POLITECNICO DI MILANO
Scuola di Architettura Urbanistica Ingegneria delle
Costruzioni
Building and Architectural Engineering Degree
DEFORMATION ANALYSIS USING LOW COST
GNSS RECEIVERS
Supervisor: Prof. Riccardo Barzaghi
Assistant supervisor: Ing. Lorenzo Rossi
Degree thesis of:
Francesca Accetta
Matr. 853040
SUMMARY
Abstract ... 1
Introduction ... 2
1. Cathedral of St. Gaudenzio ... 3
1.1. The Church ... 4
1.1.1. History and construction ... 4
1.1.2. Structure ... 4
1.2 The Bell tower ... 6
1.3. The Dome ... 6 1.3.1. Construction ... 7 1.3.2. Antonelli’s mechanism ... 10 1.3.3. Structure ... 13 1.3.4. Strengthening ... 15 2. The GPS ... 20
2.1. The GPS reference system ... 21
2.2. Principles of operation ... 23
2.3. The components of GPS system ... 24
2.3.1. The space component ... 25
2.3.2. The monitoring component ... 26
2.3.3. The user component... 26
2.4. The GPS signal ... 27 2.5. GPS measurements ... 28 2.5.1. Code measurement ... 29 2.5.2. Phase measurement... 30 2.5.3. Errors in GPS measurement ... 31 2.5.3.1. Systematic errors ... 31 2.5.3.2. Observation errors ... 35
2.5.4. Final observation equations ... 36
2.5.5. Possible linear combinations of observations ... 36
2.6. Differential phase measurements ... 37
2.7. Software for data processing ... 39
2.8. GPS receivers ... 40
3. Equipment used in the tests and in the St. Gaudenzio survey... 45
3.2. Reference stations ... 46
3.2.1 Milan ... 47
3.2.2 Monza ... 47
3.2.3. Pavia ... 48
3.2.3. Novara ... 48
3.3. Terrestrial topographic survey at St. Gaudenzio ... 49
4. Tests in a controlled scenario ... 50
4.1. Testing the precision of low cost receivers... 50
4.1.1. Milan ... 54
4.1.2. Monza ... 61
4.1.3. Pavia ... 66
4.2. Performance comparison between high quality and low-cost devices ... 71
4.2.1. Monza ... 71
4.2.2. Pavia ... 72
5. Testing the of St. Gaudenzio spire ... 73
5.1. The installed device ... 73
5.2. The processing software ... 75
5.3. Least Squares GPS Coordinate time series modeling and testing ... 77
5.3.1. The Milano data ... 80
5.3.2. The St. Gaudenzio in Novara data ... 81
6. The plumb line of the St. Gaudenzio spire... 84
6.1. The network design and survey ... 84
6.2 The point cloud acquisitions ... 86
6.3 Estimating the dome and sphere centers ... 87
6.3.1. Estimating the horizontal position of the St. Gaudenzio’s dome center ... 87
6.3.2. Estimating the horizontal position of the sphere center ... 90
6.4. Comparing the coordinates of the dome and sphere centers. ... 93
7. Conclusion ... 96 APPENDIX A ... 97 APPENDIX B ... 98 APPENDIX C ... 100 APPENDIX D ... 102 LIST OF FIGURES ... 104 LIST OF TABLES ... 107 REFERENCES... 107
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Abstract
Nowadays, to promptly safeguard the public safety and to reduce the economic costs related to unexpected structural failures, it is increasingly important to control these risks through a continuous monitoring of the structures. In this dissertation, it was considered the case study of St. Gaudenzio’s cathedral. Indeed, it was subjected to invasive renovations, which doubled its spire weight, rising several doubts regarding its stability. Therefore, it was asked to study spire deformations along with the definition of its verticality. Accordingly, after detecting the equipment resolution and a proper choice of the processing software, a series of analyses were implemented. To carry out the oscillations monitoring process, two low-cost GPS devices were installed on St. Gaudenzio’s spire. Then, thanks to a statistical analysis of the GPS coordinate time series, it was successfully verified the absence of a trend in the spire coordinates. Subsequently, for the definition of the plumb line, a traverse scheme was utilized, framed in the ETRF2000 system. Five scans were performed, that were analyzed through a MATLAB program properly implemented for the estimate of the cathedral spire of the out of vertical angle.
Riassunto
Al giorno d'oggi, per salvaguardare prontamente la popolazione e per diminuire i costi economici legati ad improvvisi cedimenti strutturali, è sempre più importante l’attivazione di sistemi di monitoraggio, i quali grazie alla determinazione continua delle coordinate di una serie di punti, attivano allarmi se viene superata una certa soglia di deformazione del sistema complessivo. In questa tesi, si prende in analisi la basilica di San Gaudenzio, poiché alcune ristrutturazioni avvenute tra il 1931 e 1937-1938, hanno causato un irrigidimento della struttura, tanto da considerare l’azione del vento critica per la sua stabilità. Pertanto, è importante studiare le deformazioni alla quale è soggetta la guglia e la definizione del fuori piombo della basilica. Di conseguenza, dopo aver verificato le accuratezze raggiungibili con la più recente strumentazione GPS e scelto il software più appropriato per il post-processamento dei dati, sono state implementate una serie di analisi. Per monitorare le oscillazioni alle quali è soggetta la guglia, sono stati utilizzati due ricevitori GPS a basso costo e, grazie a una analisi statistica dei dati, si è verificata con successo l’assenza di deformazioni plastiche, almeno nel periodo analizzato. Successivamente, per la definizione del fuori piombo, si è utilizzato lo schema della poligonale chiusa su punti noti, inquadrando i punti rilevati nel sistema di riferimento rete locale e in quello ETRF2000. Fatto ciò, si sono effettuate delle scansioni, che, analizzate attraverso l’uso di un codice MATLAB implementato ad hoc, hanno permesso la stima delle coordinate dei punti di sommità e di base della basilica. Infine, attraverso il confronto di queste coordinate si è trovato l’angolo di fuori piombo della guglia di San Gaudenzio.
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Introduction
The risks of large structure failures and those related to potential geological disasters, ask for an efficient monitoring system, so that to mitigate the consequences in terms of economical protection and life expectancy. One of the possible approach to monitor these risks consists in the continuous determination of the coordinates of a set of points and in providing a prompt alarm when, considering the entire system, a certain deformation threshold is exceeded. Nowadays, in order to monitor the deformations, it is a common practice to use geodetic techniques, particularly GPS/GNSS systems. These are typically implemented installing a network of receivers that guarantee, with daily consecutive observations, accuracies and precision around millimeters. Unfortunately, the high cost of the geodetic receivers is one of the main problem of this solution, implying a reduction in their number, and thus, a decrease in the efficiency of the monitoring system.
Indeed, by the beginning of the XXI century satellite positioning applications were limited by very expensive instruments and their restricted diffusion. In the last decade, however, many low-cost receivers, interesting for some geomatics applications, have been manufactured.
The first generation of low cost GPS receivers was conceived in order to track the points of interest in any condition. This kind of devices carried out an estimated positioning using just code data. With the second generation of low cost receivers, an early improvement in the chipset was made and it was possible to get code and the phase L1 data from these receivers.
Nowadays, for geomatics applications, one can rely on low-cost receivers far more sophisticated and engineered than the old ones. Indeed, recently it has been studied an alternative use of low-cost devices, showing the possibility of obtaining good results.
The main point of the thesis will be the study of the dynamic and static of St. Gaudenzio’s cathedral. Since after several damaging consolidation interventions, it became more fragile, resulting susceptible to external stresses. As a result, through the use of low-cost GPS devices the spire oscillations were monitored. Furthermore, using terrestrial surveying methods the plumb line of the spire was checked to verify its present situation.
