## 1. Introduction

[2] Climate hindcasts and projections are strongly affected by two key climate model parameters: climate sensitivity (*CS*) and vertical ocean diffusivity. Meridional overturning circulation, global temperature, and ocean heat accumulation that produces thermosteric sea level rise are good examples of climate variables that depend on these parameters [*Goes et al.*, 2010; *Knutti et al.*, 2002]. Better characterization of the uncertainty in these parameters is thus critical for future climate prediction.

[3] Climate sensitivity is defined as the equilibrium near-surface temperature response to a doubling of atmospheric CO_{2}. *CS*is a measure of climate feedbacks that amplify or dampen the direct response of near-surface temperature to radiative forcings [*Andronova et al.*, 2007]. Vertical ocean diffusivity is a parameter that influences heat uptake by the ocean. It parameterizes mixing processes below the grid scale of climate models. For the same climate sensitivity, at higher diffusivities the atmosphere will reach the equilibrium temperature specified by *CS* more slowly, due to more heat flux into the deep ocean [*National Academy of Sciences*, 1979].

[4] In order to estimate these parameters from climate models and observations, one needs to know past climate forcings. Both parameter estimation studies and simple theoretical considerations show that assumptions about these forcings influence climate sensitivity estimates and the uncertainty surrounding them [*Andreae et al.*, 2005; *Tanaka et al.*, 2009; *Urban and Keller*, 2010]. For example, *Andreae et al.* [2005]use a zero-dimensional climate model to illustrate that when they assume no aerosol effects, a climate sensitivity of just 1.3°C is needed to explain the observed 1940–2000 warming. On the other hand, aerosol forcing of −1.7 Wm^{−2} (a value that is within the IPCC range [*Forster et al.*, 2007]) requires a climate sensitivity of more than 10°C [*Andreae et al.*, 2005]. Out of the main climate forcings, the forcings due to aerosols are especially uncertain. A large part of this uncertainty is due to anthropogenic sulfate aerosols [*Forster et al.*, 2007].

[5] Parameters controlling climate sensitivity, vertical diffusion in the ocean, and strength of anthropogenic sulfate aerosols are commonly estimated using model runs and observations [*Knutti et al.*, 2002, 2003; *Forest et al.*, 2002, 2006; *Drignei et al.*, 2008; *Tomassini et al.*, 2007; *Edwards et al.*, 2007; *Sanso and Forest*, 2009]. Typically, an ensemble of model runs is used where the key parameters are systematically varied. The outputs from these runs are then compared with the observations, and the posterior probability distribution functions (pdfs) for model parameters are derived.

[6] One conceptually simple methodology selects only the model runs that are consistent with the observations using a broad, heuristic approach [*Knutti et al.*, 2003]. In this framework all parameter combinations that pass the consistency criterion are assigned a uniform probability, while those that do not pass it receive a zero probability. These probabilities are then used to construct the posterior pdfs.

[7] A more complex approach uses Bayesian statistics. This approach requires: (1) a model ensemble, (2) observations, (3) a statistical model that relates climate model output to the observations, and (4) prior information about the model parameters (priors). In this framework, each parameter combination is associated with a likelihood that depends on how well the corresponding model output matches the observations [*Tomassini et al.*, 2007; *Sanso and Forest*, 2009]. The likelihood, *L*(*Y*∣Θ), describes the degree of belief that the physical observations *Y*came from a climate model and a statistical model (describing the properties of data-model residuals) with unknown parameters Θ. Once the statistical model is defined, the likelihood*L*(*Y*∣Θ) can be calculated from the residuals between the model output and the observations. In the Bayesian framework, the posterior probability of the unknown parameters given the observations is proportional to *L*(*Y*∣Θ), and to the prior probability of the parameters:

[8] While the posterior probability *p*(Θ∣*Y*) can be evaluated on a grid of parameter values, this can become too computationally expensive if the parameter space is multidimensional. In such cases Markov Chain Monte Carlo (MCMC) methods [*Metropolis et al.*, 1953; *Hastings*, 1970] can be used. The MCMC generates a sequence of parameter values (a Markov chain) which may be treated approximately as samples from the posterior distribution. Hence, virtually any property of the posterior distribution can be approximated by a corresponding sample property of this sequence.

[9] Intermediate Complexity Earth System models are frequently used for this analysis [*Forest et al.*, 2002, 2006; *Knutti et al.*, 2003; *Tomassini et al.*, 2007; *Sanso and Forest*, 2009]. The appeal of these models is that they can be run at many parameter settings with relative ease. At the same time these models still represent many relevant physical processes. While the models can be run hundreds of times, many more runs at arbitrary parameter values are needed for the MCMC sampling. To approximate model output at these values, emulators (statistical approximators of climate models) can be used [e.g., *Drignei et al.*, 2008; *Holden et al.*, 2010; *Edwards et al.*, 2011]. The emulators draw on information about model outputs at known parameter settings to interpolate the output to any desired parameter setting.

[10] In this study, we use the University of Victoria Earth System Climate Model (UVic ESCM) to estimate these important climate parameters. We constrain the ensemble of model runs with atmospheric surface temperature and ocean heat content observations to present probability distribution functions for key model parameters controlling the processes described above: climate sensitivity *CS*, background vertical ocean diffusivity *K*_{bg}, and a scaling parameter for the direct effects of anthropogenic sulfate aerosols *A*_{sc}. The use of the full 3D ocean allows for the representation of the non-linear effects of*K*_{bg} on ocean dynamics and currents (e.g., on the Meridional Overturning Circulation). We present posterior joint and marginal pdfs for the parameters, and explore the sensitivity of the results to prior assumptions.