Introduction to Land Surveying EWB-UW Haiti Project Engineers Without Borders University of Wisconsin Madison Student Chapter
Editors Eyleen Chou Scott Hamel
Contributors Kira Langree Tyler Lark Maria Selk Miranda Scheiber Last Updated January 2, 2009
Table of Contents 1.0 1.1 1.2 1.3 1.4 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 5.0 6.0 7.0 8.0
Introduction to Surveying ................................................................................................................. 1 What is Surveying? ...................................................................................................................... 1 Why is Surveying is Important ..................................................................................................... 1 Surveying Concepts ..................................................................................................................... 1 What will I survey? ....................................................................................................................... 3 Surveying Methods .......................................................................................................................... 4 Basic Surveying Equipment/Tools ............................................................................................... 4 Surveying Terminology ................................................................................................................ 5 Finding Distances ........................................................................................................................ 6 Finding Elevations ....................................................................................................................... 7 Finding Horizontal Angles ............................................................................................................ 9 Putting It All Together ................................................................................................................ 10 Traverse ..................................................................................................................................... 11 Using Gathered Data ................................................................................................................. 12 Field Practices................................................................................................................................ 13 How to keep a field notebook .................................................................................................... 13 Setting up the theodolite and tripod ........................................................................................... 15 Setting Hubs and Control Points , ............................................................................................. 16 Taking a Compass Bearing ....................................................................................................... 16 Measuring Distance ................................................................................................................... 17 Graduated Rod .......................................................................................................................... 19 Auto-Level .................................................................................................................................. 19 Theodolite .................................................................................................................................. 20 Establishing other landmarks..................................................................................................... 22 Math Review .................................................................................................................................. 23 Surveying Trigonometry............................................................................................................. 23 Angle Calculations ..................................................................................................................... 25 Unit Conversions ....................................................................................................................... 25 Exercises (Eyleen) ...........................................................................Error! Bookmark not defined. Appendix A: More Stadia Information ............................................................................................ 26 Appendix B: More Field Book Techniques ..................................................................................... 28 Appendix C: Resources ................................................................................................................. 31
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1.0 Introduction to Surveying This section will present a basic understanding of the concepts and tools of Land Surveying.
1.1 What is Surveying? Surveying is a method of determining the three-dimensional position of points and objects. These positions are usually, but not always, on the surface of the Earth. The established points are sometimes used to create land maps and determine boundaries for ownership or governmental purposes. They are also used to produce a topographic model that is useful in civil engineering and construction. In order to accomplish their task, surveyors use trigonometry, physics, and practiced skill.
1.2 Why is Surveying is Important Surveying is a great skill to learn. In the United States, people go to school just to get a job in surveying. As a surveyor, you will become valuable to your community. Hopefully in the future, surveying in Haiti will be a paid job. Also, your surveying skills will be very valuable to Engineers Without Borders. Engineers Without Borders has been working with OFCB to create projects that will help the community of Bayonnais. Some of these future projects require a large amount of surveying.
1.3 Surveying Concepts As mentioned in Section 1.1, Surveying establishes the spatial position (location) of points and objects. These locations are recorded as numerical data in the form of 3-D Cartesian coordinates, which means that each point has an X,Y and Z coordinate. In general, these coordinates are aligned with the earth’s gravity. That is, the Z direction is parallel to gravity (a vertical line) and the X-Y plane is perpendicular to gravity (a horizontal plane). It should be noted that for large surveys, the coordinate system must account for the curvature of the earth (X-Y is not a flat plane), but that is beyond the scope of this document. The easiest way to find the coordinates of a point is to start with a known location and measure the difference in X, Y and Z between the known location and the new point. These measured differences in the coordinates are refered to as ΔX, ΔY and ΔZ. For measuring purposes, it is convenient to separate the Z coordinate (vertical coordinate, or elevation) from the X and Y (Horizontal Plane). In most surveying methods, the “known point” is the location at which the instrument is placed. This will be discussed further in sections 2.7 and 3.3. The following sections will give a conceptual description of how to measure the difference between a known and an unknown point in both elevation (ΔZ) and in the horizontal plane. .
1.3.1 Elevation (Z coordinates) There are two ways to measure the difference in elevation between two points: Direct Leveling and Trigonometric Leveling. Both require starting with two perpendicular lines, one that is perfectly level (horizontal) and one that is vertical (parallel with gravity). The horizontal line can be a "line of sight" through a scope (either hand held or tripod mounted), or a line projected by a laser. In Direct Leveling, once the horizontal line is established between the two points, a scale (or tape measure) is held vertically and the vertical distance between the two points is measured. This is illustrated in Fig XX. In some cases, the vertical distance cannot be measured, perhaps because the elevation of the known point is underground at the new point, or the distance is too large. In this case, Trigonometric Leveling is used. In this technique, the angle between the horizontal line and a line directly to the new point (slant line) is measured. If the slant distance between the two points is also known, simply trigonometry can be 1
used to find the vertical distance. This technique is discussed in more detail in Section 2.4 and illustrated in Fig XX.
1.3.2 Horizontal Location (X-Y coordinate) Horizontal locations, or coordinates, are determined by measuring the Polar Coordinates of a point using the known point as the origin. Polar coordinates are determined by measuring an angle and a distance. In order to measure an angle, one must have a reference point to measure from, or a “zero point”. There are two methods of determining your reference point depending on the equipment that is being used. If a magnetic compass is being used to determine your angles, then the reference point is always the same, magnetic north. If an instrument that measures horizontal angles, such as a theodolite, is being used, then usually the surveyor may choose what to use as the “zero point” In general, the surveyor will set this point as a “backsight”, which is discussed further in Sections Error! Reference source not found. and 3.3. Polar coordinates (θ,d) can be converted to Cartesian Coordinates (x,y) using the following formulas: x = d * Sin (θ) y = d * Cos (θ)
1.3.3 Global Positioning System (GPS) GPS or Global Positioning System is a system of determining a location on the surface of the earth using a handheld electronic device. The device passively receives data from satellites that orbit the earth and calculates its position. Each GPS satellite transmits data that indicates its location and the current time. The satellites are synchronized so that these repeating signals are transmitted at the same instant. The signals, moving at the speed of light, arrive at a GPS receiver at slightly different times, allowing the receiver to determine the distance to each satellite. Once the receiver knows the location of the satellites (contained in the signal) and the distance to each satellite, it can calculate its position using triangulation. Using 3 satellites, the position in 2-dimensions can be determined (location on the surface of the earth). Using 4 satellites, a GPS device can determine it position in 3-dimensions, (location and elevation). There are 30 functioning GPS satellites in orbit around the earth, owned and operated by the US Air Force. They each circle the earth two times a day (12 hour period) and arranged such that at least four satellites can be seen from anywhere on earth at all times. In some locations, up to 12 satellites can be seen at once. Because a GPS device must receive a signal from multiple satellites, they work best when there is a clear line of sight to the sky. They do not work well in canyons and dense forests. The accuracy of the device is dependent on how many satellites the device can “see”. The more signals a device is receiving, the more accurate the position. In addition, the time sent by the satellites is produced using onboard atomic clocks. Thus, your position will be most accurate if your GPS device is also synchronized to the atomic clock at the US Naval Observatory (also known as the “Master Clock”). Positions are displayed as a latitude and longitude, both in degrees, minutes, seconds and sometimes decimals of a second. Elevations are displayed as meters above sea level. Accuracy can range from a several hundred meters to with a few meters.