As for the dissertation structure, it is divided in seven chapters, where the first two are dedicated to the history of St. Gaudenzio’s church and to the GPS satellite method. Subsequently, in the third chapter, it is reported the equipment utilized for the GNSS experiments and then, thanks to a set of analyses, the achievable precisions are tested. Next, it was performed a statistical study of the data, examining the static of St. Gaudenzio’s spire and, in chapter six, the out of plumb calculation was performed. The last chapter is dedicated to the conclusions.
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1. Cathedral of St. Gaudenzio
The Cathedral of St. Gaudenzio is an important place of Catholic worship in the city of Novara, in Piedmont, famous for its dome, 121 meters high, by Alessandro Antonelli. The architectural complex consists of three main elements realized in different construction phases: the church, the bell tower and the dome.
Figure 1.1: Drowing 1:300 of L.Caselli depicting the prospect of the dome – table XV from La Cupola della Basilica di San Gaudenzio in Novara in L’ingegneria civile e le arti industriali – Turin 1877.
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1.1. The Church
1.1.1. History and construction
Since 841, at the beginning of the current avenue XX Settembre there was a temple dedicated to St. Gaudenzio. Subsequently the building was reconstructed and re-consecrated in 1298.
Between 1552 and 1554 the Spaniards of Carlo V decided to transform the city into a military stronghold, so all existing buildings outside the city walls, including the cathedral, were destroyed. Also in 1552 the "Fabbrica Lapidea della Basilica di San Gaudenzio" was established with the purpose of overseeing the reconstruction of the church.
Following the plague of 1576, which Novara was miraculously escaped, it was decided to reconstruct the cathedral at the highest point of the city, at the northwest corner of the walls. Here, since 1019, there had been a church dedicated to St. Vincenzo Martire, which was demolished to make room for the new building. Only three chapels were saved, including the one dedicated to St. Giorgio, where the remains of St. Gaudenzio were temporarily transported as a result of the destruction of the old cathedral located outside the city walls.
In 1553 the Fabbrica Lapidea assigned the project to Pellegrino Pellegrini, known as Tibaldi. The accentuated verticalism of the building and the sense of vigorous plasticism that are projecting from the facade and sides, both enlivened with niches, windows, and columns are all to be attributed to him.
The first stone was laid in May 1577 and on December 13, 1590, when the transept and presbytery had not been erected yet, the consecration was carried out by the bishop Cesare Speciano.
The worsening of the economic situation, exacerbated by plagues and wars, blocked the work that only restarted in 1626 and continued at a slow pace till the end in 1656.
Only a worthy arrangement of the relics of the patron was lacking: between 1674 and 1710 the scurolo was built. Placed in the right transept, this was a large chest of marble and bronze, in which the silver and crystal urn of the saint would be placed.
On June 11, 1711, the church could be said to have been completed with the solemn deposition in the scurolo of the relics of St. Gaudenzio, up to that moment preserved in the chapel of St. Giorgio.
1.1.2. Structure
The church has a Latin cross-shaped plant with a single nave, alongside six lateral chapels connected to each other with a large transept and a deep presbytery.
As for the chapels, the three on the right side are “Chapel of Good Death”, “Chapel of the Circumcision”, “Chapel of the Crucifix” respectively. Those on the left side are “Chapel of Our Lady of Loreto”, “Chapel of the Nativity”, “Chapel of the Guardian Angel”.
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By climbing through the bell tower, you can access the attic of the apse and therefore to the "Compass Hall". Here the ancient 11-meter-long compass is preserved, used by Antonelli to draw the 1:1 scale vaults that support the dome.
This hall was first opened to the public on January 26, 2013, to represent the first piece of a cathedral’s museum path.
Figure 1.2: Floor-plan of the Basilica of St. Gaudenzio.
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1.2 The Bell tower
In 1753, after completing the Cathedral, the Fabbrica Lapidea decided to build the Bell tower, assigning the project to Benedetto Alfieri, architect of the Savoy family, under whose kingdom Novara had passed in 1738.
The bell tower was built between 1753 and 1786. About 75 meters high, it is set apart from the church, to the left of the apse, and it is built of terracotta and Baveno granite.
Prior to its construction there was a temporary bell tower on the southwestern pillar of the church, but it was seen that, with the vibrations produced by the bells, this old arrangement could bring damage to the structure of the building. Accordingly, in 1753 it was decided to build a new bell tower, which was considered a priority with respect to the construction of a new dome.
In 1773, when only the bell tower was missing, the works were suspended for lack of funds.
The work will only be completed in 1786 with the help of architects Francesco Martinez and Luigi Michele Barberis, 33 years after the opening of the building site and 19 years after the death of its designer.
1.3. The Dome
The most important architectural element of the cathedral is its majestic 121-meter-high dome, designed by architect-engineer Alessandro Antonelli, a symbol of the city and a distinctive sign of its landscape.
After more than 50 years from the end of the bell tower, thanks to the money derived from meat tax, the Fabbrica Lapidea decided that the time was ripe for completing the cathedral and commissioned Antonelli for the realization of the dome.
On the arrival of Antonelli, Italy was in an historic period of great economic difficulty where all the noblest resources, such as steel, were channeled into the wars of independence.
Antonelli, not letting himself be affected by the adverse situation in the construction field decided to use zero-kilometer materials for the construction of the dome, such as lime, brick and local stone. In addition, during the 19th century the commission had changed from nobility to bourgeoisie, who
did not possess much land and thus they had the need to develop buildings in height.
His ability to combine the needs of the new commission with the materials at his disposal shaped his technological poetics, demonstrating the ability to reach heights with local materials. Furthermore, the materials used had the enormous advantage ofnot being subjected to custom duties and of being able to be assembled by already very experienced local workers.
The Architect became thus the greatest expert of the masonry frame, separating the bearing function from the shell one although using solid brick.
The dome of St. Gaudenzio is an example of this innovative technology, as this consists of three concentric circular frames and an outer casing, completely forcible. Finally, to give stability to the outer frame and thus create a kind of inner bracing, Antonelli used the conical shape of internal pillars.
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1.3.1. Construction
The first project was presented in 1841 while work began in 1844. The first two years were spent to redo the tambour and the eight supporting arches, being the older ones incapable of supporting the weight of the work. Immediately after, the building site was suspended since the wars of independence against Austria were being fought and the municipality was therefore forced to drastically reduce Fabbrica Lapidea funds.
Figure 1.5: The first Antonellian project of 1841.
Figure 1.4: Drowing of L.Caselli depicting the plan of the dome – table XIV from La Cupola della Basilica di San Gaudenzio in Novara in L’ingegneria civile e le arti industriali – Turin 1877.
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In 1855 Antonelli presented a second revised draft with which he increased the dome height from 65 to 75 meters since the peristyle of the columns had been raised on a pedestal. The latter change resulted necessary due to the will of the architect to hide the arches with the roof of the Cathedral. In 1858 the economic situation had improved and the work could resume but the architect, rather than setting the basis for the closing of the dome, erected a second round of 5-meter-high pillars, recovering the visual usability of the monument. In 1860, he presented the project of a dome with two orders of columns which was rejected. In May 1861, the project was resubmitted with the assurance that it would cost less than the last and, after many complaints, was eventually accepted. The construction of the dome came to an end two years later.
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At that point only the spire was missing, but disagreements between the factory and the architect again halted the works for a decade. During this time, Antonelli devoted himself to the construction of the Mole Antonelliana of Turin.
