Abstract: The dimensions of climate attractor are estimated from phase space trajectories of observed climate variables (Southern and Northern Hemisphere surface air temperature) based on the method estimating the fractal dimensions given by Grassberger and Procaccia. The results show that the fractal dimensions of climate attractor are 3.3~3.7 for the Southern Hemisphere, and 3.2~3.7 for the Northern Hemisphere. They provide the primary information for the descriptions of selt-similar properties of the climate attractor, and the four independent variables are needed at least for simulated climate system. Also, the effects of the noise on estimating dimensions of climate attractor are discussed