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1.3.4 Barometric Altimeter Surveying An altimeter survey is one in which ONLY the altitude (or elevation) is recorded. Horizontal locations are not determined or recorded. This is useful in situations where the surveyor is only interested in an elevation (such as the top of a mountain) or in a change of elevation (such as the height of a waterfall). As altitude increases, the barometric pressure (ie- air pressure) of the atmosphere decreases. A barometric altimeter measures the atmospheric pressure, and the corresponding elevation is read directly off the instrument. Normal weather patterns cause the air pressure at any altitude to fluctuate slightly throughout the day. Thus, even if the altimeter is at a point, the elevation reading may increase and decrease by several meters throughout the day. Such pressure variations must be measured and accounted for when conducting a barometric altimeter survey.
1.3.5 Latitude and Longitude Latitude and Longitude are angular measurements (in degrees, minutes and seconds) that specify a location on the surface of the earth. Latitude lines are imaginary lines parallel to the equator (usually horizontal on a map) and measure the north/south location of a place, ranging from 0° at the equator to 90° at the poles. Longitudinal lines (also called Meridians) run from the north pole to the south pole (usually vertical on maps) and measure the east/west position of a place. 0° Longitude is an (arbitrary) line that runs through Greenwich, England and is called the Prime Meridian. Longitude is measured east or west of the Prime Meridian, up to 180°, which is exactly opposite the Prime Meridian. It is important, when denoting position to also note the latitude direction from the equator (north or south) and the longitude direction from the Prime Meridian (east or west). For example, the bridge in Bayonnais is at 45°10’23” N, 45°10’23” W. The location of Madison, WI is 45°10’23” N, 45°10’23” W.
1.4 What will I survey? In general, there are two types of surveys: Geographic and Topographic. In a Geographic Survey, also called a Property Survey, the surveyor is locating objects and/or property lines. The finished product is a map of an area including whatever the surveyor recorded. This would be referred to as a 2D survey, since no topography is recorded or displayed. A Geographic survey is frequently used for government, legal or business purposes involving land management. For example, property surveys record and define the exact location, size and shape of a piece of property. This helps to alleviate land disputes and allows for accurate and fair tax assessment. In a Topographic survey (or 3D survey), much of the same information as a Geographic Survey is recorded, and in addition, elevations are measured. The finished product is a map of general features (such as roads and rivers) along with contour lines and spot elevations (the exact elevation of a point, such as a mountain top). These surveys are used more for engineering purposes. It is helpful for civil engineers to know the contour of the land when building roads, bridges, structures, canals and dams, or installing power lines, or irrigation systems. Since Topographic surveys are for a different purpose, the level of detail recorded may be lower for a topographic survey. For example, a Topographic Survey may cover an entire river valley and only record large buildings, the centerlines of roads and location of large rivers, while a Property Survey might locate every small structure on the property along with the boundaries (edges) of every roads and river or stream.
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2.0 Surveying Methods Chapter 1 provided an introduction to the concepts used to locate objects and points in space. The following sections describe some of the equipment and methods used to employ these concepts.
2.1 Basic Surveying Equipment/Tools Theodolite an optical instrument for measuring horizontal and vertical angles in land-surveying (alternate def. A Theodolite is a precision surveying instrument; consisting of an alidade with a telescope and an accurately graduated circle; and equipped with the necessary levels and optical-reading circles. The glass horizontal and vertical circles, optical-reading system, and all mechanical parts are enclosed in an alidade section along with 3 leveling screws contained in a detachable base or tribrach.) Automatic Level An automatic level is another piece of sighting equipment used in surveying. It can be used in place of the theodolite. Automatic levels are designed so that they are sighting horizontally; they cannot measure vertical angles. Data collection when using the automatic level will be easier (because you always know the vertical angle is zero). However, the shots may be closer together especially on slopes.
Abney Level An abney level is a surveying instrument that consists of a spirit level and a sighting tube. It is used to measure the angle of inclination of a line from the observer to the target point. Tripod a three-legged rack to hold the theodolite
Tape a measuring tape used to measure distances between two points (oftentimes, the theodolite and graduated rod)
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Graduated rod a telescoping pole used to measure height. The height is sighted with the theodolite.
