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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
Academic year 2016-2017
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
3.1 Rover stations ........................................................................................................................................ 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
1
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.
2
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.
3
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.
4
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”.
5
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.
Figure 1.3: Antonelli’s compass preserved at the "Compass Hall".
6
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 of not 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.
7
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.
8
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.
Figure 1.6: The third Antonellian project of 1860.
9
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.
10
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.
11
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 of arches. 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.
Figure 1.9: Antonelli’s “mechanism”.
12
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.
Figure 1.12: Walkway of the dome supported by granite corbels.
13
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
Basilica di San Gaudenzio in Novara in L’ingegneria civile e le arti industriali – Turin 1877.
15
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.
Figure 1.15: Drawing depicting Antonellian foundations dated June 1883.
16
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 sulphur action 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.
17
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.
18
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.
19
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.
20
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.
Figure 2.1: The Earth and GPS satellites.
21
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.
Figure 2.2: Geometric ellipsoid.
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.
23
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 − 𝑋𝑅)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.
24
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 − 𝑋𝑅)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 = 𝑃𝑅
𝑖 (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.
25
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.
Figure 2.5: The space component.
26
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.
Figure 2.6: The five monitoring stations of GPS system.
27
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.
28
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.
Figure 2.8: Combination of a sinusoidal with a binary code.
29
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 𝑡𝑅(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 𝑑𝑡𝑅(𝑡) represent the errors of the satellite and receiver's clocks.
The observation equation is so re-written in this way:
∆𝑇𝑅𝑆(𝑡) = 𝑡𝑅 + 𝑑𝑡𝑅(𝑡) − 𝑡𝑆 + 𝑑𝑡𝑆(𝑡) = 𝜏𝑅
𝑆 + 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 ∗ ∆𝑇𝑅
𝑆(𝑡) = 𝑐 ∗ 𝜏𝑅𝑆 + 𝑐 ∗ (𝑑𝑡𝑅(𝑡) − 𝑑𝑡𝑆(𝑡)) (2.15)
Received code
Internal repetition
Figure 2.9: Determination of the flight time ΔT.
30
where c ∙ τRS = dR
S 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( )(𝑡𝑅
𝑆 ) (2.18)
where
- ∅𝑅𝑆 (𝑡) observation of the phase difference
Figure 2.10: Measurement of phase difference.
Received carrier
Internal repetition
Phase difference
31
- ∅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 (dtR(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:
𝐿𝑅𝑆 (𝑡) = 𝜆 ∙ ∅𝑅
𝑆 (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 ∙ br ⁄ where δr represents the orbital error, b is
the base joining the two receivers and r is the radius of the orbit.
32
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
33
- 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.
34
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
35
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
36
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) = 𝑑𝑅
𝑆(𝑡) + 𝑐(𝛿𝑡𝑅(𝑡) − 𝛿𝑡𝑆(𝑡) + 𝛿𝑖𝑜𝑛 (𝑡) + 𝛿𝑡𝑟𝑜𝑝 (𝑡) + 𝛿𝜌(𝑡) + 𝛿휀(𝑡) (2.22)
Phase:
𝐿𝑅𝑆 (𝑡) = 𝑑𝑅
𝑆 (𝑡) + 𝑐(𝑑𝑡𝑅(𝑡) − 𝑑𝑡𝑆(𝑡)) + 𝜆 ∙ (∅𝑅 − ∅𝑆 + 𝑁𝑅 𝑆 (𝑡)) + 𝛿𝑖𝑜𝑛 (𝑡)
+𝛿𝑡𝑟𝑜𝑝 (𝑡) + 𝛿𝜌(𝑡) + 𝛿휀(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
- 𝛿휀 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) = 𝜌𝑅
𝑆(𝑡) + 𝑐(𝑑𝑡𝑅(𝑡) − 𝑑𝑡𝑆(𝑡)) + 𝑇𝑅𝑆(𝑡) + 𝐼1𝑅
𝑆 (𝑡) (2.24)
𝑃2𝑅𝑆 (𝑡) = 𝜌𝑅
𝑆(𝑡) + 𝑐(𝑑𝑡𝑅(𝑡) − 𝑑𝑡𝑆(𝑡)) + 𝑇𝑅𝑆(𝑡)+𝐼2𝑅
𝑆 (𝑡) (2.25)
𝐿1𝑅𝑆 (𝑡) = 𝜌𝑅
𝑆(𝑡) + 𝑐(𝑑𝑡𝑅(𝑡) − 𝑑𝑡𝑆(𝑡)) + 𝑇𝑅𝑆(𝑡) − 𝐼1𝑅
𝑆 (𝑡) + λ(𝑁1𝑅𝑆 (𝑡) + ∅1𝑅 − ∅1𝑆) (2.26)
Figure 2.15: Possible geometric configurations of the satellites
Bad satellite geometry Good satellite geometry
37
𝐿2𝑅𝑆 (𝑡) = 𝜌𝑅
𝑆(𝑡) + 𝑐(𝑑𝑡𝑅(𝑡) − 𝑑𝑡𝑆(𝑡)) + 𝑇𝑅𝑆(𝑡) − 𝐼2𝑅
𝑆 (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) = 𝛼1𝑃1𝑅
𝑆 (𝑡) + 𝛼2𝑃2𝑅𝑆 (𝑡) + 𝛽1𝐿1𝑅
𝑆 (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.
38
Single Phase Difference
Two receivers, R1 e R2 are taken into consideration. The two receivers must have observed the same
satellite S at the same time t. The 'single difference' is the difference between the observations of the
receiver placed in the unknown point and those of the receiver located in the point of reference.
LR1,R2S (t) = LR1
S (t) − LR2S (t) =
= dR1 S (t) − dR2
S (t) + c (dtR1(t) − dtR2(t)) − c (dtS(t) − dtS(t)) + λ ∙ (∅R1
− ∅R2 − ∅S + ∅S + NR1 S (t) − NR 2
S (t)) + ∆δion (t) + ∆δtrop (t)
+ ∆δρ(t) + ∆δε(t)
= dR1 S (t) − dR2
S (t) + c (dtR1(t) − dtR2(t)) + λ ∙ (∅R1 − ∅R2
+ NR1 S (t) − NR 2
S (t)) + ∆δion (t) + ∆δtrop (t) + ∆δρ(t) + ∆δε(t)
(2.29)
The single difference is a new observation where, the satellite clock error (𝑑𝑡𝑆(𝑡) − 𝑑𝑡𝑆(𝑡)) is
deleted along the orbital uncertainty (∆𝛿𝜌(𝑡)). Finally, if the length of the base is less than 30km,
also the tropospheric (∆𝛿𝑡𝑟𝑜𝑝 (𝑡)) and ionospheric (∆𝛿𝑖𝑜𝑛 (𝑡)) delays can be reduced.
The only problem is that the unknown factor linked to the receivers' clocks errors is not eliminated
(𝑑𝑡𝑅1(𝑡) − 𝑑𝑡𝑅2(𝑡)), so these single differences are not suitable for geodetic or typographical
purposes.
Starting from the two phase observations obtained considering a single receiver and two satellites, it
is possible to obtain another kind of differences where the receiver's clock error is eliminated.
Double phase-differences
Double differences are the difference of two single differences referred to the same pair of receivers
and to the same satellites S1 and S2 at the same time.
𝐿𝑅1,𝑅2𝑆1,𝑆2 (𝑡) = 𝐿𝑅1
𝑆1 (𝑡) − 𝐿𝑅2𝑆2 (𝑡) =
= 𝑑𝑅1 𝑆1 (𝑡) − 𝑑𝑅2
𝑆1 (𝑡) − 𝑑𝑅1 𝑆2 (𝑡) + 𝑑𝑅2
𝑆2 (𝑡) + 𝑐 (𝑑𝑡𝑅1(𝑡) − 𝑑𝑡𝑅2(𝑡)) − 𝑐 (𝑑𝑡𝑅1(𝑡)
− 𝑑𝑡𝑅2(𝑡)) + 𝜆 ∙ (∅𝑅1 − ∅𝑅2 − ∅𝑅1 + ∅𝑅2 + 𝑁𝑅1 𝑆1 (𝑡) − 𝑁𝑅2
𝑆1 (𝑡)
− 𝑁𝑅1 𝑆2 (𝑡) + 𝑁𝑅2
𝑆2 (𝑡)) + ∆′𝛿𝑖𝑜𝑛 (𝑡) + ∆′𝛿𝑡𝑟𝑜𝑝 (𝑡) + ∆′𝛿𝜌(𝑡) + ∆′𝛿휀(𝑡)
= 𝑑𝑅1 𝑆1 (𝑡) − 𝑑𝑅2
𝑆1 (𝑡) − 𝑑𝑅1 𝑆2 (𝑡) + 𝑑𝑅2
𝑆2 (𝑡) + 𝜆 ∙ (+ 𝑁𝑅1 𝑆1 (𝑡)
− 𝑁𝑅2 𝑆1 (𝑡) − 𝑁𝑅1
𝑆2 (𝑡) + 𝑁𝑅2 𝑆2 (𝑡)) + ∆′𝛿𝑖𝑜𝑛 (𝑡) + ∆′𝛿𝑡𝑟𝑜𝑝 (𝑡) + ∆′𝛿𝜌(𝑡)
+ ∆′𝛿휀(𝑡)
(2.30)
In these double differences the unknown factors linked to both the clocks are eliminated, the
remaining systematic errors (∆′𝛿𝑖𝑜𝑛 (𝑡), ∆′𝛿𝑡𝑟𝑜𝑝 (𝑡), ∆′𝛿𝜌(𝑡)) are also reduced in case the two
receivers are at a distance shorter that 20 km. It only remains the term of ambiguity (+ 𝑁𝑅1 𝑆1 (𝑡) −
𝑁𝑅2 𝑆1 (𝑡) − 𝑁𝑅1
𝑆2 (𝑡) + 𝑁𝑅2 𝑆2 (𝑡)) to which the so-called algorithms of estimation and of fixation may
be applied. It makes it possible to obtain a higher precision, of about some centimeters. But the
negative side is that the noise of the observed data increases more than in the case of the single
differences.
The double difference equation is widely used in GPS data processing and in the last 30 years these
became fundamental for the relative positioning.
39
Triple phase-differences
They are obtained by subtracting double phase differences at two different times t1 and t2. So, to the
double differences advantages, the elimination of the unknown factor caused by the integer phase
ambiguity can be added.
This kind of difference is not used in the calculation of geometrical parameters because of its noise,
while it is widely used in cycle slips detection.
2.7. Software for data processing
There are many softwares for the GPS data processing available on the market. These are classified
according to their aims into two categories: the scientific codes and the commercial codes. The
scientific softwares are used mainly for the geodetic network design, for the study of the Earth's crust
movements, and for the determination of point's coordinates. The commercial softwares are mainly
used for the traditional geodetic applications. Among the scientific softwares we have: the Bernese
that can work both with a differentiated approach and with the Precise Point Positioning (PPP) one,
the Gamit that is characterized by a differentiated approach, the Gipsy-Oasis that works according to
the PPP approach, goGPS and RTKLIB. Among the widely used commercial software we have: Leica
Geo Office (LGO), Trimble Geomatics Office (TGO), Trimble Total Control (TTC), Trimble
Business Center (TBC) e Magnet.
Scientific software
Firstly, we will introduce two software that use a differentiated approach.
The Bernese, developed by the Astronomical Institute, University of Berne, Switzerland (AIUB), is
characterized by power and flexibility in the analysis of GPS data and it is also the most commonly
used by many research institutions like EUREF which is in charge for the definition and realization
of the European Reference System ETRS89.
The Gamit (acronym for GPS At MIT) was developed by the MIT’s (Massachusetts Institute of
Technology) Department of Earth, Atmospheric and Planetary Sciences for the scientific analysis of
a network made up of a large number of stations heterogeneously distributed in space.
For what concerns the Gipsy-Oasis it is characterized by an undifferenced approach based on the
PPP. It tracks the position of the single receiver ex post, thanks to phase and code observations (that
are not differentiated), and the precise orbits, by analyzing a global network. In the PPP approach the
ambiguities and the clocks’ adjustments are to be considered as unknown factors and are calculated
thanks to appropriate statistical models for the correction. It is thus different from what happens in
the differentiated approach where thanks to the differentiation of the GPS observables, clocks errors,
other systematic errors and the integer ambiguities are eliminated.
Among the negative aspects we have to point out how point’s precision and accuracy depend on the
duration of the observation, that however does not mean that the PPP approach does not guarantee
the same or even higher precision than the differentiation-based analysis strategies
Among the open source scientific software, we can mention goGPS and RTKLIB.
As far as the navigation software goGPS is concerned, its development started in 2007 at the
Geomatics Laboratory of Politecnico of Milan, Como campus (Italy). Its early growth was overseen
by Dr. Mirko Reguzzoni, as a Master thesis project. In the years that followed, the project was carried
on by other two Master theses and one Ph.D. thesis. The software, born initially for kinematic
40
navigation, now is able to process raw GPS/GNSS data in differential mode with respect to a network
of permanent stations. Besides it elaborates both measures of code and measures of phase, adapting
itself to the data provided by both geodetic and low-cost receivers. In August 2009, the open source
software, goGPS code (MATLAB) was first made public under GPL license. Since then, different
teams in Italy (namely GReD and the Department of Civil and Environmental Engineering of
Politecnico di Milano) and in Japan take care about its ongoing development.
