第二章 GPS 定位相关基本知识

1. 第二章 GPS定位相关基本知识 • 围绕卫星、地面测站的位置信息需要设定参考的空间和时间系统以及对卫星轨道精确计算。坐标系统与时间系统是描述卫星运动，处理观测数据和表达定位结果的数学与物理基础。因此，了解和掌握一些常用坐标系统和时间系统，熟悉它们各自间的转换关系，对GPS用户来说，是极为重要的。

2. 基础知识 地球的形状和大小

3. 基础知识 地球的形状和大小

4. 基础知识 地球的形状和大小

5. 基础知识 地球的形状和大小

6. 基础知识 地球的形状和大小 地球形状和大小 1.地球是一个表面起伏较大的椭球 地球表面最高峰： 8844.43m 海洋底部最深处: 11022.00m 地球表面最大高差近20km 2. 地球又是一个近似光滑的水球 大陆面积: 占29% 海洋面积: 占71 % 3. 地球平均半径: 6371km

7. 地球表面虽然很不规则，有高山、平原、丘陵、海洋等。但这些起伏相对于地球本身十分微小。地球表面虽然很不规则，有高山、平原、丘陵、海洋等。但这些起伏相对于地球本身十分微小。

8. 抽象 地球的形状

9. 地球的形状 地球表面 实际地球 大地体 大地水准面 旋转椭球体 旋转椭球面

10. 重力的方向线称为铅垂线 地球的形状和大小 基本概念 1. 重力方向线 即铅垂线, 是测量工作的基准线 2. 水准面 自由静止的水面; 是等位面, 有无数个 离心力 地心引力 重力G 地心O

11. 大地水准面 • 设想当海洋处于静止均衡状态时，将它延伸到陆地内部所形成的封闭曲面。 陆地 大地水准面 静止海水面

12. 地球的形状和大小 Z 基本概念 旋转椭球 与大地体非常接近的 数学椭球 长半径为a，短半径为b 扁率 数学模型 地球平均半径 R=6371km Y X

13. 地球椭球——参考椭球体 • 旋转椭球理论上是唯一的数学球体 • 旋转椭球参数，难以全球统一确定；各国自己测定并采用的旋转椭球称为参考椭球 • 同时顾及地球几何参数和物理参数的旋转椭球称为地球椭球体，又称为参考椭球体 • 参考椭球面是测量计算和制图的基准面

14. §2-1 空间参考坐标系统介绍 • 空间系统即空间参考坐标系，需要确定原点、坐标轴指向、长度单位。 • GPS测量与应用中，常采用直角坐标系统及其相应的大地坐标系，取地球的质心作为原点。根据坐标轴取向不同： • 一类是地球坐标系，该系坐标系是固结在地球上的，随地球一起转动，故又称为地固坐标系。它是一种非惯性坐标系，对于表述点的位置和处理GPS观测结果是十分方便的。地固坐标系有多种表达形式，对GPS测量来说，最基本的是以地球质心为原点的地心坐标系。 • 第二类是天球坐标系，该类坐标系与地球自转无关，称空固坐标系，对于描述卫星的运行状态、确定卫星轨道是极其方便的。 • 坐标系 相互转换

15. Coordinate Systems • Navigation: knowing where you are, where you want to go, and how to get there • – Also useful: knowing how long it will take • – To achieve these goals in a general way; a coordinate system is needed that allow quantitative calculations • Reference Frames (describes coordinate system basis) • –Definition • –Realization (implementation of definition)

16. Coordinate system definition • Definition of a 3D set of axes requires: – An origin (3 quantities) – An orientation (3 quantities) – A scale (1 quantity) • (A “Helmert ” transformation estimates these 7 quantities to relate two reference frames). • For the Earth; terrestrial frames come in two forms: – Geometric (mathematical description) – Potential field based (gravity and magnetic)

17. Simplest Global Reference Frame • Geometric: • Origin at the center of mass of the Earth; • Orientation defined by a Z axis near the rotation axis; • one “Meridian” (plane containing the Z-axis) defined by a convenient location such as Greenwich, England. • Coordinate system would be Cartesian XYZ.

18. Simple System • The use of this type of simple system is actually a recent development and is the most common system used in GPS. • Until the advent of modern “space-based geodetic systems” (mid-1950s), coordinate systems were much more complicated and based on the gravity field of the Earth.

19. Potential based coordinate systems • The basic reason is “realization”: Until distance measurements to earth-orbiting satellites and galactic-based distance measurements, it was not possible to actually implement the simple type measurement system. • Conventional (and still today) systems rely on the direction of the gravity vector • We think in two different systems: A horizontal one (how far away is something) and a vertical one (height differences between points).

20. Conventional Systems • Conventional coordinate systems are a mix of geometric systems (geodetic latitude and longitude) and potential based systems (Orthometric heights). • The origin of conventional systems are also poorly defined because determining the position of the center of mass of the Earth was difficult before the first Earth-orbiting artificial satellite. (The moon was possible before but it is far enough away that sensitivity center of mass of the Earth was too small).

