mine surveying

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Basic Concepts of Surveying Surveying can be defined in a number of ways. Surveying is the science of determining the position in three dimensions, of natural and man made features on or beneath the surface of the earth. Alternatively surveying is the art and science of making the measurements required to enable the relative positions of points on the surface of the earth or in the subsurface to be represented on a map or plan, or alternatively in a digital model in a computer database. The art in surveying involves the selection of the most elegant and efficient method of making the required measurements. Generally one can state that the minimum number of measurements should be made in the shortest possible time to achieve the required result. The science of surveying comes from the requirement to have knowledge of the following: Trigonometry Geometry in two and three dimensions Physics with respect to optical theory, for precise measurements also meteorological aspects and other aspects of physical science Statistics and error theory Construction of instrumentation Surveying Equipment Numerous types of surveying equipment exist, the list below details some: Measuring tapes Differential levels Plane tables Theodolites Tacheometers Electronic distance measuring equipment Global positioning systems In addition, a range of ancillary equipment is used; details are in a number of surveying texts and are not included here. Measurements Measurements are undertaken to determine numeric values for: Distances Elevation or differences in elevation Angles Time 1

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  • Basic Concepts of Surveying

    Surveying can be defined in a number of ways. Surveying is the science of determining the position in three dimensions, of natural and man made features on or beneath the surface of the earth. Alternatively surveying is the art and science of making the measurements required to enable the relative positions of points on the surface of the earth or in the subsurface to be represented on a map or plan, or alternatively in a digital model in a computer database. The art in surveying involves the selection of the most elegant and efficient method of making the required measurements. Generally one can state that the minimum number of measurements should be made in the shortest possible time to achieve the required result. The science of surveying comes from the requirement to have knowledge of the following: Trigonometry Geometry in two and three dimensions Physics with respect to optical theory, for precise measurements also

    meteorological aspects and other aspects of physical science Statistics and error theory Construction of instrumentation Surveying Equipment Numerous types of surveying equipment exist, the list below details some: Measuring tapes Differential levels Plane tables Theodolites Tacheometers Electronic distance measuring equipment Global positioning systems In addition, a range of ancillary equipment is used; details are in a number of surveying texts and are not included here. Measurements Measurements are undertaken to determine numeric values for: Distances Elevation or differences in elevation Angles Time

    1

  • It must be noted that measurements are a comparison with a standard and will always be subject to error. This is not just the case in surveying and affects all aspects of measurement. The error in measurement can be defined as:

    value true theL valuemeasured theL

    tmeasuremenin error theewhere

    T

    M

    ==

    =

    = Tm LLe

    For a treatment of errors please refer to Schofield W (1993) Engineering Surveying, fourth edition, Butterworth-Heinemann Ltd, Oxford. Also note a fifth edition of this text is due to be released in 2001. Relative position of points Surveying requires points on the earths surface or in the subsurface to be related to each other. The relative position of points can be divided into two types: 1. Determining the horizontal, also called the planimetric position 2. Determining differences in height or elevation The horizontal position of points can be determined by a number of methods including: a) Triangulation b) Trilateration c) Radiation d) Rectangular offsets e) Satellite positioning systems

    2

  • Triangulation The length and direction of the line AB is known. Angles A and B are measured.

    Trilateration The length and direction of the line AB is known. The lengths a and b are measured.

    3

  • Radiation The length and direction of line AB is known, angle A and length B are measured.

    Rectangular offsets The length and direction of the line AB is known. The distance d along the line AB and the right angle offset h are measured. Satellite Methods Described later in the notes.

    4

  • Differences in Elevation Differences in elevation between points can be determined by: a) Differential levelling b) Trigonometrical levelling Differential levelling

    The levelling instrument defines a horizontal line of sight. The vertical distances above or below this line of sight are measured using a graduated vertical staff. Trigonometrical levelling In this method the vertical angle and the slope distance to the target location are measured and the difference in elevation is calculated.

    5

  • Map and Plan scales Survey results are usually presented in the form of a map, plan or computer generated output at some scale. Numerous scales are used, the scale used depends on the detail required. Scales are represented as some form of representative fraction. Usually in the form of 1 map or plan unit to X ground units, for example a scale of 1:500 means 1 mm on the plan represents 500 mm on the ground. At this point it must be stressed that such plans are a horizontal representation of the survey and are reduced to a horizontal plane for plotting purposes. In the modern world computers are used to process and store survey information. Generally a database is created from the field survey information, there may be several databases for example raw survey information and various levels of processed survey information. Such databases may form a portion of a Geographic information system (GIS). Typically the database may contain the spatial coordinates of the points, the east or E coordinate, the north or N coordinate and the Z or elevation coordinate. These coordinates will be referred to some datum. Additional information may be appended, for example the depth, dip and direction of a borehole. Trigonometry Trigonometry is an essential element in the science of surveying. To aid the student the following presents a review of the basic trigonometrical formulae that are used in surveying.

    6

  • Basic Trigonometrical ratios With reference to the right angled triangle ABC, the angles are represented by A, B and C whilst the length of the side opposite each angle is represented by a, b and c.

