In this study, we model the daily recordings of the GPS (Global Positioning System) data, and examine the noise characteristics of its residual signal (which is the difference between the real data and the model.) Here, two main problems are studied: The first one is the issue of selecting proper model parameters that are fitted to GPS data. Note that, at this step, some critical preprocessing issues are also addressed. The second issue is to assess the noise characteristics of the GPS residual signal which is very critical for accurate estimation of the parameters. It is assumed that GPS residual signals consist of time independent white noise and time-dependent colored noise components. Here, we propose a wavelet based method to estimate the amount of mixture of white and colored noise portions, plus the self-similarity index of the colored noise. Our proposed method is tested on the synthetic data and giving promising results. Later, a total of 60 GPS recordings from 20 stations are analyzed by this promising method. It is shown that the colored noise portions in the GPS residual signals can be modeled by so-called the flicker noise with the self similarity index of approximately "1".