The RTKLIB navigation software (an open source program package for GNSS positioning) is used
for the standard and accurate positioning. There is also another software, the GLOBK (GLOBal
Kalman filter) that is a compensation software developed by Professor Thomas Herring of the MIT
Department of Atmospheric Sciences. This software uses the least squares approach and thus makes
it possible to obtain a precise estimate of the coordinates of each one of the stations taken into
consideration by merging observations coming from GNSS, VLBI and SLR.
Commercial Softwares
The Leica Geo Office (LGO) software estimates (with variable precision from few centimeters to a
decimeter) the GPS receivers' coordinates thanks to the use of double differences, thus whit the help
of data recorded by one of the stations of reference.
As for Trimble Navigation, it is a company that designs and produces technological products, like,
for example, GPS devices that highly increase works’ productivity. Among the softwares that have
been created it is worth to mention: Trimble Geomatics Office (TGO). It is a software that is used for
the elaboration of GPS baseline and for the compensation for the GPS's prominent network and
conventional data; Trimble Total Control (TTC); and Trimble Business Center (TBC).
Magnet software is one of the latest software developed by Topcon, a company that has a three-
business lines organization, two of these are called 'Positioning' and 'Smart Infrastructure', that are
focused on the engineering and developing of measuring and precision positioning systems.
2.8. GPS receivers
A GPS receiver is essentially made up of four parts: the antenna, the receiver, the computer, and input
or output devices. The basic functioning of a GPS receiver can be described as follows:
1. The receiver, in order to allow positioning, receives all the satellites seen above the cut-off angle.
2. It detects the signals coming from the satellites and it decodes them using its PRN5 code.
3. It decodes navigation data contained in the D code and it stores them in the memory.
4. It measures the time-delay in the reception of satellite's signals and it calculates the distances
between each satellite and the receiver.
5. It tracks the position of the receiver and GPS Time.
The GPS receiver's functions can be achieved in different ways, according to the goal for which the
receiver is used and to the level of precision that is required.
5 Each NAVSTAR satellite has its own and unique PRN code (pseudo-random number).
41
The receivers can be of three kinds:
Sequential receivers
They are characterized by the presence of a single channel that follows the four satellites that have
been selected, one by one, in order to determine its position for a period of time of 1-2 seconds. This
kind of receivers are pretty cheap but in the cases in which the speed of data elaboration represents a
key point they cannot be used because the time they take for the observation of the four satellites is
quite long (from 4 to 8 seconds).
Multi-channel receivers
These are characterized by the presence of many channels that follow the same number of satellites
at the same time. Each channel follows one satellite, it demodulates the signal and measures the
distance. Then all the information is combined in order to obtain the position and other kind of data
like speed and direction. Given the fact that these receivers are far more complex than the previously
mentioned, they are also more expensive.
Multiplex receivers
These receivers are characterized by the presence of a single channel and they follow sequentially the
satellites scanning completely them in less than 20 seconds. Thanks to these receivers the navigation
data are received steadily, without the interruptions typical of the sequential receivers, and are also
quite cheap.
Receivers can be further classified according to the measurements that they can make. The first two
kinds are the so-called low-cost receivers.
Figure 2.16: Functioning of a GPS receiver.
Antenna Signal
acquisition
Receiver
Data recovery
Measurement of distances
Display
Calculator
Keyboard for
entry data Initialize
Satellite
selection
Data
storage Calculation of the position
42
Code measurement receivers, also called "mass-market" (pseudoranges)
These receivers can receive only the C/A code. They are the widest spread on the market and because
of their size are also called handheld. Usually they have six channels, prices currently range from 50€
to 400€. They are mostly used for terrestrial navigation thanks to their computing power and to the
fact that the display of cartography is interactive, so the user can choose the routes.
They give coordinates on the display and usually do not allow the storage of raw data. They can
assure an absolute pseudo-range positioning with a planimetric accuracy of about 100 m in presence
of the Selective Availability6 (SA) and of about 10÷20m without the SA.
Some of the receivers of this category can obtain the differential correction with the RTCM7 protocol
and reach precision that is around one meter.
Some receivers can have a limited interface or do not have with a price of about 60€. These broadcast
via Bluetooth the NMEA8 message that gives the coordinates to another device of the antenna's
position.
Another kind of receivers are the GPS OEM9 cards, they cost about 20€. The disadvantage of these
cards is related to the fact that they have to be incorporated in a more complex device. Among these
receivers we shall focus on the devices developed by U-blox, like chips, software modules and
solutions. These make it possible to track your position and to communicate this information with
your voice, with a text or an image.
U-blox is a Swiss company specialized in the realization of semi-conductor products for the
positioning systems and for the wireless devices fundamental for industrial and automotive sectors.
U-blox provides various products in order to meet specific needs: standard level, professional level
6 The Selective Availability is an artificial interference of the satellite's signal designated for that purpose by the Department of Defence
of the USA, in order to make the Global Positioning System (GPS) available for citizens by means of the Standard Positioning System
(SPS), made available in 1991. 7 RTCM stands for Radio Technical Commission for Maritime Service. It is a USA governmental organization that has set up a special
commission in order to define data format and the structure of the GPS message. The RTCM protocol is, thus, the name of a
standardized data format for the exchange of data within a network of permanent GNSS stations. 8 the NMEA is a standard of data communication used for the communication of GPS satellite data. The body managing and developing
the protocol is the National Marine Electronic Association. 9 An original equipment manufacturer (OEM) is a supplier company which manufacture an equipment that is installed in a finished
product, on which the final manufacturer affixes its brand. Often, the firm that commercializes and brands the finished product is almost
always of great dimensions of the OEM firm, from which it acquires the components and to which entrust manufacturing processes.
Figure 2.17: “Mass-market” receivers.
43
and automotive level. The standard level consists in applications for a common use, the device is
subject to normal environmental conditions; the professional level consists in apps for industrial
installations, security systems and warning systems. In these cases, mechanical durability and tool
reliability are central factors. For what concerns the automotive level, it consists in applications
suitable for the automotive and transportation sectors. These have been engineered in order to resist
in extreme conditions.
Single frequency receivers
These receivers not only can receive the C/A code, but also the carrier L1. Usually they have twelve
channels with prices that range from 800€ (in the case of the OEM version) to 5000€ (in the case of
the GIS version and with the GLONASS reception). They are mainly used for the tracking and
detailed real-time measurements, data generation and data update for the mapmaking and the
Territorial Information System. The functions are more or less the same of the receiver above
mentioned, the only difference is that this one gives the possibility to store C/A code data and L1
phase data in an internal or external storage. It makes it possible a relative or differential positioning,
with code or phase measurements. This kind of receivers are used in the case of both static or dynamic
positioning, but it has to be taken into consideration the fact that as these receivers use a single
frequency their use is limited to bases with a length shorter than 20km. Some of these receivers can
be inserted into computers that make it possible for the user to have access to the GIS functions and
to display satellite images.
Figure 2.18: Single frequency receivers.
44
Dual frequency receivers P code
These receivers are more comprehensive and can receive every part of the GPS signal (L1, L2, C/A,
P). They are mainly used for topographic and detailed surveying. In case of absence of the Anti
Spoofing10 (AS), the P code is directly available. With these receivers the positioning with (relative
or differential) phase or code measurement is possible. They can be used both for the static or dynamic
tracking, without any limit on the base length.
Dual frequency receivers Y code.
Thanks to these receivers it is possible obtain the P code even if in presence of the AS, through the
use of an auxiliary chip, Auxiliary Output Chip (AOC), that is to be placed on every receiver's
channel. The AOC was also used for the correction of the SA effect by certain authorized users of the
USA Department of Defense.
For what concerns the output data, the GPS receiver can give the user coordinates (real time data) or
raw data (code or phase data for the post processing). For the real-time data, the format used is the
international protocol NMEA, while for the code or phase data, stored in the receiver's memory during
the data acquisition, the format in which they are downloaded from the memory depends on the
receiver's company. These files must be turned into another format so that it could be used by any
data processing software. This open format is called RINEX (Receiver Independent Exchange
Format). The RINEX is the ASCII format (American Standard Code for Information Interchange), a
standard format that is used for files coming from different brands of GPS receivers. It is made up of
three different kinds of files: observation files, navigation files, and meteorological files.
10 The Anti Spoofing is an encryption of the P code. When the P code is encrypted, it is transposed in a Y code. The loss of accuracy
caused by the AS in dual frequencies, is in part caused by the inability to determine the ionospheric delay in real time. The AS is no
longer in service, since 2000.
Figure 2.19: Geodetic receivers.
45
3. Equipment used in the tests and in the St. Gaudenzio survey
With the aim of inspecting St. Gaudenzio spire, three rover stations were considered. Moreover,
during the test phase, using double differences to process the data, three reference stations were taken
into account, namely the one of Milan, Monza and Pavia. Performing the study at St. Gaudenzio, it
was made reference to the station of Novara.
As for topographical survey and scans in St. Gaudenzio, a total station and a GPS receiver were
employed.
3.1 Rover stations
As for rover stations, they were used low-cost devices, namely three antennas and three equal
receivers. As far as the antennas are concerned, the “TW3740” and “TW3742”, precision high gain
GNSS devices, and “ANN-MS active GPS”, an antenna with lower performances were used. Dealing
with their receivers, three “EVK-5T” were employed, characterized by their compactness and user-
friendly interface.
As for the “TW3740” and the “TW3742”, with a cost of approximately 160€, they were launched on
the market by Tallysman. These antennas acquire signal from the BeiDou B1, Galileo E1, GPS L1,
GLONASS L1 and SBAS (WAAS, EGNOS, QZSS & MSAS) frequency band (1557 to 1606 MHz).
The “TW3740” and “TW3742” antennas made the use of Tallysman’s unique Accutenna®
technology, with the result of a superior multipath signal rejection. Moreover, “TW3740” is
characterized by three stage Low Noise Amplifier and “TW3742” is provided by an extra strong
protection thanks to an additional pre-filter. Eventually, both “TW3740” and “TW3742” are placed
in a permanent support, made of a metal base with two nickel coated nuts and a weatherproof
cladding.
FEATURES BENEFITS
Low noise LNA: 1 dB Circular polarization all over the entire
bandwidth
High rejection SAW filter High-quality multipath signal rejection
High gain LNA: 40 dB typ Outstanding signal to noise ratio
Low current: 19 mA typ Significant out of band signal rejection
Table 3.1: TW3740/TW3742 antenna benefits and features.
Figure 3.1: “TW3740/TW3742” antenna.
46
As for “ANN-MS active GPS”, it is an antenna produced by U-blox with a cost around 50€. It has
the powerful advantage of being compact and easy to use. Besides, among its features, it worthwhile
noticing that it has an incorporated low noise amplifier with 0.9 dB noise figure and 29 dB gain and,
it boasts of a broad range of supply voltage. Finally, as well as “TW3740” and “TW3742” antennas,
its operating temperature range is between -40°C and +85°C.
As previously mentioned, it was thought to use three equal low-cost receivers, i.e. “EVK-5T”
Precision Timing Evaluation Kit, having a cost of 150€, being characterized by their ease-of-use and
compactness.
3.2. Reference stations
As far as the reference stations are concerned, the RINEX data were provided by two different
referencing services. Indeed, for Milan, Pavia and Novara it was made the use of Servizio di
Posizionamento Interregionale GNSS Piemonte - Lombardia (SPIN GNSS) and for Monza it was
made reference to SmartNet ItalPoS network.
The Servizio di Posizionamento Interregionale GNSS Piemonte – Lombardia is a network that
consists of 30 GNSS permanent stations spread evenly in Piedmont and Lombardy. These stations
are equipped with multi-constellation geodetic receivers open to the use of GPS and GLONASS
constellations and each receiver is connected to an antenna GNSS calibrated individually. This
referencing service is geo-referenced in the national geodetic system ETRF2000-RDN.
Dealing with the SmartNet ItalPoS, it is the first network of GNSS permanent stations with national
coverage that provides real time and post processing positioning services. As the Servizio di
Posizionamento Interregionale GNSS Piemonte – Lombardia, SmarNet Italpos, it is geo-referenced
in the national geodetic system ETRF2000-RDN.
Figure 3.2: “ANN-MS active GPS” by u-blox.
Figure 3.3: “EVK-5T” reciver.by u-blox.
47
3.2.1 Milan
The reference station of Milan is located at the Politecnico of Milan, on the roof of the Building 14
(historic name: Nave) located in via Bonardi 9. It is equipped with “TPSCR3_GGD CONE” antenna
linked to the receiver, “TPS NET-G3”, located in a metallic box, on the roof of the same building.
