Fundamentals of Surveying (LE/ESSE )

Transcription

1 Fundamentals of Surveying (LE/ESSE ) Lecture 2 Basics of Surveying Dr.-Ing. Jian-Guo Wang Geomatics Engineering York University Fall

2 2-1. Overview Part 1: Basics - The Earth s Shape & Size. - Coordinate Systems and Geodetic Positioning - Map Projections - The Transverse Mercator Projections and UTM - Elevations - Limitation of the use of plane instead of level surface -Maps - Field and Office Works Part 2: Theory of Errors in Observations (lecture 3) 2

3 2-2. The Earth s Shape and Size The Geoid & the Reference Ellipsoids 1. The Problem The main objective of surveying: studying the natural surface of the Earth The Characteristics of the Earth: a) Irregular surface (high mountains, hills, plains, rivers, lakes, oceans). The highest point: 8,848 m (Mount Everest or Zhumulangma Peak) The deepest point: -11,000m (Mariana Trench) 71% of the Earth s surface is covered by oceans. b) The distribution density of Earth s mass is not uniform. The Earth s gravity is as the combined effect of its gravitation and centrifugal force (next slide). c) The earth is not a SPHERE, but flattened slightly at the poles and bulging somehow at the Equator. d) Our knowledge about the Earth is getting better and better, but still limited. e) In engineering practice one has to make a lot of assumptions to approximate the real characteristics of the Earth! Need for Geodetic Datum(s) (The Model of the Earth) to define the size and shape of the earth and the origin and orientation of the coordinate systems used to map the earth (the reference for the mathematical reduction of geodetic and cartographic data, and field measurements.) 3

4 2-2. The Earth s Shape and Size 1. The Problem a). Irregular surface (high mountains, hills, plains, rivers, lakes, oceans). The highest point: 8,848 m (Mount Everest or Zhumulangma Peak) Optional for reading! 4

5 2-2. The Earth s Shape and Size 1. The Problem a. Irregular surface (high mountains, hills, plains, rivers, lakes, oceans). The deepest point: -11,000m (Mariana Trench in the western Pacific Ocean) Optional for reading! 5

6 2-2. The Earth s Shape and Size Centrifugal force Gravitation: a phenomenon through which all objects attract each other The Earth s gravitation: the Earth is surrounded by its own gravitational field, which exerts an attractive force on any object. The Earth s Centrifugal Force: due to the Earth's rotation is inseparably superimposed on the attraction The Earth s Gravity: (1). The combined effect of its gravitation and centrifugal force. (2). Direction: vertical plummet line (not towards the center of the Earth s mass). (3). Perpendicular to the gravity potential surface. 6

7 2-2. The Earth s Shape and Size 2. The Geoid (Reference Surface for Heights) a. The Canadian VERTICAL DATUM is CGVD28. CGVD2013 the Mean Sea Level (MSL) - The average level of the ocean surface halfway between the highest and lowest levels recorded, as a plane upon which we can reference or describe the heights of features on, above or below the ground. b. The sea level surface is connected with the Geoid, which is a horizontal surface shaped by the gravity field of the earth (theoretical only) - The natural extension of the MSL surface under the landmass. - The surface of the water in the trench would represent the geoid if we allowed the trench to fill with sea water. Geoid: a representation of the surface of the earth that it would assume if the sea covered the earth, also known as surface of equal equipotential surface which (approximately) coincides with the mean ocean surface (mean sea level). - sea level isn't level (and also not a mathematical surface)! - The vertical coordinate, Z (elevation), is referenced to the Geoid. - On the Geoid, the vertical axis of a properly-leveled instrument is coincided with the vertical plummet line, which is perpendicular to the Geoid. 7

8 2-2. The Earth s Shape and Size 3. The Reference Ellipsoid The earth is not a SPHERE, but flattened slightly at the poles and bulging somehow at the Equator The best approximation of the figure of the Earth is the ellipsoid (an oblate ellipsoid). An ellipsoid is mathematically defined as X 2 Y 2 a 2 Z b Positioned and oriented with the best fitting - to the entire Earth the Global Reference Ellipsoid; -toaregionoftheearth-the(local) Reference Ellipsoid. A reference ellipsoid is a mathematically-defined (ellipsoid) surface that approximates the geoid, the truer figure of the Earth. 8

