{"id":5129,"date":"2018-04-25T09:24:44","date_gmt":"2018-04-25T07:24:44","guid":{"rendered":"http:\/\/www.sigterritoires.fr\/?p=5129"},"modified":"2018-10-17T12:10:28","modified_gmt":"2018-10-17T10:10:28","slug":"accuracy-uncertainty-and-linear-altitude-of-geographic-data-1","status":"publish","type":"post","link":"https:\/\/www.sigterritoires.fr\/index.php\/en\/accuracy-uncertainty-and-linear-altitude-of-geographic-data-1\/","title":{"rendered":"Accuracy, uncertainty and linear alteration of geographic data (1)"},"content":{"rendered":"

It is not easy, in geomatics, to get a good grasp of the geodesy notions that underlie the manipulation of geographic data. Unfortunately, many users do not have a sound knowledge of these concepts. I have to admit, that this knowledge can become superfluous if, for example, all your data is in the same coordinate system.<\/p>\n

I will discuss here the main notions, but please do remember that I am not a geophisicist, and that this article is the result of all the mistakes I have made (as a geomatician) over the years. Since I am convinced that I have no yet achieved a complete understanding of this topic (otherwise life would become monotonous), the accuracy of my remarks may not be flawless. At the bottom of this page you will find a place to leave your comments, please do it, I am as much interested for me as for future readers of this article. The main problem one has to face when building a geographical information system is to determine the gap between the position of our objects in our system, and their actual position in the world. And, right now, we are faced with the dilemma of how to name this gap: error? uncertainty? precision?<\/p>\n

Each of these terms implies a different notion. And the problem becomes even more complicated when we say that the gap we are talking about is due to the combination of these three concepts (and others that we have not yet named).<\/p>\n

We can, already, say that to determine this gap we must firstly state that its determination will be useful to measure it. Yes, our problem is more complicated than others, but, on the other side, it means that we have a whole range of possibilities to express the location of an entity in the space. The two best-known methodologies is the use of latitudes\/longitudes (geographic coordinate system) or metric XY coordinates (metric coordinate system).<\/p>\n

A significant contribution to the development of GIS tools has been the introduction of the on the fly projection<\/a>: you can use the data in different systems while displaying and processing them into your workspace, the software that performs the necessary coordinates transformations. The issue with these black boxes is that the user has been relieved of the need to know how these transformations are done as well as what they might imply as problems for the quality of the data.<\/p>\n

\u00a0<\/strong>A few \u201cdefinitions\u201d<\/strong><\/h3>\n

The main characteristics of measuring instruments (or metrological properties of measuring devices) are defined within the framework of the International Vocabulary of Metrology and include, among\u00a0others:<\/p>\n

    \n
  • \u00a0the\u00a0resolution<\/li>\n
  • \u00a0the\u00a0accuracy<\/li>\n
  • \u00a0the\u00a0exactitude<\/li>\n<\/ul>\n

    Note that the term \u00ab\u00a0precision\u201d\u00a0is not used\u00a0and\u00a0is considered as a language error in metrology.\u00a0If you search the International Vocabulary of\u00a0Metrology\u00a0\u00a0\u00a0for\u00a0the term \u00ab\u00a0Precision\u00a0\u00bb you will find nothing.\u00a0The \u00a0\u00bb\u00a0precision\u00a0\u201c\u00a0does not exist in\u00a0metrology!<\/p>\n

    On the other hand\u00a0\u00a0<\/u><\/strong>\u00a0the precision<\/u><\/strong>\u00a0\u00a0<\/u>\u00a0is\u00a0very much used in computer science.\u00a0Any measurement made using a device\u00a0gives you\u00a0a result that is never the true value of the quantity being measured.\u00a0Even in the absence of precision given explicitly on\u00a0a\u00a0result, the simple numerical value implies a precision by the only number of significant digits indicated.\u00a0 The figures given in the result are jus those that make sense, which means that the next digit would not make sense in the context of the measurement. This is why we depict 6 decimal places in our geographic position values, for example on a layer in Lambert 93, would mean that the position is accurate to the micrometer!<\/p>\n

    The resolution<\/u><\/strong>\u00a0\u00a0\u00a0does not fall within the scope of\u00a0this\u00a0article.\u00a0The resolution of an apparatus\u00a0is\u00a0the smallest variation of the measured quantity that causes a perceivable variation of the instrument reading.<\/p>\n