3.2.2 Monza
As far as the reference station of Monza is concerned, it is positioned in via Enrico Arosio 3 and it is
provided by Leica devices, that is “Leica AS10” antenna and “Leica GRX1200+GNSS (GPS+GLO)”
receiver.
Figure 3.4: MILA antenna. Figure 3.5: MILA receiver.
Figure 3.6: Monza antenna. Figure 3.7: Monza receiver.
48
3.2.3. Pavia
Dealing with the reference station of Pavia, it is located at the University of Pavia, in via Ferrata 1
and it is equipped with “TPSCR3_GGD CONE” antenna, same model used in Milan, connected to
the receiver, “TPS NET-G5 (GPS+GLO)”.
3.2.3. Novara
Concerning the reference station of Novara, it is positioned at Largo don Giovanni Minzoni 1, less
than 1km away from the Basilica of St. Gaudenzio. The reference station is provided by “Leica
AR25.R3 LEIT” antenna and “LEICA GRX1200+GNSS (GPS+GLO)” receiver, model previously
mentioned for the reference station of Monza.
Figure 3.8: Pavia antenna. Figure 3.9: Pavia receiver.
Figure 3.10: Novara antenna. Figure 3.11: Novara receiver.
49
3.3. Terrestrial topographic survey at St. Gaudenzio
As for topographical survey and scans at St. Gaudenzio, a total station and a GPS receiver were used.
As far as the total station is concerned, the Leica Nova MS60 MultiStation, a superior instrument that
combines all available measurement technologies, was used. As regards the GPS receiver, Leica Viva
GS14 was used equipped with an easy-to-use and powerful GNSS smart antenna.
Looking at Leica Nova MS60 MultiStation, with the use of Captive software, it turns complex data
in workable three-dimensional models, while the information processing is carried out by Leica
Infinity. In addition, the scan, having accuracy of a millimeter, consists in the instantaneous creation
of point clouds and 3D models in one view.
Dealing with Leica Viva GS14, with its integrated design, it tracks the data in field thanks to Leica
SmartWorx Viva software, characterized by its convenience and clarity. To process the information,
also this device utilizes Leica Infinity.
Figure 3.12: Leica Nova MS60 MultiStation.
Figure 3.13: Leica Viva GS14.
50
4. Tests in a controlled scenario
4.1. Testing the precision of low cost receivers
Before starting the analysis of St. Gaudenzio spire, with the use of double differences, it was decided
to test the resolution of the equipment and a fifteen-day data acquisition was made, from 8 to 22
November 2017.
As far as the test is concerned, bearing in mind the study that will be held in Novara, and therefore,
that a short distance between the reference and rover stations will be present, it was thought to
consider short bases, that is, with inter-distances not exceeding 20km. Accordingly, three different
cases study were taken into account, using the reference stations of Milan, Monza and Pavia. It should
be noted that Pavia, 32km away from Milan, wasn’t a short base but it was considered in order to
have a better understanding of the possible problems occurring when longer distances are considered.
As regards the analysis, bearing in mind the similarity of “TW3740” and “TW3742” antennas, it was
decided to test just one of the two, the “TW3742”. Thus, two rover stations were installed in Milan,
namely the ANN-MS active GPS and the one previously mentioned. Then, with the aim of finding
their positions, the reference stations of Milan, Monza and Pavia were considered at subsequent times.
The two rover antennas were installed at the Politecnico of Milan, on the roof of the Building 14
(historic name: Nave) located in via Bonardi 9. Thereafter, by means of a coaxial cable, the antennas
were connected to their respective receivers, located inside a metallic box on the roof of the same
building. In the following figure, it is shown the position of the rover stations.
) )
20m
Figure 4.1: Politecnico of Milan, building 14. Instrumentation used for the testing phase a) ANN-MS active
GPS b) TW3742.
51
This solution for the positioning of the antennas, located as shown in the figure a) and b), resulted
particularly convenient for different reasons. Firstly, the height of the building allowed avoiding
different obstacles surrounding the place of acquisition, minimizing reflection phenomena
(multipath) and allowing an unobstructed view of the sky. Secondly, the designated posts were easily
accessible though the stairs inside the building, which allowed an easy intervention in case of
malfunctioning.
U-blox receivers were activated, thanks to the connection with a pc. This laptop had no special
features, but for its stability during the data acquisition and processing. Moreover, using a laptop, in
order to maintain constant the power, there was no need to have an extra power supply (UPS), since
the pc, when needed, could rely on its internal battery.
The data acquired from the receivers, with a sampling interval of 1 second, were stored in the memory
of the pc, in a unique proprietary file (ubx) respectively. Subsequently, the ubx files were
automatically converted in daily RINEX files and thanks to the use of teqc software they were divided
in hourly files.
Afterward, the available RINEX files of the reference and rover stations were processed with Leica
Geo Office software, estimating the coordinates of the rover stations.
After the acquisition of the data, having hourly files for a period of fifteen days, it was decided to
process data in sessions of twenty-four hours, twelve hours, eight hours and one hour.
In order to have clearer visualization of the results, solutions were referred to their mean. For each
set of observations, the sample mean was computed using the well-known formula listed in the
following
𝑚 =
1
𝑁 ∑ 𝑥𝑖
𝑁
𝑖=1
(4.1)
Figure 1: Politecnico of Milan, building 14. Instrumentation used for the testing phase
a) ANN-MS active GPS b) TW3742.
Figure 4.2: Laptop used for the data acquisition and processing.
52
where xi are the N sampled values.
Also, to have a comprehensive statistical evaluation of the data variability, for every kind of solution,
it was calculated the sample variance.
𝑆̅2 =
1
𝑁 − 1∑(𝑥𝑖 − 𝑚)2
𝑁
𝑖=1
(4.2)
Where xi are the observations, m represents the sample mean and N. is the number of sampled values.
Calculating the square root of the sample variance, the (sample) standard deviations of the different
solutions was obtained.
Finally, in order to remove outliers in the data, we rely on Tchebycheff theorem. This theorem, linking
the average (μ) and the variance (σ), enable to define anomalous observations. More in detail,
Tchebycheff theorem states that for any random variable 𝑥, which has finite variance, the following
inequality holds:
𝑃[|𝑥 − 𝜇𝑋| ≤ 𝜆𝜎𝑋] ≥ 1 − 1 𝜆2⁄ ∀ 𝜆 > 1 (4.3)
As said, this theorem allows eliminating anomalous observations. In fact, considering e.g. 𝜆 = 3, the
probability to find values of the random variable within the interval |𝑥 − 𝜇𝑋| ≤ 3𝜎𝑋 is at least of
89%. Indeed, it is possible to find external values to this range, but this is unlikely, and so, it is
reasonable to neglect them.
Bearing in mind the case study of Novara, where point variations of few millimeters are expected,
standard deviations from some tenth of millimeter to few millimeters will be considered acceptable.
53
As a preliminary analysis, the reference station of Milan was considered, located on the roof of the
Building 14, at a very short distance (around 0,02km) from the rovers. Thereafter, for a more realistic
setting, the reference station of Monza was considered. Finally, with the intention of understanding
how the length base could influence the positioning resolution, we also took into account the reference
station of Pavia.
Figure 4.3: Reference and rover stations.
Reference
Rover
54
4.1.1. Milan
As stated above, the reference station of Milan, named MILA, was firstly considered. The following
graphs illustrate the consecutive observations for a duration of twenty-four hours, where the trends
of the three coordinates (East, North, Height) are shown overlapped, so that an easier identification
of their evolution was possible. The results of the “TW3742” antenna are firstly shown.
As it was predictable, due to the long duration of the observations and to the very short baseline, the
coordinates oscillate with respect to reference within a range of about 0,8mm. Furthermore, both
North and East coordinates reveal a high; as for the peak of North coordinates, it is in correspondence
of the first day of acquisition and it can be related to the shorter duration of the observation session.
Dealing with East coordinates, a peak of -0,5mm was found on November 20.
-0,5
0,6
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21/11/2017
22/11/2017
-0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 [mm]
Time
ΔE
ΔN
Δh
Figure 4.4: “TW3742” antenna, evolution of the coordinates for consecutive
observations with a duration of twenty-four hours, Milan.
55
As far as the results of “ANN-MS active GPS” antenna are concerned, they were corrupted by the
presence on the Nave roof of telephone antennas. Indeed, even if they didn’t affect “TW3742”
antenna results, it was just due to its high-quality shield. A closer look to the results reveals that for
five times, over fifteen days, the integer ambiguities weren’t fixed and thus low accuracy values were
found. In the following graph, float solutions are highlighted by orange rectangles 11.
As for the resolution test of this device, it was made reference to calculations performed before the
installation of telephone antennas. More in depth, specular analysis considering this rover station and
the reference station of Milan were made, from the 9 to the 18 February 2014.
11 The position solution is called fixed when the processing software is able to estimate the integer values entering in the double
difference formula (see chapter 2). If this is not, i.e. only rational values are given for these integers, the solution is called float.
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21/11/2017
22/11/2017
-1000,0 -800,0 -600,0 -400,0 -200,0 0,0 200,0 400,0 600,0 [mm]
Time
ΔE
ΔN
Δh
Figure 4.5: “ANN-MS active GPS” antenna, evolution of the coordinates for consecutive observations
with a duration of twenty-four hours, from the 8 to the 22 November 2017, Milan.
56
The following graph, is reported from “ANALISI PER LA DEFINIZIONE DELLA PRECISIONE
DI UN RICEVITORE GPS U-BLOX”, dissertation that regarded the performances examination of
the low-cost antenna “ANN-MS active GPS”, used also in this study. The results shown in the
following plot concern the coordinate estimated with sessions of twenty-four hours.
As it is illustrated from the analysis, the coordinates oscillate with respect to reference within a range
of 1mm, which is reasonable considering that for “TW3742” antenna they oscillate in around 0,8mm.
Based on these results, considering the low screen provided by “ANN-MS active GPS” and the
importance of preserving the precision of the acquired data, it was decided to not consider this antenna
for the study of St. Gaudenzio spire.
08/02/2014
09/02/2014
10/02/2014
11/02/2014
12/02/2014
13/02/2014
14/02/2014
15/02/2014
16/02/2014
17/02/2014
18/02/2014
19/02/2014
-2,0 -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,0 [mm]
Time
ΔE
ΔN
Δh
Figure 4.6: “ANN-MS active GPS” antenna, evolution of the coordinates for consecutive
observations with a duration of twenty-four hours, from the 9 to the 18 February 2014, Milan.
57
Accordingly, the study was continued with the results provided by “TW3742” antenna. As for the
sessions of twelve hours, it is clear how the solutions variability of the three coordinates start to
increase, resulting in a reduction of precision.
As already anticipated, the three coordinates, for sessions of twelve hours, extend their oscillations
with respect to the mean, amplifying their range between 1mm and -1mm. As far as the solutions
instability is concerned, in the first acquisition of 8 November, two peaks are present, accounting for
2,5mm and 3mm, for North and Height coordinates respectively. A more detailed look at the two high
show that they can be clearly related to the day of acquisition; in fact, the first session covered just 1
hour and 16 minutes instead of twelve hours, causing irregularities in the results. Finally, it is
noticeable the larger instability of the Height coordinates due to the geometry of observation.
2,5 3,0
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-2,0 -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 [mm]
Time
ΔE
ΔN
Δh
Figure 4.7: Evolution of the coordinates for consecutive observations with a duration of twelve hours, Milan.
58
Examining the coordinates for consecutive observations of eight hours, for each day three solutions
are present and, despite Height coordinates that strongly accentuate their instability, the East and
North coordinates show a small increase in their oscillations that range between 1mm and -1mm.
As it was foreseeable, shortening the duration of the consecutive observation leads to a decrease in
the precision but, even if the variability of the results rises, thanks to the high performance of
“TW3742”, the precision in the positioning remains high.
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21/11/2017
22/11/2017
-3,0 -2,0 -1,0 0,0 1,0 2,0 3,0 4,0 5,0 [mm]
Time
ΔE
ΔN
Δh
Figure 4.8: Evolution of the coordinates for consecutive observations with a duration of eight hours, Milan.
59
Finally; dealing with the three coordinates estimated on hourly sessions, the number of the solutions
increases up to twenty-four in a day and, for that reason, it was decided to study just three days,
namely 9, 14 and 21 of November. As expected, the trends previously depicted are now exacerbated,
i.e. Eight and North coordinates extend their coordinate oscillations between 4mm and -6mm
overtaken by Height coordinates that also in this case reveals the largest instability.
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21/11/2017
-14,0 -12,0 -10,0 -8,0 -6,0 -4,0 -2,0 0,0 2,0 4,0 6,0 8,0 10,0 [mm]
Time
ΔE
ΔN
Δh
Figure 4.9: Evolution of the coordinates for consecutive observations with a duration of one hour,
Milan.
14/11/20
17
60
As outlined before, following the evaluation of the results, for each kind of solution the sample
standard deviation was estimated, reporting its values by means of the following table and graph.
Looking at the results, one can notice that the standard deviation of the points depends on the solution
type. In fact, as it is well known, shortening the session length implies an increase in the standard
deviation values.
