### A Modified Least Square Harmonics Estimation Method and Comparative Analysis of Established Full Periodicity Models

X. Zhou, Y. Yang, H. Chen, H. Chen, W. Ouyang, W. Fan
Published: 13 September 2019
Earth and Space Science ; doi:10.1029/2019ea000750

Abstract: Unmodelled periodicities of GNSS coordinate time series lead to colored noise and therefore, unreal estimations of uncertainties and misinterpretation of geophysical phenomena. This paper firstly conducted Least Square Harmonics Estimation (LSHE) and Lomb‐Scargle periodogram method respectively on 25 CMONOC GNSS time series in Yunnan Province, China to establish the corresponding function models for each station. However, several prominent problems emerge: (1) design matrix singularity occurs when too close alternative frequencies are introduced; (2) low frequencies would be missed due to the cutoff of alternative frequencies. Consequently, periodical variations of a station would be depicted in an incorrect way. In order to solve these problems, this paper proposes a method that takes advantages of both LSHE and Lomb‐Scargle periodogram, that is, (1) to conduct an examination on the reciprocal of condition number of design matrix to avoid singularity problem, (2) to introduce the frequency results from the periodogram as a priori candidate frequencies to include low frequencies and improve accuracy of alternative frequencies. Compared with LSHE method and Lomb‐Scargle periodogram, the modified LSHE method reduces Root Mean Square (RMS) value of residuals by 0.83 mm and 0.43 mm, and reduces absolute spectrum indices of residuals by 0.11 and 0.04. Spectrum analysis and auto‐correlation function of residuals indicates corresponding residuals are closer to white noise, indicating modified LSHE method of this paper is valid to reduce colored noise through establishing a full periodicity model.
Keywords: Function / models / singularity / square / COLORED NOISE / Design Matrix / low frequencies / Established Full Periodicity