## Abstract

This paper analyzes changes of maximum temperatures in Europe, which are evaluated using two state-of-the-art regional climate models from the EU ENSEMBLES project. Extremes are expressed in terms of return values using a time-dependent generalized extreme value (GEV) model fitted to monthly maxima. Unlike the standard GEV method, this approach allows analyzing return periods at different time scales (monthly, seasonal, annual, etc). The study focuses on the end of the 20th century (1961–2000), used as a calibration/validation period, and assesses the changes projected for the period 2061–2100 considering the A1B emission scenario. The performance of the regional models is evaluated for each season of the calibration period against the high-resolution gridded E-OBS dataset, showing a similar South-North gradient with larger values over the Mediterranean basin. The inter-RCM changes in the bias pattern with respect to the E-OBS are larger than the bias resulting from a change in the boundary conditions from ERA-40 to ECHAM5 20c3m. The maximum temperature response to increased green house gases, as projected by the A1B scenario, is consistent for both RCMs. Under that scenario, results indicate that the increments for extremes (e.g. 40-year return values) will be two or three times higher than those for the mean seasonal temperatures, particularly during Spring and Summer in Southern Europe.

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## Acknowledgements

The ENSEMBLES data used in this work was funded by the EU FP6 Integrated Project ENSEMBLES (Contract number 505539) whose support is gratefully acknowledged. We acknowledge the E-OBS data set and the data providers in the ECA&D project (http://eca.knmi.nl). R. Mínguez is indebted to the Spanish Ministry MICINN for the funding provided within the “Ramon y Cajal” program. This work was partly funded by projects “GRACCIE” (CSD2007-00067, Programa Consolider-Ingenio 2010), “AMVAR” (CTM2010-15009) and EXTREMBLES (CGL2010-21869) from Spanish Ministry MICINN, by project C3E (200800050084091) and ESCENA (200800050084265) from the Spanish Ministry MARM, and by project MARUCA (E17/08) from the Spanish Ministry MF. The authors would like to especially thank the anonymous reviewers who helped to considerably improve the former versions of our manuscript.

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## Appendix: Aggregated quantile expression derivation

### Appendix: Aggregated quantile expression derivation

This appendix explains in detail the derivation of the aggregated quantile expression (3). We use the analogy with the monthly stationary approach, which consist of the fitting of 12 GEV models, one for each month, using the maximum data associated with each month. Using these models, it is possible to calculate the probability of obtaining a maximum temperature value lower or equal to \(\bar x_q\) during each month, i.e.:

where \(f_i(\bar x_q)=\left[1+\xi_i\left( \frac{\bar x_q-\mu_i}{\psi_i } \right)\right]^{-1/\xi_i}\). Note that location, scale and shape parameters are constant for each month. This expression allows obtaining the probability *q*
_{
i
}, which corresponds to an annual probability, since each month occurs once a year.

The equivalent expression to Eq. 9 for the non-stationary approach is:

where the exponent corresponds to an average value of the function \(f(\bar x_q,t)\) over the integration interval, for this reason, it is divided by the integration interval length. Note that expression (10) is the same as Eq. 3.

If using monthly maxima and the stationary approach, we want to calculate the annual maxima cumulative distribution function, the following expression is used:

For the non-stationary approach, and considering the relationship between Eqs. 9 and 10, it becomes:

which is also the same as Eq. 3 but modifying the integration interval.

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Frías, M.D., Mínguez, R., Gutiérrez, J.M. *et al.* Future regional projections of extreme temperatures in Europe: a nonstationary seasonal approach.
*Climatic Change* **113**, 371–392 (2012). https://doi.org/10.1007/s10584-011-0351-y

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DOI: https://doi.org/10.1007/s10584-011-0351-y

### Keywords

- Return Period
- Generalize Extreme Value
- Return Level
- Generalize Extreme Value Distribution
- Nonstationary Model