Bearing in mind the application to the case study of Novara, where point variations of few millimeters
are expected, one can define the session length that should be used. Finally, in this test, another issue
worthwhile considering is the inter-distance between the antennas. In fact, in this preliminary study
held in Milan, the inter-distance accounts for approximately 0,02km, a smaller base in comparison to
the one between St. Gaudenzio spire and the reference station of Novara. As a result, the performance
of the device appears maximized and as it was foreseeable, consecutive observations up to eight hours
could be considered.
STANDARD DEVIATION [mm]
Solution type σΔE σΔN σΔh
1h 2,18 2,54 5,35
8h 0,54 0,62 2,18
12h 0,33 0,71 1,20
24h 0,23 0,25 0,22
Figure 4.10: Standard deviations of the coordinates in relation with the
type of observation, Milan.
Table 4.1: Standard deviations of the coordinates in relation with the
different type of observations, Milan.
0,0
1,0
2,0
3,0
4,0
5,0
6,0
1h 8h 12h 24h
Time
σΔE
σΔN
σΔh
[mm]
61
4.1.2. Monza
Secondly, the reference station of Monza was taken into account. In this case, the distance between
reference and rover station was approximately of 11km and thus greater in comparison to the case
study of Novara, accounting for 0,6km. The aim of this analysis was studying a less favorable
situation. In fact, if variances of tenths of millimeter were found the corresponding observations could
be undoubtedly used for Novara.
Continuing the analysis with “TW3742” antenna, the results related to sessions of twenty-four hours
are shown in the following graph.
The three coordinates oscillate with respect to their average between 2mm and -2mm. Here is already
noticeable the increase in the range of oscillation due to the longer inter-distance between the stations;
passing from 0,8mm in the correspondent observations of Milan to 4mm. The coordinate with higher
instability remains the Height followed by North coordinates, showing four and two peaks
-3,5
3,9
3,2
-3,6
-3,6
3,9
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20/11/2017
21/11/2017
22/11/2017
-4,0 -2,0 0,0 2,0 4,0 6,0 [mm]
Time
ΔE
ΔN
Δh
Figure 4.11: Evolution of the coordinates for consecutive observations with a duration of twenty-
four hours, Monza.
62
respectively. On the contrary, East coordinates reveal a stable evolution ranging between 1mm and -
1mm.
Afterwards, the calculations related to sessions of twelve hours were performed. At a first sight is
visible how halving the duration of observations the data variability starts to increase.
The three coordinates, for the consecutive observations of twelve hours, amplified their range of
oscillation between 3mm and -3mm. Dealing with their trends the previous statements are confirmed,
with the higher stability in East coordinates followed by North and Height coordinates. As far as
peaks are concerned, as already found in Milan, the first acquisition of the 8 November reveals one
peak in the Height coordinates. This once again could be caused by the lower duration of the
observation.
21,1
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-9,0 -6,0 -3,0 0,0 3,0 6,0 9,0 12,0 15,0 18,0 21,0 24,0 [mm]
Time
ΔE
ΔN
Δh
Figure 4.12: Evolution of the coordinates for consecutive observations with a duration of twelve hours, Monza.
63
As for the sessions of eight hours, East and North coordinates increase their oscillation within a range
of 8mm, a far larger amount in comparison to 2mm found when using the Milan reference station.
Considering Height coordinates they strongly emphasize their instability, ranging approximately
between 6mm and -6mm.
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-8,0 -6,0 -4,0 -2,0 0,0 2,0 4,0 6,0 8,0 10,0 12,0 [mm]
Time
ΔE
ΔN
Δh
Figure 4.13: Evolution of the coordinates for consecutive observations with a duration of eight hours, Monza.
64
Finally; dealing with the three coordinates for sessions of one hour, recalling the study performed in
Milan, specular analyses were carried out and therefore among the fifteen days of analyses, three of
them were chosen. As far as the coordinates oscillations are concerned, they range between 4mm and
-12mm, determining a very low accuracy in the positioning of the rover station.
09/11/2017
15/11/2017
21/11/2017
-28,0 -24,0 -20,0 -16,0 -12,0 -8,0 -4,0 0,0 4,0 8,0 12,0 16,0 20,0 24,0 [mm]
Time
ΔE
ΔN
Δh
Figure 4.14: Evolution of the coordinates for consecutive observations with a duration of one hour, Monza.
14/11/20
17
65
As for the standard deviation, all the values show an increase mainly caused by the larger distance
between the reference station of Monza and the rover located in Milan. Thanks to this analysis it was
found that in the case study of Novara sessions of twenty-four hours should be used. In fact, although
the variance of North coordinates exceeded one millimeter, it can be considered acceptable, on the
grounds that in Novara the inter-distance between St. Gaudenzio spire and the reference station is
about 0,6km, and not 11km as in the case of the Monza reference station.
STANDARD DEVIATION [mm]
Solution type σΔE σΔN σΔh
1h 3,37 5,48 8,02
8h 0,85 2,47 3,72
12h 0,93 2,31 4,90
24h 0,59 1,64 2,03
0,0
1,0
2,0
3,0
4,0
5,0
6,0
7,0
8,0
9,0
1h 8h 12h 24hTime
σΔE
σΔN
σΔh
Figure 4.15: Standard deviations of the coordinates in relation with the
type of observation, Monza.
Table 4.2: Standard deviations of the coordinates in relation with the
different type of observations, Monza.
[mm]
66
4.1.3. Pavia
Finally, the reference station of Pavia was considered. As anticipated, this was not the case of a short
base; in fact, the distance between reference and rover station was in around 32km, a far greater length
in comparison to the cases study of Milano and Monza. The following graph illustrates the results for
sessions of twenty-four hours.
As far as coordinates oscillation are concerned, they range between 5mm and -5mm, figures almost
doubled if compared to the corresponding observations of Monza, where the oscillations account for
4mm, instead of 10mm. As for the stability of the data, it is confirmed that East coordinates appears
the more stable, followed by North and Height coordinates.
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12/11/2017
13/11/2017
14/11/2017
15/11/2017
16/11/2017
17/11/2017
18/11/2017
19/11/2017
20/11/2017
21/11/2017
22/11/2017
23/11/2017
-20,0 -15,0 -10,0 -5,0 0,0 5,0 10,0 15,0 [mm]
Time
ΔE
ΔN
Δh
Figure 4.16: Evolution of the coordinates for consecutive observations with a duration of twenty-
four hours, Pavia.
67
Next, looking at the results based on sessions of twelve hours, also in this case, it is noticeable an
increase in the data variability. In fact, the three coordinates range between 10mm and -10mm,
showing a doubling in the oscillations when compared to sessions of twenty-four hours.
08/11/2017
09/11/2017
10/11/2017
11/11/2017
12/11/2017
13/11/2017
14/11/2017
15/11/2017
16/11/2017
17/11/2017
18/11/2017
19/11/2017
20/11/2017
21/11/2017
22/11/2017
-40,0 -30,0 -20,0 -10,0 0,0 10,0 20,0 [mm]
Time
ΔE
ΔN
Δh
Figure 4.17: Evolution of the coordinates for consecutive observations with a duration of twelve hours, Pavia.
68
Dealing with sessions of eight hours, despite Height coordinates rise significantly in their instability,
East and North coordinates show a tiny increase. In fact, although North coordinates presents a higher
number of peaks, the oscillations with respect to the reference mean remain stable, with a superior
and inferior threshold of 10mm and -10mm respectively.
08/11/2017
09/11/2017
10/11/2017
11/11/2017
12/11/2017
13/11/2017
14/11/2017
15/11/2017
16/11/2017
17/11/2017
18/11/2017
19/11/2017
20/11/2017
21/11/2017
22/11/2017
-25,0 -20,0 -15,0 -10,0 -5,0 0,0 5,0 10,0 15,0 20,0 25,0 30,0 [mm]
Time
ΔE
ΔN
Δh
Figure 4.18: Evolution of the coordinates for consecutive observations with a duration of eight hours, Pavia.
69
Finally, as far as the hourly solutions are concerned, as expected, their variability increased
considerably, showing coordinates oscillations ranging within 60mm. As for the solutions instability,
two considerable peaks are present in the second and third acquisition of November 9.
09/11/2017
15/11/2017
21/11/2017
-105 -90 -75 -60 -45 -30 -15 0 15 30 45 60 75 90 105 [mm]
Time
ΔE
ΔN
Δh
Figure 4.19: Evolution of the coordinates for consecutive observations with a duration of one hour, Pavia.
14/11/20
17
70
Furthermore, performing these analyses three outliers were found the 21 November, probably due to
the short duration of the observations. In the above graph, the outliers accounting for -481,4mm
-127,7mm and 187,7mm for East, North and Height coordinates respectively were removed.
As for previous cases studies, after the evaluation of the results, for each kind of solution the standard
deviation was computed. Once again, the large distance between the reference station of Pavia and
the rover located in Milan is reflected in the results. Indeed, standard deviations of some millimeters
were already found for the sessions of twenty-four hours. In short, it can be concluded that with the
equipment used and the expected movements of some millimeters to be estimated in St. Gaudenzio,
such a large base could not be considered.
STANDARD DEVIATION [mm]
Solution type σΔE σΔN σΔh
1h 57,80 22,41 30,03
8h 2,91 5,88 9,92
12h 2,30 5,14 9,83
24h 1,73 4,28 6,29
0,0
10,0
20,0
30,0
40,0
50,0
60,0
70,0
1h 8h 12h 24hTime
σΔE
σΔN
σΔh
Figure 4.20: Standard deviations of the coordinates in relation with the
type of observation, Pavia.
Table 4.3: Standard deviations of the coordinates in relation with the
different type of observations, Pavia.
[mm]
71
4.2. Performance comparison between high quality and low-cost devices
With the aim of detecting the differences in the stability of high quality and low-cost stations, further
analyses were carried out on the same time period, between 8 and 22 November 2017. More in depth,
the stations of Monza and Pavia were considered as reference and the reference station of Milan,
named Mila, as rover. In this way, taking into account that “TW3742” station, named COM16, was
just 0,02km away from Mila, it was possible to understand the effectiveness of a low-cost station. In
fact, for both, Monza and Pavia, the results were compared to the respective outcomes found with
“TW3742” station as rover station. Finally, as far as the sessions are concerned, twenty-four, twelve
and eight hours sessions were analyzed.
4.2.1. Monza
As for the comparison performed with the reference station of Monza, named Monz, the results are
illustrated in the following graph, where the standard deviations are plotted.
Overall, as it was predictable, being the inter-distances between the reference and the rover stations
of just 11km, the advantages of using a high-quality station were low. In fact, although the standard
deviation of Height and East coordinates seems to be more stable, on the whole the performance of
COM16 and Mila are quite similar. More in detail, using Mila station, the standard deviations of
Height coordinates decreased for eight and twelve-hour sessions, from 3,54mm and 2,88mm
respectively, instead of 3,72mm and 4,90mm when using COM16. Likewise, for East coordinates
standard deviations, where the values underwent two small decreases when considering sessions of
twelve and twenty-four hours.
MONZ -
COM16
MONZ -
MILA
MONZ -
COM16
MONZ -
MILA
MONZ -
COM16
MONZ -
MILA
σΔE 0,85 0,98 0,93 0,87 0,59 0,52
σΔN 2,47 3,14 2,31 2,69 1,64 2,29
σΔh 3,72 3,54 4,90 2,88 2,03 2,17
0,0
0,5
1,0
1,5
2,0
2,5
3,0
3,5
4,0
4,5
5,0
[mm]
Figure 4.21: Comparison between low-cost and high-quality stations in Milano Nave, with Monza as reference.
8h 12h 24h
72
4.2.2. Pavia
Looking at the comparison performed with the reference station of Pavia, named Pavi, 32km away
from Milan, the advantages of using a high-quality station were quite clear. The results are shown in
the following graph in terms of standard deviations.
As it was predictable, by increasing the baselines, the benefits of high quality station appeared in all
the consecutive observations. In fact, the standard deviations of East and North coordinates
diminished noticeably, passing from values in around some millimeters to approximately one
millimeter.
In conclusion, it can be inferred that in the case of short baselines the use of low-cost receivers should
be preferred because they are cheaper and give the same precision level. On the other hand, when
having long baselines, i.e. more than 20km, the use of high quality antenna/receiver should be adopted
in order to have high precision in the observed coordinates.
PAVI -
COM16
PAVI -
MILA
PAVI -
COM16
PAVI -
MILA
PAVI -
COM16
PAVI -
MILA
σΔE 2,91 1,74 2,30 1,59 1,73 1,38
σΔN 5,88 1,45 5,14 1,21 4,28 0,85
σΔh 9,92 9,21 9,83 7,77 6,29 6,28
0,0
1,0
2,0
3,0
4,0
5,0
6,0
7,0
8,0
9,0
10,0
11,0
[mm]8h 12h 24h
Figure 4.22: Comparison between low-cost and high-quality stations, with Pavia as reference.