9 2-2. The Earth s Shape and Size 4. The Geodetic Datum In summary, a reference ellipsoid is used to describe the Earth and a Geoid model is used to express the vertical height, the MSL height. Ellipsoid Surface of the earth P H P e O L B Geoid Ellipsoid Coordinate: Latitude B Longitude L Altitude H (Ellipsoid Height) 9

10 2-2. The Earth s Shape and Size 4. The Geodetic Datum (cont d) Some Facts in surveying: a. Measurements are made on the actual topographic surface (Earth s natural surface). b. Geodetic computations are preferably performed on the reference ellipsoid. c. The (MSL) heights are based on the Geoid. d. The properly-leveled instrument at a point on the ground has its vertical axis being coincided with the local plummet line, which will be perpendicular to the Geoid if it is located on Geoid because the equipotential surfaces at different height levels are not parallel. 10

11 2-2. The Earth s Shape and Size 4. The Geodetic Datum (cont d) e. The geodetic coordinates of a point on, above or under the Earth s surface is defined as Ellipsoid Coordinate: - Latitude - Longitude H - Altitude (Ellipsoid Height) f. The (MSL) height: P Z P H P e H O H orth N the geoidal height O Y H orth = H N X The Earth, Geoid and Ellipsoid 11

12 2-2. The Earth s Shape and Size 4. The Geodetic Datum (cont d) g. The Earth-Centered, Earth-Fixed 3D Cartesian coordinates X, Y, Z X ( R Y ( Re Z ( R H )cos cos H )cos sin 2 (1 e ) H )sin e e - the (geodetic) latitude - the (geodetic) longitude a - the semi - major axis of the reference ellipsoid b - the semi - minor axis of the reference ellipsoid e R e a 2 b 2 a 2 1 e - the first eccentricity a 2 sin 2 - the radius of curvature in the prime vertical H - the ellipsoidal height of the point 12

13 2-2. The Earth s Shape and Size 4. The Geodetic Datum (cont d) h. Some Reference Ellipsoids (Examples) Optional for reading! 13

14 2-2. The Earth s Shape and Size 4. The Geodetic Datum (cont d) i. Geodetic Datum used in Canada (Natural Resources Canada) (website: Three different horizontal reference systems were used over the years: 1). Earlier blocks ( ) were based on the North American Datum of 1927 (NAD27); 2). Blocks completed between 1980 and 1989 were based on the MAY76 adjustment of NAD27; 3). Everything completed since is on the North American Datum of 1983 (NAD83). 4). The vertical datum used is the Canadian Geodetic Vertical Datum of 2013 (Geoid based). CGVD28: based on the mean sea level determined from several tidal gauges located in strategic areas of the country (4 tide gauges at the East coast & 2 at the West coast have been assigned the height 0) The official Canadian height system: CGVD2013 (vertical datum) - Normal (orthometric) heights (MSL Height) - CGVD28 became obsolete. The Earth s model: NAD83 (horizontal datum) 14

15 2-3. The Plane Coordinate System 1. Why to need a plane coordinate system? The engineering design or plan is commonly performed on the plane so that the position of a point of interest and the map representation need to be provided in a plane coordinate system. 2. The 2D plane Cartesian coordinates in surveying y y 4 th quadrant 1 st quadrant 2 nd quadrant 1 st quadrant P(x, y) P(x, y) x x o o 3 rd quadrant 2 nd quadrant 3 rd quadrant 4 th quadrant The Geodetic 2D Cartesian coordinate system trigonometric functions unchanged applicable By swapping x and y (North America) The Mathematical 2D Cartesian coordinate system 15

16 2-3. The Plane Coordinate System 2. The 2D plane Cartesian coordinate in surveying (cont d) N x 4 th quadrant 1 st quadrant 4 th quadrant 1 st quadrant P(N, E) P(x, y) E y o o 3 rd quadrant 2 nd quadrant 3 rd quadrant 2 nd quadrant The Geodetic 2D Cartesian coordinate system trigonometric functions unchanged applicable By using N x & E y (North America) The Geodetic 2D Cartesian coordinate system All trigonometric functions unchanged applicable (Germany, China etc.) The AZIMUTH from one point to the other in a 2D plane Cartesian Coordinate system is defined as the angle from the vertical axis onwards clockwise. 16

17 2-4. The Limitation of the Use of Plane instead of Level Surface 1. The problem A point on the Earth is determined by three coordinates, e.g. the latitude, longitude and altitude (or MSL height); A point on a map is located by only two coordinates, e.g. the north and east coordinates; The engineering design and plan need the plane coordinates of points; Any point on the Earth defines its own local level surface that differs from the one at a different point in general ( why an ellipsoid is employed to describe the Earth). How large is an area on the Earth can be described as a plane without significant distortion due to the curvature of the Earth? 17

