Relative weights of GCM runs for stochastic analyses and downscaling

Various climate simulations with atmosphere ocean general circulation models (GCM) are performed to project future climate states. In addition, downscaling experiments using the GCMs to identify details of climate changes are also important for adaptations. Here, to quantify uncertainties of climate change projection covered by GCM runs, sampling problem of GCM runs should be considered, because GCM runs are performed individually by model developers and the number of provided GCM data is not sufficiently large for stochastic analyses. Therefore, relative weights for GCM run were newly proposed. The weights of GCM runs are defined as wr. To determine wr, a cost function J was proposed and defined as Eq. 1. r is the label of GCM run and the number of GCM runs is R. dx and B are bias of P x 1 column vector and its covariance of P x P matrix, respectively, where P is degree of freedom of grids and elements within an analysis domain. The control variable is wr. This cost function is based on the maximum likelihood estimation assuming the normal distribution of variables. The wr could be estimated by an iterative calculation under the assumption that the summation of wr is equal to 1. By using the relative weights, variables of GCM show approximately the normal distribution around 0. The width of uncertainty defined as the standard deviation of Gaussian function is due to the extraction of GCM samples. The weights of negative bias runs are larger than that of positive bias runs when there are less negative bias runs than positive bias runs. This cost function focuses only on biases of GCM runs in the present climate. However, this function could be extended to treat other climatological parameters such as near-past trends by using additional terms. The iteration estimation of wr includes outer and inner loops. In the inner loop of iterative estimation, B is fixed while B includes wr. In the outer loop, B is updated by the estimated wr. The two kinds of iteration loops, weighted ensemble mean of dx approaches toward 0, while ensemble mean with equivalent weights is not around 0. By using wr, many GCM run results could be used simultaneously to estimate the most possible future image and uncertainties without sample extraction by quality check. Here, weights of climatological differences of GCM runs are carried out by bias of GCM runs in the present climate. This idea should be discussed in comparison with other approaches.