# Monitoring of Structures and Infrastructures by Low-Cost GNSS Receivers
## 2 Concepts
### The a Posteriori Motion Model  
Let \(\{x(t)\},\ t=1\times T,2\times T,\ldots\) be the time series of the true values of any of the coordinates of the monitoring network. The values of the coordinates are considered at multiples of the compensation period \(T\); thus, for instance, \(t\) attains the values \(t=\) day 1, day 2,...  
Let \(\{\widehat{x}(t)\}\) be the time series of the values of \(x(t)\) estimated by compensating the observations made over the last \(T\) period before \(t\). We can write  
\[\widehat{x}(t)=x(t)+\eta(t) \tag{1}\]  
where \(\{\eta(t)\}\) is the series of the estimation errors. We assume that \(x(t)\) can be represented by  
\[x(t)=m(t)+s(t)+a(t) \tag{2}\]  
where  
* \(m(t)\) is the mean slow motion, which is a trend, and that could very well be \(m(t)\equiv 0\) when the point does not drift away from a reference mean position. * \(s(t)\) is a true signal in \(x(t)\) due to external causes that, however, has a limited extent and is often periodic; as an example, one can think of mean temperature of seasons causing cycles of expansion and contraction of a concrete structure. * \(a(t)\) represents an anomalous sudden change in the coordinate \(x\). Characteristic of this term is its persistence. For instance, if \(a(t)\) represents a slip of a crack in the direction of \(x\) at epoch \(t\), then the original position in \(x\) is not recovered in the subsequent epochs. This allows discrimination between this term and other anomalous changes that are included in \(\eta(t)\). A clear example is presented in Figure 3.  
As for the estimation error \(\eta(t)\), we can adopt the representation  
\[\eta(t)=\epsilon(t)+b(t)+o(t) \tag{3}\]where  
* \(\epsilon(t)\) is the ordinary measurement noise propagated from the observations to the estimates through the least-squares algorithm. * \(b(t)\) is that part of the model error in the observation equations that is absorbed into the parameter estimates by the least-squares algorithm. Typically, \(b(t)\) is smooth in time and can be treated as correlated (not white) noise. The presence of \(b(t)\) is, in fact, the reason why the estimation error \(\sigma_{\varepsilon}\) provided by the least-squares adjustment is systematically too optimistic. * \(o(t)\) is an outlier in the series \(\widehat{x}(t)\); namely, it is a value that at a specific epoch \(t\) is clearly far away from values at neighboring epochs and that can be smoothly interpolated. An example is given in Figure 3. Such values can be generated by a set of observations with a particularly unfavorable signal-to-noise ratio, by restricted sky visibility, and/or by some undetected cycle slip that can bias the estimates of the ambiguities and, consequently, of the coordinates, too. The example represented in Figure 3, concerning the monitoring of a point on a bridge with reduced sky visibility, shows hourly coordinates obtained by using GPS observations only (red dots) and GPS and Galileo joint observations (blue dots). When data from the GPS-only system are processed, a number of outliers due to quite unfavorable sky visibility are clearly detectable. However, the same example shows that the joint use of GPS and Galileo, doubling the number of tracked satellites and diversifying the directions of the lines of sight, can reduce, if not eliminate, the outliers, meaning that the evolving technology of low-cost receivers can significantly reduce this effect. Summarizing, we can:  
* Represent \(m(t)\) by the linear combination of some shifted base functions; e.g., we use cubic splines, \[(t=2T,3T,\ldots)\ \ \ \ \ m(t)=\sum_{i=-2}^{2}p_{i}S(t-k_{i}\tau)\,\] (4) with \(p_{i}\) parameters to be determined by interpolation and \(\tau\) as the inter-distance (in time) between the central points of two consecutive splines. For instance, for daily solutions \((T=1\) day), splines with an interdistance of \(1\) week \(=\tau\) have proved to be effective. In case of fast movements, interdistances might be shortened.