    Six common trigonometrical functions are used, the following lists the common abbreviations employed mathematically: Sine = sin Cosine = cos Tangent = tan Cotangent = cot Secant = sec Cosecant = cosec The basis of trigonometry is summarised by the following relationships:

    BabCotA

    BacACo

    ecBbcSecA

    BbaTanA

    BcbCosA

    BcaSinA

    tan

    secsec

    cos

    cot

    sin

    cos

    ====

    ========

    From the above the following can be derived:

    7

  • Ab

    Bb

    Ba

    Aac

    BaAaBcAcbBbAbBcAca

    cossincossin

    tancotsincoscottancossin

    ============

    Additionally the pythagoras theorum can be applied to a right angled triangle

    222 bac += also for a right angled triangle the sum of the angles is always 180. A + B + C = 180 In the case of oblique triangles, those where no angle is equal to 90 as illustrated on the following page the following relationships can be applied: Sine rule:

    Cc

    Bb

    Aa

    sinsinsin ==

    Cosine Rule:

    CabbacBaccabAbccba

    cos2cos2cos2

    222

    222

    222

    +=+=+=

    8

  • Half Angle Formulae

    abbsasC

    accsasB

    bccsbsA

    ))((2

    sin

    ))((2

    sin

    ))((2

    sin

    =

    =

    =

    Note: s = the semi perimeter of the triangle ABC = (a + b + c)/2 An alternative form of the half angle formulas are:

    )(2tan

    )(2tan

    )(2tan

    csKC

    bsKB

    asKA

    =

    =

    =

    where

    scsbsasK /))()(( = The following table summarises the formulas required to solve for various missing elements of a triangle. This can be any three of the six elements, angles A, B, C and lengths a,b,c in combination except the case where the three angles are given, this yields an infinite number of solutions.

    9

  • A number of other trigonometrical identities are also employed in the derivation of formula used in surveying. Compound angle relationships

    )tan1()tan2(2tan

    1cos2sin21sincoscos2sinAcosAsin2A thenBA if

    )tantan1()tan(tan)tan(

    sinsincoscos)cos(sincoscossin)sin(

    2

    22222

    AAA

    AAAAA

    BABABA

    BABABABAAABA

    ====

    ==

    ==

    =

    m

    m

    Transformation of Products and Sums

    }2/)sin{(2sinsinsinsin2)cos()cos(

    coscos2)cos()cos(sincos2)sin()sin(cossin2)sin()sin(

    BABABABABABABABABABABABABABA

    +=+=+

    =++=+=++

    2)(cos

    2)(cos2coscos

    2)(sin

    2)(sin2coscos

    2)(cos

    2)(sin2sinsin

    2)(cos

    2)(sin2sinsin

    BABABA

    BABABA

    BABABA

    BABABA

    +=+

    +=

    +=

    +=+

    Errors It must be noted that measurements are a comparison with a standard and will always be subject to error. This is not just the case in surveying and affects all aspects of measurement. The error in measurement can be defined as:

    10

  • value true theL valuemeasured theL

    tmeasuremenin error theewhere

    T

    M

    ==

    =

    = Tm LLe

    With reference to the equation above the value of e is never known. Estimates of the true value and error can be estimated by the use of statistics such as the mean and standard deviation of a repeated set of measurements, however this will only give an indication of the true value. Errors in measurements can be subdivided into two types:

    1. Random errors 2. Systematic errors

    In addition to the above types an addition variation can be introduced to measurements, these are not classed as errors and are Blunders and/or Mistakes. Random errors remain after all blunders and systematic errors have been removed from a measurement. Random errors are the result of human and instrument limitations and as such cannot be completely eliminated even following statistical manipulation. Propagation of random errors can result in large variations in calculated quantities. This can be reduced by taking the mean of repeated measurements and by applying the least square. Systematic errors result from a number of causes. These include poor adjustment of measuring equipment or changes in environmental conditions. Under the same set of conditions these errors remain constant. Systematic errors conform to physical laws and can be eliminated by applying the correct adjustments to instruments prior to taking the measurement or by the application of appropriate measurement techniques or adjustments to measured values. Accuracy and Precision These are both important to understand. Accuracy is an indication of the degree of agreement to the true value of a measurement. Precision is the degree of agreement between measurements and has nothing to do with accuracy.

    11

  • 1

    Levelling

    Levelling is also termed vertical control and refers to various heighting procedures that are used to obtain the elevation of points of interest above or below a reference datum. In Australia the most commonly used datum is the Australian Height Datum (A.H.D.)

    Engineers are generally more concerned with the relative height of one point above or below another, to obtain the difference in height between the points. As such it is not unusual for an arbitrary datum to be used locally.

    The vertical heights of points above or below a reference datum are referred to as the reduced level (RL) or more simply the level of a point.

    Levelling

    Levelling is the most widely used method of obtaining the elevations of ground points relative to a reference datum and is usually carried out separately to the determination of the planimetric position. Basically levelling involves the measurement of vertical distance relative to a horizontal line of sight

    Definitions

    Reduced Level (R.L.) The height of a point above the selected datum.