2.2 Surveying Terminology There is a large amount of terminology that is specific to surveying. Here is a brief list. Axe de visée, axe de collimation (Line of Sight): ligne passant par les foyers de l’objectifs d’une lunette et le point de mesure en correspondance avec le réticule. Basculement (Swing/Turn an angle): la lunette du théodolite est tournée de 200 gr autour de l’axe horizontal pour éliminer les erreurs instrumentales. Calage et mise en station (Locate and Level Station) : opération effectuée par l’opérateur pour amener l’axe vertical de l’appareil à l’aplomb d’un repère sur le sol. Correction : valeur algébrique à ajouter à une valeur observée ou calculée pour éliminer les erreurs systématiques connues. Croisée du réticule (Crosshairs): croix dessinée sur le réticule représentant un point de l’axe de visée. Erreur de fermeture (Error): écart entre la valeur d’une grandeur mesurée en topométrie et la valeur fixée ou théorique. Fils stadimétriques (Stadia): lignes horizontales marquées symétriquement sur la croisée du réticule. Elles sont utilisées pour déterminer les distances à partir d’une échelle graduée placée sur la station. Hauteur de l’appareil (Height of scope/instrument) : distance verticale entre l’axe horizontal de l’appareil et celle de la station. Implantation (Lay-out) : établissement de repères et de lignes définissant la position et le niveau des éléments de l’ouvrage à construire. Levé (leveled): relevé de la position d’un point existant. Lunette (scope/lens): instrument optique muni d’une croisée de réticule ou d’un réticule, utilisé pour établir un axe de visée par l’observation d’un objet de mesure. Mesurage (Measurement) : opérations déterminant la valeur d’une grandeur. Nivelle (Level, bubble): tube en verre scellé, presque entièrement rempli d’un liquide (alcool) dont la surface intérieure a une forme bombée obtenue par moulage, de sorte que l’air enfermé forme une bulle qui prend différentes positions suivant l’inclinaison du tube. Nivellement (Leveling, v.): opération consistant à mettre une ligne ou une surface dans la position horizontale, ou mesurage de différences de niveaux. 5
Repères (Reference Points) : points dont on connaît les coordonnées. Réticule (Eye piece): disque transparent portant des traits ou des échelles. Il permet d’effectuer correctement des lectures. Signal, balise (Control Point): dispositif auxiliaire pour indiquer l’emplacement d’une station (par un jalon). Station : tout point à partir duquel ou vers lequel on effectue une mesure. Cela peut être un point spécifié sur un bâtiment ou un point marqué dans la zone d’étude. Tolérance : variation admissible pour une dimension. Stadia Ratio : The ratio of the viewed length between the stadia lines using an objecta t a distance and the distance from the scope to the object. Backsight : Azimuth: a horizontal angle measured clockwise from any fixed reference plane or base line (in land navigation measured from a north base line or meridian). Zenith: the direction pointing directly above a particular location Zenith Angle: the vertical angle between an object or a point and the Zenith (90° to level or horizontal plane) Elevation: the height of a geographic location above a fixed reference point (often the reference point is sea level) Tri-brac: Triangular piece of equipment placed on a tripod beneath a theodolite used to level the instrument, Cartesian Coordinate System : A coordinate system in which the coordinates of a point are its distances from a defined set of perpendicular lines (called axes) that intersect (the intersection point is called “the origin”), In 2-d, there are two axes and two coordinates (usually denoted x and y), while in 3d, there are three axes and three coordinates (x,y, and z). Polar Coordinate System : A 2 dimensional coordinate system in which the coordinates of a point are determined by an angle and distance from a specified origin or reference point. “Pick up”: Picking up means determining the location and orientation of an object by surveying multiple points on the object as necessary. For example, if one is going to “pick up the school”, this would include taking points at the corners of the building, the height of the roof, etc. “Taking a shot”: Taking measurements to point. That is, recording the distance, horizontal angle, and possibly vertical angle of that point. This term comes from the fact that the surveying instrument, such as a theodolite, is usually referred to as a “gun”.
2.3 Finding Distances This section describes methods of determining ground or “sloping” distances. They increase in accuracy, starting with the least accurate.
2.3.1 Pacing
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Pacing is the technique of determining a distance by walking between the two points and counting the number of steps that it takes. A single pace is the distance travelled in two steps, that is if you are counting paces, you would count each time you take a step with your right foot (or if you prefer, your left). If one knows the distance travelled in a single pace, it is easy to find a distance. This method is of course not accurate, but it will provide a good estimate of a distance. The accuracy increases if the distance is paced several times and the results are averaged. See section Error! Reference source not found. for techniques on pacing,
2.3.2 Stadia Stadia is a method of determining distance by using optical magnification. Many surveying instruments have indicator lines overlayed on the lens (similar to the cross hairs) that are a preset distance from the cross hairs. Because there is a known magnification of the lens, the measured length of the stadia lines using an object at a distance is a specified ratio of the distance from the scope to the object. More simply, if you look through the scope with a 1:100 stadia ratio at a rod with markings (or a tape measure) and read a distance of 0.142 m between the stadia lines, then the rod is 14.2 meters away.
2.3.3 Tape Taping is the technique of using a calibrated measuring device to determine a distance. This is not necessarily limited to a tape measure; string, sticks, anything can be calibrated and then used to measure distance. In most cases, however, this takes the form of using a tape measure. The tape can be steel or cloth. Practices of taping are discussed further in section 3.5.3
2.3.4 Total Station (Laser) A Total Station is a device used in modern surveying that encompasses both a computerized Theodolite (see section Error! Reference source not found.), and a laser that calculates distance. A prism (reflective object) is placed on top of the rod (see section 3.6) and the Total Station, shoots a laser beam at the prism. The beam is reflected back to the station, which uses the time it took to travel to accurately calculate the distance traveled.
2.4 Finding Elevations As discussed in section 1.3.1, there are two methods of determining the difference in elevation between a known point and an unknown point. It should be noted that Direct Leveling will yield more accurate results than Trigonometric Leveling. This is because the measurement is limited only by the graduation of the scale (1 cm) and the precision of the level line. Trigonometric leveling is less accurate, but it can be very useful to determine the elevation of an inaccessible point. Trigonometric leveling is slightly more complicated than described in Section 1.3.1 because the surveyor must account for the height of the instrument HI) and the height of the rod (vertical scale). This is illustrated in Fig XX. The elevation of Point B can be determined using the following equation : ZB = ZA + HI + d*sin(a) - HR where : ZB = elevation of New Point ZA = elevation of known location 7
d = slant distance between points HI = Height of instrument HR = Height of rod a = vertical angle between points
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Figure 2. Principe du nivellement trigonométrique
2.4.1 Abney Level The Abney Level is a hand held tool that determines the vertical angle of a target that can be sighted. Trigonometric Leveling can then be used to determine the elevation of the target. The Abney is similar in principle to the theodolite vertical angle (see section 2.4.3), but far less accurate. It is useful for certain types of surveys where a high degree of accuracy is not required. The Abney is a small square tube with an eyepiece at the observer’s end and horizontal cross-hair at the objective end. Near the center of the tube is a 45 mirror, which reflects half the line-of- sight upwards through an aperture in the tube, Mounted above the aperture is a bubble level which is affixed to a movable index arm. The index arm can be adjusted and indicates an angle using gradations on the arc of the index arm. To use the Abney, the instrument is held to the eye and sighted on a target, centering the cross-hair against the target. The index arm is then adjusted until the bubble (visible in the right half of the field-of-view) is centered against the target and the cross-hair. The vertical angle can then be read off the index
2.4.2 Autolevel and Rod The Autolevel is a device that uses Direct Leveling, which is discussed in section 1.3.1. The Autolevel has a magnifying scope and is placed on a Tripod (legs) and leveled. Once properly leveled, the surveyor looking through the scope is sighting along a line that is exactly level, At the location of interest, a graduated rod is placed vertically and the surveyor at the Autolevel can then read the exact difference in elevation between the location and the scope.
2.4.3 Theodolite Vertical Angle The Theodolite is an instrument that can measure both vertical and horizontal angle. When measuring vertical angles, the Theodolite uses Trigonometric Leveling (see section 1.3.1). In the special case where the angle is set to 90° (level), the Theodolite is employing Direct Levling. See section 3.8 for more information on Theodolites.