    The accuracy<\/u><\/strong>\u00a0is\u00a0also linked to the measuring device.\u00a0The higher the coincidence between the \u201ctrue value\u201d and the instrument reading, the higher the accuracy of the measuring device. It\u00a0has to be highlighted that the accuracy is not expressed by a numerical value.\u00a0It is\u00a0a qualitative assessment of the results.\u00a0Apart from the operating conditions, the accuracy of a device is essentially related to its accuracy and fidelity.\u00a0A device\u00a0is\u00a0accurate if it is both accurate and faithful.<\/p>\n

    We can represent symbolically\u00a0\u00a0\u00a0fidelity<\/u><\/strong>\u00a0,\u00a0\u00a0\u00a0exactitude<\/u><\/strong>\u00a0and\u00a0accuracy<\/u><\/strong>\u00a0\u00a0\u00a0as\u00a0follows:<\/p>\n

    \"\"<\/p>\n

    (Fidelite=Fidelity, Justesse=Exactitude, Exactitude=Accuracy)<\/p>\n

    In the first\u00a0option, the measurements are close to each other (good fidelity) but outside the probability zone of the true value (bad accuracy).<\/p>\n

    In the second\u00a0option, the measurements in the area where the true value is located are the opposite, and the \u00ab\u00a0centre of gravity\u00a0\u00bb of the points is in the centre of the red zone (good accuracy) but although good, the measurements are scattered ( bad fidelity).<\/p>\n

    Finally, the last\u00a0option\u00a0depicts correct measures (in the true value area) and faithfulness (close to each other).\u00a0This is\u00a0the case of a good measuring device, where a correction is, a priori, not necessary and the measurements made with the device are accurate.<\/p>\n

    Our interest resides in the accuracy of the location of our objects in our information system.<\/p>\n

    To understand it, we will track the acquisition of data, from a GPS to its inclusion in\u00a0an\u00a0information system.<\/p>\n

    Astronomical coordinates<\/strong><\/h3>\n

    We\u00a0will\u00a0begin with astronomical latitude and longitude, because even though they will not be useful to us in the context of a GIS, they allow introducing some aspects that we will be using in the other systems.<\/p>\n

    The astronomical latitude (\u00a0\u03a6\u00a0) of a point\u00a0is\u00a0given by the angle formed between the direction of the vertical at this point, and the equatorial plane.<\/p>\n

    Its astronomical (or true) meridian is defined as the plane passing through the vertical\u00a0and\u00a0the Earth rotation axis:<\/p>\n

    \"\"<\/p>\n

    Conventionally, the prime meridian\u00a0is\u00a0\u00a0\u00a0the\u00a0astronomical meridian that passes through the telescope of the former Greenwich Observatory.\u00a0Therefore the longitude of a point (\u00a0\u03bb\u00a0)\u00a0is\u00a0the angle between two planes, one of which is the local meridian (or the meridian plane) and the Greenwich meridian.<\/p>\n

    As you could see in the previous figure, the vertical of the point on the Earth’s surface does not necessarily go through the\u00a0centre\u00a0of the Earth (intersection of the equatorial and meridian planes and the axis of rotation of the Earth).<\/p>\n

    It would always pass through the centre if the Earth were a perfect sphere, but as the Earth’s surface\u00a0is\u00a0a geoids (dented balloon) you can have multiple points with exactly parallel verticals and so … with the same astronomical latitude!<\/p>\n

    The following figure shows four places with the same\u00a0astronomical\u00a0latitude:<\/p>\n

     <\/p>\n

    \"\"<\/p>\n

    Then what is the point? Well, as its name suggests, it is useful in astronomical terms.\u00a0The astronomical\u00a0latitude\u00a0and\u00a0longitude is used to position the stars relative to the place of observation.\u00a0On the other hand they are of no use to locate our entities in\u00a0a GIS.<\/p>\n

    We\u00a0will\u00a0use the first figure to understand another coordinate system.<\/p>\n

    Geocentric coordinates (GPS)<\/strong><\/h3>\n

    If we take the same point on the earth’s surface and use an imaginary line connecting this point to the centre of the Earth, we can calculate the two angles formed (latitude and longitude).\u00a0On the other hand, this line will not be perpendicular to the surface where the point is located.<\/p>\n

