---
title: "Estimate geodetic time series from the Nevada Geodetic Laboratory"
output: rmarkdown::html_vignette
bibliography: REFERENCES.bib
vignette: >
  %\VignetteIndexEntry{Estimate geodetic time series from the Nevada Geodetic Laboratory}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---



```{r setup, include=FALSE}
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
)

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.
```{r}
library(gmwmx2)
```

## Download a station from Nevada Geodetic Laboratory
```{r}
station_data <- download_station_ngl("1LSU")
```

## Plot Station
```{r}
plot(station_data)
```


## Plot Northing component
```{r}
plot(station_data, component = "N")
```


## Estimate models on station data
```{r}
fit1 <- gmwmx2(station_data, n_seasonal = 2, component = "N", model = wn())
fit1
```


```{r}
plot(fit1)
```

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

```{r}
plot(fit2)
```

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


```{r}
plot(fit3)
```
```{r}
fit4 = gmwmx2(station_data, n_seasonal = 2, component = "N", model = wn() + ar1())
fit4
```


```{r}
plot(fit4)
```


# References