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2.5 Finding Horizontal Angles As discussed in section 1.3.2, horizontal angles, along with distance, are used to determine horizontal location (x, y coordinates). There are several techniques, which are covered in the sections below, beginning with the least accurate.
2.5.1 Compass The compass is a simple instrument that utilizes the earth’s magnetic field. When held level, the needle of a compass points to Magnetic North, which is not the same as True North (or the north pole). When surveying with a compass, one must account for this difference and adjust appropriately. This is discussed further in section 3.4. Since the compass always points in essentially the same direction, one can measure horizontal angles by determining the angle between the object that is being sighted, and Magnetic North. This is called a “Compass Bearing” and is illustrated in Fig XX. While this method is simple, it is only accurate to within a few degrees.
2.5.2 Autolevel Horizontal Circle The Autolevel, which is primarily for determining elevation (see section 2.4.2), usually also contains a horizontal circle with labeled angles. This circle, which rotates on the vertical axis, allows the surveyor to determine the horizontal angle between any two points. As discussed in section 1.3.2, when a compass is not being used, a surveyor can choose what to use as a zero angle (or backsight). This is the case for the Autolevel and once a backsight is chosen, the horizontal circle is rotated such that the backsight is at 0°. The angles for subsequent points can then be read off the circle once the scope is rotated to view the point or object. Since the horizontal circle does not involve any magnification or fine adjustment, it is only accurate to about 1°.
2.5.3 Theodolite Horizontal Circle As discussed in section 2.1, a theodolite accurately measures both horizontal and vertical angles. Similar to the Autolevel, the surveyor must choose a backsight from which to measure angles. Most theodolites have an adjustable scale so that the backsight may be set at 0°. Some do not allow this adjustment and the backsight angle must be recorded and later subtracted from all other angles measured from that station. Once the backsight has been established, the surveyor must simply read and record the horizontal angles of each desired point using the theodolite. Depending on the instrument used, a theodolite can be have an accuracy anywhere from 20” to 2” (1/1800 of a degree). The basic principle of the theodolite is illustrated in Fig XX.
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2.6 Putting It All Together Recall that in order to locate new points, all three coordinates must be defined (or two if elevation is not considered). To do this, a complete set of information must be gathered about each point. Combining the various methods discussed in this chapter is necessary to get the appropriate survey in the least amount of time with the equipment available. However, remember that you must use enough methods to ensure that you have gathered all the required data. For example, if you only record a horizontal angle to a point, but not the distance, you cannot locate it in the XY plane. Below is a list of combinations that can be used to complete the indicated survey. • • • • • • • • • • • •
Compass and Pacing (2d) Compass, Pacing and Abney Level (3d) Compass and Tape (2d) Compass, Tape and Abney Level (3d) Elevation Difference (Autolevel), Compass and Tape (3d) Elevation Difference (Autolevel), Compass and Stadia (3d) Elevation Difference (Autolevel), Horizontal Angle and Stadia (3d) GPS (2d or 3d) Theodolite Vertical Angle, Compass and Tape Theodolite Vertical Angle, Compass and Stadia Theodolite Vertical Angle, Theodolite Horizontal Angle, and Stadia (Theodolite) Theodolite Vertical Angle, Theodolite Horizontal Angle, and Tape (Theodolite)
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In addition, note that it is best to use methods with similar accuracy, such as compass and pacing, or Theodolite and Tape. It does not make sense to measure a horizontal angle to a precision of 10 seconds, and then pace the distance.
2.7 Traverse In the previous sections, most of the surveying methods assumed that the surveyor determines the relationship of a new point to a “known point”. This “known point” is usually the place where your instrument is set up (also known as a setup location, or Station). But what if the surveyor cannot see all the required points or objects from one setup location? Or what if the location of the Station is unknown? This is where a Traverse will come in. A Traverse is simply a series of Stations that starts from a known location and extends to wherever the surveyor needs it to go. Many times a traverse is along a road, path or river. Commonly a traverse is linear, that is the stations form a single line. Sometimes however, the traverse must branch off to cover other areas, similar to a fork in a road.
2.7.1 Hubs and Control Points As discussed further in Chapter 3, the best way to setup an instrument over a known location is to create a semi-permanent marker in the ground over which to set your tripod legs. This is usually referred to as a “hub”. Many times, hubs are given a special symbol on the map of a completed survey and a traverse is displayed as a line of hubs. See section 3.3 for more information on setting up hubs. As stated above, hubs are semi-permanent markers. They may be lost or change location. If you are setting hubs in a road, a truck’s tire may move the hub, especially if it has recently rained. A good surveyor should always double check the location of a hub if it has not been used recently. The method of doing this is by setting, and then later locating, “Control Points”. Control points are precise and permanent points (or points on an object) that can be found and relocated at a future time. Multiple control points should be placed at each hub for two reasons: accuracy of relocating your hub increases with more points, and your control points might disappear (trees can get cut down, large rocks moved, etc.). Control points are discussed further in Chapter 3, but some common ones are a nail in a tree (called a Tree Bench Mark or TBM), plate embedded in a concrete slab, or an iron pipe buried in the ground.
2.7.2 Backsights and Foresights The key to a traverse is accurately determining the location of the next “known point”. The next hub to be used in a traverse is called a Foresight. The hub that was used previous to the one now being used is called a Backsight. To locate the Foresight, determine where this point will be, install a hub (called “setting a hub”), then determine the location of the hub using the same methods used to determine other points. Special care should be taken to record the location of the foresight as accurately as possible because all future points taken from that hub will depend on it. Backsights are generally used as the “zero point” for determining Horizontal angles from a new Station. This is arbitrary, as technically any point can be used as a “zero point”, but this is a convention and can make the math easier when the survey is completed. Determining the relative location of the Backsight, also called “Sighting the Backsight”, also provides a check to the location of the hub that the Station is set up on (also called “Occupying the Hub”). Sighting the Backsight is essentially the same as locating the Foresight, just in reverse.
2.7.3 Loop One possible configuration of doing a traverse, or “running a traverse”, is to set the hubs such that the first hub can be sighted from the last Station. This is called a “Closed Loop” because it will look like a closed shape (instead of a line) on a map. There are many advantages to doing a Closed Loop, usually 11
called just a loop, but the primary one is that the error of the survey can be determined. This is done by comparing where the first hub is predicted to be, to where it is measured to be. Using all of the data from the Traverse, a surveyor should be able to predict the relative location of the first hub (angle and distance) from the Station, when occupying the last hub. (In fact, a surveyor should be able to predict the location of the first hub from ANY hub, but this may be a very large distance.) Then, if the traverse is a loop, the surveyor can MEASURE the distance from the Station to the first hub. Inevitably, these will not be the same. If the traverse has been done well, the difference in the locations will be off by a few centimeters. Traverse errors are generally expressed in centimeters or millimeters (or inches).
