Towards probabilistic projections of climate change impacts on global crop yields

Abstract

[1] There is a widely recognized need in the scientific and policy communities for probabilistic estimates of climate change impacts, beyond simple scenario analysis. Here we propose a methodology to evaluate one major climate change impact ‐ changes in global average yields of wheat, maize, and barley by 2030 ‐ by a probabilistic approach that integrates uncertainties in climate change and crop yield responses to temperature, precipitation, and carbon dioxide. The resulting probability distributions, which are conditional on assuming the SRES A1B emission scenario and no agricultural adaptation, indicate expected changes of +1.6%, −14.1%, −1.8% for wheat, maize, and barley, with 95% probability intervals of (−4.1, +6.7), (−28.0, −4.3), (−11.0, 6.2) in percent of current yields, respectively. This fully probabilistic analysis aims at quantifying the range of plausible outcomes and allows us to gauge the relative importance of different sources of uncertainty.

1. Introduction

[2] A particularly pressing need from a risk analysis standpoint is to provide probabilistic assessments of impacts of climate change. General circulation models (GCMs) are powerful tools for the analysis of future changes in climate variables, and statistical analysis of their output can provide not only point estimates, but also a rigorous evaluation of the uncertainty inherent in future projections [Tebaldi et al., 2004, 2005; Tebaldi and Sansó, 2008; R. L. Smith, Bayesian modeling of uncertainty in ensembles of climate models, submitted to Journal of the American Statistical Association, 2007]. Recent work [Lobell and Field, 2007] has quantified through statistical regression analysis the relation between observed changes in temperature and precipitation and recorded changes in agricultural yields of several major crops at the global level. In this work we seek to draw a connection between these two areas of study, by assessing the potential impacts on global yields of three important crops of changes in temperature and precipitation as they are projected in the GCM experiments archived in the Coupled Model Intercomparison Project phase 3 (CMIP3) multi‐model dataset. We choose to assess the sensitivity of crop yields to climate change through regression models rather than process‐based crop models because of our focus on the quantification of uncertainties, since we are not aware of any systematic means to quantify the dependence of the process‐based model results to the choice of a specific model and specific parameter values within each model. Our results are probabilistic projections of percent crop yield changes by 2030, compared to current yields, in the absence of adaptation practices.

2. Estimating Probabilities of Temperature and Precipitation Changes

[3] Tebaldi and Sansó [2008] estimate a joint probability distribution of temperature and precipitation change by extending the multi‐model approach developed for temperature and precipitation separately by Tebaldi et al., 2004, 2005; Smith et al., submitted manuscript, 2007]. We refer to that paper for details on the statistical model and its estimation. Here we want to underline that the development of joint projections responds specifically to the needs of many climate impact research areas, for which the change in temperature and precipitation concurrently is of much greater interest than projected changes in the two variables separately. Tebaldi and Sansó [2008] estimate seasonal decadal averages of the true and unknown temperature and precipitation signal over a given region (or globally), spanning the period 1950–2100, on the basis of observed and GCM‐simulated temperature and precipitation averages, through a Bayesian hierarchical approach. The main statistical assumptions can be summarized as follows:

[4] 1. the true climate signal time series (both temperature and precipitation) is a piecewise linear trend with an elbow at 2000, to account for the possibility that future trends will be different from current trends;

[5] 2. superimposed to the piecewise linear trends is a bivariate gaussian noise with a full covariance matrix, which introduces correlation between temperature and precipitation;

[6] 3. observed decadal averages provide a good estimate of the current series and their correlation, and of their uncertainty;

[7] 4. GCMs may have systematic additive bias, assumed constant along the length of the simulation;

[8] 5. after the bias in each GCM simulation is identified there remains variability around the true climate signal that is model specific.

[9] These assumptions are formalized in the likelihood function. We have T_o = 6 decadal averages of observed temperature and precipitation, and T = 15 decades of temperature and precipitation simulated by the M = 18 GCMs, all those that provided temperature and precipitation results under A1B to the CMIP3 archive. Let O_t, t = 1950, 1960, … , 2000 be a two‐component vector of observed decadal average temperature and precipitation, and X_jt, t = 1950, 1960, … , 2000, 2010, … , 2090 be the decadal averages of temperature and precipitation simulated by the j‐th GCM. We indicate by μ_t = the underlying, unobservable climate signal; the parameters d_j^T and d_j^P represent systematic model biases; η^T, η^P, ϕ_j^T and ϕ_j^P represent the inverse of the variances of observations and model simulated quantities, respectively. Finally, the parameters β_xo and β_xj introduce correlation between temperature and precipitation.

