such declining pattern is more evident for rainfed maize than for irrigated maize (Fig X). Suppose a county grows both rainfed and irrigated maize, and irrigated maize (point A) has a higher yield and a lower LST than rainfed maize (point D), as shown in Fig X. If there was a cooling effect (line C-D) applied to rainfed maize, the yield would increase from point D to B along its yield response curve, and the yield difference ,denoted as line B-C, thus reflects the cooling effect on yield. At this point, although rainfed maize with cooling (at point C) has the same temperature with irrigated maize (point A), its yield is still lower than that of irrigated maize, the yield difference, denoted by (line A-B), therefore, reflects the effect of water supply and other factors of irrigation.  
 
We can build two models to predict county-level maize yield with their LST and precipitation, one for irrigated maize (eq. 1) and the other for rainfed maize (eq. 2). 
Model_rainfed:  Yield_rainfed = f_rainfed(LST_rainfed, precipitation)
Model_irrigated:  Yield_irrigated = f_irrigated(LST_irrigated, precipitation)  
Show the performance of the model
the f_rainfed includes both the temperature and water effects. 
Assuming there is enough water in irrigated maize is not water limited (otherwise the yield would be declined to rainfed maize that experienced severe water limitation), yield difference between hotter and cooler counties is due to temperature effect. Therefore, the cooling contribution can be estimated as the predicted yield differences:
f_irrigated(LST_irrigated, ...) - f_irrigated(LST_rainfed, ...).
Because LST in irrigated maize is lower than that of rainfed maize, it gives the yield change caused by the cooler LST ofirrigated compared to hotter LST in rainfed maize, while other factors are kept the same.  
The high yield predicted by the model f_irrigated is due to water supply embeded in the function.  
The water supply effect 
f_irrigated(LST_irrigated, ...) - f_rainfed(LST_irrigated, ...).
yield = year + lst6^2 + lst7^2 + lst8^2 + precip6^2 + precip7^2 + precip8^2 

Results