Fast white noise estimation in GNSS time series by wavelet variance

The noise in the GNSS position time series is mainly a combination of white noise and power law noise. Noise amplitudes are estimated using variance component estimation (VCE) procedures. These methods require repeated inversion of covariance matrix, which is a computational burden for analysis of long time series. This work proposes an algorithm to estimate the white noise amplitude, through the estimation of wavelet variance based upon the Maximal Overlap Discrete Wavelet Transform (MODWT). MODWT can be used for any sample size and number of wavelet and scaling coefficients does not decrease by factor 2 for each increase in the level of the transform, so it does not decrease our ability to perform statistical analysis. To test the performance of the proposed algorithm, we used 180 synthetic daily time series with different lengths (2000, 4000 and 8000) emulating real GNSS time series. They composed of linear trends, periodic signals, offsets, transient displacements, gaps (up to 10%), and a combination of white, flicker, and random walk noises. The results of proposed method were compared to those of REstricted Maximum Likelihood (REML) approach. Biases of white noise amplitudes for the proposed and REML method indicated that results given by the two methods are in good agreement. Moreover, the proposed algorithm has computational complexity of order O(N) where N is the number of observations. Also, the results demonstrated that this proposed algorithm can be about 450-10000 times faster than REML method depending on the length of time series.  For further evaluation of the method, the time series of 19 real stations were used, and the results indicated the effectiveness of the proposed method. The low complexity of the proposed algorithm can considerably speed up the processing of GNSS time series.