2.8 Using Gathered Data It is important for a good surveyor to understand how the data that he or she will be gathering will be used. The finished product of a surveyor, that which is delivered to the client is NOT pages of numbers that represent the survey (though this is important), it is the map that the data is used to produce, In this section there are a few tips and figures to give insight into the map making process.
2.8.1 Defining Objects Object defined in a survey come in three varieties: points, linear and polygonal. Point objects are obviously those that can be defined by a single point, such as a tree, fence post, or boulder. Point objects sometimes require further notation, such as “600mm Diameter Tree”. If an object is linear, such as a road or stream, it can be defined by either determining its centerline and noting its width (easier), or by recording both its edges. Remember when defining linear objects that any points that are “picked up” will be connected by straight lines. Thus, if the stream or road is straight for a long distance, the points can be far apart, but if the object is curved, there must be enough points to define the curve. Points defining a polygonal object will also be connected using straight lines. Thus, if picking up an object with straight sides (like a building), it is best to locate the corners. For a square building, if the corners are roughly 90 angles, only three corners are necessary. For circular objects like a water tank, at least 3 points are also required to define the circle.
2.8.2 Contour Lines Contour lines are imaginary lines on a map that represent a constant elevation. They can also be defined as lines perpendicular to the flow of water (water always flows down hill in the steepest direction). Creating contour lines on a map is the easiest method to display the three dimensional topography of land. The surveyor should note that in general, a straight line will be drawn between two points (and thus the contour lines will be evenly spaced). Therefore, care should be taken to record enough points to define the topography. If a hill or grade has a constant slope, points only need to be taken at the top and bottom of the slope. But if the slope is constantly changing, topography points must be taken at regular intervals, One example of this is if a stream has a wide flat bottom and then high, steep banks. If the surveyor only records points at the stream centerline and top of the banks, then the stream will be recorded as a vshaped ravine. However, if the surveyor records the edge of the stream at the bottom and the top of the bank, then the stream will be accurately recorded. This can be seen in Fig XX. In addition, when conducting a 3D survey, it is important to remember that any objects that are picked up, such as buildings, also provide an elevation at that point. This may be necessary and useful, such as when locating the centerline of a road, or unnecessary. In some cases, the surveyor is interested in the
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elevation, such as the top of a water tank, but should not be included in the topography map. These points should be noted as such in the field notes.
3.0 Field Practices In this section, how to survey in the field is outlined. Sections include how to keep a field notebook, setting up hubs, and taking various measurements.
3.1 How to keep a field notebook You must keep accurate notes and data to get good results while surveying. Thus, it is important to keep a neat and detailed field notebook. All of your data, sketches, and notes need to be written down in your field notebook. Any notebook can be a field notebook. Be sure to follow this format: Title Page Index Page Data/Notes/Sketches/Diagrams Here is an example title page. Anything that is underlined would be changed according to what surveying work you are doing. Survey Field Notebook Location: North of the water source Project: Surveying for EWB-OFCB-Bayonnais Hydroelectric Book: 1 of 5 Instruments used: T2 Theodolite Name: Kenold
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For the Index Page, you must have columns for page number, date, and title of each of the pages in your notebook. When you begin, you will only have one line in your index. As you collect more data, you will be adding more lines to your index.
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The remaining pages in your notebook should be very detailed. You must write down as much as you can about what you are surveying. Recorded field notes consist of a combination of tabulated data, sketches, and descriptions. The total record of any survey in the field notebook should provide a clear and concise picture of the survey performed. This information will include descriptions of the starting and closing stations, a description of any principal station established, the area or locality in which the work is performed, the purpose of the survey, and general remarks on weather, terrain, or other conditions that may be factors in evacuating the results. The information in the field notes must be complete enough that anyone not familiar with that particular survey operation can take the notebook, return to the locality, and recover or reconstruct any portion of the survey. Here is an example of recorded field notes
Note that the students recorded the date, location, and weather in the upper right hand corner. In addition, each student’s name was written next to a symbol of what their job was during the survey. For example, W. Doolittle has an open book next to his name, so he was recording data. R. Rodgers was sighting (tripod symbol). D. Evans and H. Trigger were taking measurements (Φ). There are more examples of how to take field notes for different types of surveys (e.g. triangulation, traverse, etc.). However, this will be covered in a later section Appendix E: More Field Book Techniques
3.2 Setting up the theodolite and tripod Setting up the tripod and plumb bob • Unfold and extend legs to desired length; then tighten the leg screw to secure in place 15
• • • • • • • • • • • • • • • • • •
Tighten clamp nut in the middle of the leg. Remove the plumb bob from tripod accessory case. Insert bayonet socket into bridge screw. Secure by turning clockwise ¼ turn. Position tripod so that plum bob is 1/2 inch from station point. Set legs in ground, push down firmly with your foot Remove plumb bob assembly and return to tripod accessory case. Remove head cover and stow it on the bracket Remove the theodolite from the carrying case Loosen the three nuts and pull levers away from carrying case base Grasp the right-side axle bearing with one hand and place your other hand below the main housing to lift the theodolite from the carrying case base. Install the theodolite on the tripod Position theodolite on tripod head and secure it loosely with the bridge screw Center the circular level bubble using leveling screws. Look through the optical plummet eyepiece and adjust eyepiece until the circular marks are clearly visible Adjust the theodolite on the tripod head until the station point is centered on the eyepiece circular marks Tighten down the bridge screw to secure the theodolite in place. Install the illumination system Install the telescope eyepiece sunglass filter as necessary Proceed on to leveling the theodolite
NOTES: If you’re on a slope, have 2 legs on the downhill side. Once you’re close to level, plant one leg and move the 2 others while watching the optical plumb.
3.3 Setting Hubs and Control Points , As discussed in section 2.7, a surveyor must set Hubs and Control Points in order to run an effective traverse, Both Hubs and Control Points must be a precise location, such as a nail or a tack (not an object corner or an object that is 10cm wide). Requirements for a good hub location are: • • • • • •
Line-of-sight to Foresight (next hub) Line-of-sight and Backsight (last hub) View of all points and objects that need to be picked up Placed in a reasonably immobile object Placed where it can be found again in the future If located in a road, placed off to the side so that it is not disturbed by vehicles or livestock
If placing a hub in a solid object, such as a tree, root, concrete or pavement, a nail will usually be sufficient. If placed in the soil, obviously a nail will not be immobile or easy to find again. Thus, the nail is usually placed in a wooded stake after the stake has been driven into the ground.