21. Simple Geocentric Latitude and Longitude • The easiest form of latitude and longitude to understand is the spherical system: • Latitude: Angle between the equatorial plane and the point. Symbol φc • Latitude is also the angle between the normal to the sphere and the equatorial plane • Related term: co-latitude = 90o-latitude. Symbol θc. Angle from the Z-axis • Longitude: Angle between the Greenwich meridian and meridian of the location. Symbol λc

22. Geocentric relationship to XYZ • One of the advantages of geocentric angles is that the relationship to XYZ is easy. R is taken to be radius of the sphere and H the height above this radius

23. Problem with Geocentric • Geocentric measures are easy to work with but they have several serious problems • The shape of the Earth is close to an bi-axial ellipsoid (i.e., an ellipse rotated around the Z-axis) • The flattening of the ellipsoid is ~1/300 (1/298.257222101 is the defined value for the GPS ellipsoid WGS-84). • Flattening is (a-b)/a where a is the semi-major axis and b is the semi-minor axis. • Since a=6378.137 km (WGS-84), a-b=21.384 km

24. Geocentric quantities • If the radius of the Earth is taken as b (the smallest radius), then Hc for a site at sea-level on the equator would be 21km (compare with Mt. Everest 28,000feet~8.5km). • Geocentric quantities are never used in any large scale maps and geocentric heights are never used. • We discuss heights in more in next class and when we do spherical trigonometry we will use geocentric quantities.

25. Ellipsoidal quantities • The most common latitude type seen is geodetic latitude which is defined as the angle between the normal to the ellipsoid and the equatorial plane. We denote with subscript g. • Because the Earth is very close to a biaxial ellipsoid, geodetic longitude is the same as geocentric longitude (the deviation from circular in the equator is only a few hundred meters: Computed from the gravity field of the Earth).

26. Geodetic Latitude

27. Relationship between φg and XYZ • This conversion is more complex than for the spherical case.

28. Inverse relationship • The inverse relationship between XYZ and geodetic latitude is more complex (mainly because to compute the radius of curvature, you need to know the latitude). • A common scheme is iterative:

29. Closed form expression for small heights

30. Other items • A discussion of geodetic datum and coordinate systems can be found at: http://www.colorado.edu/geography/gcraft/notes/datum/datum.html • Geodetic longitude can be computed in that same way as for geocentric longitude • Any book on geodesy will discuss these quanititiesin more detail (also web searching on geodetic latitude will return many hits). • The difference between astronomical and geodetic latitude and longitude is called “deflection of the vertical”

31. Astronomical latitude and longitude • These have similar definitions to geodetic latitude and longitude except that the vector used is the direction of gravity and not the normal to the ellipsoid (see earlier figure). • There is not direct relationship between XYZ and astronomical latitude and longitude because of the complex shape of the Earth’s equipotential surface. • In theory, multiple places could have the same astronomical latitude and longitude. • As with the other measures, the values of depend on the directions of the XYZ coordinate axes.

32. 天文经度、天文纬度和天文方位角 天文经度：包含测站垂线的子午面与起始子午面的夹角； 天文纬度：测站垂线的与赤道面的夹角； 天文方位角：包含测站垂线的子午面与测站垂线和照准面所张成的垂直面的夹角； 天文天顶距：测站垂线与观测方向的夹角

33. Coordinate axes directions • The origin of the XYZ system these days is near the center of mass of the Earth deduced from the gravity field determined from the orbits of geodetic satellites (especially the LAGEOS I and II satellites). • The direction of Z-axis by convention is near the mean location of the rotation axis between 1900-1905. At the time, it was approximately aligned with the maximum moments of inertia of the Earth.

34. Heights (Altitude) • Definition of heights • –Ellipsoidal height (geometric) • –Orthometric height (potential field based) • Shape of equipotential surface: Geoid for Earth • Methods for determining heights

35. Ellipsoidal heights • Calculation of ellipsoid heights from Cartesian XYZ was covered. • The ellipsoid height is the distance along the normal to the reference ellipsoid from the surface of the ellipsoid to the point who height is being calculated. • While the geometric quantities, geodetic latitude and longitude are used for map mapping and terrestrial coordinates in general; ellipsoidal height is almost never used (although this is changing with the advent of GPS) • Why is ellipsoidal height not used?

36. Orthometric heights • The problem with ellipsoidal heights are: –They are new: Ellipsoidal heights could only be easily determined when GPS developed (1980’s) –Geometric latitude and longitude have been around since Snell (optical refraction) developed triangulation in the 1500’s. –Primary reason is that fluids flow based on the shape of the equipotential surfaces. If you want water to flow down hill, you need to use potential based heights.