18 2-4. The Limitation of the Use of Plane instead of Level Surface 2. The effect on horizontal distances by the curvature of the level surface at a point S s S S S S S t R (the arc s, 1 3 S R 2 2 t 1 3 R( 3 Rtg 2 ( AB ) 5 ( S...) Directly analyse the distance on the ellipsoid because the geodetic computation is on the ellipsoid 1 3 S R 3 2 8mm) with Level surface Geoid or Ellipsoid S 10km This distortion is smaller than the maximal tolerable error in precise distance measurement A R O t s C B h 18

19 2-4. The Limitation of the Use of Plane instead of Level Surface 3. The effect on horizontal angles by the curvature of the level surface at a point The spherical excess of a spherical polygon, the difference between the sum of their internal angles on the sphere and the sum of the projected angles on the plane, is P 2 R R - the Earth' s radius P - the area of A B C the spherical polygon 0.51 with P 100km 2 This distortion is considered only in high precise surveying for the area of 100[km 2 ] 19

20 2-4. The Limitation of the Use of Plane instead of Level Surface 4. The effect on the height difference by the curvature of the level surface at a point ( R h) h 2 R 2 t 2R h 2 t 2 2 s 2R h 8cm with s 1km Geoid or Ellipsoid A Level surface R t s R C B h O This distortion is significant even over the short distance 20

21 j The Map Projections 1. Concept of map projections The reference ellipsoid can be used as the reference surface of the surveying computations, but already very complicated and tedious. For maps of small areas, the curvature of the Earth need not be considered, but not for maps of larger areas be impossible to develop those surface areas exactly onto a plane (to flatten a section of orange peel without tearing it?); will always be some distortion, no matter whatever procedure is used to represent a large area on a map. What can be done: minimize the distortion by specifying a selected set of criteria for transforming the positions on the Earth in term of latitude (B) and longitude (L) into the scaled linear dimensions on the plane map: x y f f x y (, ) (, ) 21

22 Slide 21 j1 The mathematical derivation of the map projections are beyond the scope of this course. jgwang,

23 j The Map Projections 1. Concept of map projections - Possible Criteria: a. Conformal or orthomorphic projection correct angle/correct shape b. Equal-area projection areas in proper relative size c. Equal distant projection correct distances from one central point to other points d. Azimuthal projection correct direction/azimuth of any point vs. one central point The mathematical derivation of map projections is beyond the scope of this course and will partially be covered by ESSE3610 Geodeitc Concepts. 22

24 Slide 22 j1 The mathematical derivation of the map projections are beyond the scope of this course. jgwang,

25 2-5. The Map Projections 2. Types of map projections Orientations Azimuthal Developable surfaces Cylindrical Conic 23

26 2-5. The Map Projections 3. The requirement of map projection for topographic maps One "traditional" rule (Maling, 1992) says: a. A country in the tropics asks for a cylindrical projection. b. A country in the temperate zone asks for a conical projection. c. A polar area asks for an azimuthal projection. Optional for reading! 24

27 2-5. The Map Projections 4. Examples: Example 1: Mercator Projection 1). Conformal. 2). Used for navigation and regions near the equator. 3). History - Invented in 1569 by Gerardus Mercator (Flanders) graphically. - Standard for maritime mapping in the 17th and 18th centuries. - Used for mapping the world/oceans/equatorial regions in 19th century. - Used for mapping the world/u.s. Coastal and Geodetic Survey in 20th century. 25

28 2-5. The Map Projections 4. Examples (cont d): Example 2: Transverse Mercator Projection 1). Mercator rotated 90 degrees 2). Conformal 3). History - Invented by Lambert in Modified for ellipse by Gauss in 1822, Krüger in Used for regions with a N/S expanse - Used for topo mapping (large scale topographic map series worldwide) - Basis for State Plane Coordinate System, e.g., UTM 26

29 2-5. The Map Projections 5. Universal Transverse Mercator (UTM) Grids a. UTM is also called the UTM coordinate system, a grid-based method of specifying locations on the Earth s surface (a global system of the grid-based maps), used to identify locations on the earth, but differs from the traditional method of latitude and longitude in several respects. 27

30 2-5. The Map Projections 5. Universal Transverse Mercator (UTM) Grids b. The Global UTM Zones 28