Meanwhile the dome aroused the admiration of visitors and slowly took hold of the idea that it would be complete as long as its designer, now elderly, was alive and that therefore it was necessary to give him carte blanche.
The works resumed: between 1873 and 1874 Antonelli dedicate himself to the floral decoration in stucco of the interior dome and only in the summer of 1876 he started the spire, which was completed in 1878.
This other long interruption gave the architect a chance to conceive the doubling of the spire, in harmony with the already existing doubling of the colonnade under the dome.
On May 16 of the same year the statue of Christ the Savior by Pietro Zucchi was raised atop. The statue is made of bronze covered with gold foil and it is 5 meters high. Currently a modern fiberglass copy was placed at the top of the dome, while the original is placed inside the cathedral, in the left transept.
Counting the statue as well, the total height of the building rises to 126 meters with a total weight of over 5,500 tons.
Although the work has lasted almost 50 years, the work was never completed. In the intents of the architect, in fact, the second inner dome, which today appears white, should have been decorated with a series of murals visible from below. Similarly, also the external colonnades were to be enriched by a series of statues.
Figure 1.7: Photographic history of the dome's constructive phases performed by 1862 and photographic history of the installation of the statue of the Savior.
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As anticipated, for the construction Antonelli decided to use only local materials, to tie it more closely to its place of origin. The structure, in fact, is entirely in brick (2046 m ³) and lime, without the use of iron, making it one of the world's tallest masonry buildings.
The genius of Alessandro Antonelli was to have designed the building breaking it down into a series of many concentric circles that rise into the sky, getting smaller, gradually unloading the load on the bearing structure. In the event of structural failure, the dome would collapse on itself and not on the surrounding buildings.
1.3.2. Antonelli’s mechanism
For the construction of the dome, Antonelli thought to create a complex masonry “mechanism” apt to resist the set of vertical and horizontal forces, in the static and dynamic field.
Figure 1.8: Undated watercolor section of the Basilica of St. Gaudenzio.
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As outlined, the dome of St. Gaudenzio consists of three concentric circular frames contained by an outer casing, completely forcible. In addition, to create an inner bracing, Antonelli used a conical shape of internal pillars.
Antonelli created a dome with a thin structural shell, only one brick header thick, stiffened by a brick network of “meridians” and “parallels”. More in depth, the main skeleton of the structure is made up by a system of ribs, or meridians, stiffened by compression rings along three parallels.
On the other hand, the vault, the thin shell between two meridians, plays two different roles. Firstly, it works as a conjunction element between the ribs. Secondly, it supports the external covering. Furthermore, to give stiffness and stability to the vault, two brick rings were used: one at the top of the thin shell, and one at the base, that is the ring of the drum.
With the aim of designing the loadbearing skeleton of the entire building, Antonelli conceived, inside the dome, a stiffening structure with the shape of a truncated cone.
The inner bracing is made up by inclined brick pilasters connected by means ofarches. This complex system gave rise to a grid with a circular base that increases its diameter as it descends.
Figure 1.10: External thin shell in brick stiffened by the system of “meridians” and “parallels” and a schematic description of the system.
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As far as the connections between the internal and external structures are concerned, there are firstly, the two rings already mentioned, and secondly, a 1 m wide walkway that creating an internal chain prevents the ovalization of the dome. More in depth, the brick masonry catwalk, at the intrados, is carried by means of arches leaning against granite corbels.
Furthermore, it is important to highlight that, although the external dome and the internal truncated cone are interconnected, because of different external forces and loads carried, the two bodies have their own autonomous static behavior. Even if, being both constituted by masonry frames, and therefore transmitting their dead loads by means of a pre-established path, their behaviors are similar. In fact, the external dome, uses the ribs to transmit the loads, while the internal structure exploits the grid of pilasters and ribs.
In short, recent studies on the geometry and the mechanics of the dome revealed that the system of principal and secondary ribs, which connects the two stiffening rings, allows it to accommodate small variations of the initial state of equilibrium without the risk of causing a kinematic collapse.
Figure 1.11: Section of the exterior dome and the interior cone.
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1.3.3. Structure
The whole system of the dome weigh on four pairs of arches, from which the pendentives are inserted. The arches transmit the load to the four fundamental pillars and to the eight wings of ramming, made up of the four walls of the transept, two of the sacristies and two of the side chapels.
The upper arches, receive the fulcrums of the two-external rows of columns and, the lower arches, receive with the pendentive the internal row. It follows from a cursory examination that the two-outer row support the entire lower outer side of the monument, the colonnades of the two orders up to the attic, and instead, the inner row will support the true dome, the interior of the castle and, finally, the spire.
Clearly, to carry out the work an important role has been played by the scrupulous control of materials, the value of Italian 19th century builders and the technical perfection desired by the architect and his team.
Figure 1.13: Wooden model representing a pair of arches with its fundamental pillar.
14 Figure 1.14:Drowing 1:300 of L. Caselli the section of the dome – table XII from La Cupola della
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1.3.4. Strengthening
In the years following the completion of the church, it began to show signs of structural failure, already noticeable during the early stages of construction. It was admitted that the pillars on which the construction leaned could not withstand the new weight (the only dome weighs approximately 5572 tons) and they were failing.
It is important to stress that, in those years, there was a problem related to the binders. In fact, the cement did not exist, lime was used, bearing two main problems. First and foremost, it was very soft, second it does not have grip under water, except for very expensive hydraulic limes, and thus seldom used. For this reason, when the foundations were laid they avoided digging too deep and find water. As a result, these foundations were generally shallow, relying on the fact that the buildings were so massive and so heavy that they stuck into the ground.
So it was also for the Cathedral of St. Gaudenzio, with foundations of about three meters.
In 1881 the pillars, which had the function of supporting the dome, gave way to sinking. Antonelli attributed the structural failure to two causes. Firstly, the foundations were recognized as too shallow, secondarily, the pillars did not have a sufficient base to unloading the weight of the dome.
For Antonelli’s foundations, solid brickwork and hard concrete with hydraulic mortar were used. The architect always using masonry frames, made a hollow sub-foundation, expanding the base of the structure using inverted arches, and thereby creating a kind of pillared gallery. Antonelli’s foundations pushed under the old foundations of the cathedral (about three meters below the floor of the church) up to the hard layer of solid clay called ferretto (five meters below the floor). In addition, new foundations interested not only the four pillars, but also the eight wings of ramming.
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As for the pillars, these were linked to new foundations and the system of steps within them was redone, in order to let them pass where the load was lighter. The work ended in early 1887, just in time for the celebration of the patron saint, January 22.
Subsequently, concrete reinforcement to the spire and other work at the Antonelli’s arches were carried out by Arturo Danusso.
On August 29, 1902, Danusso, graduated with honors in civil engineering and, immediately after graduation, was hired by the technical office of the company Porcheddu Ing. G.A. Danusso became a pupil of Giovanni Porcheddu, who had the intuition to appreciate immediately the validity of the «Systéme Hennebique», which is concrete reinforced with iron profiles placed and strengthened with special brackets. This technique was invented and patented in 1892 by the French engineer François Hennebique and Porcheddu, in the same year, obtained the exclusive license for the patent application in Italy. Moreover, after a few years, the Italian engineer, under the guise of collaborating with the Parisian studio, learned to calculate the reinforced concrete.
As introduced, the cultural context in which Danusso grew was the one of reinforced concrete construction, a technique which he applied, wrongly, even for the strengthening of St. Gaudenzio’s spire. This was externally made of iron and white granite, one of the less sensitive stones to atmospheric action. On the other hand, the old core was a sort of brick tube that brought cantilever staircases.
