A difficult question in estimating multi-model ensembles is always how to weight different models

These studies include a wide range of process-based crop models as well as empirical papers, published between the late 1990s and 2012, and they vary substantially in the geographic regions examined as well as the extent to which they include on-farm adaptations. In this paper we focus on maize, rice, soy and wheat, four crops that make up a major part of the scientific literature on climate impacts on crops. These crops collectively account for approximately 20% of the value of global agricultural production, 65% of harvested crop area, and just under 50% of calories directly consumed . For these four crops, the database contains 1010 data-points . Of these, 451 are reported as including some form of onfarm, within-crop agronomic adaptation. The majority of these adaptations involve adjusting either planting date , cultivar or both . In total, 28 models are represented in the 56 studies used for the estimation, made up of 17 process-based model families and 11 statistical models. For this analysis we have complemented the existing database in two ways. Firstly, we coded each study based on whether a process-based or empirical approach was used. Secondly we added baseline growing-season temperatures to the database. To do this, each data point was assigned to a country. 86% of points come from studies located in a single country. For the remaining 14% coming from studies with an international scope the assigned country was the country with the highest production of the relevant crop. Average growing season temperatures were calculated using planting and harvest dates from Sacks et al and gridded monthly temperature from the Climate Research Unit . We treat the database of studies as a kind of ‘ensemble of opportunity’ . The benefits of this approach are that predictions from multi-model ensembles have been shown to consistently out-perform individual models in both climate modeling and, increasingly, in agricultural modeling .

Though not derived from a formal ensemble modeling project,growing raspberries in pots the universe of individual studies contained in the database can be thought of as draws from a set of possible models, each of which captures the response of crops to changing climate conditions with some error. The sampling of models is not systematic or random, but instead has emerged from scientific research and associated peer review of work on climate effects on crop yields over the last two decades. The approach described here takes advantage of an implicit weighting derived from representation of models within the scientific literature. To the extent that this representation reflects researchers’ judgements about the best model to use for particular crops in particular locations, it may be that this implicit weighting emerging from the scientific literature is preferred to simpler one-model one-vote aggregation schemes . We use multiple regression to aggregate the results from individual studies to an ensemble average response function. This approach allows us to estimate common response functions at an appropriate level of aggregation. For example, every study in the database examines the effect of change in temperatures on yields, allowing us to estimate separate yield response functions by crop and by baseline temperature, as well as by type of study. Fewer studies examine the effect of CO2 fertilization or adaptation, limiting our ability to model heterogeneity in response to changes in these variables.The effect of adaptation on crop yields is modelled with both an intercept term . This is prompted by the observation that in many studies that report including on-farm agronomic adaptations, adaptation is represented by changing management practices that would improve yields even in the current climate . Failing to include an adaptation intercept in this context will lead to an over-estimation of the potential of the adaptation actions included in these studies to reduce the negative impacts of a warming climate. We therefore include an adaptation intercept in the estimating equation but then subtract it out to produce a damage function that goes through the origin. The true effect of adaptation is captured by the interaction with temperature change, given by the b8 term in equation .

This term reflects the effect of management changes that are not beneficial today but that will be beneficial under a changed climate, the standard definition of adaptation. To estimate the impact of local warming for a prespecified increase in global mean temperature we use pattern-scaling between local and global temperature changes based on the CMIP5 multi-model mean for RCP8.5 . The multi-model mean was calculated using the Climate Explorer tool using methods documented in van Oldenborg et al . For each grid cell we take the change in temperature between a future and baseline period and divide by the mean global warming over this time period. Local warming is greater than global average warming over land areas and is larger at high latitudes and in continental interiors. Gridded local temperature changes are combined with the response functions estimated using equation and baseline growing season temperatures based on CRU and Sacks et al to give projected changes in yield with warming on a 0.5 degree grid. Global average yield changes are calculated by production-weighting the gridded data using production data for the relevant crop in the year 2000 . In presenting results, we focus on global temperature changes ranging from 1 to 3 °C and use 4 cases to examine the importance of different variables. The reference case is based on the temperature response curves for process-based models, including CO2 and adaptation. The ‘No CO2’ case is the same as the reference except without CO2 fertilization. ‘No Adaptation’ is the same as the reference except excluding adaptation. And finally, the ‘Statistical’ case is the same as the reference except the temperature response comes from statistical studies. In order to assess the economic implications of alternative studies of climate impacts on crop yields, we use the Global Trade Analysis Project model . GTAP is a global, computable general equilibrium model which seeks to predict changes in bilateral trade flows, production, consumption, intermediate use and welfare, owing to changes in technology, policies or other exogenous shocks. In this case, we treat the climate-induced yield changes as Hicks-neutral productivity changes. Thus, a 10% yield loss would mean that, if farmers did not alter their practices in the face of the changing climate, application of the same inputs to the same amount of land would result in 10% less output. The economic model does allow for changes in area planted, as well as input intensities, in response to changing relative prices, so actual yields will not change by 10%.

