On the use of observations in assessment of multi-model climate ensemble

Abstract

The Bayesian weighted averaging (BWA) method is commonly used to integrate over multi-model ensembles of climate series. This method relies on two criteria to assign weights to individual outputs: model skill in reproducing historical observations, and inter-model agreement in simulating future period. Observations are generally thought to be relevant for correcting biases in model outputs in the BWA framework. However, they concurrently may introduce unpredictable impacts in the context of the downscaling process, in particular, when model output on precipitation is of interest. Specifically, the posterior distribution may excessively depend on few ‘outlier models’ being close to the observation, when all other models fail to capture observation of the historical period—a common situation for precipitation metrics. Another issue emerges for climates with very dry months: the inclusion of observation in BWA may result in a significant spread of the posterior distribution into the negative region. To address these problems, a modified version of the BWA method that removes observations in the initial phase of downscaling (computation of Factors of Change) and adds them in the estimation of posterior distributions is explored in this work. Comparisons of simulation results for the locations of Miami (FL), Fresno (CA), and Flint (MI) between the modified BWA and the traditional BWA demonstrate consistent outcomes with regards to the effect of observation in the Bayesian framework. Further, the modified BWA approach generally reduces uncertainty, as compared to ‘simple averaging’ in the Bayesian context, which assigns equal weights to all model outputs.

Appendix

The BWA framework of Tebaldi et al. (2005) relies on several statistical variables, which are defined in the following:

$X_{i}$ and $Y_{i}$ represent model output of climate variables (e.g., mean monthly precipitation or temperature) for the CTL and FUT periods respectively, which are assumed to follow Normal distribution.

$$X_{i} \sim N\left( {\mu ,\lambda_{i}^{ - 1} } \right),Y_{i} \sim N\left( {\nu ,(\theta \lambda_{i} )^{ - 1} } \right),$$

(3)

The parameters $\mu$ and $\upsilon$ are the corresponding the true values for the past and future, which are common for all GCMs. These two parameters have uniform prior density on the true values. The $\lambda_{i}$ is a “precision” parameter for each model i = 1…N = 18, a reciprocal of the variance of the climate variable, which is used as a model-specific weight in the Markov Chain Monte Carlo (MCMC) simulation, when outputs are combined to estimate the posterior distribution for a variable of interest. The weight related parameters have Gamma prior densities (e.g., Ga(a, b)). The values of a and b are set to be 0.01 to make sure the prior distributions for $\lambda_{i}$ are diffuse/non-informative (Tebaldi et al. 2004). Parameter $\lambda_{0}$ represents a measure of natural variability for the climate variable of interest, it introduces a weight for observation in the MCMC simulation. The $\theta$ is the parameter that allows GCMs to have different precisions, when simulating future climate conditions, as compared to the historic period. $\beta$ introduces a regression parameter between the model outputs representing past and future (Eq. 11 of Tebaldi et al. 2005). The process of estimation of the posterior distributions is facilitated by the MCMC simulation, as the joint posterior distribution cannot be treated analytically (Tebaldi et al. 2005). The distributions of each parameter with remaining parameters fixed are derived in Tebaldi et al. (2005) and are summarized here:

$$\lambda_{i} | \cdots \sim Ga\left( {a + 1,b + \frac{{(X_{i} - \mu )^{2} + \theta [Y_{i} - \nu - \beta (X - \mu )]^{2} }}{2}} \right),$$

(4)

$$\mu | \cdots \sim N\left( {\tilde{\mu },\left( {\sum\limits_{i = 1}^{18} {\lambda_{i} + \theta \beta^{2} \sum\limits_{i = 1}^{18} {\lambda_{i} + \lambda_{0} } } } \right)^{ - 1} } \right),$$

(5)

$$\nu | \cdots \sim N\left( {\tilde{\nu },\left( {\theta \sum\limits_{i = 1}^{18} {\lambda_{i} } } \right)^{ - 1} } \right),$$

(6)

$$\beta | \cdots \sim N\left( {\tilde{\beta },\left[ {\theta \sum\limits_{i = 1}^{18} {\lambda_{i} (X_{i} - \mu )^{2} } } \right]^{ - 1} } \right),$$

(7)

And $\tilde{\mu },\;\tilde{\nu },\;\tilde{\beta }$ are computed as:

$$\tilde{\mu } = \frac{{\sum\nolimits_{i = 1}^{18} {\lambda_{i} X_{i} - \theta \beta_{x} \sum\nolimits_{i = 1}^{18} {\lambda_{i} (Y_{i} - \nu - \beta_{x} X_{i} ) + \lambda_{0} X_{0} } } }}{{\sum\nolimits_{i = 1}^{18} {\lambda_{i} + \theta \beta_{x}^{2} \sum\nolimits_{i = 1}^{18} {\lambda_{i} + \lambda_{0} } } }},$$

(8)

$$\tilde{\nu } = \frac{{\sum\nolimits_{i = 1}^{18} {\lambda_{i} \{ Y_{i} - \beta_{x} (X_{i} - \mu )\} } }}{{\sum\nolimits_{i = 1}^{18} {\lambda_{i} } }},$$

(9)

$$\tilde{\beta } = \frac{{\sum\nolimits_{i = 1}^{18} {\lambda_{i} (Y_{i} - \nu )(X_{i} - \mu )} }}{{\sum\nolimits_{i = 1}^{18} {\lambda_{i} (X_{i} - \mu )^{2} } }}.$$

(10)

In the above equations, i = 1…N = 18, where 18 represents the number of model outputs used in this study.

Of particular interest are the parameters $\lambda_{i}$ that are effectively used as weights for individual model outputs. The posterior distribution of the parameter $\lambda_{i}$ is based on the performance of the corresponding GCM model: its bias $(X_{i} - \tilde{\mu })^{2}$, i.e., the deviation of modeled past period from observations, and convergence $(Y_{i} - \tilde{\upsilon })^{2}$, i.e., a penalty if the modeled future/projected climate variable lies far from the center of model ensemble cluster. To remove the effect of observation in the current BWA framework, we can set $\lambda_{0} = 0$. This modification leads to the change of estimate of $\mu$ with all the other parameters remaining the same:

$$\mu | \cdots \sim N\left( {\tilde{\mu },\left( {\sum\limits_{i = 1}^{18} {\lambda_{i} + \theta \beta^{2} } \sum\limits_{i = 1}^{18} {\lambda_{i} } } \right)^{ - 1} } \right),$$

(11)

$$\tilde{\mu } = \frac{{\sum\nolimits_{i = 1}^{18} {\lambda_{i} X_{i} - \theta \beta_{x} } \sum\nolimits_{i = 1}^{18} {\lambda_{i} (Y_{i} - \nu - \beta_{x} X_{i} )} }}{{\sum\nolimits_{i = 1}^{18} {\lambda_{i} + \theta \beta_{x}^{2} \sum\nolimits_{i = 1}^{18} {\lambda_{i} } } }}.$$

(12)

Once the samples for posterior distributions are estimated from MCMC process, additive FOC (for temperature) and product FOC (for precipitation) are computed from $\upsilon - \mu$ and $\frac{\upsilon }{\mu }$ respectively. The corresponding marginal CDF is constructed using the FOC samples, and FOC PDF is subsequently estimated based on the CDF with the Monte Carlo method.