# Monitoring of Structures and Infrastructures by Low-Cost GNSS Receivers
## 2 Concepts
### Warnings and Alarms  
We design two procedures, each based on a two-step level, warning and alarm, intended to answer two questions:  
* Q1: Is the deviation of \(\widehat{x}(t)\) from the slowly varying and predictable motion model statistically significant? * Q2: Is the shift in \(\widehat{x}(t)\) assuming a critical value larger than a given structurally critical threshold \(\overline{S}\)? In both cases, the answer comes from evaluating the probability that \(\widehat{u}(\overline{t}+T)\) attains a value larger than some critical threshold \(c\) under some hypothesis \(H\) on average. The value of \(c\), as typical of statistical inference, is chosen so that  
\[P(\widehat{u}(\overline{t}+T)>c|H))=\alpha \tag{9}\]  
where \(\alpha\) is some significance probability parameter, the value of which is discussed below. We make the hypothesis, justified by experience, that the variable \(\widehat{u}(t)\) is normally distributed, with the anomalous motion \(a(t)\) as an average, and  
\[\sigma_{u}^{2}=\sigma_{v}^{2}+\sigma_{pr}^{2} \tag{10}\]  
as variance; here, \(\sigma_{pr}^{2}\) is the variance of the predicted value \(\widehat{x}_{pr}(\overline{t}+T)\), which can be derived from the covariance of the parameters \(\widehat{p}_{\mu},\widehat{\lambda}_{i}\) estimated by least squares applied to the model (7). Since practice shows that \(\sigma_{pr}^{2}\ll\sigma_{v}^{2}\), we continue by assuming that \(\sigma_{u}^{2}\approx\sigma_{v}^{2}\), so we put  
\[\widehat{u}(t)\sim\mathcal{N}[a(t),\sigma_{v}^{2}]\Rightarrow\frac{\widehat{ u}(t)-a(t)}{\sigma_{v}}=\mathcal{Z}\, \tag{11}\]  
with \(\mathcal{Z}\) a standard normal variable. Therefore, we have [33]:  
\[P(\widehat{u}(\overline{t}+T)>c|H)=P\bigg{(}Z>\frac{c-a(t)}{\sigma_{v}}\bigg{)} =\alpha\, \tag{12}\]  
so that we must have  
\[\frac{c-a(t)}{\sigma_{v}}=Z_{\alpha}\Rightarrow c=a(t)+Z_{\alpha}\sigma_{v}. \tag{13}\]  
Let us remark that we are reasoning here on only one tail of the normal distribution, as if \(c,\widehat{u}(\overline{t}+T)>0\); yet it is very easy to generalize our conclusion to cases using two tails. Now, if our \"observed\" \(\widehat{u}(\overline{t}+T)>c\) and \(\alpha\) are chosen to be conveniently small, Equation (12) says that it is likely that the hypothesis \(H\) is wrong. On the other hand, \(\alpha\) cannot be too small because we do not want to risk failing to give an alarm. Therefore, we decide \(\alpha=1\%\). Moreover, a particularly high value of \(\widehat{u}(\overline{t}+T)\) might be due to the presence of an outlier in \(\widehat{x}(\overline{t}+T)\), which most likely disappears after that epoch. Thus, when \(c\) is overtaken by \(\widehat{u}(\overline{t}+T)\), we do not immediately raise an alarm, but we only declare a warning, i.e., a state of pre-alarm requiring one further step before the alarm is confirmed. This second step consists, essentially, of producing a two-step ahead prediction \(\widehat{x}_{pr}(\overline{t}+2T)\) and seeing whether, again, at time \(\overline{t}+2T\), the residual  
\[\widehat{v}(\overline{t}+2T)=\widehat{x}(\overline{t}+2T)-\widehat{x}_{pr}( \overline{t}+2T) \tag{14}\]  
confirms the warning because  
\[\widehat{v}(\overline{t}+2T)>c. \tag{15}\]  
The advantage of this approach is that now we have  
\[P(\widehat{u}(\overline{t}+T)>c\ ;\ \widehat{v}(\overline{t}+T)>c|H)=\alpha^{2} \tag{16}\]  
so that the probability that we give an alarm when, on the contrary, \(H\) is true, i.e., a false alarm, is now diminished significantly. For instance, for a series of daily solutions, with \(\alpha=0.01\), we expect to have three false alarms per year, while with \(\alpha^{2}=0.0001\), the false alarms, on average, decrease to \(1\) in \(27\) years. Moreover, if the event \(\widehat{u}(\overline{t}+T)>c\) is an isolated outlier, then with the two-step residuals, we expect \(\widehat{v}(\overline{t}+T)<c\), so the alarm will not be raised in vain. Indeed, this happens at the cost of two-step latency of the alarm; on the other hand, the decrease in false alarms clearly has a favorable economical implication.