The Annals of Applied Statistics

Temperatures in transient climates: Improved methods for simulations with evolving temporal covariances

Andrew Poppick, David J. McInerney, Elisabeth J. Moyer, and Michael L. Stein

Full-text: Open access


Future climate change impacts depend on temperatures not only through changes in their means but also through changes in their variability. General circulation models (GCMs) predict changes in both means and variability; however, GCM output should not be used directly as simulations for impacts assessments because GCMs do not fully reproduce present-day temperature distributions. This paper addresses an ensuing need for simulations of future temperatures that combine both the observational record and GCM projections of changes in means and temporal covariances. Our perspective is that such simulations should be based on transforming observations to account for GCM projected changes, in contrast to methods that transform GCM output to account for discrepancies with observations. Our methodology is designed for simulating transient (nonstationary) climates, which are evolving in response to changes in CO$_{2}$ concentrations (as is the Earth at present). This work builds on previously described methods for simulating equilibrium (stationary) climates. Since the proposed simulation relies on GCM projected changes in covariance, we describe a statistical model for the evolution of temporal covariances in a GCM under future forcing scenarios, and apply this model to an ensemble of runs from one GCM, CCSM3. We find that, at least in CCSM3, changes in the local covariance structure can be explained as a function of the regional mean change in temperature and the rate of change of warming. This feature means that the statistical model can be used to emulate the evolving covariance structure of GCM temperatures under scenarios for which the GCM has not been run. When combined with an emulator for mean temperature, our methodology can simulate evolving temperatures under such scenarios, in a way that accounts for projections of changes while still retaining fidelity with the observational record. The emulator for variability changes is also of interest on its own as a summary of GCM projections of variability changes.

Article information

Ann. Appl. Stat., Volume 10, Number 1 (2016), 477-505.

Received: July 2015
Revised: January 2016
First available in Project Euclid: 25 March 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Climate change climate variability transient climate observation-driven simulation nonstationary processes evolutionary spectrum


Poppick, Andrew; McInerney, David J.; Moyer, Elisabeth J.; Stein, Michael L. Temperatures in transient climates: Improved methods for simulations with evolving temporal covariances. Ann. Appl. Stat. 10 (2016), no. 1, 477--505. doi:10.1214/16-AOAS903.

