Modeling trends and periodic components in geodetic time series: a unified approach

Geodetic time series are usually modeled with a deterministic approach that includes trend, annual, and semiannual periodic components having constant amplitude and phase-lag. Although simple, this approach neglects the time-variability or stochasticity of trend and seasonal components, and can potentially lead to inadequate interpretations, such as an overestimation of global navigation satellite system (GNSS) station velocity uncertainties, up to masking important geophysical phenomena. In this contribution, we generalize previous methods for determining trends and seasonal components and address the challenge of their time-variability by proposing a novel linear additive model, according to which (i) the trend is allowed to evolve over time, (ii) the seasonality is represented by a fractional sinusoidal waveform process (fSWp), accounting for possible non-stationary cyclical long-memory, and (iii) an additional serially correlated noise captures the short term variability. The model has a state space representation, opening the way for the evaluation of the likelihood and signal extraction with the support of the Kalman filter (KF) and the associated smoothing algorithm. Suitable enhancements of the basic methodology enable handling data gaps, outliers, and offsets. We demonstrate the advantage of our method with respect to the benchmark deterministic approach using both observed and simulated time series and provide a fair comparison with the Hector software. To that end, various geodetic time series are considered which illustrate the ability to capture the time-varying stochastic seasonal signals with the fSWp

Tracking hurricanes Harvey and Irma using GPS tropospheric products

The 2017 Hurricanes season was one of the most powerful severe weather events producing catastrophic socio-economic and environmental effects on the east coast of the United States. Therefore, tracking their path accurately is extremely useful. Today Global Navigation Satellite Systems (GNSS) tropospheric products, such as Zenith Wet Delays (ZWD), and Integrated Water Vapor (IWV) are used as complementary data sets in Numerical Weather Prediction (NWP) models. In this study, we employed GPS-derived IWV and horizontal tropospheric gradient information to monitor and investigate the complicated characteristics of hurricane events in their spatial and temporal distribution using a dense ground network of GPS stations. Our results show that a surge in GPS-derived IWV occurred several hours prior to the manifestation of the major hurricanes Harvey and Irma. We used the derived GPS-derived IWV information as input to spaghetti lines weather models, allowing us to predict the paths of Harvey and Irma hurricanes. As such, a parameter directly estimated from GPS can provide an additional resource for improving the monitoring of hurricane path

On the noise characteristics of time series recorded with nearby located GPS receivers and superconducting gravity meters

International audienc

Total Impact of Periodic Terms and Coloured Noise on Velocity Estimates

The uncertainties of velocity estimates for position time series of Global Navigation Satellite System (GNSS) stations are mainly affected by a misfit of the deterministic model applied to this data. Insufficiently modelled seasonal signals will propagate into the stochastic model and falsify the results of the noise analysis besides the velocity estimates and their uncertainties. In this presentation we derived the General Dilution of Precision (GDP) of velocity uncertainties. We define this dilution as the ratio between the uncertainties of velocities determined when different deterministic and stochastic models are applied. In this way we discuss, referring to previously published results, how insufficiently modelled seasonal signals influence station velocity uncertainties with white and coloured noise. Using simulated and real data from selected (115) IGS (International GNSS Service) stations we show that the noise character affects GNSS data more than seasonals for time series longer than 9 years

On the combined effect of periodic signals and colored noise on velocity uncertainties

The velocity estimates and their uncertainties derived from position time series of Global Navigation Satellite System stations are affected by seasonal signals and their harmonics, and the statistical properties, i.e., the stochastic noise, contained in the series. If the deterministic model in the form of linear trend and periodic terms is not accurate enough to describe the time series, it will alter the stochastic model, and the resulting effect on the velocity uncertainties can be perceived as a result of a misfit of the deterministic model. The effects of insufficiently modeled seasonal signals will propagate into the stochastic model and falsify the results of the noise analysis, in addition to velocity estimates and their uncertainties. We provide the general dilution of precision (GDP) of velocity uncertainties as the ratio of uncertainties of velocities determined from to two different deterministic models while accounting for stochastic noise at the same time. In this newly defined GDP, the first deterministic model includes a linear trend, while the second one includes a linear trend and seasonal signals. These two are tested with the assumption of white noise only as well as the combinations of power-law and white noise in the data. The more seasonal terms are added to the series, the more biased the velocity uncertainties become. With increasing time span of observations, the assumption of seasonal signals becomes less important, and the power-law character of the residuals starts to play a crucial role in the determined velocity uncertainties. With reference frame and sea level applications in mind, we argue that 7 and 9 years of continuous observations is the threshold for white and flicker noise, respectively, while 17 years are required for random-walk to decrease GDP below 5% and to omit periodic oscillations in the GNSS-derived time series taking only the noise model into consideration

The Combined Effect of Periodic Signals and Noise on the Dilution of Precision of GNSS Station Velocity Uncertainties

Station velocity uncertainties determined from a series of Global Navigation Satellite System (GNSS) position estimates depend on both the deterministic and stochastic models applied to the time series. While the deterministic model generally includes parameters for a linear and several periodic terms, the stochastic model is a representation of the noise character of the time series in form of a power-law process. For both of these models the optimal model may vary from one time series to another while the models also depend, to some degree, on each other. In the past various power-law processes have been shown to fit the time series and the sources for the apparent temporally-correlated noise were attributed to, for example, mismodelling of satellites orbits, antenna phase centre variations, troposphere, Earth Orientation Parameters, mass loading effects and monument instabilities

Statistical significance of trends in Zenith Wet Delay from re-processed GPS solutions

Long series of Zenith Wet Delay (ZWD) obtained as part of a homogeneous re-processing of Global Positioning System solutions constitute a reliable set of data to be assimilated into climate models. The correct stochastic properties, i.e. the noise model of these data, have to be identified to assess the real value of ZWD trend uncertainties since assuming an inappropriate noise model may lead to over- or underestimated error bounds leading to statistically insignificant trends. We present the ZWD time series for 1995–2017 for 120 selected globally distributed stations. The deterministic model in the form of a trend and significant seasonal signals were removed prior to the noise analysis. We examined different stochastic models and compared them to widely assumed white noise (WN). A combination of the autoregressive process of first-order plus WN (AR(1) + WN) was proven to be the preferred stochastic representation of the ZWD time series over the generally assumed white-noise-only approach. We found that for 103 out of 120 considered stations, the AR(1) process contributed to the AR(1) + WN model in more than 50% with noise amplitudes between 9 and 68 mm. As soon as the AR(1) + WN model was employed, 43 trend estimates became statistically insignificant, compared to 5 insignificant trend estimates for a white-noise-only model. We also found that the ZWD trend uncertainty may be underestimated by 5–14 times with median value of 8 using the white-noise-only assumption. Therefore, we recommend that AR(1) + WN model is employed before tropospheric trends are to be determined with the greatest reliability