Irrigated and rainfed crops differ in their responses to temperature. Although crop yields generally decline with increasing temperature, the declining yield pattern is more evident for rainfed than irrigated maize, implying a higher temperature sensitivity of the former (Figure \ref{249304}a). Suppose a county grows both irrigated and rainfed maize in Figure \ref{249304}b, the irrigated maize at point A would have a higher yield and a lower LST than the rainfed maize from the same county at point D. For rainfed maize, if a hypothetical cooling effect was applied (line D-C), its yield would move along its temperature response curve to increase from point D to point B, and the yield difference, denoted by line B-C, quantifies the cooling effect on yield. Although rainfed maize at point B has the same lower temperature as irrigated maize (point A), there is still a yield gap between them as denoted by line A-B. The yield gap under this condition is not caused by their temperature difference, but reflects the water supply effect of irrigation. Therefore, the yield effect of irrigation (line A-C) can be effectively decomposed into the contribution from cooling (line B-C) and water supply (line A-B, it may include other factors, see discussion).
The above idea can be implemented rigorously using statistical models. The statistical model was constructed using monthly LST and precipitation from June to August as independent variables to predict county yield. The model configuration is shown below:
\(yield=a\cdot year+\sum_{_{m=June}}^{Aug}\left(b_m\cdot LST_m+c_m\cdot LST_m^2+d_m\cdot P_m+e_m\cdot P_m^2\right)\ +c_0\) Eq.1
where a, b, c, ..., e, and C0 are estimated coefficients whose subscripts m denote month. The “year” predictor was included to account for the long-term increase yield trends due to improvements in management and technology. By training the model with yield data of rainfed and irrigated maize respectively, we would have two models, one for rainfed maize (Eq. 2) and another one for irrigated maize (Eq. 3):
Rainfed maize model: \(Yield_{rain}=f_{rain}(LST_{rain}, P)\) Eq. 2
Irrigated maize model: \(Yield_{irr}=f_{irr}(LST_{irr}, P)\) Eq. 3
where \(f_{rain}\) and \(f_{irr}\) are the fitted functions (i.e., the right-hand side of Eq. 1) for rainfed and irrigated maize, respectively. The water supply effect embeds the function \(f_{irr}\) intrinsically which gives rise to a higher yield. These statistical models serve as a tool to emulate temperature response curves in Figure \ref{249304}b mathematically. The models after training can explain about 85% and 46% of spatiotemporal yield variations of rainfed and irrigated maize from 2003 to 2016, respectively (Figure \ref{282712}). The relatively lower explanation power of irrigation model was expected as it reflected the fact that irrigated crop yield is more stable and less sensitive to climate variability \cite{Troy_2015,Li_2019a,Shaw_2014}. The predicted yield difference (irrigated versus rainfed) showed high correlation with their actual yield differences (r=0.86). This good model performance enabled us to separate the irrigation effect on crop yield into cooling and water supply.
The cooling effect on yield, \(\Delta Yield_{cooling}\ \), is defined as the hypothetical yield increase in rainfed maize if the same cooling as irrigated maize was applied. \(\Delta Yield_{cooling}\ \) can be calculated by Eq.4 as the yield predicted by rainfed maize model with irrigated LST minus the yield predicted by rainfed model with rainfed LST. Similarly, the water supply effect on yield, \(\Delta Yield_{water}\), is defined as the yield increase in rainfed maize if additional water was added as irrigated maize. It is can be calculated by Eq.5 as the yield predicted by irrigated maize model with irrigated LST minus the yield predicted by rainfed maize model with the same irrigated LST.
\(\Delta Yield_{cooling} = f_{rain}(LST_{irr}, P) - f_{rain}(LST_{rain}, P)\) Eq.4
\(\Delta Yield_{water} = f_{irr}(LST_{irr}, P) - f_{rain}(LST_{irr}, P)\) Eq.5
3. Results
3.1 Irrigation effects on LST, EVI, and maize yield