8. 坐标参考系

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目标:

了解坐标参考系。

关键词:

坐标参考系(CRS),地图投影,动态投影,纬度,经度,北距,东距

8.1. 概述

Map projections 试着在一张纸或电脑屏幕上描绘地球表面或地球的一部分。用外行人的话说,地图投影试图将地球从球形(3D)转变为平面(2D)。

A coordinate reference system (CRS)然后定义您的地理信息系统中的二维投影地图如何与地球上的真实位置相关。决定使用哪种地图投影和CRS取决于您要处理的区域的区域范围、您要进行的分析,而且通常还取决于数据的可用性。

8.2. 详细的地图投影

表示地球形状的一种传统方法是使用地球仪。然而,这种方法存在一个问题。尽管地球仪保留了地球的大部分形状,并展示了大陆大小的特征的空间形状,但它们很难放在口袋里。它们也只能在极小的比例(例如1:1亿)下方便地使用。

在地理信息系统中常用的专题地图数据大多具有相当大的比例尺。典型的地理信息系统数据集具有1:25万或更大的比例,具体取决于细节级别。这种大小的地球仪制造起来既困难又昂贵,而且更难随身携带。因此,制图员开发了一套名为 map projections 旨在以合理的精确度显示二维的球形地球。

近距离观察时,地球似乎相对平坦。然而,当我们从太空中观察时,我们可以看到地球是相对球形的。正如我们将在接下来的地图制作主题中看到的那样,地图是现实的表现。它们不仅被设计成代表特征,而且还代表了它们的形状和空间安排。每个地图投影都有 advantagesdisadvantages 。地图的最佳投影取决于 scale 地图,以及它将被用于的目的。例如,如果用于绘制整个非洲大陆的地图,投影可能具有不可接受的扭曲,但可能是一个很好的选择 large-scale (detailed) map 你的国家。地图投影的属性也可能影响地图的某些设计特征。有些投影适用于小区域,有些适用于绘制东西向范围较大的区域,有些则更适合绘制南北范围较大的区域。

8.3. 地图投影的三个家族

The process of creating map projections is best illustrated by positioning a light source inside a transparent globe on which opaque earth features are placed. Then project the feature outlines onto a two-dimensional flat piece of paper. Different ways of projecting can be produced by surrounding the globe in a cylindrical fashion, as a cone, or even as a flat surface. Each of these methods produces what is called a map projection family. Therefore, there is a family of planar projections, a family of cylindrical projections, and another called conical projections (see 图 8.16)

../../_images/projection_families.png

图 8.16 地图投影的三个家族。它们可以由a)柱面投影、b)圆锥投影或c)平面投影来表示。

当然,今天,将球形地球投影到一张平板纸上的过程是利用几何和三角的数学原理完成的。这重建了光在地球上的物理投影。

8.4. 地图投影的准确性

Map projections are never absolutely accurate representations of the spherical earth. As a result of the map projection process, every map shows distortions of angular conformity, distance and area. A map projection may combine several of these characteristics, or may be a compromise that distorts all the properties of area, distance and angular conformity, within some acceptable limit. Examples of compromise projections are the Winkel Tripel projection and the Robinson projection (see 图 8.17), which are often used for producing and visualizing world maps.

../../_images/robinson_projection.png

图 8.17 罗宾逊投影是一种折衷方案,其中面积、角度一致性和距离的扭曲都是可以接受的。

通常不可能在地图投影中同时保留所有特征。这意味着当您想要执行准确的分析操作时,您需要使用为您的分析提供最佳特征的地图投影。例如,如果需要在地图上测量距离,则应尝试对数据使用可提供高精度距离的地图投影。

8.4.1. 角度一致的地图投影

使用地球仪时,指南针的主要方向(北、东、南、西)将始终彼此成90度角。换句话说,东面总是与北面成90度角。保持正确 angular properties 也可以保存在地图投影上。保留角度一致性这一属性的地图投影称为 conformalorthomorphic projection

These projections are used when the preservation of angular relationships is important. They are commonly used for navigational or meteorological tasks. It is important to remember that maintaining true angles on a map is difficult for large areas and should be attempted only for small portions of the earth. The conformal type of projection results in distortions of areas, meaning that if area measurements are made on the map, they will be incorrect. The larger the area the less accurate the area measurements will be. Examples are the Mercator projection (as shown in 图 8.18) and the Lambert Conformal Conic projection. The U.S. Geological Survey uses a conformal projection for many of its topographic maps.

