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Estimating daily climatological normals in a changing climate

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Climatological normals are widely used baselines for the description and the characterization of a given meteorological situation. The World Meteorological Organization (WMO) standard recommends estimating climatological normals as the average of observations over a 30-year period. This approach may lead to strongly biased normals in a changing climate. Here we propose a new method with which to estimate daily climatological normals in a non-stationary climate. Our statistical framework relies on the assumption that the response to climate change is smooth over time, and on a decomposition of the response inspired by the pattern scaling assumption. Estimation is carried out using smoothing splines techniques, with a careful examination of the selection of smoothing parameters. The new method is compared, in a predictive sense and in a perfect model framework, to previously proposed alternatives such as the WMO standard (reset either on a decadal or annual basis), averages over shorter periods, and hinge fits. Results show that our technique outperforms all alternatives considered. They confirm that previously proposed techniques are substantially biased—biases are typically as large as a few tenths to more than 1\(^{\circ }\text{C}\) by the end of the century—while our method is not. We argue that such “climate change corrected” normals might be very useful for climate monitoring, and that weather services could consider using two different sets of normals (i.e. both stationary and non-stationary) for different purposes.

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The authors acknowledge Météo-France for supporting this study along with the climate modeling groups involved in CMIP5 for producing and sharing their simulations. They also wish to thank the two anonymous referees for their constructive comments which were of great value in improving the quality of the paper.


Also funded by Météo France.

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Correspondence to Alix Rigal.

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Jean-Marc Azaïs: Member of the federal University “Université de Toulouse”.

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The analysis in this article has been performed using the statistical software R.

1.1 Computational and simulation details

The 21 simulations used for daily mean temperature were: ACCESS1-0, ACCESS1-3, CCSM4, CESM1-BGC, CMCC-CMS, CNRM-CM5, CSIRO-Mk3-6-0, CanESM2, GFDL-CM3, GFDL-ESM2G, GFDL-ESM2M, IPSL-CM5A-LR, IPSL-CM5A-MR, IPSL-CM5B-LR, MIROC-ESM-CHEM, MIROC-ESM, MPI-ESM-LR, MPI-ESM-MR, MRI-CGCM3, NorESM1-M, inmcm4.

The simulations used for annual mean temperature were: ACCS0 _ r1i1p1, ACCS3 _ r1i1p1, BCCl _ r1i1p1, BCCm _ r1i1p1, BNU _ r1i1p1, CCCMA _ r1i1p1, CCCMA _ r2i1p1, CCCMA _ r3i1p1, CCCMA _ r4i1p1, CCCMA _ r5i1p1, CNRM _ r10i1p1, CNRM _ r1i1p1, CNRM _ r2i1p1, CNRM _ r4i1p1, CNRM _ r6i1p1, CSIRO _ r10i1p1, CSIRO _ r1i1p1, CSIRO _ r2i1p1, CSIRO _ r3i1p1, CSIRO _ r4i1p1, CSIRO _ r5i1p1, CSIRO _ r6i1p1, CSIRO _ r7i1p1, CSIRO _ r8i1p1, CSIRO _ r9i1p1, GFDLc _ r1i1p1, GFDLg _ r1i1p1, GFDLm _ r1i1p1, GISSr _ r1i1p1, IAPg _ r1i1p1, IAPs _ r1i1p1, IAPs _ r2i1p1, IAPs _ r3i1p1, INGVc _ r1i1p1, INGVe _ r1i1p1, INGVs _ r1i1p1, INM _ r1i1p1, IPSLal _ r1i1p1, IPSLal _ r2i1p1, IPSLal _ r3i1p1, IPSLal _ r4i1p1, IPSLam _ r1i1p1, IPSLb _ r1i1p1, MIROC5 _ r1i1p1, MIROC5 _ r2i1p1, MIROC5 _ r3i1p1, MIROCc _ r1i1p1, MIROCe _ r1i1p1, MPIMl _ r1i1p1, MPIMl _ r2i1p1, MPIMl _ r3i1p1, MPIMm _ r1i1p1, MRI _ r1i1p1, NCARc _ r1i1p1, NCARc _ r2i1p1, NCARc _ r3i1p1, NCARc _ r4i1p1, NCARc _ r5i1p1, NCARc _ r6i1p1, NCARe _ r1i1p1.