In order to weld the iron belts, in the absence of welding torch, sulphur was set fire. This however, in the most hidden parts, did not receive enough oxygen and was not completely burnt. Then, already in the 20s, with the arrival of precipitation, the remaining sulphur gave rise to a sulfuric solution. This, penetrating the granite, weakened the adhesion of the granules that make it up (quartz, feldspar and mica). This disintegration continued uninterruptedly, causing the fall of fragments of granite on the roofs of the church.
Thus, it was in 1940 that all the damaged parts of granite were removed for prevention.
The sulphuraction resulted more harmful in correspondence to the edges of the bases, on the capitals and generally on the more elaborated and protruded parts.
In 1931 the work of consolidation of the spire began. Danusso did not realize the risk brought by welds made with sulphur. In fact, he thought that the breaking of the capitals of the spire was due to instability of the structure, insufficiently rigid, therefore attributing to the wind forces the deformation of the spire and the inevitable fall of the pieces on the roof of the church. Danusso, did not understand that the cause was not physics but chemistry. On this misunderstanding, he based the intervention of the 30s. As a result, by acting on the upper part of the spire (second stylobate and peristyle, cusp), he decided to maintain the exterior unchanged and built a concrete core cable internally, in which he would have positioned the ladder. In doing so, Danusso wanted to give rigidity to an externally unchanged structure.
At this point, the first stylobate and peristyle were weighted by the oscillations of the monolith above. Clearly, the regime of new oscillations was very different from the first one, since, if before each gust of wind caused small deflections on each node, now all the stresses flowed into the base of the second stylobate, increasing moments at that point.
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So it was that between 1932 and 1934, subsequent to the first intervention, the fractures of the inner capitals of the first stylobate of the spire, dividing them into two parts, were determined.
In the second building site, 1937-1938, Danusso was summoned once again to continue the consolidation of the dome he acted on several fronts. Firstly, to absorb the high moments of the spire, he continued the concrete core cable throughout the lower part of the spire, including three little domes inside the dome. Secondly, he improved the existing pillars that, because of concrete reinforcements had to work at double of the load. Finally, the voids behind pendentive were filled. Having completed the intervention of 1937, the weight of the spire, above the dome, was almost doubled. If before, at the wind's action, the Antonelli’s spire reacted with a deformation work now instead, the rigid spire, laid on the elastic lower structure, as well as transmitting the increased weight multiplies the wind blow. Practically, once the interventions were completed, the structure gravitational center rose and a greater instability in the structure was created.
Also in 1937, the Basilica was closed for almost a decade due to a possible structural failure of the four pairs of arches, that were carrying the dome. In reality, the convergence of several factors seems to best justify what happened.
Figure 1.16: Early works of consolidation of the spire, 1931.
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Firstly, more than ninety years after the beginning of the cathedral, the brick culture disappeared, and in this way also the confidence of technicians in masonry buildings. In fact, with the prevalent use of reinforced concrete and steel the trust in the models of computation was increasing more and more. Secondly, it is possible to suppose a political disagreement that can be attributed to the bishop of Novara, little inclined to the civil rhetoric of the time dominated by fascism.
Finally, the technicians worried about the presence of multiple cracks formed already before the intervention of 1881, which stopped their formation.
The culture of the brick was overcome and so the crack phenomenon, considered earlier as a joint, was interpreted as a symptom of instability and not of settling, more in harmony with the monolithic culture of concrete, which saw a dangerous discontinuity in the cracks.
As expected, during these ten years of closure, the work was very little and, if analyzed more closely, not effective to address the structural failure of the four pairs of arches. More in detail, at the presbytery, a giant wooden scaffold was erected, which would have to support the arches. The propping, which did not come directly to the Antonelli’s arch, pushed under the intrados of the old Tibaldi’s arch. In addition, in the interior, the old arch was welded to the Antonelli’s arch with masonry, and instead, in the outer part they were spaced by a void, which was initially not taken into account for the transmission of the loads. Subsequently, it was discovered that the structure would not be able to support Antonelli’s arches but only the vault arches, which had no bearing function. Thus, to properly transfer the load on the scaffold, little pillars were placed, some in bricks and others in wood between Antonelli’s arches and the vault arches.
The plaster, placed at the central key of each arch, emphasized the lack of credibility of this intervention. This in fact remained intact, while it would have raised alarm if their failure had been close.
Figure 1.18: Plastered Antonellian arch and little brick pillars. Figure 1.17: Wooden scaffold that would have to support the dome.
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The last consolidation operations took place in 1946-1947, when several interventions took place. More in depth on this occasion, in correspondence to the four pendentives, the high relief of the evangelists were taken away and a concrete cast was made. Furthermore, at the ring of sets of the first inner dome a reinforced concrete ring was built, hiding the existing decorations. Eventually, as far as the four pairs of Antonellian steel rods are concerned, new rods with a bigger diamenter were added, fifteen for each pair.
In recent years, after the damaging interventions of 1931 and 1937-1938, several doubts regarding St.Gaudenzio spire stability rose. In fact, even though structural failure not yet occurred, after these invasive consolidations the structure became fragile and, wind action started having a critical role. Accordingly, it was asked to study spire oscillations along with the definition of its verticality degree; in order to carry out the monitoring process they were used a series of GPS (Global Positioning System) devices.
Figure 1.19: Church of San Gaudenzio before the interventions of 1946-1947, on the left, and in the situation of the eighties of the twentieth century, to the right.
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2. The GPS
In order to know the point positions, has been created a network of artificial satellites, helping the users in the navigation, namely the Global Navigation Satellite System (GNSS).
The GNSS is a complex of different satellite systems: - The American Global Positioning System (GPS);
- The Russian Global Navigation Satellite System (GLONASS); - The BeiDou Navigation Satellite System (BDS);
- The Japanese Quasi-Zenith Satellite System (QZSS);
- The GALILEO System that is under development in Europe.
The NAVSTAR GPS (Navigation Satellite Timing And Ranging Global Positioning System) had been developed during the 1970s by the Department Of Defense (DOD) of the United States of America.
From 1978 to 1985 eleven satellites were put in orbit, those of the so-called Bloc 1, but only six of them were effectively in use.
In February 1989 the satellites of Bloc II were put in orbit followed by Bloc IIA, IIR, IIR-M, IIF (the last one in 2010).
In 1993 the GPS system started to be operative twenty-four hours a day (in the experimental stage), then from 1995 on, it was officially declared up and running.
The GPS is a global satellite positioning system which works thanks to the decodification of complex signals emitted from the satellites in orbit. These signals enable us to get information about the distances between the satellite and the receiver. The use of these tools, the reception and interpretation of those signals make the three-dimensional positioning in real time possible anywhere on Earth. In our case study, for coordinates determination, we will always refer to the GPS satellite system.
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2.1. The GPS reference system
The GPS is based on a geocentric Cartesian reference system which is valid all around the world. It is a Conventional Terrestrial Reference System, characterized as follows:
- individuated by a triplet of Cartesian axes (𝑋𝑡, 𝑌𝑡 , 𝑍𝑡);
- having origin in the terrestrial barycenter;
- the 𝑍𝑡 axis coincides with the mean axis of terrestrial rotation;
- the 𝑋𝑡 axis is defined by the intersection between terrestrial equator with the mean plane of
Greenwich.