../../_images/mercator_projection.png

图 8.18 例如,墨卡托投影用于角度关系很重要,但面积关系扭曲的地方。

8.4.2. 等距离地图投影

If your goal in projecting a map is to accurately measure distances, you should select a projection that is designed to preserve distances well. Such projections, called equidistant projections, require that the scale of the map is kept constant. A map is equidistant when it correctly represents distances from the centre of the projection to any other place on the map. Equidistant projections maintain accurate distances from the centre of the projection or along given lines. These projections are used for radio and seismic mapping, and for navigation. The Plate Carree Equidistant Cylindrical (see 图 8.19) and the Equirectangular projection are two good examples of equidistant projections. The Azimuthal Equidistant projection is the projection used for the emblem of the United Nations (see 图 8.20).

../../_images/plate_carree_projection.png

图 8.19 例如,当精确的距离测量很重要时,就使用托盘等距柱面投影。

../../_images/azimuthal_equidistant_projection.png

图 8.20 联合国徽标使用方位等距投影。

8.4.3. 面积相等的投影

When a map portrays areas over the entire map, so that all mapped areas have the same proportional relationship to the areas on the Earth that they represent, the map is an equal area map. In practice, general reference and educational maps most often require the use of equal area projections. As the name implies, these maps are best used when calculations of area are the dominant calculations you will perform. If, for example, you are trying to analyse a particular area in your town to find out whether it is large enough for a new shopping mall, equal area projections are the best choice. On the one hand, the larger the area you are analysing, the more precise your area measures will be, if you use an equal area projection rather than another type. On the other hand, an equal area projection results in distortions of angular conformity when dealing with large areas. Small areas will be far less prone to having their angles distorted when you use an equal area projection. Alber's equal area, Lambert's equal area and Mollweide Equal Area Cylindrical projections (shown in 图 8.21) are types of equal area projections that are often encountered in GIS work.

../../_images/mollweide_equal_area_projection.png

图 8.21 例如,Mollweide等面积圆柱形投影确保所有贴图区域与地球上的区域具有相同的比例关系。

请记住,地图投影是一个非常复杂的主题。全世界有数百种不同的投影,每一种都试图在一张扁平的纸上尽可能忠实地描绘地球表面的某一部分。在现实中,选择使用哪种投影通常会为您做出选择。大多数国家都有常用的预测,当数据交换时,人们会遵循 national trend

8.5. 详细介绍坐标系(CRS)

在坐标参考系(CRS)的帮助下,地球上的每个地方都可以用一组三个数字来指定,称为坐标。一般说来,CRS可以分为 projected coordinate reference systems (也称为笛卡尔或直角坐标系)和 geographic coordinate reference systems

8.5.1. 地理坐标系

地理坐标参考系的使用非常普遍。它们使用纬度和经度,有时还使用高度值来描述地球表面的位置。最受欢迎的是叫 WGS 84

Lines of latitude run parallel to the equator and divide the earth into 180 equally spaced sections from North to South (or South to North). The reference line for latitude is the equator and each hemisphere is divided into ninety sections, each representing one degree of latitude. In the northern hemisphere, degrees of latitude are measured from zero at the equator to ninety at the north pole. In the southern hemisphere, degrees of latitude are measured from zero at the equator to ninety degrees at the south pole. To simplify the digitisation of maps, degrees of latitude in the southern hemisphere are often assigned negative values (0 to -90°). Wherever you are on the earth’s surface, the distance between the lines of latitude is the same (60 nautical miles). See 图 8.22 for a pictorial view.

../../_images/geographic_crs.png

图 8.22 地理坐标系,其纬线平行于赤道,经线以本初子午线穿过格林威治。

Lines of longitude, on the other hand, do not stand up so well to the standard of uniformity. Lines of longitude run perpendicular to the equator and converge at the poles. The reference line for longitude (the prime meridian) runs from the North pole to the South pole through Greenwich, England. Subsequent lines of longitude are measured from zero to 180 degrees East or West of the prime meridian. Note that values West of the prime meridian are assigned negative values for use in digital mapping applications. See 图 8.22 for a pictorial view.