3.4 Taking a Compass Bearing Here’s how you take a compass bearing (Tyler) A compass is primarily used to take bearings. A bearing is the horizontal angle measured clockwise from north (Note: this can be either magnetic north or true north. The needle on the compass points to magnetic North. For our main purpose of measuring a horizontal angle between two points, it is easiest to just use magnetic north. Since both bearings will be compared to the same reference point (magnetic north), this will not affect accuracy of the horizontal angle between points.)
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To use a compass to take bearings off a real point on the landscape (as opposed to a map) use your compass to measure the angle clockwise from magnetic north to this point on the landscape. This is called magnetic bearing. Remember that the bearing is measured clockwise. Thus, if you think of north as 12 O’clock on a watch, then a bearing to the right of that (say 1 o’clock) is greater than north, and a bearing to the left of north (like 11 o’clock) is less than north. To determine the horizontal angle between two points, simply subtract the two bearings. This difference will be the number of degrees between the two points, referenced from where you are standing.
3.5 Measuring Distance 17
In this section, you will learn how to measure ground distance using measuring tape and pacing.
3.5.1 Pacing As discussed in Error! Reference source not found., pacing is using one’s normal steps to measure distances. The following paragraph outlines how to do this. Accurately measure a pacing course on level ground. Repeatedly pace off the course, counting off the number of paces it takes to complete the distance. Keep a natural comfortable pace that can be held all day. Don’t try to adjust to even standard, but try to count your pace to the course distance. It is easier to measure the number of paces per meter (for example, 46 paces per 100 meters) than measuring number of meters per pace. Measure your pace against different terrains and at different times of day, because your pace may change between the morning and afternoon or when walking on different terrains There are many methods to keep track of the distance traveled when using the pace count. Some of these methods are: put a pebble in your pocket every time you have walked 100 meters according to your pace count; tie knots in a string; or put marks in a notebook. Do not try to remember the count; always use one of these methods or design your own method. Certain conditions affect your pace count in the field, and you must allow for them by making adjustments. a) Slopes. Your pace lengthens on a downslope and shortens on an upgrade. Keeping this in mind, if it normally takes you 120 paces to walk 100 meters, your pace count may increase to 130 or more when walking up a slope. b) Winds. A head wind shortens the pace and a tail wind increases it. c) Surfaces. Sand, gravel, mud, snow, and similar surface materials tend to shorten the pace. d) Elements. Falling snow, rain, or ice cause the pace to be reduced in length. e) Clothing. Excess clothing and boots with poor traction affect the pace length. f) Visibility. Poor visibility, such as in fog, rain, or darkness, will shorten your pace.
3.5.2 Stadia The extra two cross hairs in the scope can tell you a horizontal distance. The stadia distance (SD) is the distance between these two cross hairs. In this example, SD= 15-13.45=1.55 ft. To get horizontal distance (HD ) the formula is SD=100 HD. (This will be off by more if the vertical angle is very large) Then there is the formula that sin^2 of the zenith * SD gives you your vertical angle.
3.5.3 Measuring Tape One Person holds the metal clip at the end of the measuring tape at the first location. Another person takes the body of the measuring tape and walks to the next location. The second person reads the number on the measuring tape and records the distance. Make sure to keep the measuring tape taut and measure the distance from the ground to where the tape is being held. When reading the tape, be sure to check the feet and inches markers twice. The "foot" markers are close to the "11th inch" marker and sometimes an extra foot is mistakenly added to the measurement.
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3.6 Graduated Rod You must be able to sight readings on the graduated rod with the theodolite to take vertical measurements. You can make the graduated rod taller by extending or telescoping the inner tubes of it. Be sure to extended the largest diameter telescoping parts before the smaller diameter telescoping parts, otherwise the numbers will be off. You can try this on your own and notice the difference between measurements when you extend different parts of the rod. From the vertical angle set up, the number on the graduated rod that falls between the crosshairs of the viewer is the vertical height. Both the number read from the rod and the height of the eyepiece of the scope should be recorded.
3.7 Auto-Level 3.7.1 Nivellement direct ou géométrique Les méthodes de nivellement direct constituent l’arsenal le plus efficace pour déterminer l’altitude de points particuliers. La précision des déterminations dépend du matériel employé (cf. §Error! Reference source not found.) mais aussi et surtout, des méthodes, ce que nous allons aborder maintenant : Nivellement par rayonnement : la première mesure est effectuée sur un point d’altitude connue, de • façon à déterminer l’altitude du plan de visée. A partir de là, toutes les altitudes sont déterminées par différence par rapport à ce plan. Cette méthode permet de lever rapidement un semis de points matérialisés (sondages, points de berges, de fonds…). Elle présente néanmoins l’inconvénient de n’offrir aucun contrôle sur les déterminations : toute erreur de lecture est indétectable et fatale. Nivellement d’itinéraires par cheminement : • c’est la méthode la plus couramment employée pour déterminer les altitudes de points matérialisés, non situés à une même distance d’une seule station d’appareil. Elle est également plus sûre, quant aux éventuelles erreurs de lecture, et plus intéressante du point de vue de la précision des déterminations : on dispose de méthodes de compensation des erreurs très efficaces. Plusieurs règles sont appliquées pour minimiser l’influence des erreurs systématiques et accidentelles : les portées équidistantes, les contrôles de marche, le contrôle sur fermeture… Nivellement de franchissement : cette • méthode est beaucoup plus difficile à mettre en œuvre et s’applique dans le cas de franchissement de vallées, où le principe des portées équidistantes est inapplicable. On travaille dans ce cas simultanément avec deux appareils, de part et d’autre de l’obstacle (le cas idéal étant de pouvoir les aligner avec les mires), afin de minimiser les erreurs instrumentales et atmosphériques. Nivellement d’auscultation : cette dernière méthode a pour objectif de déterminer la cote d’un repère • et ses variations dans le temps (barrage, pont, bâtiment). Elle nécessite l’application de tous les principes énoncés précédemment, et plus encore : équidistance, réglage optimal du niveau, mires en invar, contrôles, problèmes de réfraction accidentelle (intérieur/extérieur d’un bâtiment), sûreté des repères… Pour résumer : Le principe du nivellement géométrique est la mesure d’une différence d’altitude, ou d’une succession de différences, par rapport à un plan ou un point connu. Il est réalisé au niveau, et la précision des mesures peut aller de 1/10ème de mm à quelques mm, selon les matériels et protocoles mis en œuvre. De manière générale, la relation entre l’altitude du point de départ R1 et le point d’arrivée R2
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d’un nivellement par cheminement est donné par la relation :
Z R2 =Z R1 +∑(AR− AV) Eq. 1
où AR représente les lectures Arrière (en rapport à la direction de l’itinéraire), et AV les lectures Avant. Lorsque les altitudes des points de départ et d’arrivée sont connus, on peut alors calculer la fermeture du cheminement :
f =ΔH obs −ΔH th
Eq. 2
3.8 Theodolite 3.8.1 Theodolite Usage Tips Convenient Horizontal Angle Measurement Method for Theodolites equipped with dual swiveling horizontal plates: 1. align the top part of the theodolite with the lower swiveling plate such that an angle of zero appears in the viewing window. 2. Tighten the upper plate such that the theodolite is locked to the lowering swiveling plate and proceed to point this to the reference object which you wish to measure the horizontal angle from. 3. Tighten the lower swivel plate such that it locks the coordinate plane with zero aligned on the reference point. 4. Loosen the upper plate and rotate the theodolite, aiming it at the desired object to measure the horizontal angle between the object and reference point. To use the horizontal fine adjustment knobs, the rough adjustment knobs must first be tightened Once the theodolite is appropriately placed the person at the theodolite will need to read the vertical angle, the horizontal angle, the vertical height, and use the stadia to get the horizontal distance.