31 2-5. The Map Projections 5. Universal Transverse Mercator (UTM) Grids (cont d) c. UTM divides the globe into sixty zones (6 degree zones), each spanning six degrees of longitude. Each zone is mapped by the Transverse Mercator projection; Each zone has its own central meridian (in the center of the zone) from which it spans 3 degrees west and 3 degrees east. UTM zone numbers designate 6 degree longitudinal strips extending from 80 degrees South latitude to 84 degrees North latitude. Zones are numbered from 1 to 60. Zone 1 is bounded by longitude 180 to 174 W and is centered on the 177 th West meridian, for example. UTM zone characters designate 8 degree zones extending north and south from the equator segments each longitude zone into 20 latitude zones starting from the letter C to X omitting the letters "I" and "O". Eastings are measured from the central meridian (with a 500km false easting to insure positive coordinates). Northings are measured from the equator (with a 10,000km false northing for positions south of the equator). 29

32 2-5. The Map Projections 5. Universal Transverse Mercator (UTM) Grids d. A single UTM Zone - 6 degrees of longitude wide ~ 672 km on the Equator, becoming narrower towards north and south. - Central meridian at false easting m. - Secant longitude 180 km from the central meridian on the Equator. - Scale factor = 1 along secant meridians and along the central meridian. - Scale = 1 along secant meridians and constant along the central meridian. - Some FMU s fall between UTM zones problems with file projections and map orientations! - UTM zones can be projected into other (neighbouring) zones. Secant meridians UTM Zone 30

33 2-5. The Map Projections 5. Universal Transverse Mercator (UTM) Grids e. UTM Zones in Canada Canada is covered with 16 zones, 7-22 Ontario is covered with 4 zones,

34 2-5. The Map Projections 5. Universal Transverse Mercator (UTM) Grids e. Map sheets in BC, Canada Optional for reading! 32

35 2-5. The Map Projections 5. Universal Transverse Mercator (UTM) Grids f. Ontario Digital Topographic Database Optional for reading! 33

36 2-5. The Map Projections 5. Universal Transverse Mercator (UTM) Grids f. Ontario Digital Topographic Database Distribution of Maps in Ontario Optional for reading! 34

37 2-5. The Map Projections 5. Universal Transverse Mercator (UTM) Grids g. Projection in Canada (Natural Resources Canada) (website: All the ASDB (Aerial Survey Database) coordinates have been computed using the Universal Transverse Mercator (UTM) projection. The Canadian landmass extends from UTM Zone 7 in the west to UTM Zone 22 in the east. Each zone is six degrees of Longitude in width. When a database block spans more than one zone, the predominant zone is used for the entire block. Should the client require the data in the adjacent zone, personnel from the Aerotriangulation Unit will transform the entire block to the desired zone. COMMENT: MTM Modified Transverse Mercator (3 o Zones) 1) 10TM Alberta 10 degree Transverse Mercator is used to cover the whole Alberta within a single zone. 2) 1.5, 3 degree zone, even some specific zoning maps can also be used in practice 3) Arbitrary zoning maps can be required by urban planning, civil engineering projects The central axis of the working area can be chosen to be the central meridian line of the zone and the average height plane as the projection plane (not on ellipsoid) in order to minimize the projection distortion. 35

38 2-5. The Map Projections 5. Universal Transverse Mercator (UTM) Grids h. MTM Modified Transverse Mercator (3 o Zones) 1) is another map projection, similar to the UTM 2) is based on completely on the original transverse Mercator with a secant cylinder 3) has a region divided into zones of 3 of longitude each (i.e., 1.5 on each side of the zone s central meridian). Each zone is projected separately, which leads to a small distortion 4) utilize a scale factor of along the zone s central meridian to at the zone boundary 5) introduce the false northing and false easting of 0.0m and 304,800m in Canada to avoid negative coordinates 36

39 2-5. The Map Projections 5. Universal Transverse Mercator (UTM) Grids h. MTM Modified Transverse Mercator (3 o Zones) 37

40 2-6. The Azimuth Angles 1. Three Basic Directions The true north direction: The direction from a point of interest along the tangent line of the true meridian north (determined by observing the sun, stars or gyroscopic theodolite). The grid north direction: the positive direction of the vertical axis in the grid (the north direction in a UTM grid) The magnetic north direction: the magnetic north direction (not a fixed direction), not coincided with the true north because the magnetic north pole is not exactly the earth s north pole. Comment: The North Magnetic Pole is slowly drifting across the Canadian Arctic. The Geological Survey of Canada keeps track of this motion by periodically carrying out magnetic surveys to redetermine the Pole's location. The most recent survey, completed in May, 2001, determined an updated position for the Pole and established that it is moving approximately northwest at 40 km per year. The observed position for 2001 and estimated positions for 2002 to 2005 are given in the table. [Geological survey of Canada, 2005] 38

