This thesis gives an analytic approach for estimating the best number of modes in the spectral optimal averaging (SOA) formula established by Samuel Shen and his colleagues in 1994. The formulas of SOA and the spectral optimal interpolation (SOI) formula are described in detail. The importance of the best mode selection is explained from the point of view of optimal modeling. The Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) are used as the best mode selection tools, which are derived via an maximum likelihood estimate. Using a sample set of R's cherry data, a simple numerical example of AIC and BIC is presented. Also, it will visualize both information criteria. Based on either AIC or BIC, an empirical criterion for the mode selection in the SOA formula is given. The empirical optimal averaging information criteria (OAIC) is applied to the temperature data from the US Historical Climatology Network version 2 (USHCN v2) in the period of 1895-2010. The results are compared to the those obtained by the variance criteria. By using the OAIC, the number of modes is greater for the USHCN data set. The variance criterion gave a constant number of modes for the whole period from 1895 - 2010. Now, the values changes of the time and ranges between 12 and 14. Because of the flat structure of the OAIC values between 12 and 20, for some years we obtained higher values. This problem is discussed further.