73
5. Testing the of St. Gaudenzio spire
5.1. The installed device
In order to proceed with the monitoring phase, on January 3, 2018, “TW3742” and “TW3740”
antennas and u-block receivers were installed on St. Gaudenzio spire. The two antennas are shown in
the following pictures.
As illustrated above, with the aim of respecting Antonelli’s architecture, made of white granite, it was
decided to fix the antennas though two metallic supports. Therefore, the two bases where fixed in
presence of concrete, which is in correspondence of the subsequent work of consolidation realized by
Ing. Danusso. Accordingly, the metalworking company Caccia Snc was commissioned to design an
antenna support allowing the best possible antenna deployment, in order to minimize the effects of
multipath.
For the realization of the antenna support, particular importance was given to the choice of its
material, since this should have a very low thermal expansion coefficient, so not to add thermal noise
to the collected data. The physical and mechanical characteristics of the metal used are shown below.
Stainless steel – AISI 304 X5CrNi 18-10
Young’s module E 196000 N/mm2
Ultimate tensile strength Rmax 515 N/mm2
Specific weight psp 7,91 kg/dm3
Thermal expansion coefficient λ 0,0165 mm/m/°C
Specific heat csp 0,12 kcal/kg
Electrical resistance Ω 0,714 ohm/mm2 *m
Thermal conductivity k 12,9 kcal/m*°C
Table 5.1: AISI 304 X5CrNi 18-10, physical and mechanical. characteristics.
Figure 5.1: “TW3742” antenna, East positioned. Figure 5.2: “TW3740” antenna, West positioned.
74
For our application, we considered, during the whole year, a possible variation of temperature equal
to ∆𝑇 = 50°𝐶, and so, considering the geometrical dimension of the antenna support, 𝑙0 = 0,65𝑚,
the maximum elongation will be equal to ∆𝑙 = 0,536𝑚𝑚, that can be considered acceptable, on the
grounds that the expected oscillation of St. Gaudenzio spire are a few millimeters.
∆𝑙 = 𝜆 ∗ 𝑙0 ∗ ∆𝑇 = 0,536𝑚𝑚 (5.1)
The antenna support plot is represented in Appendix A.
Moreover, for the positioning of the antennas, the trajectory of GPS satellites was considered. The
following picture illustrates the sky plot of Novara over the twenty-four hours, i.e. all the visible
satellite in the given position of the Novara reference station.
As a result, with the aim of being barely affected by the lower coverage of the North quadrant, it was
decided to locate “TW3742” antenna towards East and “TW3740” towards West. In the following
pictures the sky plots of the two antennas are shown taking into account the existing obstruction of
St. Gaudenzio spire. Performing a qualitative analysis of satellites distribution, the sky plot of
“TW3740” is more uniform in comparison to “TW3742”. In fact, the latter is characterized by a lower
coverage of the satellite paths towards South.
Figure 5.3: Sky plot of Novara.
Figure 5.4: Sky plot of “TW3742” antenna. Figure 5.5: Sky plot of “TW3740” antenna.
75
5.2. The processing software
At this stage, it was important to manage the obstruction created by St. Gaudenzio spire, and thus,
finding a suitable software to work out accurate solutions. To this end, the performances and limits
of a commercial software, such as LGO, in comparison to the scientific ones, goGPS and Bernese,
were studied.
Four analyses were carried out, three in the controlled scenario and one with the presence of the spire.
For the analyses in a controlled scenario, they were performed with LGO and goGPS, utilizing the
data acquired between 8 and 22 November 2017. As for the LGO analyses, the standard deviations
found in chapter 4.1. were considered while for goGPS analyses, the same raw data were used in
order to have a homogeneous comparison.
As for the study on the coordinate precision in Novara, always using double differences, the rover
antenna “TW3742” placed on the East side of the St Gaudenzio spire was taken into account. The
analysis was performed with LGO, goGPS and Bernese, using data collected between 4 and 19
January 2018, considering the reference station of Novara.
Thus, to understand the softwares behaviour in presence of the spire, the standard deviations of the
coordinates estimated on twenty-four hours sessions in Novara were compared with those obtained
in the controlled experiment.
Dealing with the settings, the following inputs were utilized. As for elevation cut-off angle, it was
used 15° for LGO and goGPS and 10° for Bernese. To model the tropospheric effect, LGO and goGPS
utilized a standard model, named Saastamoinen model, based on hypothesis of ideal conditions for
the atmosphere. Instead, as for Bernese analyses, a new empirical function was used, that takes into
account numerical weather model data, that is the Global Mapping Function (GMF). Looking at the
ionospheric models, Klobuchar model was applied for LGO and goGPS analyses, while for one
implemented by Bernese software no model was adopted. In fact, the points were considered so close
that their ionospheric effect could be considered negligible. Lastly, the ambiguity was fixed with
LAMBDA method in goGPS, SIGMA method in Bernese while in LGO, being a commercial
software, there were no specific indications on the utilized method.
In the following, the outputs of the analyses are summarized.
76
If the controlled scenario is considered, one can see that, based on the same data and session length
(i.e. twenty-four hours), LGO gives always better solutions. Even though the performance of LGO
and goGPS appears to be similar for the baseline Mila – COM16, goGPS produces worse results in
the other two tests. Indeed, looking at the analyses of Monza and Pavia, the results carried out by
LGO are more accurate. A closer look at the figures shows that, for the baseline Monz – COM16, the
standard deviations of LGO are roughly speaking half of those estimated with goGPS. Similarly, for
Pavi – COM16 where, unless East standard deviation, North and Height coordinates are estimated
with higher precision.
However, in St. Gaudenzio station, LGO is not able to properly process the data. In fact, apart from
January 19, the commercial software was not able to fix the phase ambiguity and thus finding
acceptable solutions. Furthermore, comparing goGPS and Bernese, their performance appears
comparable, with goGPS performing even a little better than Bernese.
One can conclude that, even though goGPS and Bernese results are less accurate than those obtained
in the controlled scenario, differently from LGO, they are able to find solutions also with noisy data,
resulting more suitable for our case study.
Furthermore, the St. Gaudenzio results show how strong is the impact of the environment around the
antenna. Considering that for this station standard deviations between the ones of Milano and Monza
(controlled scenario) were expected, it was clear that the input data disturbed by the presence of the
spire were not suitable to reach the requested accuracies. In fact, recalling that the expected
fluctuations of St. Gaudenzio spire were in about few millimeters, standard deviations of 6mm or
more could not be accepted. This means that raw data acquires in St. Gaudenzio must be pre-
processed in order to improve the solution and get standard deviations of the order of few millimeters,
as required for the analysis of the St. Gaudenzio spire oscillations. As a result, it was decided to refer
LGO goGPS LGO goGPS LGO goGPS LGO goGPS Bernese
σΔE 0,22 0,29 2,03 5,66 6,29 4,54 0 6,91 12,04
σΔN 0,25 0,22 1,64 5,94 4,28 6,47 0 6,47 7,46
σΔh 0,23 0,19 0,59 2,61 1,73 5,49 0 11,18 8,63
0,0
1,0
2,0
3,0
4,0
5,0
6,0
7,0
8,0
9,0
10,0
11,0
12,0
13,0
14,0
[mm]
-
-
-
MILA - COM16 MONZ - COM16 PAVI - COM16 NOVA- COM16
Figure 5.6: Comparison between commercial and scientific software: LGO, goGPS and Bernese.
77
to an external company that, with the use of Bernese software, was able to perform a preprocessing
of the data. In this way the original input files, severely disturbed by the presence of the spire, were
clean up and then a successful improvement of the solution was possible. As far as the preprocessing
of the data and their analyses with Bernese GPS Software 5.2, the results were provided by Geomatics
Research & Development srl in collaboration with Softeco Consulting spa. In fact, these companies
deliver the GeoGuard service that, by means of low-cost GNSS receivers, provides an end-to-end
service for the constant monitoring of critical events. GeoGuard offers flexible solutions for different
application scenarios and it was first available on the market in 2014.
The entire recorded data set, from 4 January to 19 February 2018 was processed in sessions of twenty-
four hours. The statistics of the coordinate time series are listed in the following table.
SOFTWARE: BERNESE
Solution type Base line σΔE σΔN σΔh
[mm] [mm] [mm]
24h NOVA - COM16 1,85 1,40 2,09
NOVA - COM9 1,03 0,96 2,32
Table 5.2: Standard deviations of COM16 and COM9 carried out with Bernese in Novara.
As it is visible from the table 5.1, coordinate precisions showed standard deviations of few millimeters
and thus suitable for the monitoring of the spire, which has expected movements up to 1-2 cm.
5.3. Least Squares GPS Coordinate time series modeling and testing
In order to properly estimate possible deformations occurring in a monitored structure, a statistical
analysis must be carried out. In the applications devised in this thesis the GPS coordinate time series
are the data to be considered and modeled. To this aim, least squares adjustment and testing can be
applied.
In fact, using the least squares principle, the model interpolating the data can be estimated, and then,
a statistical test can be carried out on the model coefficients to e.g. verify the static of the spire.
In our case study, we wanted to fit a straight line in each time series coordinates, i.e. fit a line either
in the East, North and Up time series coordinates of the two antennas deployed on the St. Gaudenzio
spire. The model is thus defined as
𝛼(𝑡𝑖) = 𝑎 + 𝑏𝑡𝑖 (5.2)
where 𝛼(𝑡𝑖) is one of the three coordinates observed at time ti and the two unknown coefficient of the
model are the slope and the intercept of the straight line. This model is represented in the general
form
𝛼 = 𝐴�̂� + 𝐿 (5.3)
where 𝛼 is the vector containing the observed coordinates
78
𝛼 = [𝛼(𝑡1)
⋮𝛼(𝑡𝑛)
]
being n the number of observed values and the 𝐴 matrix is given by
𝐴 = [1 𝑡1
⋮ ⋮1 𝑡n
]
�̂� is the vector of unknowns constituted by �̂�, the intercept, and �̂�, the slope.
�̂� = [�̂�
�̂�]
The least-squares method can be adopted to estimate the two parameters. This principle writes as
(𝛼 − �̂�𝛼)𝑡𝑄−1(𝛼 − �̂�𝛼) = 𝑚𝑖𝑛 (5.4)
�̂�𝛼 = 𝐴�̂� + 𝐿 (5.5)
where the quadratic form is minimized with respect to the estimator of the mean �̂�𝛼.
The matrix Q is proportional to the covariance matrix of the observation, that is
𝐶𝛼𝛼 = 𝜎02𝑄 (5.6)
Moreover, hypothesizing the observed values independent and with the same accuracy, it results that:
𝐶𝛼𝛼 = 𝜎02𝑄 = 𝜎0
2𝐼 (5.7)
Based on the previous assumptions, one can write the least squares principle as:
(𝛼 − �̂�𝛼)𝑡(𝛼 − �̂�𝛼) = 𝑚𝑖𝑛 (5.8)
�̂�𝛼 = 𝐴 �̂� + 𝐿 (5.9)
By applying the minimum principle with the given linear constraint, one gets:
�̂� = [�̂�
�̂�] = (𝐴𝑇𝐴)−1𝐴𝑇𝛼 (5.10)
Subsequently, the residuals can be estimated
�̂� = 𝛼 − 𝐴�̂� (5.11)
and, based on that, the estimated value of 𝜎02 is obtained as
79
�̂�0
2 =�̂�𝑡�̂�
𝑛 − 𝑚 (5.12)
where, in our model, m=2.
Given this estimated model, in order to verify the static of the spire, a proper test can be devised,
namely the test on the slope coefficient. More specifically, one can test the hypothesis that the
estimated slope �̂� is significantly equal to zero (i.e. no trend is present in the spire coordinates).
𝐻0: �̂� = 0
It can be proved that the proper test statistic in this case is the t of Student. Having assumed that, the
hypothesis is true if the following statement was valid:
𝑡(𝑛−𝑚) =�̂� − 0
�̂�02√[(𝐴𝑡𝐾−1𝐴)−1]22
(5.13)
|𝑡(𝑛−𝑚)| ≤ 𝑡𝛼(𝑛−𝑚)
where 𝛼 is the significance level (that in our tests was set equal to 5%) and, the 𝑡𝛼(𝑛−𝑚), value can
be found in the t Student’s table.
This procedure has been implemented in a MATLAB code that is listed in Appendix B.
It has been applied to the time series collected in the controlled environment and in St. Gaudenzio.
This allowed comparisons between the two setting and a tuning of the procedure.
80
5.3.1. The Milano data
Before starting the statistical analysis of Novara, it was decided to verify the effectiveness of the test
in Milan. Accordingly, Mila was considered as reference station, and COM16 as rover. As for the
analyses, the estimated daily coordinate time series from the 8 to 22 November 2017 were considered.
Dealing with the results of the statistical analysis, only the test performed on the East coordinate time
series led to the acceptance of the null hypothesis 𝐻0: �̂� = 0. In fact, for North and Height coordinates
the straight lines interpolating the observations could not be considered horizontal. In the light of
these results, it is worthwhile reminding that the instability of the Height coordinates is due to the
geometry of observation and, that their values are not in the interest of case study, since point
variations are expected in the plane. As for the North component, the null hypothesis for the slope
coefficient was nearly verified and thus its behavior is to be considered sufficiently stable.