    Here is\u00a0a\u00a0diagram with the two types of positioning we have just discussed (\u00a0\u03a6\u00a0: astronomical latitude and\u00a0\u03a6\u00a0‘: geocentric latitude)<\/p>\n

     <\/p>\n

    \"\"<\/p>\n

    Later in this article, we will discuss the third type of latitude (geodesic latitude).<\/p>\n

    The GPS provides geocentric coordinates.<\/p>\n

    What to remember?\u00a0First of all, contrary to common belief, the coordinates are constituted by three values \u200b\u200band not\u00a0two:\u00a0latitude, longitude and height.\u00a0The geocentric\u00a0latitude\u00a0and\u00a0longitude defines the\u00a0dotted\u00a0line (\u00a0\u03a6\u00a0‘) of the figure.\u00a0The height (in meters from the centre of the Earth) defines the position of the point on this line.<\/p>\n

    The second thing to remember is that the whole system\u00a0is\u00a0based on the position of the centre of the Earth (intersection of the rotation axis and the equatorial plane).\u00a0As the earth is not a perfect sphere, the definition of\u00a0this\u00a0centre has to be calculated.\u00a0Inevitably,\u00a0there\u00a0has been a plethora of different calculations made by scientists from different countries.<\/p>\n

    So the GPS system has chosen\u00a0a\u00a0centre of the Earth.\u00a0The WGS84 has been chosen\u00a0and\u00a0 adopted as the standard system for the so-called\u00a0\u00a0\u00a0global systems<\/em><\/strong>\u00a0.\u00a0Global systems, as opposed to local systems, can be used anywhere on Earth.<\/p>\n

    This coordinate system is perfect when defining the position of a point, but presents a problem when used in an information system: it is not possible to accurately calculate a distance between two places on the Earth’s surface, because it would be necessary to know all the hollows and bumps of the geoids between these two points, or at least to know a priori all the heights (third parameter)of the points that can be part of the information system.<\/p>\n

    Since this is very complicated, we have adopted a simpler solution: replace the real surface of the geoids by a smooth and regular surface: an\u00a0ellipsoid.<\/p>\n

    \u00a0<\/strong>Geodetic coordinates<\/strong><\/h3>\n

    An ellipsoid\u00a0is\u00a0as follows:<\/p>\n

     <\/p>\n

    \"\"<\/p>\n

    The geodesic position of a point on the Earth\u00a0surface is\u00a0defined by the ellipsoidal coordinates of the projection of that point on the surface of a reference ellipsoid, along the normal to that ellipsoid.\u00a0The geodesic latitude (\u00a0\u03a6\u00a0)\u00a0is\u00a0defined as the inclination of the equatorial plane normal to ellipsoidal, and the geodesic meridian as the plane passing through the normal axis and the minor axis of the reference ellipsoid.\u00a0Finally, the geodesic longitude (\u00a0\u03bb\u00a0) of a point\u00a0is\u00a0the angle between its meridian plane and the reference meridian.<\/p>\n

     <\/p>\n

    \"\"<\/p>\n

    Unlike the astronomical coordinates of the points on the Earth’s surface, no point of projection on the reference ellipsoid can have identical geodesic coordinates.\u00a0Therefore,\u00a0the same applies for the geodesic coordinates of two points on the Earth surface, unless they are placed on the same normal as the ellipsoid.\u00a0In addition, as with the plane\u00a0and\u00a0surface of the sphere, one can develop formulas and calculate the distance (ellipsoidal) and azimuth between two points whose geodesic coordinates (ellipsoid) are known.\u00a0The reason for using a rotating ellipsoid rather than a plane, sphere or cube as a geodesic reference surface is a matter of convenience (and not theoretical rigour), since the ellipsoid of rotation is the regular geometrical form best fitted to the real shape of the Earth that helps us solve the two basic geodesic problems.\u00a0In other words, one can develop formulas for calculating the distance\u00a0and\u00a0the azimuth between two points of known coordinates and the coordinates of a second point whose azimuth and the distance of a first known point are given.\u00a0The size (that is, the major axis,\u00a0\u00a0<\/em><\/strong>a<\/em><\/strong>\u00a0, and flattening,\u00a0\u00a0\u00a0f<\/em><\/strong>\u00a0) of the reference ellipsoid can be chosen arbitrarily, but the positioning of the ellipsoid relative to the solid Earth\u00a0is\u00a0of crucial importance.\u00a0In classical geodetic practice, this is done by arbitrarily adopting values \u200b\u200bof geodesic latitude, longitude, and height above the ellipsoid for a station of origin (the Paris Pantheon Cross in the case of the New French Triangulation, NTF), and using mathematical formulas that maintain the parallelism between the minor axis of the ellipsoid and the mean axis of rotation of the Earth.<\/p>\n