在赤道,并且只在赤道,一条经线表示的距离等于由一度纬度表示的距离。随着你向两极移动,经线之间的距离逐渐变小,直到在电线杆的确切位置,所有360度的经度由一个你可以把手指放在上面的点表示(尽管你可能想戴上手套)。使用地理坐标系,我们有一个线网,将地球分成几个正方形,在赤道处覆盖大约12363.365平方公里-这是一个很好的开始,但对于确定正方形内任何东西的位置来说,用处不是很大。

To be truly useful, a map grid must be divided into small enough sections so that they can be used to describe (with an acceptable level of accuracy) the location of a point on the map. To accomplish this, degrees are divided into minutes (') and seconds ("). There are sixty minutes in a degree, and sixty seconds in a minute (3600 seconds in a degree). So, at the equator, one second of latitude or longitude = 30.87624 meters.

8.5.2. 投影坐标参考系

A two-dimensional coordinate reference system is commonly defined by two axes. At right angles to each other, they form a so called XY-plane (see 图 8.23 on the left side). The horizontal axis is normally labelled X, and the vertical axis is normally labelled Y. In a three-dimensional coordinate reference system, another axis, normally labelled Z, is added. It is also at right angles to the X and Y axes. The Z axis provides the third dimension of space (see 图 8.23 on the right side). Every point that is expressed in spherical coordinates can be expressed as an X Y Z coordinate.

../../_images/projected_crs.png

图 8.23 二维和三维坐标参考系。

南半球(赤道以南)的投影坐标参考系通常以赤道上的特定位置为原点 Longitude 。这意味着Y值向南增加,X值向西增加。在北半球(赤道以北),原点也是赤道在一个特定的位置 Longitude 。然而,现在Y值向北增加,X值向东增加。在接下来的部分中,我们将描述一种投影坐标参考系,称为通用横坐标墨卡托(UTM),通常用于南非。

8.6. 万能横向墨卡托(UTM)CRS详细介绍

The Universal Transverse Mercator (UTM) coordinate reference system has its origin on the equator at a specific Longitude. Now the Y-values increase southwards and the X-values increase to the West. The UTM CRS is a global map projection. This means, it is generally used all over the world. But as already described in the section 'accuracy of map projections' above, the larger the area (for example South Africa) the more distortion of angular conformity, distance and area occur. To avoid too much distortion, the world is divided into 60 equal zones that are all 6 degrees wide in longitude from East to West. The UTM zones are numbered 1 to 60, starting at the antimeridian (zone 1 at 180 degrees West longitude) and progressing East back to the antemeridian (zone 60 at 180 degrees East longitude) as shown in 图 8.24.

../../_images/utm_zones.png

图 8.24 宇宙横向墨卡托地带。对于南非,使用UTM区域33S、34S、35S和36S。

As you can see in 图 8.24 and 图 8.25, South Africa is covered by four UTM zones to minimize distortion. The zones are called UTM 33S, UTM 34S, UTM 35S and UTM 36S. The S after the zone means that the UTM zones are located south of the equator.

../../_images/utm_for_sa.png

图 8.25 UTM带33S、34S、35S和36S及其中央经度(子午线)用于高精度地预测南非。红十字会显示了一个感兴趣的区域(Aoi)。

Say, for example, that we want to define a two-dimensional coordinate within the Area of Interest (AOI) marked with a red cross in 图 8.25. You can see, that the area is located within the UTM zone 35S. This means, to minimize distortion and to get accurate analysis results, we should use UTM zone 35S as the coordinate reference system.

The position of a coordinate in UTM south of the equator must be indicated with the zone number (35) and with its northing (Y) value and easting (X) value in meters. The northing value is the distance of the position from the equator in meters. The easting value is the distance from the central meridian (longitude) of the used UTM zone. For UTM zone 35S it is 27 degrees East as shown in 图 8.25. Furthermore, because we are south of the equator and negative values are not allowed in the UTM coordinate reference system, we have to add a so called false northing value of 10,000,000 m to the northing (Y) value and a false easting value of 500,000 m to the easting (X) value. This sounds difficult, so, we will do an example that shows you how to find the correct UTM 35S coordinate for the Area of Interest.

8.6.1. 北距(Y)值

我们要查找的位置是赤道以南3,550,000米,因此北距(Y)值得到一个 negative sign 和-3,550,000米。根据UTM的定义,我们必须添加一个 false northing value 这意味着我们坐标的北距(Y)值是6,450,000米(-3,550,000米+10,000,000米)。

8.6.2. 东距(X)值

First we have to find the central meridian (longitude) for the UTM zone 35S. As we can see in 图 8.25 it is 27 degrees East. The place we are looking for is 85,000 meters West from the central meridian. Just like the northing value, the easting (X) value gets a negative sign, giving a result of -85,000 m. According to the UTM definitions we have to add a false easting value of 500,000 m. This means the easting (X) value of our coordinate is 415,000 m (-85,000 m + 500,000 m). Finally, we have to add the zone number to the easting value to get the correct value.