Figure 5.7: Evolution East coordinates with their interpolating
model, Milan.
Figure 5.8: Evolution North coordinates with their interpolating
model, Milan.
�̂� = 0 �̂� ≠ 0
Figure 5.9: Evolution Height coordinates with their interpolating
model, Milan.
�̂� ≠ 0
81
5.3.2. The St. Gaudenzio in Novara data
As already anticipated, always using double differences, coordinates based on sessions of twenty-
four hours from the 4 January to 19 February 2018 were collected. Data acquisition has,
unfortunately, a gap from 3 to 13 of February, owed to an unexpected turning off of the receivers.
As for the analyses, “TW3742” and “TW3740” antennas were considered as rover stations, and Nova
as the reference one. For simplicity, “TW3742” and “TW3740” were renamed, as COM16 and COM9
respectively.
The statistics of the coordinate time series coming from the Bernese data analysis already presented
in section 5.2 are listed in the table 5.1.
SOFTWARE: BERNESE
Solution type Base line σΔE σΔN σΔh
[mm] [mm] [mm]
24h NOVA - COM16 1,85 1,40 2,09
NOVA - COM9 1,03 0,96 2,32
Table 5.2: Standard deviations of COM16 and COM9 carried out with Bernese in Novara.
Thanks to this comparison it was possible to highlight the accuracy of the devices together with the
impact of the existing obstruction. Indeed, as it was foreseeable from the sky plots, even though
COM9 has lower performances, it provides better results. This is due to the better satellite
configuration seen from this antenna with respect to the other.
As for the data collected in Milan, these observed data were analyzed using the approach presented
above. As a result, the straight-line parameters fitting the East, West and Height observations were
estimated. The testing procedure was then applied to the slope of the straight line in order to check
for the spire static. The null hypothesis
𝐻0: �̂� = 0
was tested. If the 𝐻0 is true, no significant movements of the spire are evidenced.
During the analyzed thirty-six days, the null hypothesis was verified. In fact, the slope of the six
models could be considered significantly equal to zero. In the following pictures, the observations
with their interpolating lines are shown.
82
As for COM16:
Figure 5.10: Evolution East coordinates with their interpolating
model, COM16, Novara.
Figure 5.11: Evolution North coordinates with their interpolating
model, COM16, Novara.
�̂� = 0 �̂� = 0
Figure 5.12: Evolution Height coordinates with their interpolating
model, COM16, Novara.
�̂� = 0
83
As for COM9:
As a final remark on the data collected at St. Gaudenzio, one can say that the obtained results are only
provisional since a very short data segment is considered. Nevertheless, the coordinate time series
have the required precisions and the testing procedure could than lead to significant results allowing
a reliable analysis of the spire deformations in time caused by e.g. temperature variations.
Figure 5.13: Evolution East coordinates with their interpolating
model, COM9, Novara.
Figure 5.14: Evolution North coordinates with their interpolating
model, COM9, Novara.
Figure 5.15: Evolution Height coordinates with their interpolating
model, COM9, Novara.
�̂� = 0
�̂� = 0 �̂� = 0
84
6. The plumb line of the St. Gaudenzio spire
Besides the problem of detecting possible deformation in time of the St. Gaudenzio spire, another
problem is the plumb line of the spire. As a matter of facts, it has not been recently tested the position
of the vertex of the spire with respect to the center of the dome. Since in recent years subsidence
occurred in the foundations of the dome, this point can be considered critical for the static of the spire.
With the aim of defining the plumb line of St. Gaudenzio spire, it was decided to estimate the
horizontal position of dome center and to compare it with the horizontal position of the spire vertex.
In order to do so, we considered this vertex as represented by the center of the sphere located at the
top of the spire (see the figure 6.1).
In order to properly compare the two horizontal positions, a survey has been performed connecting
points inside and outside the St. Gaudenzio cathedral. Then, scans of the dome and of the sphere have
been performed from proper station points that allowed collecting a wide range of data related to the
internal dome and to the external sphere. Lastly, thanks to the use of a program implemented in
MATLAB, the estimates of the horizontal coordinates of the dome and the sphere centers were
computed and compared.
6.1. The network design and survey
The network design is represented in the next figure. It is a traverse consisting of twelve points, three
of which are inside the St. Gaudenzio church. The survey was carried out on the 26 and 28 of February
2018, starting from points 100, 200 and 300 inside the Church. They have been surveyed using the
Leica Nova MS60 total station (for technical details, see paragraph 3.3 of this thesis). Additionally,
in points S1, S2, 500, 900, 1000, NRTK survey has been performed in order to frame the survey to the
ETRF2000 system (epoch 2008.0). These points were measured with the Leica Viva GS14 GNSS
receiver using the facilities of the SPIN GNS network (see paragraph 3.3 for technical details).
Figure 6.1: The Christ statue on top the St. Gaudenzio spire and the sphere.
85
Thus, the surveyed points are framed to this system and, as for the horizontal coordinates, North and
East are derived using the UTM32 mapping. Also, by rotation in the plane, horizontal coordinates in
the local (x,y) system are given, with origin in point 100 and x axis along the line connecting points
100 and 300. The surveyed network is shown in the following picture, where the red markers indicate
the points from witch the scans were performed.
The basic formulas to be used for estimating the coordinates of the surveyed points are given in the
following.
To explain the concept on which the traverse surveying is based, we can consider a simple example,
where 𝐴 and 𝑃1 has known coordinates and 𝑃2 is a point of the network that need to be calculated.
As it can be easily proved,
𝑥2 = 𝑥1 + 𝑙1sin 𝜗12 (6.1)
Figure 6.1:The survey network.
Figure 6.2:Polygonal scheme with two known points.
x
y
86
𝑦2 = 𝑦1 + 𝑙1cos 𝜗12 (6.2)
where 𝑙1 and 𝛼1 are the observed quantities, 𝜗12 = 𝛼1 + 𝜗1 and 𝜗1 = 𝑎𝑟𝑐𝑡𝑎𝑛 (𝑥𝐴−𝑥𝑃1
𝑦𝐴−𝑦𝑃1
) is a known
angle.
In general, the coordinates of a point along the traverse are computed as
𝑥𝑖+1 = 𝑥𝑖 + 𝑙𝑖sin 𝜗𝑖,𝑖+1 (6.3)
𝑦𝑖+1 = 𝑦𝑖 + 𝑙𝑖cos 𝜗𝑖,𝑖+1 (6.4)
In case the traverse ends or contains other known points, besides A and P1, it can be adjusted by least
squares. This was the case of the surveyed traverse where GNSS points were included and used for
adjusting it.
6.2 The point cloud acquisitions
As already pointed out three scan sessions took place from points 100, 200 and 300 to acquire points
in the dome interior. Two scan sessions were then performed outside the church from points S1 and
S2 in order to get points on the spire and the sphere on top of it. 1711992 points were acquired in the
dome interior while 6176 points were obtained in the two scan sessions that map the upper part of the
spire and the sphere. The points clouds acquired in the five scan sessions are represented in the
following figures.
Figure 6.4: Points cloud of the dome interior.
87
6.3 Estimating the dome and sphere centers
As stated in the beginning of this chapter, the survey was designed in order to estimate the plumb line
of the spire. The estimate was then carried out by implementing two dedicated MATLAB codes. They
are based on least squares adjustment and are detailed in the following.
6.3.1. Estimating the horizontal position of the St. Gaudenzio’s dome center
The estimate of the horizontal coordinates of the dome center has been performed over three
horizontal sections extracted from the point clouds collected inside the church.
The three clouds were merged and three sections were sampled at 24, 48 and 50 meters. The amplitude
of the three sections was set to 5 mm. The three sections consist of 1262, 356 and 225 points. The
three sections were modeled as circumferences: their centers and radii were estimated via least
squares adjustment.
In each section, the model is then given by
(𝑥𝑖 − 𝑥0)2 + (𝑦𝑖 − 𝑦0)2 = 𝑟2 i=1,…, n (6.5)
Figure 6.5: Points cloud acquired by
external sessions.
88
where: n is the number of observations; 𝑥𝑖 and 𝑦𝑖 are the observations coming from the clouds
sections; 𝑥0, 𝑦0 and 𝑟 are the coordinate of the center and the radius of the circumference.
Dealing with a non-linear problem, the first thing to do was the linearization of the equation. In the
following expressions, it is shown the linearization of the observation equation with respect to the
observations and the parameters
If we set
𝑓(𝑥, 𝛼) = (𝑥𝑖 − 𝑥0)2 + (𝑦𝑖 − 𝑦0)2 − 𝑟02 = 0 (6.6)
the linearization is given by
(�̃�𝑖 − �̃�0)2 + (�̃�𝑖 − �̃�0)2 − �̃�02 +
+2(�̃�𝑖 − �̃�0)𝛿𝑥𝑖 + 2(�̃�𝑖 − �̃�0)𝛿𝑦𝑖 +
−2(�̃�𝑖 − �̃�0)𝛿𝑥0 − 2(�̃� − �̃�0)𝛿𝑦0 − 2�̃�0𝛿𝑟0 = 0
where
- �̃�𝑖 and �̃�𝑖 are the approximate positions of the observations
- �̃�0, �̃�0 �̃�0 the approximate position of the center and the radius of the circumference
- 𝛿𝑥𝑖 and 𝛿𝑦𝑖 are the increments to the approximate positions of the observations
- 𝛿𝑥0, 𝛿𝑦0 and 𝛿𝑟0 are the increments to the approximate position of the center and the radius.
The previous equation has to be set in the form,
𝐵𝛿�̂� = 𝐴𝛿�̂� + 𝐿 (6.7)
i.e. the standard observation equation in least squares applications.
To this end, the linearized observation equation of the circumference can be rewritten as
2(�̃�𝑖 − �̃�0)𝛿�̂�𝑖 + 2(�̃�𝑖 − �̃�0)𝛿�̂�𝑖 =
= 2(�̃�𝑖 − �̃�0)𝛿�̂� + 2(�̃�𝑖 − �̃�0)𝛿�̂� + 2�̃�0𝛿�̂� +
−(�̃�𝑖 − �̃�0)2 − (�̃�𝑖 − �̃�0)2 + �̃�02
Based on this last expression, it was possible to identify the various matrices of the standard model
as:
𝐵 = [
2(�̃�1 − �̃�0) 2(�̃�1 − �̃�0) 0
0 0 2(�̃�2 − �̃�0)⋯0
⋯0
⋯0
02(�̃�2 − �̃�0)
⋯0
⋯⋯⋯⋯
00⋯
2(�̃�𝑛 − �̃�0)
00⋯
2(�̃�𝑛 − �̃�0)
]
𝛿�̂� = [𝛿�̂�𝑖
𝛿�̂�𝑖] i=1,…,n
89
𝐴 = [2(�̃�1 − �̃�0) 2(�̃�1 − �̃�0) 2�̃�0
⋯ ⋯ ⋯2(�̃�𝑛 − �̃�0) 2(�̃�𝑛 − �̃�0) 2�̃�0
]
𝛿�̂� = [
𝛿�̂�0
𝛿�̂�0
𝛿�̂�0
]
𝐿 = [−(�̃�1 − �̃�0)2 − (�̃�1 − �̃�0)2 + �̃�0
2
… … … …−(�̃�𝑛 − �̃�0)2 − (�̃�1 − �̃�0)2 + �̃�0
2]
For solving the least square problem, the approximate positions of the observations and of the
parameters are needed so to compute the above described matrices.
The approximate values for the observations are usually considered equal to the observations
themselves. As for the approximate parameters, the values have been obtained based on three points
selected among the points in the section.
Having done that, one can obtain the parameters estimate as
𝛿�̂� = [
𝛿�̂�0
𝛿�̂�0
𝛿�̂�0
] = (𝐴𝑇𝐾−1𝐴)−1𝐴𝑇𝐾−1∆ (6.8)
where
𝐾 = 𝐵𝑄𝐵𝑡 = 𝐵𝐵𝑡
∆= 𝐵𝛿𝛼0 − 𝐿 = −𝐿
Finally, the estimated positions of the center and the radius of the circumference were carried out
thanks to the following equation:
�̂� = 𝛿�̂�0 + �̃�0 (6.9)
Then, the residuals and the estimate of the variance can be computed:
�̂� = −𝐿 − 𝐴𝛿𝑥 (6.10)
�̂�2 =�̂�𝑡𝐾−1�̂�
𝑛 − 𝑚 (6.11)
where 𝑚 is the number of unknown, which in our case is equal to three.
This procedure was implemented in a MATLAB script that is listed in Appendix C.
90
By applying this procedure to the three selected sections, named C1, C2 and C3 we obtained the
following estimates. One iteration was sufficient for getting stable parameter estimates.
DOME COORDINATES SOLUTIONS
Circumference
centres/radii
x0 y0 r
[m] [m] [m]
C1 9,0110 11,6821 7,2036
C2 9,0791 11,6288 5,4609
C3 9,0791 11,6131 4,0936
Table 6.1: The estimated dome section parameters.
