# Monitoring of Structures and Infrastructures by Low-Cost GNSS Receivers

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## Abstract

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## 1. Introduction

## 2. Concepts

#### 2.1. What We Want to Monitor

#### 2.2. Absolute and Relative Positioning

#### 2.3. Electronic Noise, Model Error and Estimation Errors

#### 2.4. The a Posteriori Motion Model

- $m\left(t\right)$ is the mean slow motion, which is a trend, and that could very well be $m\left(t\right)\equiv 0$ when the point does not drift away from a reference mean position.
- $s\left(t\right)$ is a true signal in $x\left(t\right)$ 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.
- $a\left(t\right)$ represents an anomalous sudden change in the coordinate x. Characteristic of this term is its persistence. For instance, if $a\left(t\right)$ 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 $\eta \left(t\right)$. A clear example is presented in Figure 3.

- $\epsilon \left(t\right)$ is the ordinary measurement noise propagated from the observations to the estimates through the least-squares algorithm.
- $b\left(t\right)$ is that part of the model error in the observation equations that is absorbed into the parameter estimates by the least-squares algorithm. Typically, $b\left(t\right)$ is smooth in time and can be treated as correlated (not white) noise. The presence of $b\left(t\right)$ is, in fact, the reason why the estimation error ${\sigma}_{\epsilon}$ provided by the least-squares adjustment is systematically too optimistic.
- $o\left(t\right)$ is an outlier in the series $\widehat{x}\left(t\right)$; 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.

- Represent $m\left(t\right)$ by the linear combination of some shifted base functions; e.g., we use cubic splines,$$(t=2T,3T,\dots )\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}m\left(t\right)=\sum _{i=-2}^{2}{p}_{i}S(t-{k}_{i}\tau )\phantom{\rule{4pt}{0ex}},$$
- Model $s\left(t\right)$ as a linear response to one (or more) exogenous forces $\xi \left(t\right)$ possibly shifted in time$$s\left(t\right)=\sum _{i=0}^{m}{\lambda}_{i}\xi (t-iT)$$
- Not model $a\left(t\right)$, verifying a posteriori whether there are jumps in the trajectory of $\widehat{x}\left(t\right)$ that persist after the discontinuity epoch.
- Not model $o\left(t\right)$ for the same reason as for the previous point.
- Model$$\epsilon \left(t\right)+b\left(t\right)=\nu \left(t\right)$$

#### 2.5. Warnings and Alarms

- Q1: Is the deviation of $\widehat{x}\left(t\right)$ from the slowly varying and predictable motion model statistically significant?
- Q2: Is the shift in $\widehat{x}\left(t\right)$ assuming a critical value larger than a given structurally critical threshold $\overline{S}$?

- (a1)
- In this case, the hypothesis H is$$H:a(\overline{t}+T)=0$$In this case, the critical threshold is simply$$c={Z}_{\alpha}{\sigma}_{\nu}\phantom{\rule{4pt}{0ex}}.$$It is clear that if we want to go with two tails, the test must verify whether$$|\widehat{u}(\overline{t}+T)|<c={Z}_{\alpha /2}{\sigma}_{\nu}\phantom{\rule{4pt}{0ex}}$$
- (a2)
- In this case, the hypothesis H we want to test is$$H:\widehat{a}(\overline{t}+T)\ge \overline{S}\phantom{\rule{4pt}{0ex}},$$We then have to fix a critical value $c<\overline{S}$ (when $\overline{S}>0$), such that if$$\widehat{u}(\overline{t}+T)>c$$$$P(\widehat{u}(\overline{t}+T)<c|H)=\alpha \phantom{\rule{4pt}{0ex}}.$$Let us start with a simple hypothesis H$$\widehat{a}(\overline{t}+T)=\overline{S}\phantom{\rule{4pt}{0ex}};$$$$P(\widehat{u}(\overline{t}+T)<c|H)=P(Z<-\frac{\overline{S}-c}{{\sigma}_{\nu}})=\alpha $$$$\frac{\overline{S}-c}{{\sigma}_{\nu}}={Z}_{\alpha}\Rightarrow c=\overline{S}-{Z}_{\alpha}{\sigma}_{\nu}\phantom{\rule{4pt}{0ex}},$$Now it is clear that if we change H from (23) to$$H:\widehat{a}(\overline{t}+T)>\overline{S}$$$$P(\widehat{u}(\overline{t}+T)<c|H)<\alpha \phantom{\rule{4pt}{0ex}},$$Just to make a small example with realistic numbers, assume there is a series of hourly solutions with ${\sigma}_{\nu}=3$ mm, and $\overline{S}$ is fixed to $\overline{S}=4$ cm; then ${H}_{0}(a(\overline{t}+T)=0)$ is refused if$$\widehat{u}(\overline{t}+T)\phantom{\rule{4pt}{0ex}},\phantom{\rule{4pt}{0ex}}\widehat{v}(\overline{t}+T)>7\mathrm{mm}$$$$\widehat{u}(\overline{t}+T)\phantom{\rule{4pt}{0ex}},\phantom{\rule{4pt}{0ex}}\widehat{v}(\overline{t}+T)>33\mathrm{mm}.$$

## 3. The Experience

#### 3.1. Many Examples

#### 3.2. The Structure of the Bridge and Its Problems

#### 3.3. The Network and the Installation on the Bridge

#### 3.4. Hardware and Software

- Tallysman GNSS antenna and cables;
- Solar panel (30 W, 50 W, or 80 W) to provide electric power when it is not available on site;
- Box containing a battery (providing autonomy for 5–7 days), a GNSS receiver, memory and a transmission apparatus (based on u-blox SARA-U201 module);
- Low-cost, single-frequency, multi-constellation GNSS receiver (u-blox NEO-M8T) or low-cost, dual-frequency, multi-constellation GNSS receiver (u-blox ZED-F9T).