Export citation


  • Barnes, E. A. (2013). Revisiting the evidence linking Arctic amplification to extreme weather in midlatitudes. Geophys. Res. Lett. 40 4734–4739.
  • Bhat, K. S., Haran, M., Olson, R. and Keller, K. (2012). Inferring likelihoods and climate system characteristics from climate models and multiple tracers. Environmetrics 23 345–362.
  • Castruccio, S. and Stein, M. L. (2013). Global space-time models for climate ensembles. Ann. Appl. Stat. 7 1593–1611.
  • Castruccio, S., McInerney, D. J., Stein, M. L., Liu Crouch, F., Jacob, R. L. and Moyer, E. J. (2014). Statistical emulation of climate model projections based on precomputed GCM runs. J. Climate 27 1829–1844.
  • Chang, W., Haran, M., Olson, R. and Keller, K. (2014). Fast dimension-reduced climate model calibration and the effect of data aggregation. Ann. Appl. Stat. 8 649–673.
  • Collins, W. D., Bitz, C. M., Blackmon, M. L., Bonan, G. B., Bretherton, C. S., Carton, J. A., Chang, P., Doney, S. C., Hack, J. J., Henderson, T. B. et al. (2006). The community climate system model version 3 (CCSM3). J. Climate 19 2122–2143.
  • Dahlhaus, R. (1997). Fitting time series models to nonstationary processes. Ann. Statist. 25 1–37.
  • Dahlhaus, R. (2000). A likelihood approximation for locally stationary processes. Ann. Statist. 28 1762–1794.
  • Dahlhaus, R. (2012). Locally stationary processes. In Handbook of Statistics, Time Series Analysis: Methods and Applications 30 351–408. North-Holland, Amsterdam.
  • Decker, M., Brunke, M. A., Wang, Z., Sakaguchi, K., Zeng, X. and Bosilovich, M. G. (2012). Evaluation of the reanalysis products from GSFC, NCEP, and ECMWF using flux tower observations. J. Climate 25 1916–1944.
  • Francis, J. A. and Vavrus, S. J. (2012). Evidence linking Arctic amplification to extreme weather in mid-latitudes. Geophys. Res. Lett. 39 L06801.
  • Gotway, C. A. and Young, L. J. (2002). Combining incompatible spatial data. J. Amer. Statist. Assoc. 97 632–648.
  • Guinness, J. and Stein, M. L. (2013). Transformation to approximate independence for locally stationary Gaussian processes. J. Time Series Anal. 34 574–590.
  • Hawkins, E., Osborne, T. M., Ho, C. K. and Challinor, A. J. (2013). Calibration and bias correction of climate projections for crop modelling: An idealised case study over Europe. Agric. For. Meteorol. 170 19–31.
  • Ho, C. K., Stephenson, D. B., Collins, M., Ferro, C. A. T. and Brown, S. J. (2012). Calibration strategies: A source of additional uncertainty in climate change projections. Bull. Am. Meteorol. Soc. 93 21–26.
  • Holmes, C. R., Woollings, T., Hawkins, E. and de Vries, H. (2015). Robust future changes in temperature variability under greenhouse gas forcing and the relationship with thermal advection. J. Climate 2015.
  • IPCC (2001). Climate Change 2001: The Scientific Basis. Contribution of Working Group I to the Third Assessment Report of the Intergovernmental Panel on Climate Change (J. T. Houghton, Y. Ding, D. J. Griggs, M. Noguer, P. J. van der Linden, X. Dai, K. Maskell and C. A. Johnson, eds.). Cambridge Univ. Press, New York.
  • IPCC (2007). Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change (S. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tignor and H. L. Miller, eds.). Cambridge Univ. Press, New York.
  • IPCC (2013). Climate Change 2013: The Physical Science Basis. Contribution of Working Group I to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change (T. Stocker, D. Qin, G.-K. Plattner, M. Tignor, S. K. Allen, J. Boschung, A. Nauels, Y. Xia, V. Bex and P. M. Midgley, eds.). Cambridge Univ. Press, New York.
  • Leeds, W. B., Moyer, E. J. and Stein, M. L. (2015). Simulation of future climate under changing temporal covariance structures. Advances in Statistical Climatology Meteorology and Oceanography 1 1–14.
  • Morice, C. P., Kennedy, J. J., Rayner, N. A. and Jones, P. D. (2012). Quantifying uncertainties in global and regional temperature change using an ensemble of observational estimates: The HadCRUT4 data set. J. Geophys. Res. 117. D08101.
  • Neumann, M. H. and von Sachs, R. (1997). Wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra. Ann. Statist. 25 38–76.
  • Ombao, H., Raz, J., von Sachs, R. and Guo, W. (2002). The SLEX model of a non-stationary random process. Ann. Inst. Statist. Math. 54 171–200.