The following figure represents the obtained results. It clearly shows the very good fit obtained by
the three estimated circumferences, which properly model the observed data in the sections
Particularly, the fit is extremely accurate in the lower section, which is observed on a smooth surface.
The circumferences at 48 and 50 m height are taken on the decorated part of the dome and are thus
more scattered. Nevertheless, the circumference model is able to fit them in a very careful way. The
values that testify the goodness of fit are the standard deviations of the estimated parameters that are
in the range of few millimeters. In the last paragraph of this chapter, a testing procedure is set up in
order to prove if the estimated horizontal coordinates are significantly different.
6.3.2. Estimating the horizontal position of the sphere center
Similarly, the problem of the sphere was analyzed. In fact, the least square principle was again applied
to find the four unknown variables for the center and the radius of the sphere: 𝑥0, 𝑦0, 𝑧0 and 𝑟. Also
in this case, accurate estimates where reached just with one iteration.
The equation of sphere interpolating the data is shown in the following equation:
(𝑥𝑖 − 𝑥0)2 + (𝑦𝑖 − 𝑦0)2 + (𝑧𝑖 − 𝑧0)2 = 𝑟2 (6.12)
Figure 6.6: Dome sections representing the observations and models.
91
where
- n number of observations
- 𝑥𝑖, 𝑦𝑖 and 𝑧𝑖 observations
- 𝑥0, 𝑦0, 𝑧0 and 𝑟 are the position of the sphere center and its radius.
In the same way the sphere equation has to be linearized, always considering the approximate values
of the observations and of the parameters (they were found based on a subset of values and
considering known information on the sphere diameter coming from documents on the statue).
As done for the circumference, one can write
𝑓(𝑥, 𝛼) = (𝑥𝑖 − 𝑥0)2 + (𝑦𝑖 − 𝑦0)2 + (𝑧𝑖 − 𝑧0)2 − 𝑟02 = 0 (6.13)
(�̃�𝑖 − �̃�0)2 + (�̃�𝑖 − �̃�0)2 + (𝑧𝑖 − 𝑧0)2 − �̃�02 +
+2(�̃�𝑖 − �̃�0)𝛿𝑥𝑖 + 2(�̃�𝑖 − �̃�0)𝛿𝑦𝑖 + 2(�̃�𝑖 − �̃�0)𝛿𝑧𝑖 +
−2(�̃�𝑖 − �̃�0)𝛿𝑥0 − 2(�̃� − �̃�0)𝛿𝑦0 − 2(�̃� − �̃�0)𝛿𝑧0 − 2�̃�0𝛿𝑟0 = 0
Next, to formulate the least square problem, the linearized models is set to:
𝐵𝛿�̂� = 𝐴𝛿�̂� + 𝐿 (6.14)
where the matrices are expressed in the following:
𝐵 = [2(�̃�1 − �̃�0) 2(�̃�1 − �̃�0) 2(�̃�1 − �̃�0)
⋯ ⋯ ⋯0 0 0
0⋯0
0⋯
2(�̃�𝑛 − �̃�0)
0⋯
2(�̃�𝑛 − �̃�0)
0⋯
2(�̃�𝑛 − �̃�0)]
𝛿�̂� = [
𝛿�̂�𝑖
𝛿�̂�𝑖
𝛿�̂�𝑖
]
𝐴 = [2(�̃�1 − �̃�0) 2(�̃�1 − �̃�0) 2(�̃�1 − �̃�0)
⋯ ⋯ ⋯2(�̃�𝑛 − �̃�0) 2(�̃�𝑛 − �̃�0) 2(�̃�𝑛 − �̃�0)
2�̃�0
⋯2�̃�0
]
𝛿�̂� = [
𝛿�̂�0
𝛿�̂�0
𝛿�̂�0
𝛿�̂�0
]
𝐿 = [−(�̃�1 − �̃�0)2 − (�̃�1 − �̃�0)2 + �̃�0
2
… … … …−(�̃�𝑛 − �̃�0)2 − (�̃�1 − �̃�0)2 + �̃�0
2]
92
By applying least squares, assuming the independency of the observed values, the correction for the
approximate position of the coordinate center and of the radius are obtained as follows.
𝛿�̂� = [
𝛿�̂�0
𝛿�̂�0
𝛿�̂�0
𝛿�̂�0
] = (𝐴𝑇𝐾−1𝐴)−1𝐴𝑇𝐾−1∆ (6.15)
Thus, parameters estimate are given as
�̂� = 𝛿�̂� + �̃�0
Residuals are then obtained as
�̂� = −𝐿 − 𝐴𝛿𝑥
and, finally, the a posteriori estimate of the variance is obtained as
�̂�2 =�̂�𝑡𝐾−1�̂�
𝑛 − 𝑚
where m is equal to 4.
The procedure detailed above has been implemented in another MATLAB script that is listed in
Appendix D.
Using this script, the computed values for the estimated parameters have been obtained and listed in
the table below.
SPHERE COORDINATES SOLUTIONS
Estimated
center/radius
x0 y0 z0 r0
[m] [m] [m] [m]
9,1741 11,4911 118,9294 0,4034
Table 6.2: The estimated coordinates of the sphere center.
93
The plot representing the data and the fitted sphere is shown in the following picture.
The standard deviations of the estimated parameters, particularly those related to the sphere center
coordinates, are in the order of few millimeters. So, the given estimates are comparable with the
estimates of the dome center given in paragraph 6.3.1 and thus suitable for testing the plumb line of
the spire. The testing procedure enabling this comparison is presented in the following paragraph.
6.4. Comparing the coordinates of the dome and sphere centers.
Having determined the centers and radius related to the dome and the sphere, it was conducted a test
to verify whether the center coordinates could be considered significantly equal. For this purpose, it
was made the hypothesis that the average of the difference between two equal components of the
center coordinates was equal to zero. For instance, taking into account the 𝑥-components of the first
and second circumference identified at different height, one can compute the difference
∆𝑜𝑏𝑠= �̂�𝐶1 − �̂�𝐶2
and set the null hypothesis as
𝐻0: 𝜇∆ = 0
Based on the assumptions that have been made, it can be shown that the test statistic is a t Student.
Namely, it can be proved that
𝑡𝑛−𝑚0 =
∆𝑜𝑏𝑠 − 𝜇∆
σ̂∆=
∆𝑜𝑏𝑠 − 0
𝜎𝐶1√[(𝐴𝐶1𝑡 𝐾𝐶1
−1𝐴𝐶1)−1]11 + 𝜎𝐶2√[(𝐴𝐶2𝑡 𝐾𝐶2
−1𝐴𝐶2)−1]11
(6.16)
Figure 6.7: Sphere three-dimensional view representing observations and model.
94
where
- 𝛼 = 5% significance level
- 𝑛 = 𝑛𝐶1 + 𝑛𝐶2 number of observations
- 𝑚 = 2 number of unknown variables.
Then, if
|𝑡(𝑛−𝑚)0 | ≤ 𝑡𝛼
(𝑛−𝑚) (6.17)
it can be stated that 𝐻0 holds.
For given α values, 𝑡𝛼(𝑛−𝑚) can be found thanks to the use of the Excel function INV.T.
More in depth, taking into account the four centres, twelve tests were performed, where the null
hypothesis is tested in all the coordinates differences of the estimated circumference/sphere centres.
Thus, the possible tests are:
𝑥 − 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡: 𝑦 − 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡:
𝑇𝑒𝑠𝑡 1. 𝜇∆ = �̂�𝐶1 − �̂�𝐶2 = 0 𝑇𝑒𝑠𝑡 7. 𝜇∆ = �̂�𝐶1 − �̂�𝐶2 = 0
𝑇𝑒𝑠𝑡 2. 𝜇∆ = �̂�𝐶1 − �̂�𝐶3 = 0 𝑇𝑒𝑠𝑡 8. 𝜇∆ = �̂�𝐶1 − �̂�𝐶3 = 0
𝑇𝑒𝑠𝑡 3. 𝜇∆ = �̂�𝐶1 − �̂�𝐶4 = 0 𝑇𝑒𝑠𝑡 9. 𝜇∆ = �̂�𝐶1 − �̂�𝐶4 = 0
𝑇𝑒𝑠𝑡 4. 𝜇∆ = �̂�𝐶2 − �̂�𝐶3 = 0 𝑇𝑒𝑠𝑡 10. 𝜇∆ = �̂�𝐶2 − �̂�𝐶3 = 0
𝑇𝑒𝑠𝑡 5. 𝜇∆ = �̂�𝐶2 − �̂�𝐶4 = 0 𝑇𝑒𝑠𝑡 11. 𝜇∆ = �̂�𝐶2 − �̂�𝐶4 = 0
𝑇𝑒𝑠𝑡 6. 𝜇∆ = �̂�𝐶3 − �̂�𝐶4 = 0 𝑇𝑒𝑠𝑡 12. 𝜇∆ = �̂�𝐶3 − �̂�𝐶4 = 0
where the coordinates of the sphere centre are named �̂�𝐶4 and �̂�𝐶4.
With the exception of Test 4, the null hypothesis is not verified. This can be considered reasonable
since �̂�𝐶2 and �̂�𝐶3 have the same values up to the fourth decimal digit. As for the remaining tests,
considering the 𝑥 and 𝑦 components of the three sections of the dome, they differ for a maximum of
7cm. As regards the dome and the sphere center coordinates, the maximum differences in the 𝑥 and
𝑦 components are around16cm and 19cm respectively. The results are reported in figure 6.8.
Figure 6.8: Circumferences and sphere centers coordinates.
16cm
19cm
95
Thus, in term of distances, the maximum displacement is between 𝐶1 and 𝐶4, that is
∆𝑆𝐶1= √(𝑋𝐶1 − 𝑋𝐶4
)2
+ (𝑌𝐶1 − 𝑌𝐶4)
2= 0,25m (6.18)
Therefore, being for the sphere �̂�0 = 118,93𝑚, the angle α, representing the out of vertical, is equal
to:
𝛼 = arctan (
∆𝑆𝐶1
�̂�0) = 0° 7′ 13′′, 58 (6.19)
Further investigations are to be made in order to state if thsis is due to contruction problems or to the
subsidence of one of the pillars in the basement of the dome.
𝐶4
𝑧1 �̂�1
𝐶1
∆𝑆𝐶1
Figure 6.9: Out of plumb.
96
7. Conclusion
Thanks to the performed studies, the dynamic and the static of the St. Gaudenzio structure was
investigated.
Though GPS equipment installation on St. Gaudenzio’s spire, the analysis of the spire deformations
in time due to, for instance, wind action and temperature variations started. In fact, with a statistical
analysis of the GPS coordinate time series, it was estimated the model interpolating the data, and
next, tested the absence of a trend in the spire coordinates. As for the results, throughout the entire
period of analysis, the positions of the TW3742” and “TW3740” antennas satisfied the null hypothesis
of the test, showing that the spire wasn’t significantly affected by external stresses.
Dealing with the analysis of the geometry of the structure, the scans carried out internally and
externally to the Cathedral allowed the preliminary metric representation of some part of the dome
and of the spire, so that to highlight whether were present discrepancies between the original project
and the current situation. More in depth, it was important to detect the possible effects of the recent
subsidence phenomenon occurred in the foundations of the dome. Indeed, it wasn’t recently tested
the spire vertex position with respect to the center of the dome, namely the plumb line. Thus, the
horizontal position of dome center, selecting three different sections, was computed with the
horizontal position of the spire vertex, represented by the sphere center located at the top of the spire.
To compare the horizontal positions, it was firstly designed a traverse connecting twelve points inside
and outside the St. Gaudenzio cathedral and then, through the scans, it was possible to implement a
program in MATLAB that estimates the horizontal coordinates of the center of the dome and of the
sphere. Finally, having available these center coordinates, the out of vertical angle between the two
points was estimated.
Thanks to the combination and further development of these control and survey methods it is possible
to define if the displacements to which the structure is subjected are properly withstand by the
structure itself.
This research should be intended as the beginning for future developments. In fact, increasing the
number of GPS devices and prosecuting with internal scans of the spire, it will be possible to carry
out a detailed structural analysis, understanding deeply the dynamic behavior of the dome and of the
spire.
97
APPENDIX A
H.P.
V.P. L.P
.
APPENDIX A: Orthogonal projection of the antenna support, units [mm].