1.2 Another system of constraints for model (9)

Once we have obtained the decomposition of model (9), it is possible to make it more interpretable. Let \(\tilde{g}=g-g(1),\tilde{f}=f+g(1)\cdot h\). Then, the decomposition of model (9) can be rewritten as:

$$\begin{aligned} f(d)+g(y).h(d)&=(f(d)+g(1)\cdot h(d))+(g(y)-g(1))\cdot h(d)\\&=\tilde{f}(d)+\tilde{g}(y)\cdot h(d). \end{aligned}$$

Thus, \(\tilde{f}\) represents the annual reference cycle of the first year of the considered period and \(\tilde{g}\) quantifies the annual mean temperature evolution. Therefore the first value, \(\tilde{g}(1)\), is zero.

1.3 Alternating least squares

Addition of a few steps to the sequential algorithm permitting an iterative procedure:

  1. 4

    Re-estimation ofg():

    We now fix \(\hat{f}, \hat{h}\) and estimate g once again, the goal of the procedure being minimization of the total sum of squares

    i.e \(RSS=\sum _{y,d}(T_{y,d}-\hat{f}_{d}-\hat{g}_{y}\cdot \hat{h}_{d})^{2}\).

    For a fixed y, let us define:

    \(RSS_{y}=\sum _{d}((T_{y,d}-\hat{f}_{d})-{g}_{y}\cdot \hat{h}_{d})^{2}=\sum _{d}(\widetilde{T}_{y,d}-{g}_{y}\cdot \hat{h}_{d})^{2}\) where \(\widetilde{T}_{y,d}=T_{y,d}-\hat{f}_{d}\)

    let \(g_{0,y}\) the mean square estimator \(g_{0,y}=\frac{\sum _{j=1}^{365}\hat{h}_{d}.T_{y,d}}{\sum _{j=1}^{365}\hat{h}_{d}^{2}}.\)

    Also by the Pythagorean theorem:

    $$\begin{aligned} &\sum _{d=1}^{365}(\widetilde{T}_{y,d}-g_{y}\cdot \hat{h}_{d})=\sum _{d=1}^{365}(\widetilde{T}_{y,d} -g_{0,y}\cdot \hat{h}_{d}+(g_{0,y}-g_{y})\cdot \hat{h}_{d})^{2}\\&=\sum _{d=1}^{365}(\widetilde{T}_{y,d}-g_{0,y}\cdot \hat{h}_{d})^{2}+\sum _{d=1}^{365}((g_{0,y}-g_{y})\cdot \hat{h}_{d})^{2}\\&=\sum _{d=1}^{365}(\widetilde{T}_{y,d}-g_{0,y}\cdot \hat{h}_{d})^{2}+(g_{0,y}-g_{y})^{2}\cdot \sum _{d=1}^{365}\hat{h}_{d}^{2}. \end{aligned}$$


    $$\begin{aligned} RSS&=\sum _{y=1}^{n}RSS_{y}\\&=\sum _{d,y}(\widetilde{T}_{y,d}-g_{0,y}\cdot \hat{h}_{d})^{2}+\sum _{y=1}^{n}(g_{0,y}-g_{y})^{2}\cdot \sum _{d=1}^{365}\hat{h}_{d}^{2}. \end{aligned}$$

    Then, we compute the smoothing spline estimate \({\hat{g}}()\) of \(g_{0,y}\), with the given \(df_g\).

  2. 5

    We iterate steps 3 and 4 to minimize sum of squares RSS.

1.4 Annual scoring for normals

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The three plots illustrate scoring on the yearly mean temperature at San Francisco, for each year normal prediction occurs on all CMIP5 models. The horizontal axis represents the end of the training period and for each method, prediction occurs the following year. The upper line shows, for each score, the winning method for predicting the next year. The different calculations are WMO (black), WMO reset (grey), OCN (yellow), hinge (light green), hinge fit reset (green) and model (9) (red). The upper figure shows the evolution of the bias, the middle one represents the variance of the prediction and the bottom plot illustrates the evolution of the mean square prediction error (MSE)

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Rigal, A., Azaïs, JM. & Ribes, A. Estimating daily climatological normals in a changing climate. Clim Dyn 53, 275–286 (2019).

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