Estimate geodetic time series from the Nevada Geodetic Laboratory

In gmwmx2: Estimate Functional and Stochastic Parameters of Linear Models with Correlated Residuals and Missing Data

knitr::opts_chunk$set(
  fig.width = 10, # Set default plot width (adjust as needed)
  fig.height = 8, # Set default plot height (adjust as needed)
  fig.align = "center" # Center align all plots
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knitr::opts_chunk$set(eval = FALSE)

Model and estimator

This vignette builds on the general framework described in the vignettes "Estimate linear models with dependent errors" and "Estimate linear models with dependent errors and missing observations". We do not repeat the full model, missingness mechanism, or estimation details here. Please refer to these vignettes for the formal setup, and use this one as a hands-on guide for geodetic time series from the Nevada Geodetic Laboratory.

Estimating tectonic velocities and crustal uplift

While the GMWMX, as described above and in more detail in @voirol2024inference, is a general method for estimating large linear models with complex dependence structures in presence of missing observations, the gmwmx2 R package allows to estimate tectonic velocities and crustal uplift from position time series in graticule distance (GD) coordinates provided by the Nevada Geodetic Laboratory [@blewitt2024improved; @blewitt2018harnessing].

Trajectory model

To estimate the trajectory model (see, e.g., @bevis2014 for more details), we construct the design matrix $\boldsymbol{X}$ such that the $i$-th component of the vector $\mathbf{X} {{\boldsymbol{\beta}}}$ can be described as follows, with $t_i$ representing the $i^{\text{th}}$ ordered time point (epoch) in Modified Julian Date and $t_0$ representing the reference epoch located exactly in the middle of the start and end points of the time series:

\begin{split} \mathbb{E}[\mathbf{Y}i] &= \mathbf{X}_i^T {{\boldsymbol{\beta}}} \ &= a + b \left(t{i} - t_0\right) + \sum_{h=1}^{m}\left[c_{h} \sin \left(2 \pi f_{h} t_{i}\right) + d_h \cos \left(2 \pi f_h t_i\right)\right] + \& \sum_{j=1}^{r}e_j H\left(t_i - t_j\right) + \sum_{k = 1 }^{s} l_k \left[1- \exp\left(\frac{-(t_i-t_k)}{\tau_k}\right)\right]H\left(t_{i}-\tau_k\right) \end{split}

where $a$ is the initial position at the reference epoch $t_0$, $b$ is the velocity parameter, and $c_h$ and $d_h$ are the periodic motion parameters ($h = 1$ and $h = 2$ represent the annual and semi-annual seasonal terms, respectively with $f_1 = \frac{1}{365.25}$ and $f_2 = \frac{2}{365.25}$). The offset terms model earthquakes, equipment changes, or human intervention, in which $e_j$ is the magnitude of the step at epochs $t_j$, $r$ is the total number of offsets, and $H$ is the Heaviside step function defined as $H(x):= \begin{cases}1, & x \geq 0 \ 0, & x<0\end{cases}$. The last term allows us to model post-seismic deformation (see, e.g., @sobrero2020logarithmic) where $s$ is the number of post-seismic relaxation times specified, $t_k$ is the time when the relaxation $k$ starts in Modified Julian Date (MJD), $\tau_k$ is the relaxation period in days for the post-seismic relaxation $k$, and $l_k$ is the amplitude of the transient. Note that by default the estimates of the functional parameters are provided in units/day.

When loading data from a specific station using gmwmx2::download_station_ngl(), we extract from the Nevada Geodetic Laboratory the position time series in GD coordinates, the time steps associated with equipment or software changes, and the time steps associated with earthquakes near the station. All these objects are stored in an object of class gnss_ts_ngl.

When applying the function gmwmx2::gmwmx2() to an object of class gnss_ts_ngl, we construct the design matrix $\boldsymbol{X}$ by considering an offset term for all equipment or software change steps and all earthquakes indicated by the NGL. We also specify a post-seismic relaxation term for all earthquakes indicated by the NGL. If no relaxation time is specified in the argument vec_earthquakes_relaxation_time, we use a default relaxation time of $365.25$ days.

Stochastic model

It is generally recognized that noise in GNSS time series is best described by a combination of colored noise plus white noise [@he2017review; @langbein2008noise; @williams2004error; @bos2013fast], where the white noise generally models noise at high frequencies and the colored noise models the lower frequencies of the spectrum.

In gmwmx2, you can pass any valid stochastic model to the estimator: either a single time-series model (e.g., wn(), ar1(), pl(), matern(), rw(), flicker()) or a composite sum model built with + (e.g., wn() + pl() or wn() + ar1() + rw()). This makes the stochastic specification very general and easy to adapt to your application.

Example of estimation

Let us showcase how to estimate tectonic velocity for one specific component (North, East, or Vertical) of one station.

First, load the gmwmx2 package.

library(gmwmx2)

Download a station from Nevada Geodetic Laboratory

station_data <- download_station_ngl("1LSU")

Plot Station

plot(station_data)

Plot Northing component

plot(station_data, component = "N")

Estimate models on station data

fit1 <- gmwmx2(station_data, n_seasonal = 2, component = "N", model = wn())
fit1

plot(fit1)

fit2 <- gmwmx2(station_data, n_seasonal = 2, component = "N", model = wn()+ pl())
fit2

plot(fit2)

fit3 = gmwmx2(station_data, n_seasonal = 2, component = "N", model = wn() + flicker())
fit3

plot(fit3)

fit4 = gmwmx2(station_data, n_seasonal = 2, component = "N", model = wn() + ar1())
fit4

plot(fit4)

References

gmwmx2 documentation built on June 10, 2026, 5:06 p.m.