In the above-mentioned system, the coordinates can be defined as the three Cartesian components (𝑋, 𝑌,𝑍). Otherwise, coupling to the triplet Cartesian axes a geometric ellipsoid, can be found the geographic coordinates: latitude1, longitude2 and the ellipsoidal height, 𝜑, 𝜆, ℎ respectively. Over the
time, has been defined various terrestrial ellipsoid, characterized by different pairs of ellipsoidal parameters. By way of example, Hayford ellipsoid parameters are reported:
Semi-major axis: 𝑎 = 6378388𝑚 Semi-minor axis: 𝑏 = 6356911,946𝑚
In figure 2.2 it is shown the geometric ellipsoid.
In order to pass from the Cartesian to the geographic coordinates, the following equations can be applied: 𝜆 = 𝑎𝑟𝑐𝑡𝑎𝑛 (𝑌 𝑋) (2.1) 𝜑 = 𝑎𝑟𝑐𝑡𝑎𝑛 (𝑍 + 𝑒 ′2𝑏𝑠𝑖𝑛3𝜗 𝑝 − 𝑒2𝑏𝑐𝑜𝑠3𝜗) (2.2)
1 The latitude 𝜑 of a point P is the angle formed between the equatorial plane and the perpendicular line passing through P.
2 The longitude 𝜆 of a point P is the angle in the equatorial plane formed between the meridian plane passing through P and a reference meridian plane, i.e. Greenwich meridian.
22 ℎ = 𝑝 𝑐𝑜𝑠𝜑− 𝑁 (2.3) where 𝜗 = 𝑎𝑟𝑐𝑡𝑎𝑛 (𝑍𝑎 𝑝𝑏) 𝑝 = √X2+ Y2 𝑒′= √a2+ b2 b2 𝑁 = 𝑎 √1 − 𝑒2𝑠𝑖𝑛2𝜑
Once defined the conventional system, it necessary to make it available for the users. The GPS global reference system is realized operatively by means of a network of permanent stations distributed all over the world. Among the various reference systems realized, the World Geodetic System 1984 (WGS84) can be considered. It is a worldwide geocentric coordinate system, based on the reference ellipsoid elaborated in 1984. The realization of WGS84 is defined though the station network of the American Department of Defense, and after its realization, it has been updated in the years. This reference system is usually represented with the Universal Transverse Mercator 3(UTM)
representation.
Moreover, it is also possible to consider a local reference system, in fact, referring to a certain point it will be possible to find local coordinates, i.e. North, East and Up, see figure 2.2. The equations 2.4, 2.5 and 2.6 illustrate how to transform local coordinates to Cartesian coordinates.
𝑋 = 𝑁𝑜𝑟𝑡ℎ + 𝑠𝑒𝑛𝜑𝑐𝑜𝑠 𝜆𝑁𝑜𝑟𝑡ℎ − 𝑠𝑒𝑛𝜆𝐸𝑠𝑡 + 𝑐𝑜𝑠𝜑𝑐𝑜𝑠𝜆𝑈𝑝 (2.4)
𝑌 = 𝐸𝑠𝑡 + 𝑠𝑒𝑛𝜑𝑐𝑜𝑠 𝜆𝑁𝑜𝑟𝑡ℎ + 𝑐𝑜𝑠𝜆𝐸𝑠𝑡 + 𝑐𝑜𝑠𝜑𝑠𝑒𝑛𝜆𝑈𝑝 (2.5)
𝑍 = 𝑈𝑝 − 𝑐𝑜𝑠𝜑𝑁𝑜𝑟𝑡ℎ + 𝑠𝑒𝑛𝜑𝑈𝑝 (2.6)
3 The Universal Transverse Mercator is a conformal projection, that uses a two-dimensional Cartesian coordinate system to locate points on the Earth surface. As the traditional method of latitude and longitude, it is a plane position representation but the two methods differ for several aspects.
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2.2. Principles of operation
The GPS system is based on the hypothesis that the position of the satellites in the space is known in every moment. Thanks to this assumption it is possible to track the coordinates of a point by measuring the distance between the unknown point and a complex of satellites.
In order to estimate the number of satellites that are needed to determine the location of a certain point, different cases were analyzed:
If only the distance between a single satellite and a receiver, whose coordinates are not defined, is known, that receiver could be located anywhere on the sphere that has the satellite as its center. That distance is the radius that has the same length of the distance between the satellite and the receiver.
If the distance between two satellites is known, the receiver could be located in anyone of the points of intersection between the circumferences of the two spheres.
A third distance circumscribes the position of the receiver in two points, that are the two intersections of the three spheres. Only one of these two positions is useful because the other one is located in space.
Then the simultaneous measuring of the distances between the receiver and the three different satellites is needed: 𝑑𝑅𝑖 = √(𝑋𝑖 − 𝑋 𝑅)2+ (𝑌𝑖− 𝑌𝑅)2+ (𝑍𝑖 − 𝑍𝑅)2 𝑖 = 1,3 (2.7) (a) (b) (c)
Figure 2.3: Number of satellites necessary to determine the position of the desired point: a) presence of one satellite b) presence of two satellites c) presence of three satellites.
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where
- 𝑑𝑅𝑖 distance between the satellite i-th and the receiver R (measured) - (Xi , Yi , Zi ) coordinates of the i-th satellite, i=1,2,3 (known)
- (XR, YR, ZR) coordinates of the receiver R (unknown).
The determination of the distance is given by the difference from the receiving time (obtained from the clock of the receiver) and the broadcasting time (linked to the signal of the satellite). All clocks must be perfectly synchronized, otherwise the solution would be wrong.
Satellite's clocks can be considered as properly synchronized with each other, but the receiver's clock, being of a lower quality, can cause some problems. Considering this, it will be convenient to add another unknown quantity to the problem, in order to take into consideration the time lag between the satellite clocks' time scale and the receiver's time scale.
The presence of an additional unknown quantity gives rise to the need of at least four satellites in order to make the real-time positioning possible.
The equation system of observations is rewritten using the distance calculated considering the error of synchronization (pseudo-distance), which is different from the geometric distance (𝑑𝑅𝑖 ) :
𝑃𝑅𝑖 = √(𝑋𝑖 − 𝑋
𝑅)2+ (𝑌𝑖− 𝑌𝑅)2+ (𝑍𝑖− 𝑍𝑅)2+ 𝑐∆𝑇 𝑖 = 1,4 (2.8)
where
- 𝑃𝑅𝑖 pseudo distance between the satellite i-th and the receiver R (measured) - (Xi , Yi , Zi ) coordinates of the i-th satellite, i=1,2,3,4 (known)
- (XR, YR, ZR) coordinates of the receiver R (unknown factors)
- ΔT the receiving time lag (unknown factor) - c propagation signal speed (known).
It follows that the distance between the receiver and the satellites is measured this way:
𝑑𝑅𝑖 = 𝑃𝑅𝑖(𝑡) − 𝑐∆𝑇 (2.9)
2.3. The components of GPS system
The GPS system needs different components in order to work properly:
- The space component: it consists in a constellation made up of satellites that broadcast radio signals.
- The monitoring component: it consists in a construction set on Earth that is able to track the position of satellites and manage the whole system.
- The user component: made up of specific tools engineered in order to receive and interpret the signals so as to do the positioning.
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2.3.1. The space component
The constellation of GPS satellites is made up of twenty-four satellites plus two spare satellites. They are located on six circular orbits, inclined at about 55° on the equatorial axis and the distance between them is about 60° in longitude. The length of the radius of the orbit is about 27.000 km, the time of revolution is of 12 sidereal hours. The distance of the orbits from the surface of the Earth is equal to about 20.200 km. This makes it possible to track the satellites, from each one of the observers set on earth, for five sidereal hours. Moreover, such a height, ensures that the satellites are out of the influence of the atmospheric drag and that their obits are barely affected by the anomalies of the earth's gravitational pull, fundamental factor for the precise determination of the orbits.