  • Piani, C. and Haerter, J. O. (2012). Two dimensional bias correction of temperature and precipitation copulas in climate models. Geophys. Res. Lett. 39 L20401.
  • Poppick, A., McInerney, D. J., Moyer, E. J. and Stein, M. L. (2016). Supplement to “Temperatures in transient climates: Improved methods for simulations with evolving temporal covariances.” DOI:10.1214/16-AOAS903SUPP.
  • Priestley, M. B. (1981). Spectral Analysis and Time Series. Vol. 2. Academic Press, New York.
  • Rougier, J., Sexton, D. M., Murphy, J. M. and Stainforth, D. (2009). Analyzing the climate sensitivity of the HadSM3 climate model using ensembles from different but related experiments. J. Climate 22 3540–3557.
  • Saha, S., Moorthi, S., Pan, H.-L., Wu, X., Wang, J., Nadiga, S., Tripp, P., Kistler, R., Woollen, J., Behringer, D. et al. (2010). The NCEP climate forecast system reanalysis. Bull. Am. Meteorol. Soc. 91 1015–1057.
  • Salazar, E., Sansó, B., Finley, A. O., Hammerling, D., Steinsland, I., Wang, X. and Delamater, P. (2011). Comparing and blending regional climate model predictions for the American Southwest. J. Agric. Biol. Environ. Stat. 16 586–605.
  • Sansó, B. and Forest, C. (2009). Statistical calibration of climate system properties. J. R. Stat. Soc. Ser. C. Appl. Stat. 58 485–503.
  • Sansó, B., Forest, C. E. and Zantedeschi, D. (2008). Inferring climate system properties using a computer model. Bayesian Anal. 3 1–37.
  • Santer, B. D., Wigley, T. M. L., Schlesinger, M. E. and Mitchell, J. F. B. (1990). Developing Climate Scenarios from Equilibrium GCM Results. Max-Planck-Institut für Meteorologie, Hamburg.
  • Schneider, T., Bischoff, T. and Płotka, H. (2015). Physics of changes in synoptic midlatitude temperature variability. J. Climate 28 2312–2331.
  • Screen, J. A. and Simmonds, I. (2013). Exploring links between Arctic amplification and mid-latitude weather. Geophys. Res. Lett. 40 959–964.
  • Semenov, M. A. and Barrow, E. M. (1997). Use of a stochastic weather generator in the development of climate change scenarios. Clim. Change 35 397–414.
  • Teutschbein, C. and Seibert, J. (2012). Bias correction of regional climate model simulations for hydrological climate-change impact studies: Review and evaluation of different methods. J. Hydrol. 456 12–29.
  • Trenberth, K. E. (2011). Changes in precipitation with climate change. Clim. Res. 47 123.
  • Vrac, M. and Friederichs, P. (2015). Multivariate—intervariable, spatial, and temporal—bias correction. J. Climate 28 218–237.
  • Wang, W., Xie, P., Yoo, S.-H., Xue, Y., Kumar, A. and Wu, X. (2011). An assessment of the surface climate in the NCEP climate forecast system reanalysis. Clim. Dyn. 37 1601–1620.
  • Wheeler, T. R., Craufurd, P. Q., Ellis, R. H., Porter, J. R. and Vara Prasad, P. V. (2000). Temperature variability and the yield of annual crops. Agric. Ecosyst. Environ. 82 159–167.
  • Wilks, D. S. and Wilby, R. L. (1999). The weather generation game: A review of stochastic weather models. Prog. Phys. Geogr. 23 329–357.
  • Williamson, D., Goldstein, M., Allison, L., Blaker, A., Challenor, P., Jackson, L. and Yamazaki, K. (2013). History matching for exploring and reducing climate model parameter space using observations and a large perturbed physics ensemble. Clim. Dyn. 41 1703–1729.
  • Wood, A. W., Leung, L. R., Sridhar, V. and Lettenmaier, D. P. (2004). Hydrologic implications of dynamical and statistical approaches to downscaling climate model outputs. Clim. Change 62 189–216.
  • Yeager, S. G., Shields, C. A., Large, W. G. and Hack, J. J. (2006). The low-resolution CCSM3. J. Climate 19 2545–2566.

Supplemental materials

  • Supplement A: Details on inference and computation, and additional figures. Additional details on estimating the mean changes in temperatures used in model (6) and the simulation (4); details on estimating the components $\delta_{l0}$ and $\delta_{l1}$ in model (6) and their associated standard errors; details on computing the proposed simulation; additional figures exploring the GCM projected variability changes and comparing the GCM output with the observational record; and a description of an animation of the proposed simulation.
  • Supplement B: Animation of global simulation. An animation of the proposed simulation.