[10] Then, the likelihood of the statistical model by Tebaldi and Sansó [2008] is formulated as:

where the notation N[a, b] refers to a Gaussian distribution with mean a and variance b⁻¹, and the superscripts T and P refer to temperature and precipitation components respectively. In (1) the true climate processμ_t consists of a piecewise linear trend in both components:

where the indicator χ_t is zero when t < T_o.

[11] The unknown parameters in (1) and (2) are modeled as having diffuse prior distributions and their joint posterior PDF is estimated by a Markov Chain Monte Carlo (MCMC) algorithm. From the joint estimation of all the uncertain parameters we also derive a posterior predictive distribution (distinct from the “posterior” distribution in that it pertains to a new piece of data, rather than to the random parameters whose prior is being updated), interpretable as the best probabilistic guess for temperature and precipitation changes from a hypothetical new GCM. The predictive distribution allows us to perform leave‐one‐out cross‐validation with the dataset and thus validate the statistical model assumptions [Tebaldi and Sansó, 2008]. In addition, we propose to use the predictive distribution to characterize the uncertainty in climate projections for our specific impact study. We do so because of the common belief within the community that produces and utilizes GCM projections that a robust representation of the uncertainty cannot simply consist of the uncertainty around the “central tendency” of all the models, as is sometimes represented by the ensemble average or, in our Bayesian approach, by the posterior PDF of the climate change signal (2). Rather, the range of model projections is considered a useful depiction of the uncertainty, as if the change that we will experience was expected to look like a projection from a new GCM from the same family. One can also think of posterior and predictive distributions as two opposite approaches to the issue of GCM dependency: the posterior distribution width is inversely proportional to the number of GCMs in the ensemble [Lopez et al., 2006], as if each of them provided an independent piece of information, an assumption that is challenged by shared components and schemes among these GCMs, and even more openly by the inclusion in these ensembles of the same GCM run at different resolutions [Tebaldi and Knutti, 2007]. At the opposite end of the spectrum, the width of the predictive distribution does not depend on the number of GCMs in the ensemble, and we take it to be close to an upper bound on the actual uncertainty represented in these ensembles.To be conservative, we use the predictive distribution for our final estimate of yield changes in this paper. Specifically, for the purpose of our analysis only a simplified summary from the full multivariate PDF of the time series is needed: the two‐dimensional probability distribution of average temperature and precipitation changes at a given time in the future, which we define for our analysis as the differences between temperature and precipitation averages over 1980–1999 and 2020–2039. We note here that the focus on mid‐range future projections is consistent with the use of an empirically estimated relation between yields and climate changes, which would not consent a safe extrapolation of the relation outside the range of changes experienced in the observed record.

3. An Empirical Statistical Model of Yield Response to Climate Variation

[12] Lobell and Field [2007] fitted a regression of crop yield on temperature and precipitation for a number of crops, by using average global yields data provided by the Food and Agriculture Organization (FAO statistical databases, 2006, http://faostat.fao.org) and temperature and precipitation observations, registered to a global grid by the Climate Research Unit of the University of East Anglia [Mitchell and Jones, 2005]. The climate data was aggregated into weighted‐area averages and growing‐season averages, both crop‐specific. It has been shown [Hu and Buyanovsky, 2003; Schlenker, 2006] how intra‐seasonal variability may affect crop yields in important ways, and this method does not explicitly account for its possibly significant effects, which would contribute to the scatter around the linear relation fitted to the aggregated data. However, the method's definition of growing season is optimized with respect to yield changes, and it could consist of a short period around a critical time in plant development, if the data showed the strongest of the climate signal being of this “intraseasonal” nature. First‐difference time series of the predictand (annual crop yield) and predictors (minimum and maximum temperature and precipitation, seasonally and spatially averaged) were used in order to minimize the effect of slowly changing factors, like agricultural practices, that introduce a trend over time. We choose the three crops that in Lobell and Field [2007] showed a stronger relation to the predictors, barley, maize and wheat, for which the R² are 0.65, 0.47 and 0.41 respectively. These three crops alone represent more than 27% of global crop area (see http://faostat.fao.org). For these, we estimate a regression over the two predictors for which we can derive probabilistic projections, ΔT and ΔP. Only the R² for wheat yield drops significantly (by 20%) as a consequence of using average temperature instead of minimum and maximum. However, the regression coefficients for average temperature are highly significant, and so are the F‐statistics for all three models. The first two panels in each row of Figure 1 show scatterplots of yield change versus temperature and precipitation change separately, together with the fitted regression lines, for the three crops under exam: barley, maize and wheat respectively.