3.8.2 Vertical Angles 1. 2. 3. 4.
Unlock the vertical clamp and tilt the eyepiece until the point of interest is aligned on horizontal lines. Lock the clamp in place. The point of interest will be whatever number on graduated rod is easiest to read. Looking through the small eyepiece, use the minutes and seconds adjuster to align one of degrees on the vertical scale with the double lines just below it. The reading is the degree that has been aligned and the minutes and seconds is read from right hand scale. See accompanying figure. To complete the reading, it may be necessary to measure the distance from the theodolite to point of interest.
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the the the the the
3.8.3 Horizontal Angles Using a Theodolite 1. 2. 3. 4. 5.
Unlock the upper horizontal clamp. Rotate the theodolite until the arrow in the upper or lower rough sight points to the feature of interest and lock the clamp. Look through the main eyepiece and use the upper horizontal adjuster to align the vertical lines on the feature of interest. The reading is taken by looking through the small eyepiece. Using the minutes and seconds adjuster set the one of the degrees on the horizontal scale so the single vertical line on the bottom scale is between the double vertical lines under the selected degree. The reading is the degree which has been aligned and the minutes and seconds read from the right hand scale and is the horizontal angle from the reference line.
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This view through the small eyepiece shows the vertical and horizontal degree scales and the minutes and seconds scale.
3.8.4 Theodolite Maintenance •
It is important to keep the theodolite dry and free from humidity. It should not be used during wet weather.
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Make sure the theodolite is clean and free from dirt and dust before storing the device.
•
Before adjusting, turning, or twisting any component of the theodelite, be sure that the appropriate adjustment knob is loosened. If adjustment is difficult, do not forcefully twist the device as this may strip or damage the tightening screws. Check again to ensure the adjustment knob is loose.
•
Keep the theodolite stored in its case whenever it is not in use.
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Always store the device with all adjustment knobs loosened in order to prevent stripping or damage while placing and removing the device from the case.
3.9 Establishing other landmarks Using a double prism optical square is a good way to establish landmarks that are at right angles to each other. • First, you will want to establish a landmark directly in front of you. This will be your reference point to find objects which are at relative right angles. • Hold the optical square close to your face and set this object in the center of the center window. • The landmarks that line up in the top and bottom windows are at right angles to the first landmark. • Things to Remember
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• •
Look for landmarks that will be permanent, you don’t want to include something in your survey data that will not be able to be referenced later on. Pegging trees is possible, the peg will not move up while tree grows.
4.0 Math Review After surveying data has been collected, one still needs to translate it into useful numbers. This sections outlines how to do the calculations to find the X, Y, and Z coordinates of points found in a survey.
4.1 Surveying Trigonometry If you were point A (0,0,0), the horizontal angle to point B, the vertical angle to point B, and the ground distance between points A and B, could you calculate (x,y,z) of point B? Horizontal Angle
Vertical Angle x-distance
z-distance
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x-distance
17.6 y-distance
Ground Distance: 50m
Vertical Angle Horizontal Distance: Ground Distance x cos(θ), x-distance = 50m x cos(60°) = 25 m Vertical Distance: Ground Distance x sin(θ), z-distance = 50m x sin(60°) = 42.86 m Horizontal Angle Vertical Distance: Horizontal Distance x sin(θ) / cos(θ), y-distance = 25 m x sin(17.6°)/ cos(17.6°) = 8.00 m -Also- Vertical Distance: Horizontal Distance x tan(θ), y-distance = 25 m x tan(17.6°) = 8.00 m
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4.2 Angle Calculations Recording data in surveying requires precise angle measurements. Angles are usually expressed in degrees, minutes and seconds (57° 42’ 30”). Sometimes, a precise angle is expressed in degrees with many decimals (57.70833°), or as degrees, and minutes with decimals (57° 42.5’). If minutes are expressed, the degrees must be an integer, and if seconds are expressed, the minutes must be an integer. Seconds are typically expressed in integers. One degree( ° ) is divided into 60 minutes ( ′ ) One minute ( ′ ) is divided into 60 seconds ( ″ ). One second ( ″ ) is normally divided into decimals (when divided). 1° (degree) = 60’ (minutes) = 3600″ (seconds) It is important to be able to convert degrees in both directions. To convert from degrees, minutes and seconds (dd.mm.ss), to degrees (dd.dddd): Decimal Degrees = Degrees + Minutes / 60 + seconds / 3600 d = D + M/60 + s/3600 Example: D = 58°, M = 39′, s = 13″ 58° 39’ 13” = 58° + 39’/60 + 13”/3600 = 58° + 0.65° + 0.00361° = 58.6536° To convert decimal degrees (dd.dddd) to degrees, minutes and seconds (dd.mm.ss), subtract the integer part and multiply the remainder by 60 to get minutes, then repeat for seconds. Minutes= [Degrees - Int(Degrees)]*60 Seconds = [Minutes – Int(Minutes)]*60 Example: D = 49.5125° Minutes = (49.5125 – 49)*60 = 30.75 minutes Seconds = (30.75 – 30)*60 = 45 seconds 49.5125° = 49° 30’ 45”
4.3 Unit Conversions There are 3.28084 feet (ft) in 1 meter (m). To convert meters (m) to feet (ft), multiply by 3.28084. To convert feet (ft) to meters (ft), divide by 3.28084 or multiply by 0.30479999
2 feet
X
0.3048 meters 1 foot
=
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0.6096 meters
