Harmonic analysis

Abstract

The theoretical foundation of the harmonic analysis and its calculation procedure are presented. An example is studied, in which the variance was not independent of the mean, and thus the data were not normally distributed. In spite of this, the contrasts necessary to the harmonic analysis have shown homogeneity of variance. This has permitted the direct processing of the data (monthly precipitations at Pelotas, Rio Grande do Sul, Brazil, 1900/51). In another example, the data were submitted to an earlier transformation, to normalize, presumably, the distribution, then the transformed data were analysed (monthly precipitations at Morro Velho, Minas Gerais, Brazil, 1855/1951). The stability of the harmonic components of the monthly precipitations at Pelotas was tested by the variance analysis method. The waves with periods of six, four and twelve months were significant in regard to their variation between years. The second and the third harmonics were predominant in the yearly course of the monthly precipitations (maxíma at February and September). The completeness test reveals that 91.2% .of the variation of the monthly precipitation means was accounted for those three waves. Since yearly data concerning the monthly precipitations at Morro Velho were not available to the author, it was impossible to submit them to a stability test. Practically all the yearly variation of the monthly precipitation means (99,3%) resulted from the yearly wave, which was to be antecipated in a continental tropical climate. Two original contributions are made in this paper: the establishment of the distribution of the difference between the squares of two amplitudes and the introduction of the concept of cyclic correlation coefficient. The latter is not to be confounded with the known serial correlation coefficient. That new concept is applied in the adaptation of the Fuhrich's periodographycal method to the harmonic analysis, permitting the isolation of the harmonics in the order of their importance, as measured by their amplitudes.