Objectives  In order to analyze the independent physical contribution of the post-seismic relaxation term on global positioning system (GPS) observing displacements, the regional post-seismic relaxation time and the characteristics of its spatial distribution are studied.

  Methods  The logarithmic function is used to describe the post-seismic relaxation term, and the logarithmic relaxation time is estimated from post-seismic GPS coordinate time series by using nonlinear weighted least squares method. Steps are as follows: Remove the gross errors; fix the step items; remove the constant term and the steady-state velocity term; the logarithmic relaxation terms are modeled for the preprocessed GPS coordinate time series.

  Results  Taking the 2009—2019 continuous coordinate time series of the global navigation satellite systems earth observation network stations in Japan as an example, the logarithmic relaxation time after 2011 Tohoku-Oki Mw 9.0 earthquake is estimated. It is found that the post-seismic relaxation time is significantly related to the epicenter distance and obeys the Gaussian distribution. Hence, a Gaussian function plus constant model is constructed to reflect the law of post-seismic relaxation time and epicentral distance. The peak value, peak position, half-width information, and minimum position of the Gaussian distribution curve of relaxation time after the earthquake are 3.5 a, 0 km, 262 km, and 0.5 a, respectively.The relaxation range with relaxation time more than 0.5 a is mainly concentrated in the region of about 524 km from the epicenter. The regional distribution of post-seismic relaxation effect is significantly different. The longer the relaxation time, the larger the average horizontal displacement of the relaxation term in the corresponding region, and there are two central regions with an average relaxation time of 2.5 years, which were consistent with the central region and time of the afterslip.

  Conclusions  The proposed algorithm on estimating the post-seismic logarithmic relaxation time can help to analyze the independent physical contribution of post-seismic relaxation term, the afterslip and viscoelastic relaxation.