## Abstract

Quantifying model uncertainty and internal variability components in climate projections has been paid a great attention in the recent years. For multiple synthetic ensembles of climate projections, we compare the precision of uncertainty component estimates obtained respectively with the two Analysis of Variance (ANOVA) approaches mostly used in recent works: the popular Single Time approach (STANOVA), based on the data available for the considered projection lead time and a time series based approach (QEANOVA), which assumes quasi-ergodicity of climate outputs over the available simulation period. We show that the precision of all uncertainty estimates is higher when more members are used, when internal variability is smaller and/or the response-to-uncertainty ratio is higher. QEANOVA estimates are much more precise than STANOVA ones: QEANOVA simulated confidence intervals are roughly 3–5 times smaller than STANOVA ones. Except for STANOVA when less than three members is available, the precision is rather high for total uncertainty and moderate for internal variability estimates. For model uncertainty or response-to-uncertainty ratio estimates, the precision is low for QEANOVA to very low for STANOVA. In the most unfavorable configurations (small number of members, large internal variability), large over- or underestimation of uncertainty components is thus very likely. In a number of cases, the uncertainty analysis should thus be preferentially carried out with a time series approach or with a local-time series approach, applied to all predictions available in the temporal neighborhood of the target prediction lead time.

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## Acknowledgements

We thank the three anonymous reviewers for their constructive suggestions which helped to significantly improve the content of our manuscript.

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### Contributions

BH designed the analysis, developed the local-QEANOVA and the synthetic simulations. JB derived the theoretical expressions for unbiased estimators of uncertainty components and wrote the appendixes. All authors contributed to write the manuscript and discuss results.

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## Appendices

### Appendix 1: QEANOVA Estimates of uncertainty components

We summarize the theoretical developments of the QEANOVA approach to achieve unbiased estimators of uncertainty components for a simplified configuration where the trend model is a simple linear function of time. The full developments are given in Hingray and Blanchet (2018) for the general configuration where the trend model is a linear combination of *L* functions of time. Here, we first consider the case where the number of members differs from one chain to the other; the simplified equations obtained when all chains have the same number are then given. For the sake of conciseness, we omit the subscript “\(QE\)” related to the QEANOVA approach.

*Model.*

We first consider the raw projections *Y*(*g*, *m*, *t*) with \(M_g\) members for each of the *G* chains, assuming that, for all \(t_s \le t \le t_f\):

where \(\lambda (g,t)\) is the trend model expressed as a linear function of time: \(\lambda (g,t)=\varLambda _{g1}+\varLambda _{g2}(t-t_s)\) and where the \(\nu (g,m,t)\) are independent and homoscedastic random variables (with variance \(\sigma _{\nu _g}^2\)). Second let consider the change variable at future prediction lead time \(t \in [t_c,t_f]\):

where \(t_c\) is the reference period. We have thus \(\alpha (g,t)=\varLambda _{g2}(t-t_c)\) and \(\eta (g,m,t)=\nu (g,m,t)-\nu (g,m,t_c)\).

*Unbiased estimation of the model parameters for the raw variable **Y*.

We discretize \([t_s,t_f]\) into *T* time steps (from \(t_s=t_1\) to \(t_f=t_T\)) and write \(t_c\) as the *K*th time step (i.e. \(t_c=t_K\)). We are interested in the future prediction lead time \(t_k \in [t_K,t_T]\). Let consider the regression model (10) for a particular *g*. Unbiased estimators of the regression parameters \((\varLambda _{g1},\varLambda _{g2})\) are given by the least square estimates

where “”’ denotes the transpose , \({\mathbb {R}}\) is the \(T \times 2\) matrix of covariates whose *k*th row is \((1,t_k-t_1)\), for \(1 \le k \le T\), and \(\mathbb V= ({\mathbb {R}}' {\mathbb {R}})^{-1}\). Covariance matrix of the estimators \(({\hat{\varLambda }}_{g1}, {\hat{\varLambda }}_{g2})\) is given by \(\widehat{\sigma _{\nu _g}^2} M_g^{-1} {\mathbb {V}}\) where \(\widehat{\sigma _{\nu _g}^2}\) is an unbiased estimator of \(\sigma _{\nu _g}^2\) given by

where \(L=2\) and \({\hat{\lambda }}_{QE}(g,t_k)={\hat{\varLambda }}_{g1}-{\hat{\varLambda }}_{g2}(t_k-t_1)\).

In particular, an unbiased estimator of \(\varLambda _{g2}^2\) is

where \(V_{22}\) is the element (2, 2) of \({\mathbb {V}}\). Considering \(t_1,\ldots ,t_T\) regularly spaced on \([t_s; t_f]\), we have \(V_{22}=12(T-1)/\{T(T+1)(t_T-t_1)^2\}\).

*Unbiased estimation of the sample variance of the *\(\alpha\)’*s in the change variable.*

Given (11) and (13), an unbiased estimator of the sample variance of \(\alpha (g,t_k)\), i.e. of \(s_\alpha ^2(t_k)=\frac{1}{G-1}\sum _{g=1}^G \{\alpha (g,t_k)\}^2\) , is

where

When all GCMs have the same number of runs (*M*), this expression reduces to:

where \({\widehat{\sigma }}_{\eta }^2\) is an unbiased estimator of internal variability variance for *X*

and where

### Appendix 2: Estimates with a local QEANOVA approach

We here summarize the expressions of the different uncertainty estimators obtained with a local-QEANOVA approach. The full developments, similar to those presented in appendix A for the QEANOVA approach, are detailed in Hingray and Blanchet (2018).