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#### 3.5. Results of Relative Motion

- C3–C1: very little motion of at most ∼1 cm over the 4 years in all the coordinates, a long-term very small negative trend in X and Y;
- C3–C2: marked seasonal signal with an amplitude of ∼2 cm in X (longitudinal) with clear annual periodicity;
- C3–C2: very small negative trend in Y up to March 2020, followed by a descent during springtime, a period in which ASPI had been working on the bridge. Although it is not covered by the displayed data, it is possible to prove stabilization of the descent after about 3 months;
- C3–C2: essential stability in Z with a pattern very similar to that of C3–C1.

#### 3.6. Results of the Absolute Motion

## 4. Discussion and Perspectives

#### 4.1. Discussion

#### 4.2. Perspectives

- We strive to shorten the time unit of adjustments by going to 15–30 min in such a way as to improve latency while still remaining with RMS under 1 cm.
- Parallel experiments show that it is possible to combine GNSS observations with other sources of data, primarily inclinometers, to improve the monitoring capability of the system. Further, an external source of information, such as SAR, can be integrated into a unique solution.
- We expect that future data from monitoring sites will be endowed with specific expert systems that also combine completely different information, such as temperatures (of the air and the structure), the level of rain by pluviometers placed near the structure, and other ancillary variables such as the level of water of a river the bridge is crossing, or the level of water of an artificial lake barring the monitored dam.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ASPI | Autostrade per l’Italia |

GNSS | Global Navigation Satellite System |

GPS | Global Positioning System |

GReD | Geomatics Research & Development |

RMS | Root Mean Square |

SAR | Synthetic Aperture Radar |

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**Figure 1.**The star-shaped design of a monitoring network; M master station; ${P}_{1},{P}_{2},{P}_{3}$ monitored points; and ${S}_{1},{S}_{2}$ permanent stations of a global positioning service.

**Figure 2.**The difference of geometry-free combination between two closed GNSS stations after subtracting the average. GNSS receivers are u-blox.

**Figure 3.**Hourly estimates of the coordinates of a point on a bridge: red dots are GPS-only solutions; blue dots are GPS and Galileo joint solutions. Outliers are quite evident.

**Figure 9.**Conceptual scheme showing the position of permanent station S with respect to the monitoring network on the bridge. The aspect ratio is not maintained.

**Figure 13.**X and Y components on the baseline S–C3 with the X-axis oriented along the steepest descent direction of the slope.

Time Interval | Modality | Latency |
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1 h (24 solutions per day) | Relative | 15 min after the end of the hour, for early warning |

2 h (12 solutions per day) | Relative | 15 min after the end of 2 h, to confirm the warning |

24 h (daily solution) | Relative/Absolute | Delivered after midnight |

**Table 2.**Root mean squares (RMSs) of $XYZ$ with respect to the mean motion (splines every 14 days) for the 2 baselines over the 6-year period.

Baseline | Length | $\mathit{\sigma}$X (Longitudinal) | $\mathit{\sigma}$Y (Transversal) | $\mathit{\sigma}$Z (Up) |
---|---|---|---|---|

C3–C1 | 69.1 m | 0.4 mm | 0.5 mm | 0.7 mm |

C3–C2 | 67.4 m | 1.1 mm | 0.6 mm | 0.7 mm |

**Table 3.**Root mean squares (RMSs) of $XYZ$ with respect to the mean motion (splines every 14 days) for the 2 baselines over the 1-month period.

Baseline | Length | $\mathit{\sigma}$X (Longitudinal) | $\mathit{\sigma}$Y (Transversal) | $\mathit{\sigma}$ Z (Up) |
---|---|---|---|---|

C3–C1 | horiz.: 69.1 m, vertic.: −0.4 m | 0.7 mm | 0.8 mm | 1.4 mm |

C3–C2 | horiz.: 67.4 m, vertic.: −0.4 m | 0.9 mm | 0.7 mm | 1.1 mm |

**Table 4.**Root mean squares (RMSs) of $XYZ$ with respect to the mean motion (splines every 14 days) for the 3 baselines over the monitoring period.

Baseline | Length | $\mathit{\sigma}$X (Longitudinal) | $\mathit{\sigma}$Y (Transversal) | $\mathit{\sigma}$Z (Up) |
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S–C1 | horiz.: 2390.5 m, vertic.: −214.6 m | 0.6 mm | 1.0 mm | 2.7 mm |

S–C2 | horiz.: 2390.3 m, vertic.: −214.2 m | 1.1 mm | 1.0 mm | 2.7 mm |

S–C3 | horiz.: 2405.2 m, vertic.: −214.2 m | 0.6 mm | 1.0 mm | 2.6 mm |

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**MDPI and ACS Style**

Caldera, S.; Barindelli, S.; Sansò, F.; Pardi, L.
Monitoring of Structures and Infrastructures by Low-Cost GNSS Receivers. *Appl. Sci.* **2022**, *12*, 12468.
https://doi.org/10.3390/app122312468

**AMA Style**

Caldera S, Barindelli S, Sansò F, Pardi L.
Monitoring of Structures and Infrastructures by Low-Cost GNSS Receivers. *Applied Sciences*. 2022; 12(23):12468.
https://doi.org/10.3390/app122312468

**Chicago/Turabian Style**

Caldera, Stefano, Stefano Barindelli, Fernando Sansò, and Livia Pardi.
2022. "Monitoring of Structures and Infrastructures by Low-Cost GNSS Receivers" *Applied Sciences* 12, no. 23: 12468.
https://doi.org/10.3390/app122312468