Figure 1

The empirical relation between yields and climate variables is shown in the two scatterplots on each row of the figure (for barley, maize and wheat, respectively), for average temperature and precipitation during the crop's growing season (GS). Each dot represents a pair of yield/temperature change (in the panels along the first column) and yield/precipitation change (along the second column). The contours of the posterior predictive PDFs of temperature and precipitation change are represented in the third panel of each row. Contour lines span the interval [0.015, 0.15] in 0.015 increments. Units are degrees Celsius for temperature and percent of current average for precipitation.

4. Drawing the Connection: Future Climate and Yield Changes and Their Uncertainty

[13] From the CMIP3 dataset we extract temperature and precipitation output from the historic (20C3M) and SRES A1B experiments run by 18 GCMs. The temperature and precipitation output is aggregated into areal (by aggregating global maps of crop‐specific weights, as described by Lobell and Field [2007], seasonal (using crop‐specific growing seasons) and decadal averages, and probabilistic projections of temperature and precipitation change are derived separately for the three crops from the MCMC samples of the respective posterior predictive PDFs. The third panel in each row of Figure 1 shows contours of the distribution. A sample of 1000 pairs of (ΔT, ΔP) is used to supply predictors in the linear regressions and produce correspondingly 1000 values of crop yield changes. In addition, since we want to include the uncertainty in the estimated relation between climate and crop variations, we implement a bootstrap analysis by resampling 1000 times the first difference time series of yield and climate and re‐estimating the regression coefficients. Note that these frequentist‐derived distributions would coincide with posterior PDFs for these coefficients under uninformative priors. We then apply each of these coefficient pairs to the sample from the predictive PDF of climate change and the result is a distribution of yield changes that integrates both uncertainty sources. Last, to account for the direct physiological response of crop yields to CO₂, we utilize a recent meta‐analysis of Free Air CO₂ Enrichment (FACE) experiments representing the most current assessment of yield responses to CO₂ and arguably consistent with previous chamber experiments [Long et al., 2006]. Assuming an average CO₂ level of 352ppm for 1980–2000, an average of 449 ppm for A1b in 2020–2040 [Intergovernmental Panel on Climate Change, 2007], and a linear response of yields to CO₂ in this range, we interpolate the values in Long et al. [2006] to a mean response of +7% for wheat and barley and −0.5% for maize with a standard deviation of 1 and 2%, respectively. Here as well we note that the original analysis of Long et al. [2006] gives results in terms of maximum likelihood estimates and confidence intervals. We can appeal to Bayesian reference analysis [Bernardo and Smith, 1994] and translate these straightforwardly into means and standard deviations of Gaussian distributions to characterize the uncertainty in the response of crops to CO₂.

[14] The three panels in Figure 2 show PDFs of crop yield change, based on the predictive PDF of temperature and precipitation change only, on the distribution of the crop model's coefficients determined by the bootstrap method, and a representation of the combined effects of crop, climate and CO₂ uncertainty. Each PDF is represented by a box‐and‐whisker graph, which indicates by a box the inter‐quartile range, by a thick line within the box the median, and by the extent of the whiskers the range up to 1.5 times the interquartile range of the distribution. Results across the three crops are strikingly similar, notwithstanding a change in the absolute numbers. Both the uncertainty in the climate change magnitude (first boxplot) and the uncertainty in the crop yield change magnitude (second boxplot) obtained by applying the bootstrapped sample of regression coefficient to the median change in temperature and precipitation are significant, and the PDF that integrates the two sources of uncertainty is noticeably wider than both as a result (third boxplot). Adding the (uncertain) effects of CO₂ fertilization does not substantially increase the width of the distributions but only shifts their location, as can be assessed by comparing the fourth boxplot to the boxplot to its left. This indicates a substantially lower contribution to the overall uncertainty from CO₂ effects than from climate or crop model projections. An average yield enhancement of +7% for wheat is slightly larger than the median expected loss from climate change, resulting in a positive net median projection of 1.6%. In contrast, CO₂ effects are smaller in magnitude than median projected losses from climate change in barley, producing a net median projection of −1.8%. For maize, inclusion of CO₂ does not appreciably affect results, because the FACE experiments indicate a very small effect of CO₂ fertilization on maize crops. Of course, all of our results are under the hypothesis of no adaptation, which may in reality mitigate some of the adverse effects of climate change even over the relatively short‐time frame of the next two decades. Estimates of adaptation effects could readily be incorporated into our probabilistic framework.