Model: a regression model is still considered to estimate the response function \(\lambda (g, t)\) for *Y* in Eq. 10 but \(\lambda (g, t)\) is assumed to be only locally linear in time, in the neighborhoods of \(t_c\) and \(t_e\) respectively, i.e. on \([t_c-\omega ,t_c+\omega ]\) and \([t_e-\omega ,t_e+\omega ]\), where \(t_e \in [t_c,t_f]\) is the future prediction lead time under consideration. \(\lambda (g, t)\) can thus be expressed as

The change variable *X*(*g*, *m*, *t*), for \(t_e-\omega \le t \le t_e+\omega\), in Eq. 11 is such that \(\alpha (g,t)=(\varLambda _{g1,e}-\varLambda _{g1,c})+(t-t_e)\varLambda _{g2,e}\).

Each interval \([t_c-\omega ,t_c+\omega ]\) and \([t_e-\omega ,t_e+\omega ]\) is discretized into \(T^{\star }\) regular periods of length \(dt=2 \omega /(T^*-1)\), with \(T^{\star }\) odd, giving respectively the sequences \(t_1,\ldots ,t_{T^*}\) and \(t_{T^*+1},\ldots ,t_{2T^*}\). The values of *Y* for these different times are further considered to estimate the regression coefficients of the linear trend models in Eq. 17. For the illustration given in Sect. 5.3, \(dt = \omega =20yrs\) and \(T^*=3\).

*Unbiased estimators of model uncertainty and internal variability variance.*

Following Hingray and Blanchet (2018), an unbiased estimator of model uncertainty variance, i.e. of the sample variance of \(\alpha (g,t_e)\), is

where \(T=2T^*\) is the total number of time steps considered in the analysis and where \(\widehat{\sigma _{\nu _g}^2}\) is an unbiased estimator of \(\sigma _{\nu _g}^2\) given by

with \({\hat{\lambda }}_{c}(g,t_j)={\hat{\varLambda }}_{g1,c}+{\hat{\varLambda }}_{g2,c}(t_j-t_c)\) and \({\hat{\lambda }}_{e}(g,t_j)={\hat{\varLambda }}_{g1,e}+{\hat{\varLambda }}_{g2,e}(t_j-t_e)\) where \({\hat{\varLambda }}_{g1,c}, {\hat{\varLambda }}_{g2,c}, {\hat{\varLambda }}_{g1,e}\) and \({\hat{\varLambda }}_{g2,e}\) are the regression coefficients of the two linear models in Eq. 17.

When all GCMs have the same number of runs (*M*), the expression of model uncertainty in Eq. 18 reduces to

where \({\widehat{\sigma }}_{\eta }^2\) is an unbiased estimator of internal variability variance for *X*

### Appendix 3: Simulation of MMEs

Each MME is simulated for *Y*(*g*, *m*, *t*) assuming that, for all \(t_s \le t \le t_f\):

where the climate response function \(\lambda (g,t)\) is a linear function of time and where the \(\nu (g,m,t)\) are independent and homoscedastic random variables (with variance \(\sigma _{\nu _g}^2\)). For convenience we further assume that \(\lambda (g,t)\) can be decomposed as \(\lambda (g,t)=w(t)+d(g,t)\), for \(t=t_1,\ldots ,T\) where the mean climate response *w*(*t*) of the *G* chains and the deviations *d*(*g*, *t*) of chain *g* are linear functions of time, expressed as: \(w(t)=B+P.(t-t_1)/(t_e-t_C)\) and \(d(g,t)=D(g).(t-t_1)/(t_e-t_C)\) with the constraint \(\sum _{g=1}^{ G }D(g)=0\).

For graphical simplification purposes, each MME for *Y* is constructed so that the parameters \((\sigma ^2_\nu , P, D(g), g=1,\ldots ,G )\) lead, for the change variable *X* at time \(t=t_e\), to a prescribed value of the Response-to-Uncertainty ratio [R2U(\(t_e\))] and to a prescribed value of the fractional variance \(F_\eta (t_e)\) due to internal variability.

For *X*, we have by definition \(\varphi (g,t)=\lambda (g,t)-\lambda (g,t_C)\) and \(\eta (g,m,t) = \nu (g,m,t)-\nu (g,m,t_C)\). We have thus for the change variable \(\mu (t)\) = \(w(t)-w(t_C)\), \(\alpha (g,t) = d(g,t)-d(g,t_C)\) and in turn \(\mu (t) = P.(t-t_C)/(t_e-t_C)\) and \(\alpha (g,t) = D(g).(t-t_C)/(t_e-t_C)\).

The theoretical values for \(\mu (t_e)\) and \(s^2_\alpha (t_e)\) are thus as follows : \(\mu (t_e) = P\) and \(s^2_\alpha (t_e)={\mathbb V\text {ar}}(D(g))\). Fixing *P* to 1, we thus simply require in turn

For each MME simulation, the deviations of the different chains, \(D(g), g=1,\ldots ,G\), are obtained from a sample of *G* realizations in a normal distribution. These realizations are scaled so that their mean is zero and their variance corresponds to the prescribed value \(s^2_\alpha (t_e)\).

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Hingray, B., Blanchet, J., Evin, G. *et al.* Uncertainty component estimates in transient climate projections.
*Clim Dyn* **53, **2501–2516 (2019). https://doi.org/10.1007/s00382-019-04635-1

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### Keywords

- Uncertainty sources
- Climate projections
- ANOVA
- Internal variability
- Model uncertainty
- Scenario uncertainty
- Precision of estimates