41 2-6. The Azimuth Angles 2. The Meridian convergence P ( 0)sin sin P P - the latitude of P P - the longitude of P - the central longitude of 0 3. The Magnetic Declination not a constant angle. 4. The Azimuths A Am A A m P A- the true azimuth A - the magnetic azimuth m - the grid azimuth - the convergence of the meridian - the magnetic inclination P the zone 39

42 2-6. The Azimuth Angles 5. The Grid Azimuth and Its Reverse Grid Azimuth 1,2 2,

43 2-7. Units, Significant Figures and Field Notes 1. The Types of Measurements - Horizontal angles; - Horizontal distances; - Vertical (Zenith) angles; - Vertical Distances; - Slope Distances Need Units for length, area, volume and angle - Metric system (international system of Units); - English system 2. Measurement Canada and the Metric System in Canada Measurement Canada is the agency responsible for enforcing federal laws that protect both the Canadian public and Canadian industry when goods and services are traded on the basis of measurement. The use of the metric system within its mandate because metric is not only a recognized system for measurement, it is a legal system for measurement in Canada 41

44 2-7. Units, Significant Figures and Field Notes 3. Significant Figures Definition: The number of significant figures in any observed value includes the positive digits plus one digit that is estimated or rounded off, and therefore questionable. For Example: A distance measured with a tape whose smallest graduations are mm, and recorded as m It has 5 significant figures; The first 4 digits are certain; The last one is rounded off and therefore questionable but still significant. Don t confuse the number of significant figures with the number of decimal places! How to handle the figures in a mathematical process?

45 2-7. Units, Significant Figures and Field Notes 3. Significant Figures (cont d) Understanding of significant figures in surveying practice a. Field measurements are given to some specific number of significant figures, thus dictating the number of significant figures in answers derived from them commonly carry at least one more digit than required, and then round off the final answer to the correct number of significant figures. b. There may be an implied number of significant figures. For example, a distance is specified as 100m. In laying out this length, such a distance would probably be measured to the nearest cm or mm upon the accuracy requirement. A good practice is to provide the distance as m or m, even multiple ZEROS at the end of the number after the decimal point. c. Each factor may not cause an equal variation. Example 1: a steel tape m long to be corrected for a change in temperature of 25 o C m has five significant figures while 25 o C has only two. In the view of design, one has to analyse how good the temperature should be Example 2: the computation of a horizontal distance from a slope distance and a measured vertical angle d s tg s d 43

46 2-7. Units, Significant Figures and Field Notes 3. Significant Figures (cont d) d. Measurements are recorded in one unit or in one system of units, but may have to be converted to another. Example 1: convert an angle 123 o into degrees. Which of the following numbers should be given? 1) ) ) Answer: 1). 123 o o o Example 2: How many seconds or radians of latitude or longitude are corresponding to one cm long ground distance? 4. Rounding off numbers - the process of dropping one or more digits so the answer contains only those digits that are significant. Rule 1: The number is written without the digit, when the digit to be dropped is lower than 5: ; ; Rule 2: The number is written with the preceding digit increase by 1, when the digit to be dropped is greater than 5: Rule 3: The nearest even number is used for the preceding digit, when the digit to be dropped is exactly 5: ;

47 2-7. Units, Significant Figures and Field Notes 5. Field Notes - The records of work done in the field, including any hard and soft records. For example, Measurements, sketches, descriptions, other items of miscellaneous information The original field notes should be kept and remain unchanged. 45

48 2-7. Units, Significant Figures and Field Notes 5. Field Notes (cont d) 46

49 2-8. Field and Office Work The Essential Procedures in a surveying project: a. Collection of Existing Information: maps; control points. b. Planning and Design: specifications; datum; map projection and coordinate system; field observation outlines/procedures c. Equipment: selection; testing, calibration and adjustment; d. Field operation: control Surveying: horizontal, vertical mapping setting out survey final as-built survey deformation monitoring e. Field and Office Data QA and QC, data processing, results management COMMENTS: 1. Refer to Chapter 3 in the text book for more details 2. Your understanding will become better and better with the time being through your lab assignments and your work in the future 47