a,b, Provide hydrological and crop modelling (a) and climate and weather modelling (b). c, Input from a and b is used to produce grid-to-province PDFs of yield loss captured by weather indices, conditional on large-scale interannual climate processes. d, The grid-level yield loss PDFs and yield response functions subject to GHG and technological scenarios from c are used to derive regional-level risk profiles of production loss. e, If the region matches an economic administrative unit (for example, province, country), the input from d is used to to derive distributions of province-level economic losses. f, Uses the input from d and/or, if relevant, e, to determine optimum combinations of risk mitigation and transfer instruments to minimize risk of climate-driven losses.
Data sources.
Daily observed weather data on precipitation, radiation, and maximum and minimum temperatures were used. The data set was provided by the National Climate Centre (NCC) of the China Meteorological Administration (CMA) on a 0.25° × 0.25° longitude–latitude grid, available from 1961 to 2012; it covered the two northeastern provinces of Shandong and Hebei, and the two southern provinces of Guangxi and Guangdong. Grid-level maize and rice yields were simulated in those northeastern and southern provinces, respectively, using a mechanistic crop model called DSSAR-CERES.
Random-forest-based selection of indices.
We selected the most effective pixel-level pairs of indices to capture the effects of deficit precipitation and excess temperature on yield variability by a random-forest algorithm. This algorithm uses ensemble-based recursive partitioning and thus permits one to circumvent the issues of cross-correlation between indices and of a large number of variables versus a small sample size.
Extreme-value multivariate modelling.
Robust stochastic characterization of the interannual variability of the optimum grid-level weather indices was carried out using univariate distributions of mixed, exponential—generalized Pareto distribution (GPD)—type. The latter allows one to accurately estimate the risk of occurrence of events that are both rare and extreme, within a modified GPD framework across the whole gridded domain studied. The stochastic dependence of deficit precipitation and excess temperature is characterized by coupling their univariate mixed distributions FX and FY within a Gumbel–Hougaard copula model, as described in the equations (1) and (2) below.
Here Cθ is the Gumbel–Hougaard Archimedean extreme-value copula,
The coefficient of dependence is θ ≥ 1, where θ = 1 characterizes independence of the uniform transforms uX and uY of the mixed univariate FX and FY distributions of precipitation and heatwave grid-level indices, respectively.
The Gumbel–Hougaard Archimedean copula enables us to characterize dependence in both the upper and lower tails without assuming independence of extreme-value occurrences, as is the case in Gaussian copulas. An example of stochastic dependence of two weather indices, at the same location and subject to a technological scenario, is presented in Supplementary Fig. 2.
Nonhomogeneous Hidden Markov Model ‘weather-within-climate’ modelling.
Historical univariate or multivariate distributions of weather indices are derived by adopting a ‘weather-within-climate’ modelling framework. The distributions are modelled conditionally on hidden regional weather states, St, that capture seasonal variability. These states are conditioned themselves on observed or simulated continental- and planetary-scale climate drivers that capture interannual modes of variability. A Nonhomogeneous Hidden Markov Model (NHMM) is used to achieve this two-step conditioning and enable the introduction of non-stationarity, as illustrated in Supplementary Fig. 1 across a gridded domain and equation (3) below.
The weather index distributions, P(O1: T, S1: T|λ, z1: T), thus use continental-scale climate variables, z1: T; these covariates can be observed, as done here, or be simulated by high-end general circulation models, subject to future greenhouse gas scenarios.
The non-stationary univariate distributions of pixel-level precipitation and excess heat, O1: T, follow the mixed GPD-exponential univariate framework presented above. The copula-characterized stochastic dependency between marginals is considered stationary across weather states.
Here 1961 ≤ t ≤ 2012, St are the hidden states of the two-state Markov chain, zt is the non-stationary Niño-3.4 index acting as covariate, λ = {ai, πi}i={1,2} contains the transition parameters ai and initial probabilities πi of the NHMM, and bSt is the distribution of the observed weather indices at time t, depending on the state St as follows:
where aij(zt) is the transition probability from state i at time t to j at time t + 1 of a first-order Markov chain as a function of the non-stationary covariate zt, πi(z1) is the probability that the initial hidden state at t = 1 is i, S1 = i, and bSt(Ot+1|zt+1) is a component of the vector of observed weather indices characterized by mixed densities FX and FY cited above, and dependent on the value of the non-stationary covariate zt+1.
Generalized additive mixed crop response modelling.
To model the vulnerability functions of crop yield to the combined or individual effects of precipitation variability and excess temperature exposure, generalized additive mixed models (GAMMs) are used (see equation (4)). The use of a GAMM g(μi) enables one to capture the nonlinear response of crop yield μi to varying values of a single or several weather indices (see Fig. 2f),
Here μi ≡ E(Yi), with Yi the rice or maize yield response variable following an exponential-family probability distribution function, and Xi is the ith row of the model matrix with its corresponding θ parameter vector.
Also, to model the univariate model of rice or maize yield response to heatwaves or deficit precipitation, a smoothing basis composed of natural cubic splines is used. Ultimately, the convolution of the GAMM-based yield response function with the distribution of the corresponding grid-level indices results in the distribution of yield loss as a function of index values.
Input–output-based economic impact modelling.
An input–output modelling approach is used to assess direct and indirect province-level economic impacts due to weather-driven maize production shortfall. Further details concerning the methodology can be found in the Supplementary Information.