On board of the satellites there are four high-precision oscillators (two of them are made of cesium while the other two are made of rubidium). Satellite rockets are used to make corrections to the orbit, and two solar panels are used as source of energy. They weigh several hundred kg and they are engineered to last no more than ten years.
The main functions of these satellites are the following: - They broadcast information through signals to the users;
- The information sent by the monitoring component are received and broadcast by the usage component;
- The atomic oscillators on board make it possible to maintain a precise time-signal; - They can perform a correction-of-the-orbit maneuver using the rockets on board.
Figure 2.4: The three component that compose the GPS system.
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2.3.2. The monitoring component
The monitoring component was originally made up of five stations placed in American military bases located on the equator line; one of them functions as the Master Control Station (MCS). This MCS is located in Colorado Springs while the other ones are to be found:
- In Kwajalein (Pacific Ocean), - In Diego Garcia (Indian Ocean), - In Ascension (Atlantic Ocean), - In Hawaii (Pacific Ocean).
These stations monitor the satellites in order to track their position, they control the errors in the synchronization of the clocks and their operating state. All the four stations send their data to the Master Control Station where the clocks of the satellites are controlled and compared to each other in order to derive the analytic models used to correct the satellites' clocks. Moreover, the ephemeris data are also estimated: these are parameters that make it possible to fix the orbital position of the satellites in order to predict their position for the following 15 minutes.
Actually, the control stations are eighteen: other thirteen stations have been added to the early five, where one of them has the task of supporting the functions of the Master Control Station.
2.3.3. The user component
The user component consists in any of the users that can receive the signals broadcast by satellites thanks to an antenna and a receiver. There are many kinds of receivers that can be distinguish thanks to their technical characteristics, to the precision in the positioning, and to the technique they use in order to decode the signal they receive.
GPS antenna
receiver
calculator for the post-processing
Figure 2.7: The user component.
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Any receiver is equipped with an oscillator, but its performance characteristics differ from those of the satellites. These oscillators' function is to emit a continuous electromagnetic signal called 'replica' because it is similar to the one coming from space.
The receiver has different channels, one for each satellite, in order to make the replica of the signal coming from different satellites at the same time. The consequence is that the number of the channels represents the maximum number of satellites that can be simultaneously used.
Receivers can memorize the data coming from the satellites and the positioning, calculated in real time. The data are stored and used subsequently in a post-process that makes it possible to give a more precise positioning.
2.4. The GPS signal
The oscillators onboard the satellites emit a frequency signal f0=10.23MHz (fundamental frequency),
characterized by high stability over time.
Other frequencies derive from that fundamental one (f0):
- Two carrier / sinusoidal frequency:
L1 characterized by: L2 characterized by:
f1= 154f0 f2= 120f0
λ1= 19 cm in the vacuum λ2= 24cm in the vacuum
- Two binary codes:
A binary code is a series of pulses with values equal to +1 and - 1, the sequence of pulses transmission consists in the content of the signal.
C/A (Coarse acquisition) P (Precise) characterized by: characterized by: f C/A = 1/10f0 fp= f0
λ C/A = 300m in the vacuum λp= 30m in the vacuum
T C/A = 1ms Tp= 37 weeks
characteristic for each satellite Common to every satellite - The Navigation Message D
The navigation message is another binary code structured in order to send a message that is characterized by a f=50Hz frequency. More specifically it contains the predicted ephemeris, the offset parameters of satellite's clock, the descriptive parameters of the ionosphere, and information about the system status.
At this point, we want to identify the complete signal emanated by the satellite determined from the modulation of a carrier frequency with a binary code.
The signal obtained from the combination of a carrier frequency (pure oscillatory phenomena) and a code (sequence of pulses +1 and -1) reproduces the carrier, but in case of a state transition of the code a leap of 180° during the emission of the sound occurs.
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The analytical representation of the GPS signal can be expressed by this mathematical relationship: 𝑆𝐺𝑃𝑆= 𝑆𝑃+ 𝑆𝐶 = 𝐴𝐿1∗ 𝑃(𝑡) ∗ 𝐷(𝑡) ∗ cos(2𝜋𝑓𝐿1𝑡 + ∅𝐿1) +
+𝐴𝐿2∗ 𝑃(𝑡) ∗ 𝐷(𝑡) ∗ cos(2𝜋𝑓𝐿2𝑡 + ∅𝐿2) + 𝐴𝐿1∗ 𝐶(𝑡) ∗ 𝐷(𝑡) ∗ sin(2𝜋𝑓𝐿1𝑡 + ∅𝐿1) (2.10) where
- C(t), P(t), D(t) modulation codes (respectively C/A, P and D)
- AL1, AL2, 𝑓𝐿1, 𝑓𝐿2, ∅1, ∅2 amplitudes, frequencies and phases of the carrier waves.
So L1 is emitted by two replicas, with a phase shift of 90° between them. The first replica is modulated
by the code P, while the second one by the code C/A. The carrier L2 is emitted by a single replica,
which is modulated only by the code P. All the signals are, at the end, modulated by the D message.
2.5. GPS measurements
The signal emitted by the satellites is picked up by the receivers that replicate it using their oscillators. This replica is different from the signal they receive because of the time lag. In order to calculate the distance, it is necessary to use the observations of code and/or of phase.
Thanks to the study of the code it is possible to calculate the flying time ΔT, which is required by the signal of the satellite to reach the receiving station. The flight time is then multiplied by the propagation signal speed so as to obtain the pseudo-range from the satellite to the receiver.
The phase observation is based on the calculation of the number of GPS signal cycles that occur in the transmission from the satellite to the receiver. At the end the distance between the satellite and the receiver can be found by multiplying the number of cycles (plus the discrepancy) by the length λ of the emitted wave.
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2.5.1. Code measurement
The determination of the flight time ΔT is given by the receiver. After having identified the satellite through the C/A code (Coarse Acquisition Code) or the P code (Precise Code) - which currently switched into Y (EncrYpted P code) - it correlates the code given by the receiver's oscillator with the one emitted by the satellite. In this way it is possible to measure the signal transmission delay from the signal received and the one coming from the receiver.
As the emission of the signal is represented by 𝑡𝑆(𝑆), and the moment of reception which is at the same time the moment in which the replica starts is represented by 𝑡𝑅(𝑅), we can represent the flight
time as the period of time the signal takes to cover the unknown distance. The observation equation may be expressed by the formula:
∆𝑇𝑅𝑆(𝑡) = 𝑡𝑅(𝑅)− 𝑡𝑆(𝑆) (2.11)
The fact that 𝑡𝑆(𝑆) and 𝑡
𝑅(𝑅) are linked to the clocks of the satellite and the receiver, makes them
subject to errors. It is thus necessary to synchronize the clocks to the GPS time.
𝑡𝑆(𝑆) = 𝑡𝑆+ 𝑑𝑡𝑆(𝑡) (2.12)
𝑡𝑅(𝑅) = 𝑡𝑅 + 𝑑𝑡𝑅(𝑡) (2.13)
𝑡𝑆 and 𝑡𝑅 represent the GPS time of emission from the satellite S and the time of reception of the
receiver R. 𝑑𝑡𝑆(𝑡) and 𝑑𝑡
𝑅(𝑡) represent the errors of the satellite and receiver's clocks.
