ⁄ \sum_{h,c^\prime} \{ a_{h,c^\prime} \} \), or re-arranged:

#### Equation 8 IRRISHARE_

\begin{equation} r_{r,c}^{irri} \cdot \sum_{h,c^\prime}\{a_{h,c^\prime} \cdot f_{h,c^\prime}^{irri} \} \cdot \sum_h \{a_{h,c} \} = \sum_h \{a_{h,c} \cdot f_{h,c}^{irri} \} \cdot \sum_{h,c^\prime} \{ a_{h,c^\prime} \} \quad \forall h \end{equation}

\(s_{f_h^{irri}}\), Standard deviation of the irrigation share of crop c in spatial unit h. As this is unknown it is calculated from a coefficient of variation of 2.5% according to \(s_{f_h^{irri}}=max(0.001,0.025 \cdot \hat{f_h^{irri}}) \)

\(s_{r_{r,c}^{irri}}\) Standard deviation of the relative irrigation share of crop c in region r. As this is unknown it is calculated from a coefficient of variation of 2.5% according to \(s_{r_{r,c}^{irri}} =max(0.001,0.025 \cdot \hat{r_{r,c}^{irri}})\)

\(s_{y_{h,c}}\) Standard deviation of the yield of crop c in spatial units h. As this is unknown it is calculated from a coefficient of variation of 2.5% according to \(s_{r_{r,c}^{irri}} = max(0.001,0.025 \cdot Y_{r,c})\)

The ‘hats’ \(\hat{x} in Equation YIELDHPD_ indicate //prior values// of the variables, determined as explained below. This completes the set of equations required to set up the model minimizing the objective function under the given constraints: ===Equation 9 MODEL m_hpdYieldIrri=== \begin{equation} m_{hpdYiedIrri} : min [ hdp_{irri} ] cf. TOTYIELD\_ ,MCACTYIELD\_ ,UAARIRRI\_ ,IRRISHRE\_ \end{equation}

### Downscaling of other CAPRI regional data

No information on time series of irrigation shares and currently no time series of PESETA yields for irrigated and rainfed situation are available. Therefore, we apply the principle of the primacy of stable yield spatial distributions.

Yield data are simply scaled to match the regional average yield in CAPRI.

Before doing so, the data are checked if any crop is grown in a region where it is not available in the prior data set.

### Data sets

Update pending

SAPM 2010 – Irrigation shares by crop and region

FAO irrigation map

PESETA water limited and potential yields

### Data preparation (disyield.gms)

#### Consistency between SAPM (2010) and FAO irrigation map

Irrigation shares from SAPM (2010) are obtained from farmers’ surveys, while the FAO irrigation map is obtained from remote sensing data. We consider therefore the data from SAPM (2010) more reliable and scale FAO data, so that the average irrigation share of all crops and spatial units in a region matches the data from SAPM (2010) to obtain the priors for the irrigation share per spatial unit:

\begin{equation} f_r^{irri,FAO}=\frac{\sum_h \left( \sum_c [a_{h,c}] \cdot f_h^{irri,FAO}\right)} {\sum_{h,c} [a_{h,c}] } \end{equation}

\begin{equation} f_r^{irri,SAPM}=\sum_c \frac{ \left[ A_{r,c} \cdot f_{r,c}^{irri,SAPM} \right] } {\sum_{c} [A_{r,c}] } \end{equation}

\begin{equation} \hat{f_h^{irri}}= f_h^{irri,FAO} \cdot \frac{f_r^{irri,SAPM} } {f_r^{irri,FAO}} \end{equation}

#### Calculation of relative irrigation shares

Priors for the relative irrigation shares per crop are obtained from the weighted average of crop-specific irrigation shares

\begin{equation} \hat{r_{r,c}^{irri}}= \frac{f_{r,c}^{irri,SAPM} } {\sum_{c^\prime} \left[A_{c^\prime} \cdot f_{r,c^\prime}^{irri,SAPM}\right] / \sum_{c^\prime} \left[A_{c^\prime} \right] } \end{equation}

#### Relative water limited yields and relative potential to water limited yields

The calculation of relative yields is done with various steps:

(i) Ensuring complete data set covering all PESETA grids in the region and all PESETA crops by filling eventual data ‘gaps’ with the average water limited and potential yield for all PESETA crops available in the region

(ii) Calculation of the relative potential – to – water limited yield

\begin{equation} y_{g,p}^{reply}= \frac{y_{g,p}^{py}} {y_{g,p}^{wly}} \end{equation}

\(y_{g,p}^{relpy}\) = Relative yield increase [parameter, dimensionless] for PESETA crop *p* in PESETA grid *g* if removing water limitation by full irrigation as from PESETA simulations.

\(y_{g,p}^{py}\) = Potential yield [parameter, kg/ha] for PESETA crop *p* in PESETA grid *g*

\(y_{g,p}^{wy}\) = Water limited yield [parameter, kg/ha] for PESETA crop *p* in PESETA grid *g*

(iii) Mapping from PESETA grids to spatial units.

\begin{equation} y_{h,p,y}= \sum_g \{ f_{h,g} \cdot y_{g,p,y} \} \sum_g \{ f_{h,g} \} \end{equation}

\(y_{h,p,y}\) = Yields simulated with PESETA [parameter, kg/ha] on PESETA grid *g* and for PESETA crop *p* under water limited and potential conditions \(y \in \{wyl,relpy\}\).

\(f_{h,g}\) = Fraction of spatial unit *h* in PESETA grid *g* [parameter, dimensionless]. The sum of \(f_{h,g}\) = over all spatial units *h* gives 1 unless a few occasions where spatial inaccuracies leave small shares of a spatial unit outside the land area covered by the PESETA grid.

(iv) Calculation of relative water limited yield per spatial unit

\begin{equation} r_{h,p}^{wly}= y_{h,p}^{wly} / \overline{y_{h,p}^{wly}} \end{equation}

\(r_{h,c}^{wly}\) = Relative water limited yield [parameter, dimensionless] of PESETA crop *p* in spatial unit *h*.

\(y_{h,p}^{wly}\) = Water limited yield [parameter, kg/ha] of PESETA crop *p* in spatial unit *h*

\(\overline{y_{h,p}^{wly}}\) =Averge water limited yield [parameter, kg/ha] of PESETA crop *p* over all spatial units *h* in region *r*

If the water limited yield for a PESETA crop is missing for a spatial unit, but available in other spatial units in the same region, the area-weighted water limited yield for the PESETA crop over the region is assigned to the spatial unit.

(v) Mapping from PESETA crop *p* to CAPRI crop *c*

In a final step, PESETA crops are mapped to CAPRI crops.

Data for CAPRI crops without corresponding PESETA crop are assigned the values of the crop which is assumed to be most similar as defined in the set mmarsyieldgroups(*,*)