5.0 Appendix A: More Stadia Information From http://mysite.du.edu/~jcalvert/astro/stadia.htm (edit this) The Stadia is a method of measuring distances rapidly with a telescope (usually on an engineer's transit or an alidade) and a graduated rod. When the telescope is focused on the rod, the distance s intercepted on the vertically-held rod between two stadia hairs seen in the eyepiece gives the distance D as D = ks, where k, the stadia constant is often made to be 100. Therefore, if 6 ft is intercepted on the rod, then the distance from the telescope to the rod is 600 ft. There are small corrections to this that will be mentioned below. If the line of sight is inclined, the vertical angle is also measured and can be used to reduce the results to horizontal and vertical distances. Stadia can give results correct to about 1 ft under the best conditions, which is often sufficient, and can also serve as a check on more precise measurements. The term stadia comes from the plural of the Greek stadion, the word for a distance of 185 to 192 metres (607-630 ft). A very similar length is the modern furlong, or eighth of a mile, 660 ft. A "stadion" was also an athletic venue, with lengths laid out for competition and seats for spectators. The Latin stadium, stadia was a direct borrowing with the same meaning. Distance over the ground was traditionally measured by long poles or rods laid successively end to end. The ancient Egyptians used rope for the same purpose. This practice is reflected in the traditional rod, pole or perch of 16.5 feet. This odd length came from dividing down an English mile of 5280 ft, first into furlongs of 1/8 mile or 660 ft, then into tenths, or chains, of 66 ft, and finally into quarters of this, or 16.5 feet. Four rods make a chain, ten chains a furlong, and 80 chains a mile. The Gunter's chain of 100 iron links and length 66 ft was much easier to use and carry than an ungainly pole, and gave more accurate results. 10 square chains is an acre, so Gunter's chain was closely related to traditional measures of distance and land areas. The engineer's chain of 100 links, each 1 ft long, replaced Gunter's chain, and was itself replaced by the 100 ft steel tape, which is an excellent and easily handled way to measure distances. Doing this is still called "chaining," however, and the people who do it are called chainmen. Accurate chaining is subject to many errors, which are largely systematic, but with care they can be overcome. These errors include thermal expansion and elasticity of the tape, as well as ground irregularities. Distances are now conveniently measured by timing modulated laser beams returned by retroreflectors. Large distances can be covered at one leap, and the intervening ground does not have to be traversed on foot. Stadia shares these advantages. Microwaves were first used for this purpose, but have now been superseded by lasers. The main errors are in estimating propagation conditions, temperature and humidity, which affect the velocity of light, and are often poorly known or vary over the path. Even without consideration of these uncertainties, laser ranging is more accurate than stadia, but is also much more expensive. We also have Global Positioning System location, which is accurate to roughly 1 metre (with special care, centimetre accuracy is possible, but it requires work). In spite of these excellent alternatives, it is still interesting to know the stadia method, which is often applicable in unusual circumstances. The stadia method is an application of paraxial optics. The telescope consists of an objective (usually one achromatic lens, but sometimes more) that produces an image of the distant scene close to its focal plane, which is then examined by the eyepiece. We will be concerned only with the objective. The action of the telescope objective is described by principal planes, nodal planes and focal lengths. Since the final and initial media are the same, the nodal planes coincide with the principal planes, and the primary and secondary focal lengths are equal. The telescope is mounted so that the outer principal plane of the lens is a distance c from the axis of the instrument, that is vertically over the occupied location. If the distance of the stadia rod from the instrument axis is D, then the object distance is D - c. The corresponding image distance d behind the other principal plane is then given by 1/d + 1/(D - c) = 1/f. Fine lines are etched on a glass reticle placed approximately at the focal point of the telescope objective. These were once crosshairs made of spider web, and are still called crosshairs for that reason. There are vertical and horizontal crosshairs for sighting purposes, and two shorter stadia hairs at equal distances above and below the horizontal crosshair. The separation of the stadia hairs is denoted by i. The eyepiece is adjusted so that the crosshairs are sharp, and the telescope then focused so that the object viewed is also sharp, so that their images occur at the same point. We now make use of the unit angular magnification property of the nodal points to establish that the angle s/(D - c) of the rod intercept as seen from the outer nodal 26
point is equal to the angle i/d at the inner nodal point, or 1/d = s/i(D - c). The relations are illustrated in the diagram. When this is substituted in the lens equation, the result is f + fs/i = D - c, or D = (f/i)s + (f + c), which is the fundamental stadia formula. The derivation is confused in Breed and Hosmer; the principles are not clearly stated, and reference is made to a different diagram than the one appearing on the page, possibly one from an earlier edition. I hope that this derivation will make things clear, since they really aren't very difficult. Now, f/i will be a constant determined by the construction of the telescope and reticle, and is usually called k, the stadia constant of the instrument. It is commonly 100, but a more accurate value can be established by experiment if necessary. k = 100 corresponds to an angle of 0.01 radian, or 0.573°. The correction (f + c) is to be added to ks to find D. If f = 200mm and c = 100 mm, then (f + c) is 30 cm, or about 1 ft. This correction is sometimes ignored. The formula just derived applies to a horizontal sight on a vertical rod, or to an inclined sight on a rod held perpendicular to the direction of view. It is not easy to hold a rod perpendicular to the line of sight, so it is held accurately vertical. If s is the intercept on a vertical rod, then s cos α would be the intercept, approximately, if the rod were held perpendicular to the line of sight. The slant distance is, then, D' = ks cos α = (f + c). Now it is easy to find the horizontal distance D = D' cos α and the vertical distance V = D' sin α. At one time, tables were prepared for performing these calculations, but with pocket calculators they are no longer necessary. A pocket calculator can reduce the data quickly and accurately, including the correction (f + c) without any approximation. References C. B. Breed and G. L. Hosmer, The Principles and Practice of Surveying, 11th ed. (New York: John Wiley & Sons, 1977). pp. 100-108.
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6.0 Appendix B: More Field Book Techniques
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7.0 Appendix C: Resources http://www.globalsecurity.org/military/library/policy/army/fm/3-34-331/ch5.htm http://www.globalsecurity.org/military/library/policy/army/fm/6-2/Ch4.htm (field notes) http://www.oregon.gov/ODOT/HWY/GEOMETRONICS/docs/BasicManual2000_02.pdf http://www.scribd.com/doc/2963203/BASIC-LAND-SURVEYING-GLOSSARY-ePalmetto http://books.google.com/books?id=2gB7w9XlNJAC&pg=PA80&dq=surveying+horizontal+vertical+ angles&source=gbs_toc_r&cad=0_0 http://jb.henry.free.fr/cours/Chapitre2.htm
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