The observation equation is so re-written in this way: ∆𝑇𝑅𝑆(𝑡) = 𝑡
𝑅 + 𝑑𝑡𝑅(𝑡) − 𝑡𝑆+ 𝑑𝑡𝑆(𝑡) = 𝜏𝑅𝑆+ 𝑑𝑡𝑅(𝑡) − 𝑑𝑡𝑆(𝑡) (2.14)
where 𝜏𝑅𝑆 represents the signal travel-time from the satellite to the receiver.
The pseudo-range equation is given multiplying the travel-time by the electromagnetic signal propagation speed in a vacuum:
𝑃𝑅𝑆(𝑡) = 𝑐 ∗ ∆𝑇𝑅𝑆(𝑡) = 𝑐 ∗ 𝜏𝑅𝑆 + 𝑐 ∗ (𝑑𝑡𝑅(𝑡) − 𝑑𝑡𝑆(𝑡)) (2.15)
Received code
Internal repetition
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where c∙ τRS = dRS represents the space distance between the satellite and the receiver.
Each one of the satellites observed gives a pseudo-range equation. It is thus possible to track the position of a GPS antenna solving a system made up of four of these equations that include four unknown factors: three of them are about the position of the antenna, the last one is about the time offset. The precision of the study of the code depends on the length of the wave with which the measuring is made, so it depends on the type of the code.
Thus, at least theoretically, we have:
- code C/A λ ≅ 300m theoretical uncertainty of positioning ≅ 3-6 m - code P λ ≅ 30m theoretical uncertainty of positioning ≅ 30-60 cm.
2.5.2. Phase measurement
During the phase measurement the receiver, in order to calculate the distance between the satellite and the receiver, carried out an observation of phase difference in cycles between the carrier frequency, coming from the satellite, and a sine wave of the same frequency generated by the oscillator inside the receiver.
The satellite-receiver distance can be express as the multiple of the wave-length. This is then multiplied by the number of cycles that have taken place between the two extreme points and it usually consists in an integer number and a fractional part:
𝑑 = 𝐾 ∗ 𝜆 (2.16)
𝐾 = 𝐼𝑛𝑡 + 𝐹𝑟 (2.17)
The observation equation at time t is:
∅𝑅𝑆(𝑡) = ∅𝑅(𝑡) − ∅𝑅( )(𝑡𝑅𝑆 ) (2.18)
where
- ∅𝑅𝑆(𝑡) observation of the phase difference
Figure 2.10: Measurement of phase difference.
Received carrier
Internal repetition
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- ∅𝑅(𝑡) phase generated within the receiver, which is not directly observable
- ∅𝑅( )(𝑡𝑅𝑆 ) phase received in R from the satellite S, which is not directly observable.
Since the receiver's correlator measures only the fractional part of the phase difference (𝐹𝑟 = ∅R − ∅S ) and not the whole number of cycles that take place during the period of time
from the signal emission to the reception (internal ambiguity NR S (t)), another term must be added. Finally, without forgetting the errors of the clocks, it can be demonstrated that the following observation equation is valid:
∅𝑅𝑆(𝑡) = 𝑓𝜏𝑅 𝑆 (𝑡) + 𝑓 (dt
R(t) − dtS(t)) + ∅R − ∅S + NR S (t) (2.19)
Multiplying the observation equation by the signal wavelength, we obtain the observation in metric units:
𝐿𝑆𝑅(𝑡) = 𝜆 ∙ ∅𝑅𝑆(𝑡) =
= 𝑐𝜏𝑅 𝑆 (𝑡) + 𝑐 (dtR(t) − dtS(t)) + λ ∙ (∅
R − ∅S + NR S (t)) (2.20)
It must be said that also for the phase measuring the precision depends on the length of the wave which is used, so it depends also on the type of the carrier.
- L1 λL1=19cm the theoretical uncertainty about the positioning is of about 2-4 mm
- L2 λL2=24cm the theoretical uncertainty about the positioning is of about 2,5-5 mm
Thus, theoretically, phase measurements are more precise than the code measurements.
2.5.3. Errors in GPS measurement
The high number of systematic effects that affect the GPS system are to be considered as the main problem that limits the system to reach its full potential. The sources of errors in GPS measurement can be divided in different categories: device random errors (equal to 1-2% of the length of the wave), systematic errors, observation errors.
2.5.3.1. Systematic errors
These errors can be, for example: orbital errors, atmospheric delay errors, errors related to the eccentricity of the antenna phase center and those linked to the clocks.
Orbital errors
The broadcast ephemerides' precision (thanks to which it is possible to track the position of the main satellites used for GPS positioning) is of few feet and this is not sufficient enough for the precision positioning. If the positioning is absolute, the effect of the error reflects directly on the antenna's position. However, if the positioning is relative the effects of the error are weaker.
This error can be expressed by the equation: δb = δr ∙ b r ⁄ where δr represents the orbital error, b is the base joining the two receivers and r is the radius of the orbit.
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Atmospheric delay errors
There are three main phenomena caused by the propagation of the electromagnetic signal through the atmosphere that can change the distance between the satellite and the receiver dR S . Here we will
analyse the bending of the signal's path, the ionospheric delay, and the tropospheric delay. - The bending of the signal's path:
According to the Fermat principle, any electromagnetic signal crossing a medium, follows the shorter path, for what concerns time, and it does not necessary coincide with the geometrical distance.
The error coming from the bending of the electromagnetic signal is caused by the difference between the physical path of the electromagnetic wave and the straight-line distance.
This error is often neglected because it is considered as insignificant for the detection angles exceeding 15°-20° over the antenna horizon, below which the GPS observation usually are not considered.
Figure 2.11: The orbital errors.
Figure 2.12: Bending of the electromagnetic signal. X, Y, Z Satellite of the ephemerides
X, Y, Z Correct satellite Receiver position estimated Receiver position correct Atmosphere
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- Ionospheric delay
Sun's ultraviolet radiation in the ionosphere (part of the atmosphere that stretches from 50 to 1000km of altitude) causes the ionization of the gas molecules and these interfere with the propagation of the GPS signal. The effect of the ionospheric disturbance is different for the code measurement or the phase measurement but in both cases, it is significant. In the code measurements the ionosphere causes the increase of the distance between the satellite and the receiver, while for what concerns the phase measurements it causes the decrease of the distance between them. These effects depend on the electronic density, which varies according to how intense is the influence of solar radiation on the atmosphere. It is thus evident that the ionospheric disturbance is very variable. For the modelling of this effect, GPS receivers employ the synthetic model of Klobuchar. Its index of fallibility is of 5-10% of the total disturbance, it depends on eight numerical benchmarks that the NIMA4 monitoring network estimates and sends daily to the
satellites that, send them to the receivers.
But the ionospheric effect, as the ionosphere is a dispersive means, also depends on the frequency of GPS signal. So, the most effective method to reduce the ionospheric effect is the employment of signals with different frequencies, i.e. the two L1 and L2 signals.
- Tropospheric delay
The tropospheric disturbance is generated in the strata between the ground and the first 40km of altitude, and it is made up of two components: wet and dry. The dry component is more predictable and depends on the pressure and on the temperature in the atmosphere. The wet component, instead, depends on the quantities of precipitable water vapor that is present in the atmosphere. There are two standard methods of estimating this disturbance, but they cannot however represent exactly the real weather condition at the moment and in the place of measurement. The tropospheric effect is weaker in quantity than the ionospheric one, but as it does not depend on the GPS signal frequency it cannot be eliminated using the two frequencies L1 and L2.