Figure 2

For each crop (barley, maize and wheat, respectively) we compare boxplots of projected climate impacts on yield, accounting for either only climate change uncertainties (first box) or only crop model uncertainties (second box) or integrating the uncertainties from the two (climate change and crop yield change) in the third box. The fourth and last boxpot shows the result of combining estimates of CO₂ fertilization (and their uncertainty) with the distribution of the third boxplot (crop uncertainty + climate change uncertainty). The first three boxes are thus accounting for only climate impacts, while the fourth represents climate and CO₂ impacts combined.

5. Conclusions

[15] We believe that our analysis represents one of the first attempts to integrate probabilistically the uncertainties from both the empirical relation between climate change and its impacts and the uncertainties in the forecast of future climate change at the global scale. One of the foci of our study is indeed comparing the relative importance of the different sources of uncertainty.

[16] We have shown how rigorously quantified uncertainties in short‐term climate change, in the form of PDFs of temperature and precipitation changes, can be propagated into an impact model of crop yield changes and, as a result, projected changes in three important crops ‐ barley, maize and wheat ‐ can be obtained with their uncertainty quantified. Projected changes in temperature and precipitation negatively affect the three crops' yield, by causing a decrease in yield that is significant for all ‐ by about 9% (1.7%–17%) for barley, by 13% (5%–25%) for maize and by 5% (1%–10%) for wheat. Including CO₂ fertilization reduces projected losses by an average of 7% for wheat and barley but does not change significantly the impact on maize, and has relatively small effects on overall uncertainty.

[17] The recent Fourth Assessment Report by the Intergovernmental Panel on Climate Change synthesized results from previous studies with quantitative but non‐probabilistic statements. For example, one of the main conclusions in the report is that “550 ppm CO₂ (associated with approximately 2°C of warming) increases C3 crop yield by 17%; this increase is offset by temperature increase of 2°C assuming no adaptation and 3°C with adaptation” [Easterling et al., 2007, p. 276]. Our study considered the shorter time frame of 2030 when CO₂ levels are expected to reach ≈450 ppm, and proposes a probabilistic assessment of the expected changes, on the basis of which we estimate at most a 75% chance that CO₂ and climate effects will cancel by 2030 for wheat, at most a 30% chance for barley, and 0% for maize. Given these results, and the diminishing response of crops to further increases in CO₂, the statement that global yields of C3 crops will be unaffected at 550ppm thus appears optimistic, although within our fairly wide uncertainty bounds. In the IPCC nomenclature, we estimate the chance that global losses from climate change by 2030 will outweigh gains from CO₂ as unlikely for wheat (<33% chance), likely for barley (>66% chance) and virtually certain for maize (>99% chance). In addition, we estimate larger than 80% chance that net losses for maize will exceed 10% over this relatively short time period.

[18] We do not claim that our projections are a definitive quantification of the changes in store. Rather, we strive to present a methodology that can integrate different sources of uncertainty, conditional on the information available to us through the most recent concerted climate modeling effort, an empirical evaluation of crop yield sensitivities to climate and the most up‐to‐date meta‐analysis of effects of CO₂ fertilization on crop yields. Any of these components are likely to change and improve with additional efforts of the modeling communities and further observational studies, but we think that in the meantime there is value in proposing a rigorous approach to a transparent integration of the best information currently available and its uncertainties.

Acknowledgments

[19] The authors would like to thank Linda O. Mearns for her valuable comments on the manuscript and two anonymous reviewers. Claudia Tebaldi is grateful to Chris Field and the Department of Global Ecology, Carnegie Institution, Stanford, for the hospitality.