37. Orthometric heights • Orthometric heights are heights above an equipotential surface • The equipotential surface is called the geoid and corresponds approximately to mean sea level (MSL). • The correspondence is approximately because MSL is not an equipotential surface because of forces from dynamic ocean currents (e.g., there is about 1m drop over the Gulf stream which is permanently there but change magnitude depending on the strength of the current)

38. Mean Sea Level (MSL) • Ocean tides also need to be considered but this can be averaged over time (signal is periodic with semi-diurnal, diurnal and long period tides. Longest period tide is 18.6 years) • Another major advantage of MSL is that is has been monitored at harbors for many centuries in support of ocean going vessels • Also poses a problem because dredging of harbors can change the tides. • Land-locked countries had to rely on other countries to tell them the heights at the border. •MSL is reasonably consistent around the world and so height datums differ by only a few meters (compared to hundreds of meters for geodetic latitude and longitude.

39. Height determination • For article on the development of the US height system see: http://www.pobonline.com/CDA/ArticleInformation/features/BNP__Features__Item/0,2338,13022,00.html • Height measurements historically are very labor intensive • The figure on the next page shows how the technique called leveling is used to determine heights. • In a country there is a primary leveling network, and other heights are determined relative to this network. • The primary needs to have a monument spacing of about 50 km.

40. Leveling • The process of leveling is to measure height differences and to sum these to get the heights of other points. Orthometric height of hill is Δh1+Δh2+Δh3 N is Geoid Height. Line at bottom is ellipsoid

41. Leveling • Using the instrument called a level, the heights on the staffs are read and the difference in the values is the height differences. • The height differences are summed to get the height of the final point. • For the primary control network: the separation of the staffs is between 25-50 meters. • This type of chain of measurements must be stepped across the whole country (i.e., move across the country in 50 meter steps: Takes decades and was done).

42. Leveling problems • Because heights are determined by summing differences, system very prone to systematic errors; small biases in the height differences due to atmospheric bending, shadows on the graduations and many other types of problem • Instrument accuracy is very good for first-order leveling: Height differences can be measured to tens of microns. • Accuracy is thought to about 1 mm-per-square-root-km for first order leveling. • Changes in the shapes of the equipotential surface with height above MSL also cause problems. • The difference between ellipsoidal height and Orthometric heightis the Geoid height

43. Trigonometric Leveling • When trying to go the tops of mountains, standard leveling does not work well. (Image trying to do this to the summit of Mt. Everest). • For high peaks: A triangulation method is used call trigonometric leveling. • Schematic is shown on the next slide • This is not as accurate as spirit leveling because of atmospheric bending.

44. Trigonometric Leveling schematic • Method for trigonometric leveling. Method requires that distance D in known and the elevation angles are measured. Trigonometry is used to compute Δh

45. Trigonometric Leveling • In ideal cases, elevation angles at both ends are measured at the same time. This helps cancel atmospheric refraction errors. • The distance D can be many tens of kilometers. • In the case of Mt. Everest, D was over 100 km (the survey team was not even in the same country; they were in India and mountain is in Nepal). • D is determined either by triangulation or after 1950 by electronic distance measurement (EDM) discussed later • The heights of the instruments, called theodolites, above the ground point must be measured. Note: this instrument height measurement was not needed for leveling.

46. Geoid height • Although the difference between ellipsoidal and orthometric height allows the geoid height to be determined, this method has only be been used since GPS became available. • Determining the geoid has been historical done using surface gravity measurements and satellite orbits. • Satellite orbit perturbations reveal the forces acting on the satellite which if gravity is the only effect is the first derivative of the potential (atmospheric drag and other forces can greatly effect this assumption)

47. Geoid height • The long wavelength part of geoid (greater than 1000km) is now determined from satellite orbit perturbations. • The <1000km wavelength use surface gravity and solve a boundary value problem where the derivative of the function which satisfies Laplace’s equation is given on the boundary, and the value of the function is needed. • Most of the great mathematicians worked on field theory trying to solve the Earth boundary value problem (Laplace, Legendre, Green, Stokes) • The standard method of converting gravity measurements to geoid height estimates is called Stokes method. • This field is called physical geodesy

48. 一、天球坐标系 1．定义 • 概念：赤道、黄道、春分点、天极 • 天文学上为了与人们的直观感觉相适应，把天空假想成一个巨大的球面，这便是天球。所谓天球，指的是以地球质心为球心，以无限大长度为半径的一个假想球体。 • 天球只是人们的一种假设，是一种“理想模型”，引入天球这一概念，只是为了确定天体位置等方面的需要。 • 地球自转轴的延长线与天球的两个交点称为天极，分为北天极和南天极。 • 天顶：观察者所在位置垂直上方在天球上的点

49. 通过地球质心与天轴垂直的平面称为天球赤道面。通过地球质心与天轴垂直的平面称为天球赤道面。 • 地球公转的轨道面与天球相交的大圆称为黄道，黄道与天球赤道有两个交点，其中，太阳的视位置由南到北的交点称为春分点。