The antenna phase center eccentricity
The distance between the satellite and the receiver, measured by the GPS, has as its extreme points the instantaneous antenna phase centers. They can vary according to the elevation and the azimuth of the satellites, and to the frequency of the signal. The error consists in two vectors, the one that joins the antenna's point of reference and the mean phase center, and the one that joins the mean phase center and the instantaneous phase center.
4 In 1996 it was instituted the National Imagery and Mapping Agency (NIMA), thanks to the consolidation of some divisions of the Department of Defense and other agencies. The main role of NIMA is to support the military machine through image interpretation, the creation of soil maps and the production of geospatial Intelligence. In the field of cartography, this agency takes a leading position in the world; in fact, its services are the most required for the maritime and air navigation and also for the construction industry.
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The determination of the first vector is possible knowing the physical point of which we want to track the position (ARP Antenna Reference Point) while the mean phase center is given by the manufacturer. For what concerns the second vector it is necessary to calibrate the antenna in order to find a map of the variations of the instantaneous phase center (PVC Phase Center Variation) that depend on the position and of the frequencies emitted by the satellites. Thanks to this calculation it is possible to obtain high-precision positioning (around few millimeters).
Figure 2.13: Variation of the center of the phase of the antenna. Instantaneous phase centre
Mean phase centre
Point of reference
Mean phase centre offset
Phase centre variation
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2.5.3.2. Observation errors
Multipath
The GPS signal during his path can face multiple reflections caused by reflective surfaces surrounding the antenna. In this case the signal follows a longer path that can differ from the direct one by some meters for what concerns the codes and some centimeters for the phases.
So, in order to minimize the multipath effect, it would be useful to shield the antennas.
Cycle slips
The cycle slip is a slip in the enumeration of the number of cycles and it is caused by the interruption of the data acquisition. The use of the combination of the carriers and the diversification of the observations make it possible to fill the gaps left by the cycle slips when these are not too large, otherwise it would be necessary to introduce another unknown factor that is similar to the initial phase ambiguity.
Geometrical configuration of the satellites
The Dilution of Precision factor expresses the effect of the satellites' geometrical configuration, it is expressed as the ratio:
𝐷𝑂𝑃 = 𝜎
𝜎0 (2.21)
where
- σ positioning error
- σ0 the measurement error.
So, in order to minimize the error in the positioning the DOP must be less than 6, this can be verified when the satellites are disposed in the 'open umbrella' configuration, see figure 2.15.
Figure 2.14: Multipath. Receiver position estimated Receiver position correct Reflective surface
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2.5.4. Final observation equations
Considering the measurement errors analyzed above, it is possible to re-write the observation equations of code and phase:
Code: 𝑃𝑅𝑆(𝑡) = 𝑑𝑅𝑆(𝑡) + 𝑐(𝛿𝑡𝑅(𝑡) − 𝛿𝑡𝑆(𝑡) + 𝛿𝑖𝑜𝑛 (𝑡) + 𝛿𝑡𝑟𝑜𝑝 (𝑡) + 𝛿𝜌(𝑡) + 𝛿𝜀(𝑡) (2.22) Phase: 𝐿𝑆𝑅(𝑡) = 𝑑𝑅 𝑆 (𝑡) + 𝑐(𝑑𝑡𝑅(𝑡) − 𝑑𝑡𝑆(𝑡)) + 𝜆 ∙ (∅𝑅 − ∅𝑆 + 𝑁𝑅 𝑆 (𝑡)) + 𝛿𝑖𝑜𝑛 (𝑡) +𝛿𝑡𝑟𝑜𝑝 (𝑡) + 𝛿𝜌(𝑡) + 𝛿𝜀(𝑡) (2.23) where
- 𝛿𝑖𝑜𝑛 error caused by the ionospheric delay - 𝛿𝑡𝑟𝑜𝑝 error caused by the tropospheric delay
- 𝛿𝜌 error caused by the orbit uncertainty - 𝛿𝜀 systematic errors.
2.5.5. Possible linear combinations of observations
It is possible to minimize the errors taking into consideration that at any time any receiver can make two observations of code and two observations of phase for each satellite; so we have at least four observations at any time:
𝑃1𝑅𝑆(𝑡) = 𝜌𝑅𝑆(𝑡) + 𝑐(𝑑𝑡 𝑅(𝑡) − 𝑑𝑡𝑆(𝑡)) + 𝑇𝑅𝑆(𝑡) + 𝐼1𝑅𝑆(𝑡) (2.24) 𝑃2𝑅𝑆(𝑡) = 𝜌𝑅𝑆(𝑡) + 𝑐(𝑑𝑡𝑅(𝑡) − 𝑑𝑡𝑆(𝑡)) + 𝑇 𝑅𝑆(𝑡)+𝐼2𝑅𝑆(𝑡) (2.25) 𝐿1𝑅𝑆(𝑡) = 𝜌𝑅𝑆(𝑡) + 𝑐(𝑑𝑡 𝑅(𝑡) − 𝑑𝑡𝑆(𝑡)) + 𝑇𝑅𝑆(𝑡) − 𝐼1𝑅𝑆(𝑡) + λ(𝑁1𝑅𝑆(𝑡) + ∅1𝑅 − ∅1𝑆) (2.26) Figure 2.15: Possible geometric configurations of the satellites
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𝐿2𝑅𝑆(𝑡) = 𝜌𝑅𝑆(𝑡) + 𝑐(𝑑𝑡
𝑅(𝑡) − 𝑑𝑡𝑆(𝑡)) + 𝑇𝑅𝑆(𝑡) − 𝐼2𝑅𝑆(𝑡) + λ(𝑁2𝑅𝑆(𝑡) + ∅2𝑅 − ∅2𝑆) (2.27)
Starting from the code and phase observations of the two frequencies emitted by the receiver to the satellite at a given time, this kind of combination can be built:
𝑂𝑅𝑆(𝑡) = 𝛼1𝑃1𝑅𝑆(𝑡) + 𝛼2𝑃2𝑅𝑆(𝑡) + 𝛽1𝐿1𝑅𝑆(𝑡) + 𝛽2𝐿2𝑅𝑆(𝑡) (2.28)
Where α e β are two appropriate coefficients, chosen for the purpose for which the combination is built.
In GPS history many combinations for the GPS data elaboration have been suggested; each one has its advantages and disadvantages. Firstly, we will analyze the four phase combinations, then the ones concerning the codes.
Geometry Free L4 combination
The L4 combination takes into account only the ionospheric disturbance and a combination of the ambiguities, but it does not include the geometry. It will not be possible to use it, thus, to track the receiver's position, but it will be useful to estimate the local model of ionospheric disturbance, that has to be substituted with the Klobuchar model during the final processing of the observations. Wide Lane L5 combination
In the L5 combination the ionospheric disturbance and the electromagnetic interference pay a greater role, but we can note a great advantage in the detection of the ambiguities.
Narrow Lane L6 combination:
It is the Wide Lane specular combination and it is used along with the L5 in the process of detection of the ambiguities.
Ionospheric Free L3 combination:
The L3 does not include the Ionospheric disturbance.
For what concerns the codes the only combination that can be used is the Ionospheric Free. Nowadays this combination is not in use because only six satellites have implemented this latest combination P3.
2.6. Differential phase measurements
In order to achieve the greatest precision, the absolute positioning, that is affected by errors of different kind, has to be replaced with the relative positioning. The main point is that if two observations are affected by systematic errors, in the difference between them they will disappear. Differentiating the observations, the effects of the systematic errors will be minimized, but it won't be possible to determine the coordinates of the single points. The result will be the components of the vector that joins together the vertices on which two receivers are based at the same time. There are three differential methods of measurement.