    It’s\u00a0complicated, but let’s get to the basics, because geodesic coordinates are the coordinates used in GIS.<\/p>\n

    First of all, each geodesic system (and there are plenty)\u00a0is\u00a0characterized by the choice of a centre of the Earth, then by a model of ellipsoid.<\/p>\n

    Take three common systems used for France: the WGS84, the RGF93\u00a0and\u00a0the NTF.\u00a0The first one\u00a0is\u00a0used for UTM a map, the second one is used by Lambert 93 coverage, the third by Lambert NTF projection on IGN (National French Institute) maps.<\/p>\n

     <\/p>\n

    \"\"<\/p>\n

    As you can see, the ellipsoids are different.\u00a0The world according to NTF\u00a0is\u00a0more flattened.\u00a0In the figure we do cannot appreciate the second difference, which\u00a0is\u00a0the position of the centre of the Earth.<\/p>\n

    The two systems locate the centre of the Earth in two different places:<\/p>\n

     <\/p>\n

    \"\"<\/p>\n

    They are not \u201c\u00a0too \u201c far away\u00a0 but still they are separated by 168m in X, 60m in Y and 320m in Z. This difference makes that the same position on the Earth surface (same Latitude and Longitude) be separated by a distance of up to a hundred meters .<\/p>\n

    But\u00a0now, we have another problem.\u00a0Using\u00a0an\u00a0ellipsoid means that we will only have two parameters to define the position of a point: latitude and longitude. The third parameter, the Height (that we used with the geocentric coordinates) disappears because any position\u00a0is\u00a0brought back to the surface of the ellipsoid, the point P of the following figure.<\/p>\n

     <\/p>\n

    \"\"<\/p>\n

    In\u00a0a\u00a0geodesic coordinate system, the position of the point M becomes the point P. We move from a three-dimensional system (XYZ) to\u00a0a\u00a0two-dimensional system (XY).\u00a0To work in 3D\u00a0it is\u00a0necessary to give a third element that can be either the ellipsoidal height or the altitude.\u00a0Altitudes are measured relative to the surface of the geoids.\u00a0In order to know them,\u00a0you\u00a0have to measure them on the spot. On the other hand\u00a0a\u00a0GPS provides the ellipsoidal height H, because it can calculate the difference between the Z measured in geocentric coordinates and the terrestrial surface formula adopted with the ellipsoid.<\/p>\n

    In the previous figure\u00a0it\u00a0should be understood that H = h + N.\u00a0But if you work with a GPS only H can be measured.<\/p>\n

    In the following article we will focus on the factors that affect the quality of data in\u00a0a\u00a0geographic information system.<\/p>\n","protected":false},"excerpt":{"rendered":"

    It is not easy, in geomatics, to get a good grasp of the geodesy notions that underlie the manipulation of geographic data. Unfortunately, many users do not have a sound knowledge of these concepts. I have…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"give_campaign_id":0,"_bbp_topic_count":0,"_bbp_reply_count":0,"_bbp_total_topic_count":0,"_bbp_total_reply_count":0,"_bbp_voice_count":0,"_bbp_anonymous_reply_count":0,"_bbp_topic_count_hidden":0,"_bbp_reply_count_hidden":0,"_bbp_forum_subforum_count":0,"sfsi_plus_gutenberg_text_before_share":"","sfsi_plus_gutenberg_show_text_before_share":"","sfsi_plus_gutenberg_icon_type":"","sfsi_plus_gutenberg_icon_alignemt":"","sfsi_plus_gutenburg_max_per_row":"","_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[1260],"tags":[],"class_list":["post-5129","post","type-post","status-publish","format-standard","hentry","category-non-classe-en"],"aioseo_notices":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p6XU0A-1kJ","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/posts\/5129","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/comments?post=5129"}],"version-history":[{"count":0,"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/posts\/5129\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/media?parent=5129"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/categories?post=5129"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/tags?post=5129"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}