# Monitoring of Structures and Infrastructures by Low-Cost GNSS Receivers
## 2 Concepts
### Electronic Noise, Model Error and Estimation Errors  
We will not discuss here the observation equations of the GNSS observables because this is a well-known matter that has been well-described in many books together with different analysis strategies [24, 25, 26, 27]. However, we will use the list of the following very well-known factors that enter into a carrier-phase measurement: the distance between satellite and receiver antennas, variation of the velocity of the electromagnetic wave through the shield of the atmosphere (ionospheric and tropospheric effects), electronic errors from the beat of the oscillators  
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. and the circuits of the transmitter and receiver (also known as clock errors and biases), the electronic error in the correlation of the incoming signal with a local replica, an initial integer ambiguity when the local replica is locked to the receiver signal (this is an integer number of wavelengths of the carrier), and finally, electromagnetic interference of the environment where the receiver is situated, also called multipath. As for the error due to the clock of the transmitters, one can observe that this is an atomic clock with great stability, so that in this context, it can be considered approximately zero. Further, the biases from the satellite circuits are monitored by the International GNSS Service [28; 29] and are made available to users. As for the initial integer ambiguities, we know that they are constant so long as the received signal and replica are locked to each other. However, either by degradation of the signal in crossing the atmosphere or simply because the line of sight between the two antennas is obstructed by some obstacle, the lock can be lost, and when it is re-established, a new, different integer ambiguity has to be used--we have a jump in the ambiguity called a cycle slip. Such cycle slips can be very large, and are therefore easily identifiable, but they can also as small as a single unit, corresponding to a jump of \(\sim\)20 cm in a specific measurement, which is much harder to detect. All software for the estimation of position from GNSS observations have algorithms that identify cycle slips, and indeed, this negatively affects the estimation of some coordinates. Since nowadays low-cost receivers can track two frequencies, it is possible to form the so-called geometry-free combination, namely the difference between the phases in metric form, which depends only on the electronic noise, the initial ambiguities and the ionospheric effect. If, in addition, we subtract two such combinations from two stations separated only by tens of meters, the ionospheric effect is eliminated, and we are left with a usually constant ambiguity and the electronic noise only, apart from maybe the presence of a multipath effect. In Figure 2, we present the plot for the above combination on 720 epochs, namely 1 h with observation lag of 5 s. Since the index is constructed by combining four observations, assuming that the electronic noise is at the same level in the two receivers and for the two frequencies, we see that the s.q.m. of the noise is roughly \(\sim\)2 cm. By itself, this figure might not be sufficient to monitor the position of a point at millimeter-level, also considering that beyond coordinates of the stations one has to estimate many other parameters, especially clock errors and ambiguities. Nevertheless, one has to take into account that by using, for instance, two frequencies and two or three constellations at every measurement epoch, one has between 20 and 40 observ  
Figure 2: The difference of geometry-free combination between two closed GNSS stations after subtracting the average. GNSS receivers are u-blox. importantly, since we are looking for a position that is only slowly changing, we can assume that for a certain period the coordinates of the stations are constant.