98
APPENDIX B
% ESTIMATION OF VECTOR X - DELTA E
load MILA24h_COM16.txt;
time=MILA24h_COM16(:,1);
E=MILA24h_COM16(:,2);
N=MILA24h_COM16(:,3);
H=MILA24h_COM16(:,4);
m=size(time,1);
n=2;
gdl=m-n;
for k=(1:m)
A(k,1)=1;
A(k,2)=time(k,1);
end
x1=inv(A'*A)*A'*E;
pred1=A*x1;
a1=x1(1,1);
b1=x1(2,1);
% TEST b1 - Hp. b1=0 - aplha=5% - t975
u1=E-pred1;
o1=(u1'*u1)/(m-2);
f1=inv(A'*A);
f21=f1(2,2);
t1=(b1-0)/(o1*sqrt(f21));
load Student_Distribution.txt;
ti1=Student_Distribution(gdl,2);
if abs(t1)<ti1
99
disp('verified')
elseif abs(t1)>ti1
disp('not verified')
end
100
APPENDIX C
load San_Gaudenzio_Interno.pts;
A=San_Gaudenzio_Interno(:,1:3);
z=San_Gaudenzio_Interno(:,3);
z1=min(z);
z2=max(z);
delta=0.0025;
% SECTION 1
b=24;
L1=b-delta;
U1=b+delta;
B=A(A(:,3)>L1 & A(:,3)<U1 ,:);
xb=B(:,1);
yb=B(:,2);
Xab=[9; 11.7; 7.2];
xab=Xab(1,1);
yab=Xab(2,1);
rab=Xab(3,1);
mb=size(xb,1);
for k=(1:mb)%[A]
Ab(k,1)=2*(xb(k,1)-xab);
Ab(k,2)=2*(yb(k,1)-yab);
Ab(k,3)=2*rab;
end
fb=Ab(:,1);
gb=Ab(:,2);
for k=(1:mb)%[L]
Lb(k,1)=rab^2-(xb(k,1)-xab)^2-(yb(k,1)-yab)^2;
end
Deltab=-Lb;%[Delta]
for k=(1:mb)%[B]
Bb(k,2*k-1)= fb(k,1);
Bb(k,2*k)= gb(k,1);
end
Kb=Bb*Bb';%[K]
101
dxb=inv(Ab'*(inv(Kb))*Ab)*Ab'*(inv(Kb))*Deltab;
Xb=dxb+Xab;
Xb1=Xb(1,1);
Xb2=Xb(2,1);
Xb3=Xb(3,1);
ub=-Ab*dxb-Lb;
ob=(ub'*inv(Kb)*ub)/(mb-3); % VARIANZA (^2)
102
APPENDIX D
load San_Gaudenzio_Sfera.txt;
V=San_Gaudenzio_Sfera(:,1:3);
z=San_Gaudenzio_Sfera(:,3);
z1=min(z);
z2=max(z);
% SPHERE
L4=117;
U4=120;
B=V(V(:,3)>L4 & V(:,3)<U4 ,:);
xs=B(:,1);
ys=B(:,2);
zs=B(:,3);
zm=mean(zs);
Xas=[9.17; 11.5; 119; 0.4];
xas=Xas(1,1);
yas=Xas(2,1);
zas=Xas(3,1);
ras=Xas(4,1);
ms=size(xs,1);
for k=(1:ms)%[A]
As(k,1)=2*(xs(k,1)-xas);
As(k,2)=2*(ys(k,1)-yas);
As(k,3)=2*(zs(k,1)-zas);
As(k,4)=2*ras;
end
fs=As(:,1);
gs=As(:,2);
hs=As(:,3);
for k=(1:ms)%[L]
Ls(k,1)=ras^2-(xs(k,1)-xas)^2-(ys(k,1)-yas)^2-(zs(k,1)-zas)^2;
end
Deltas=-Ls;%[Delta]
Bs=0;
for k=(1:ms)%[B]
Bs(k,3*k-2)= fs(k,1);
103
Bs(k,3*k-1)= gs(k,1);
Bs(k,3*k)= hs(k,1);
end
Ks=Bs*Bs';%[K]
dxs=inv(As'*(inv(Ks))*As)*As'*(inv(Ks))*Deltas;
Xs=dxs+Xas;
Xs1=Xs(1,1);
Xs2=Xs(2,1);
Xs3=Xs(3,1);
Xs4=Xs(4,1);
us=-As*dxs-Ls;
os=(us'*inv(Ks)*us)/(ms-4);
104
LIST OF FIGURES
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……………………………………………………………………………………. 3
1.2 Floor-plan of the Cathedral of St. Gaudenzio………………………………………………... 5
1.3 Antonelli’s compass preserved at the "Compass Hall"………………………………………. 5
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…………………………………………………………………………………………... 7
1.5 The first Antonellian project of 1841..……………………………………………………….. 7
1.6 The third Antonellian project of 1860..………………………………………………………. 8
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. …………………………. 9
1.8 Undated watercolor section of the Cathedral of St. Gaudenzio. …………………………….. 10
1.9 Antonelli’s “mechanism”……………………………………………….……………………. 11
1.10 External thin shell in brick stiffened by the system of “meridians” and “parallels” and a
schematic description of the system..……………………………………………………….... 11
1.11 Section of the exterior dome and the interior cone...…………………………………………. 12
1.12 Walkway of the dome supported by granite corbels...……………………………………….. 12
1.13 Wooden model representing a pair of arches with its fundamental pillar……………………. 13
1.14 Drowing 1:300 of L. Caselli the section of the dome – table XII from La Cupola della
Basilica di San Gaudenzio in Novara in L’ingegneria civile e le arti industriali – Turin
1877…………………………………………………………………………………………... 14
1.15 Drawing depicting Antonellian foundations dated June 1883. .……………………………… 15
1.16 Early works of consolidation of the spire, 1931.……………………………………………... 17
1.17 Wooden scaffold that would have to support the dome.……………………………………... 18
1.18 Plastered Antonellian arch and little brick pillars.…………………………………………… 18
1.19 Church of St. Gaudenzio before the interventions of 1946-1947, on the left, and in the
situation of the eighties of the twentieth century, to the right.……………………………….. 19
2.1 The Earth and GPS satellites.………………………………………………………………… 20
2.2 Geometric ellipsoid…………………………………………………………………………... 21
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…………………….. 23
2.4 The three component that compose the GPS system…………………………………………. 25
2.5 The space component………………………………………………………………………… 25
2.6 The five monitoring stations of GPS system…………………………………………………. 26
2.7 The user component………………………………………………………………………….. 26
2.8 Combination of a sinusoidal with a binary code……………………………………………... 28
2.9 Determination of the flight time ΔT………………………………………………………….. 29
2.10 Measurement of phase difference…………………………………………………………….. 30
2.11 The orbit errors……………………………………………………………………………….. 32
2.12 Bending of the electromagnetic signal……………………………………………………….. 32
105
2.13 Variation of the center of the phase of the antenna…………………………………………... 34
2.14 Multipath……………………………………………………………………………………… 35
2.15 Possible geometric configurations of the satellites…………………………………………… 36
2.16 Functioning of a GPS receiver………………………………………………………………... 41
2.17 “Mass-market” receivers……………………………………………………………………... 42
2.18 Single frequency receivers……………………………………………………………………. 43
2.19 Geodetic receivers……………………………………………………………………………. 44
3.1 TW3740/TW3742 antenna…………………………………………………………………… 45
3.2 ANN-MS active GPS by u-blox……………………………………………………………… 46
3.3 EVK-5T reciver.by u-blox……………………………………………………………………. 46
3.4 MILA antenna………………………………………………………………………………… 47
3.5 MILA receiver………………………………………………………………………………... 47
3.6 Monza antenna………………………………………………………………………………... 47
3.7 Monza receiver……………………………………………………………………………….. 47
3.8 Pavia antenna…………………………………………………………………………………. 48
3.9 Pavia receiver…………………………………………………………………………………. 48
3.10 Novara antenna……………………………………………………………………………….. 48
3.11 Novara receiver………………………………………………………………………………. 48
3.12 Leica Nova MS60 MultiStation……………………………………………………………… 49
3.13 Leica Viva GS14……………………………………………………………………………… 49
4.1 Politecnico of Milan, building 14. Instrumentation used for the testing phase
a) ANN-MS active GPS b)
TW3742………………………………………………………………………………………. 50
4.2 Laptop used for the data acquisition and
processing…………………………………………………………………………………….. 51
4.3 Reference and rover
stations………………………………………………………………………………………… 53
4.4 “TW3742” antenna, evolution of the coordinates for consecutive observations with a
duration of twenty-four hours, Milan…………………………………………………………. 54
4.5 “ANN-MS active GPS” antenna, evolution of the coordinates for consecutive observations
with a duration of twenty-four hours, from the 8 to the 22 November 2017, Milan…………. 55
4.6 “ANN-MS active GPS” antenna, evolution of the coordinates for consecutive observations
with a duration of twenty-four hours, from the 9 to the 18 February 2014, Milan…………… 56
4.7 Evolution of the coordinates for consecutive observations with a duration of twelve hours,
Milan………………………………………………………………………………………….. 57
4.8 Evolution of the coordinates for consecutive observations with a duration of eight hours,
Milan………………………………………………………………………………………….. 58
4.9 Evolution of the coordinates for consecutive observations with a duration of one hour,
Milan………………………………………………………………………………………….. 59
4.10 Standard deviations of the coordinates in relation with the type of observation; Milan……… 60
4.11 Evolution of the coordinates for consecutive observations with a duration of twenty-four
hours, Monza………………………………………………………………………………….. 61
106
4.12 Evolution of the coordinates for consecutive observations with a duration of twelve hours,
Monza…………………………………………………………………………………………. 62
4.13 Evolution of the coordinates for consecutive observations with a duration of eight hours,
Monza…………………………………………………………………………………………. 63
4.14 Evolution of the coordinates for consecutive observations with a duration of one hour,
Monza…………………………………………………………………………………………. 64
4.15 Standard deviations of the coordinates in relation with the type of observation, Monza…….. 65
4.16 Evolution of the coordinates for consecutive observations with a duration of twenty-four
hours, Pavia…………………………………………………………………………………… 66
4.17 Evolution of the coordinates for consecutive observations with a duration of twelve hours,
Pavia…………………………………………………………………………………………... 67
4.18 Evolution of the coordinates for consecutive observations with a duration of eight hours,
Pavia…………………………………………………………………………………………... 68
4.19 Evolution of the coordinates for consecutive observations with a duration of one hour,
Pavia…………………………………………………………………………………………... 69
4.20 Standard deviations of the coordinates in relation with the type of observation, Pavia……… 70
4.21 Comparison between low-cost and high-quality stations in Milano Nave, with Monza as
reference………………………………………………………………………………………. 71
4.22 Comparison between low-cost and high-quality stations in Milano Nave, with Pavia as
reference………………………………………………………………………………………. 72
5.1 “TW3742” antenna, East positioned………………………………………………………….. 73
5.2 “TW3740” antenna, West positioned…………………………………………………………. 73
5.3 Sky plot of Novara……………………………………………………………………………. 74
5.4 Sky plot of “TW3742” antenna……………………………………………………………….. 74
5.5 Sky plot of “TW3740” antenna……………………………………………………………….. 74
5.6 Comparison between commercial and scientific software: LGO,goGPS and Bernese………. 76
5.7 Evolution East coordinates with their interpolating model, Milan…………………………… 80
5.8 Evolution North coordinates with their interpolating model, Milan………………………….. 80
5.9 Evolution Height coordinates with their interpolating model, Milan…………………………. 80
5.10 Evolution East coordinates with their interpolating model, COM16, Novara………………... 82
5.11 Evolution North coordinates with their interpolating model, COM16, Novara………………. 82
5.12 Evolution Height coordinates with their interpolating model, COM16, Novara………………. 82
5.13 Evolution East coordinates with their interpolating model, COM9, Novara………………..... 83
5.14 Evolution North coordinates with their interpolating model, COM9, Novara………………... 83
5.15 Evolution Height coordinates with their interpolating model, COM9, Novara………………. 83
6.1 The Christ statue on top the St. Gaudenzio spire and the sphere………………....................... 84
6.2 The survey network…………………………………………………………………………… 85
6.3 Polygonal scheme with two known points……………………………………………………. 85
6.4 Points cloud of the dome interior……………………………………………………………... 86
6.5 Points cloud acquired by external sessions…………………………………………………… 87
6.6 Dome sections representing the observations and models……………………………………. 90
107
6.7 Sphere three-dimensional view representing observations and model……………………….. 93
6.8 Circumferences and sphere centers coordinates……………………………………………… 94
6.9 Out of plumb………………………………………………………………………………….. 95
LIST OF TABLES
3.1 TW3740/TW3742 antenna benefits and features……………………………………………. 45
4.1 Standard deviations of the coordinates in relation with the different type of observations,
Milan…………………………………………………………………………………………. 60
4.2 Standard deviations of the coordinates in relation with the different type of observations,
Monza.……………………………………………………………………………………….. 65
4.3 Standard deviations of the coordinates in relation with the different type of observations,
Pavia………………………………………………………………………………………….. 70
5.1 AISI 304 X5CrNi 18-10, physical and mechanical………………………………………….. 73
5.2 Standard deviations of COM16 and COM9 carried out with Bernese in Novara…………… 77
6.1 The estimated dome section parameters……………………………………………………... 90
6.2 The estimated coordinates of the sphere center……………………………………………… 92
REFERENCES
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108
[13] Geomatica le radici del futuro,
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[37] https://it.wikipedia.org/wiki/WGS84.
109
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[52] Maximum likelihood estimation,
https://en.wikipedia.org/wiki/Maximum_likelihood_estimation.
