Functional model

More precisely, for the functional model, we consider a linear model which can be expressed as:

\[\begin{equation} \mathbf{Y} = \mathbf{A} {{\bf x}}_0 + \boldsymbol{\varepsilon}, \end{equation}\]

where \(\mathbf{Y} \in {\rm I\!R}^n\) denotes the response variable of interest (i.e., vector of GNSS observations), \(\mathbf{A} \in {\rm I\!R}^{n \times p}\) a fixed design matrix, \({{\bf x}}_0 \in \mathcal{X} \subset {\rm I\!R}^p\) a vector of unknown constants and \(\boldsymbol{\varepsilon} \in {\rm I\!R}^n\) a vector of (zero mean) residuals.

The gmwmx package allows to estimate functional models for which the \(i\)-th component of the vector \(\mathbf{A} {{\bf x}}_0\) can be described as follows:

\[\begin{equation} \mathbb{E}[\mathbf{Y}_i] = \mathbf{A}_i^T {{\bf x}}_0 = a+b\left(t_{i}-t_{0}\right)+\sum_{h=1}^{2}\left[c_{h} \sin \left(2 \pi f_{h} t_{i}\right)+d_{h} \cos \left(2 \pi f_{h} t_{i}\right)\right] + \sum_{k=1}^{n_{g}} g_{k} H\left(t_{i}-t_{k}\right), \end{equation}\]

where \(a\) is the initial position at the reference epoch \(t_0\), \(b\) is the velocity parameter, \(c_k\) and \(d_k\) are the periodic motion parameters (\(h = 1\) and \(h = 2\) represent the annual and semi-annual seasonal terms, respectively). The offset terms models earthquakes, equipment changes or human intervention in which \(g_k\) is the magnitude of the change at epochs \(t_k\), \(n_g\) is the total number of offsets, and \(H\) is the Heaviside step function. Note that the estimates of the parameters of the functional model are provided in unit/day.

Stochastic model

Regarding the stochastic model, we assume that \(\boldsymbol{\varepsilon}_i=\mathbf{Y}_i-\mathbb{E}[\mathbf{Y}_i]\) is a strictly (intrinsically) stationary process and that

\[\begin{equation} \boldsymbol{\varepsilon} \sim \mathcal{F} \left\{\mathbf{0}, \boldsymbol{\Sigma}(\boldsymbol{\gamma}_0)\right\} , \label{eq:model:noise} \end{equation}\]

where \(\mathcal{F}\) denotes some probability distribution in \({\rm I\!R}^n\) with mean \({\bf 0}\) and covariance \(\boldsymbol{\Sigma}(\boldsymbol{\gamma}_0)\).

We assume that \(\boldsymbol{\Sigma}(\boldsymbol{\gamma}_0) > 0\) and that it depends on the unknown parameter vector \(\boldsymbol{\gamma}_0 \in \boldsymbol{\Gamma} \subset {\rm I\!R}^q\). This parameter vector specifies the covariance of the observations and is often referred to as the stochastic parameters.

Hence, we let \(\boldsymbol{\theta}_0 = \left[\boldsymbol{{\bf x}}_0^{\rm T} \;\; \boldsymbol{\gamma}_0^{\rm T}\right]^{\rm T} \in \boldsymbol{\Theta} = \mathcal{X} \times \boldsymbol{\Gamma} \subset {\rm I\!R}^{p + k}\) denote the unknown parameter vector of the model described above.

The gmwmx allows to estimate parameters of a specified functional model as well as parameters of a stochastic model (i.e. \(\hat{\boldsymbol{\theta}} = \left[\boldsymbol{\hat{\boldsymbol{x}}}^{T} \;\; \hat{\boldsymbol{\gamma}}^{T}\right]\)) defined by a combinations of

1. White noise (wn)
2. Matérn process (matern)
3. Fractional Gaussian noise (fgn) and
4. Power Law process (powerlaw).

Note that only the gmwmx current version accepts only one process of each kind.