Send to

Choose Destination
J Geophys Res Solid Earth. 2016 Mar;121(3):2054-2068. Epub 2016 Mar 4.

MIDAS robust trend estimator for accurate GPS station velocities without step detection.

Author information

Nevada Bureau of Mines and Geology University of Nevada, Reno Reno Nevada USA.
IGN LAREGUniversity of Paris Diderot, Sorbonne Paris CitéParisFrance; Institut de Physique du Globe de Paris, PRES Sorbonne Paris CitéParisFrance.


Automatic estimation of velocities from GPS coordinate time series is becoming required to cope with the exponentially increasing flood of available data, but problems detectable to the human eye are often overlooked. This motivates us to find an automatic and accurate estimator of trend that is resistant to common problems such as step discontinuities, outliers, seasonality, skewness, and heteroscedasticity. Developed here, Median Interannual Difference Adjusted for Skewness (MIDAS) is a variant of the Theil-Sen median trend estimator, for which the ordinary version is the median of slopes vij  = (xj-xi )/(tj-ti ) computed between all data pairs i > j. For normally distributed data, Theil-Sen and least squares trend estimates are statistically identical, but unlike least squares, Theil-Sen is resistant to undetected data problems. To mitigate both seasonality and step discontinuities, MIDAS selects data pairs separated by 1 year. This condition is relaxed for time series with gaps so that all data are used. Slopes from data pairs spanning a step function produce one-sided outliers that can bias the median. To reduce bias, MIDAS removes outliers and recomputes the median. MIDAS also computes a robust and realistic estimate of trend uncertainty. Statistical tests using GPS data in the rigid North American plate interior show ±0.23 mm/yr root-mean-square (RMS) accuracy in horizontal velocity. In blind tests using synthetic data, MIDAS velocities have an RMS accuracy of ±0.33 mm/yr horizontal, ±1.1 mm/yr up, with a 5th percentile range smaller than all 20 automatic estimators tested. Considering its general nature, MIDAS has the potential for broader application in the geosciences.


Theil‐Sen; automation; breakdown; geodesy; time series; uncertainty

Supplemental Content

Full text links

Icon for PubMed Central
Loading ...
Support Center