In this sense, all of the economic results reported here allow for economic adaptation . In order to implement the yield shocks under the different climate scenarios, we aggregate the gridded impacts for each of the four crops to the level of the 140 countries/regions in the version 9 GTAP data base . Since maize and soybeans are part of larger crop aggregates in the GTAP data base , the climate-induced yield shocks are diluted by multiplying them by the share of the country-commodity aggregate made up of maize and soybeans, respectively. Thus in a region which does not produce soybeans, the climate shock would be zero, whereas a country in which maize was the only coarse grain produced would experience precisely the productivity shock specified by the aggregated maize results for that geographic region. To be consistent with these incomplete agricultural yield shocks,plant pot with drainage when it comes to reporting the welfare losses, we report the losses as a share of that incomplete production value .Figure 2 shows the temperature response functions estimated from equation . Warming has a negative impact on yields that is worse for maize and wheat than for the more heat-tolerant rice. It is striking that we find very little evidence for any yield benefits from warming over most growing areas—our point estimates are negative even for warming less than 1 °C and even in the 25th percentile of growing season temperatures . This negative impact is generally statistically significant for process based model results at warming above 2 °C. Standard errors for results from statistical models are much larger and bracket zero in almost all cases. The interaction with baseline temperature is in the expected direction: warming is less damaging for crops in cooler locations. Figure S1 shows the gridded yield responses to 2 °C of global average warming for each crop. While many areas see negative impacts, there are some positive effects in the boreal zone and in cooler temperate areas. In figure S2 we show a comparison between our estimated response to a 1 °C warming and the mean of multiple process-based crop models calibrated to specific locations as part of the Agricultural Modeling Inter-comparison and Improvement Project for the three crops that are available: maize , wheat , and rice . The results from these two very different methods are close for both wheat and maize while the findings for rice show more variability. In all cases the AgMIP data are well within the 95% confidence interval produced in this study. Figure 3 shows how the type of study, inclusion of adaptation , and the CO2 fertilization effect affect the climate change response. Point estimates for b3 and b4 suggest that on average results from statistical studies are slightly more optimistic than results from process-based models for small amounts of warming and more pessimistic for higher levels of warming. Error bars for the statistical model are extremely large though, particularly for warming beyond 2 °C, which is perhaps unsurprising given the concentration of empirical results at 1° warming documented in figure 1. The point estimate for the effect of adaptation is in the opposite direction from what would be expected , but the error bars are large and the effect is not distinguishable from zero.

Studies that include adaptation do have more positive yield outcomes than studies that don’t, but those benefits are captured entirely in the adaptation intercept term . Therefore, these findings suggest that most within-crop agronomic adaptation measures represented in process based modeling studies would provide the same benefit under the current climate as they would under future climates. In other words, they are actions that shift the supply curve out to the right but do not change the marginal impacts of future warming, consistent with the ‘adaptation illusion’ identified by Lobell . Finally, the CO2 response functions show statistically-significant benefits of CO2 fertilization that a symptote at 17.3% for C3 crops and 10.6% for C4 crops. Given the functional form assumption, this translates to yield gains of 11.5% and 8.5% for a doubling of CO2 from pre-industrial levels. For C3 crops, this value is close to that obtained from FACE experiments which range between 12 and 14% yield gains for a doubling of CO2 . and therefore this may not be a fully independent validation of the meta-analysis results.Fewer FACE experiments have been performed for C4 crops but available experimental data, as well as theory, suggest C4 crops will benefit less from CO2 except under water-stressed conditions. Figure 4 shows global production-weighted yield losses for a global temperature change of 1 °C–3 °C for four cases. Except for soybeans, the reference case that includes CO2 fertilization and adaptation shows positive effects on yields at low levels of warming, becoming negative between 2 °C–3 °C of warming. Variation between the different cases reflects what might be expected given the response curves shown in figure 3. CO2 fertilization is most important– excluding the CO2 effect produces substantial losses for all crops ranging from 14% to 25% at 3 °C of warming. The effect of excluding adaptation is very small. The effect of statistical as opposed to process-based studies is small and slightly positive for 1 °C–2 °C, becoming slightly negative at 3 °C. At higher levels of temperature change we would expect this effect to become more negative . The 95% confidence intervals are large and mostly bracket zero, with the exception of the No CO2 case at 3 °C of global average warming. Uncertainties are particularly large for soybeans and for the statistical case at 3 °C of warming—both instances where the number of data points in the meta-analysis are limited.