因此,我们的坐标 Point of Interest ,投影于 UTM zone 35S 将写成: 35 415,000 m E / 6,450,000 m N 。在某些GIS中,如果定义了正确的UTM带35S并在系统内将单位设置为米,则坐标也可能简单地显示为 415,000 6,450,000

8.7. 即时投影

正如您可能想象的那样,可能会出现这样一种情况,即要在GIS中使用的数据投影到不同的坐标参考系中。例如,您可能会获得一个显示以UTM 35S投影的南非边界的矢量图层,以及另一个包含地理坐标系WGS 84中提供的有关降雨的点信息的矢量图层。在地理信息系统中,这两个矢量层被放置在地图窗口的完全不同的区域,因为它们具有不同的投影。

为解决此问题,许多GIS都包含一个名为的功能 on-the-fly 投影。这意味着,你可以 define 启动GIS时的某个投影和随后加载的所有层,无论它们具有什么坐标系,都将自动显示在您定义的投影中。此功能允许您在GIS的地图窗口内叠加图层,即使它们可能位于中 different 参考系。在QGIS中,默认情况下应用此功能。

8.8. 需要注意的常见问题/事情

该主题 map projection 是非常复杂的,即使是研究过地理学、大地测量学或任何其他与地理信息系统相关的科学的专业人员,也经常在正确定义地图投影和坐标参考系方面遇到问题。通常,当您使用GIS时,您已经有了投影数据。在大多数情况下,这些数据将被投影到某个CRS中,因此您不必创建新的CRS,甚至不必将数据从一个CRS重新投影到另一个CRS。也就是说,了解地图投影和CRS的含义总是很有用的。

8.9. 我们学到了什么?

让我们总结一下我们在此工作表中介绍的内容:

  • Map projections 在一张二维的平面纸或计算机屏幕上描绘地球表面。

  • 有全球地图投影,但大多数地图投影是创建的,并且 optimized to project smaller areas 地球表面的。

  • 地图投影从来都不是球状地球的绝对准确表示。他们展示了 distortions of angular conformity, distance and area. 在地图投影中同时保留所有这些特征是不可能的。

  • A Coordinate reference system (CRS)定义了在坐标的帮助下,二维投影地图如何与地球上的真实位置相关。

  • 有两种不同类型的坐标参考系: Geographic Coordinate SystemsProjected Coordinate Systems

  • On the Fly 投影是GIS中的一项功能,它允许我们覆盖层,即使它们是在不同的坐标参考系中投影的。

8.10. 现在你来试试吧!

以下是一些建议,可供您尝试与您的学习者:

  1. 启动QGIS

  2. 在……里面 Project ► Properties... ► CRS 检查 No projection (or unknown/non-Earth projection)

  3. 加载同一区域但具有不同投影的两个层

  4. 让你的学生找出两层纸上几个地方的坐标。您可以向他们表明,不可能覆盖这两个层。

  5. 然后将坐标系定义为地理/WGS 84 Project Properties 对话框

  6. 再次加载同一区域的两个层,并让您的学生了解如何为项目设置CRS(因此,启用“即时”投影)。

  7. 您可以打开 Project Properties 对话框并向您的学生展示许多不同的坐标参考系,以便他们对本主题的复杂性有一个了解。您可以选择不同的CRSS来在不同的投影中显示相同的层。

8.11. 一些值得思考的事情

如果你没有可用的计算机,你可以向你的学生展示三个地图投影族的原理。拿出一个地球仪和一张纸,演示柱面、圆锥和平面投影一般是如何工作的。借助透明工作表,您可以绘制一个显示X轴和Y轴的二维坐标参考系。然后,让您的学生定义不同位置的坐标(X和Y值)。

8.12. 进一步阅读

Books

  • 张康宗(2006)。地理信息系统概论。第三版。麦格劳·希尔。ISBN:0070658986

  • 迈克尔·N·德默斯(2005)。地理信息系统基础。第三版。威利。ISBN:9814126195

  • Galati,Stephen R.(2006):地理信息系统揭开神秘面纱。Artech House Inc.ISBN:158053533X

Websites

《QGIS用户指南》还提供了有关在QGIS中使用地图投影的更多详细信息。

8.13. 下一步是什么?

在接下来的部分中,我们将更仔细地了解 Map Production