Hello. This part of the lesson of Geodesy is dedicated to geodetic references and coordinate systems for the Earth. We can pose the problems as follows: A coordinate system is a theoretical definition, it is composed of an origin, and axes with direction in space. If we look at Earth we can naturally say that the axis of rotation of Earth already provides a referential direction and equator also provides a reference plane for a coordinate system. The question is: how do we achieve a system knowing that Earth is so large and knowing that the surface thereof, the different tectonic plates move. The first part of the solution is to find a geometrical shape, which best fits Earth. In this case, we will choose here an ellipse that we rotate on the main axis, thus we have an ellipsoid of revolution, which will give the mathematical form that represents Earth. This ellipsoid have a large axis <i>a</i> and a small axis <i>b</i>, and for Earth, <i>a</i> is equal to 6377 kilometers. and <i>b</i> is equal to 6355 kilometers, a difference of about 22 kilometers, so, it's therefore relatively small compared to the total size of Earth. What coordinates for the ellipsoid? Like the sphere, we consider the geographical coordinates with the latitude, which is the angle between the plane of the equator... and the normal to the surface. We see on this figure the normal here on the surface, with my plane of the equator and the latitude here. For the longitude, like for the sphere, it is the angle between the prime meridian, here, and the meridian, which passes through the point of interest. I have here my longitude. Finally, I have the height on the ellipsoid... that I find here on my figure, along the normal of the surface. A coordinate system is a theoretical definition. To use it, we have to complete the system through a framework of coordinates and I have here an example of the measurement framework in Switzerland with one materialization of different points of the frame. So I have here an example with a pin, we also find terminals and for each point, we have an identifier and the coordinates. I invite you now to try to find the coordinate frame of the measurement and explore some points with your documentation. The second issue with geodetic references concern the vertical dimension. In fact, the gravity field affect many human activities, for example in hydraulic constructions. What references do we use for altimetry. I take in this example here, three cases of figures: a simple object, a building, and I draw, here, the vector g representing the gravitational force of the building. So we have a reference here, simple, unique, for a specific object. I will now move to a larger context in a portion of a territory, where I may be next to a mountain a vector here,<i>g1</i>, and at the edge of a lake, here, I have a vector here, gravity <i>g2</i>. We can already here pose the question: is <i>g1</i> parallel to <i>g2</i>? Do we have the same vertical reference in these two parts of the territory? If we look at a global level, it is evident that if I am here on the American continent or even in this region here in Europe, the <i>g</i> here, <i>A</i>, and the <i>g</i> here, <i>E</i>, is evidently not parallel. The solution to this issue, the vertical dimension pass through a reference surface in physics is called the geoïd. We can imagine the geoid like the ocean surface mean, which is extended under the continents. We can draw the geoïd here... which is our surface, here, the reference. It is an equipotential gravity field and it is our reference here, zero for our altitudes. Above the geoïd, I have my surface, here, topographic. And the altitude here, at a point <i>A</i> will be on the vertical line here, of my geoid with this height relative to the physical surface that we call here the altitude. If I take a point <i>B</i> here, that I descend down here to the reference surface, I have here, and altitude <i>HB</i>. Knowing that <i>A</i> and <i>B</i>, the direction of the gravity field is not necessarily parallel. As the Earth is not a uniform solid, there are masses with different densities, the reference surface, the geoïd, will vary in space. In this image, we see Earth, with its correct form, namely this reference surface, with on the other hand, bumps, with here, for example, a little over 80 meters, and then we have pits here, approximately 100 meters deep. We talk here about the geoid undulations, that should not be neglected, in our altimetric reference model. The geoid is influenced by the surrounding masses. We can see in the left image, a typical landscape with a lake, mountains, and it is clear that the masses will influence the position of the geoïd. We see in this example here, at the first location, 1, with the lake we have here a density of masses that is lower and, in this case, the geoïd would go down slightly. In the second case, we are in the presence of a mountain and here, we have one density, which is relatively strong, and on the contrary, the mass is placed above the reference surface so we will tend to attract the geoid and the geoid in this case here move up. In the third case, here with a body of very high density that is in the subsoil, so that will increase here the gravity field and the geoid in this case will also be attracted by this present mass. What is the relationship between the geoid, the physical surface, and the surface of the mathematical reference, the ellipsoid? We mainly define two geometric quantities. The first, that we call the geoïd separation, which means the separation between the two surfaces. In the example here, I have a dotted line ellipsoid and a full line geoïd, so here I find my spot height. The second geometric element is the angle that creates the vertical on the surface, do the geoïd, with the normal of the reference surface, the ellipsoïd. We have here what we call the deviation of the vertical, which is the angle between these two directions. The relationship between geoid and ellipsoid is something that is documented in the different offices of topography. We have this spot height which separate geoïd and ellipsoïd and finally, what interests us for our topographical work is the usual altitude, which is equal in this case the height in the ellipsoid minus the spot height <i>h</i> is the height of the reference surface. In this way we may set up the geoid map, whether it is on a global or local level. We see here in this example taken from the GOCE sattelite, so a mission by the European Space Agency, which ended in 2013, we see here this example of a geoïd world map. We see especially the areas here, very low, apporximately 100 meters deep, and the higher areas, a little more than 80 meters. The definition of the geoïd is one of the tasks of the national geographic institutes. In Switzerland, it is Swisstopo that has this responsibility. They established a geoid map based on the geodetic reference, in the case of the ellipsoid Bassel for Switzerland. If we look on this map, we see for example that the region here of Geneva have a geoid separation of about two meters. And in the east of Switzerland, in the region called "les Grisons", I see that I have about four meters of geoid separation. So we see here a little bit of the geoid amplitude, its variation throughout Switzerland. It exists in multiple geodetic references. In general, we consider the distance between the reference surface, the ellipsoid, and the surface level, the geoïd. We search to minimize this gap and whether we want a model for all of Earth, we apply here a global ellipsoid, or as well a local ellipsoid if we are interested in one portion of the territory. Thus we have these two categories, the worldwide or gobal systems, and the national or local systems. Here we have the example of the international system, ITRS, with an ellipsoid, GRS80, and then for the Swiss system, we have CH1903+ with its ellipsoid of Bessel. Attention, in this figure the geometry is greatly exaggerated to illustrate this principle. We have for example here, between the center of the global ellipsoïd and the center of the local ellipsoïd, only a few hundred meters. This is not at all the scale shown here in this figure. To summarize this part on geodetic references, we recall that planimetry and altimetry are two different concepts. We define a mathematical reference, the ellipsoid, and we define a physical reference for altimetry, called the geoid. Each country has its own geodetic reference associated with a framework, namely a series of materialized points, and known coordinates. Then, when receiving a set of coordinated, we will always have to pose the question: what is the geodetic reference hiding behind?