    Mean Sea Level (M.S.L.) This is the datum most frequently used. Datum for Australia is the Australian Height Datum (A.H.D.)

    Backsight (B.S.) The first reading taken by an observer at every instrument station.

    Foresight (F.S.) The last reading taken at an instrument station

    Intermediate sight (I.S.) Any reading taken at an instrument station which is NOT a permanent stable reference point.

    Bench Mark (B.M.) A point of known Reduced Level (R.L.), usually a permanent stable reference point.

    Temporary Bench Mark (T.B.M.) Where only relative heights are required or a bench mark is required within area of work T.B.M. is established.

    Change Point The point at which the position of the instrument is changed, making it both a backsight and foresight.

  • 2

    Basic Principle

    To determine the difference in height between different points it is necessary to produce a horizontal line of sight. This requires the use of an instrument. In surveying, this instrument is known as a surveyor's level, often generically known as a dumpy level although this is not strictly correct. All levels consist of these Features: a telescope with a cross hair, mounted on a device that enables the line of sight to be oriented in a horizontal plane or line.

    Levels can be used for:

    Determining the height of a particular point Determining differences in height between points Determining the contours of a land profile Providing data for road cross-sections Providing data to calculate volumes for earthworks Setting out level surfaces for construction Setting out inclined surfaces for construction

    Factors influencing the use of Levels:

    Clear lines of sight are needed between points to be measured. Height precision is dependant upon the precision of the instrument used and

    the length of the line of sight. Height accuracy is maintained through proper adjustment of the level and

    correct field procedures.

    Models

    There are three types of levels, defined by different design principles:

    Dumpy - levelled using three foot screws to define a horizontal plane. Tilting - manually tilted to give a horizontal line of sight only. Automatic - automatically tilted to give a horizontal line of sight only.

    Precision

    The different types of levels can be further divided into different classes defined by their precision:

    Precise

    These are very accurate instruments for geodetic networks, monitoring surveys or any other very precise levelling. Instruments in this class are tilting or automatic types and can be levelled within approximately +/-0.2" of a horizontal line.

    Medium Accuracy

    These are used for engineering surveys and small control networks. Instruments in this class are tilting or automatic types and can be levelled within approximately +/-1" of a horizontal line.

  • 3

    Builders

    Used for low accuracy, short range levelling such as setting out on building sites. Although described as being low accuracy instruments, this is relative to the other classes of levels and the results obtained with this class of instruments will be well within the tolerances required on the majority of construction sites. Instruments in this class are tilting or dumpy types and can be levelled within approximately +/-10" of a horizontal line or plane.

    Auxiliary Equipment Required:

    Levelling staff to determine height above a ground point or below an overhead point.

    Staff bubble to ensure the levelling staff is vertical during measurements. Change plate to ensure the levelling staff is stable during measurements.

    Figure 1 Staff Bubble

    The Staff bubble is attached to the level staff while taking level readings. The staff bubble should be centred in the circle as shown in the picture here. A centred staff bubble indicates that the staff is being held vertical and accurate staff readings can be made at this time. If a staff bubble is not available the staff can be rocked slowly backwards and forwards and the minimum reading on the staff booked as this is the reading when the staff is vertical. Levelling staffs are made of wood, metal or glass fibre and are graduated into metres and decimals as illustrated in figures 2, 3 and 4. The alternate metre lengths are black and red on a white background. Most staffs are telescopic or socketed in three-sections for easier carrying. The graduations come in numerous forms, the E Pattern type, as defined in BS 4484 is commonly applied. The smallest graduation on the staff is 0.01m with readings estimated to the nearest millimetre.

  • 4

    Figure 2

    Figure 3

  • 5

    Figure 4

  • 6

    Figure 5 Change plate Types of level Dumpy level The dumpy level is a small cheap optical level found on many construction sites, it is not usually used by surveyors. For the line of sight to be horizontal the bubble tube axis must be made perpendicular to the vertical axis - the direction of gravity. This is achieved by the precise levelling of the foot screws.

    Figure 6: Dumpy level

  • 7

    Tilting Level These are a more modern instrument than the dumpy level, where the telescope can be rotated slightly through the vertical plane. When the telescope is set onto the staff, the line of sight is set precisely horizontal using the highly sensitive tubular bubble. A tilting screw is used to raise or lower one end of the telescope to centre the tubular bubble - the line of sight at this point is horizontal. This instrument was used extensively in the past by Surveyors but has been largely superseded by the Automatic Level.

    Figure 7: Tilting level The Automatic level

    The telescope is rigidly fixed to the tribrach and the line of sight is set horizontally by a vertical compensator set inside the telescope. Gravity acts on the compensator (a series of prisms, forcing the line of sight through the telescope into the horizontal plane).

    The combination of low cost and simplicity of operation of the automatic level means that it is the most popular type of level used in the Surveying Industry.

  • 8

    Figure 8: Automatic level

    Digital Level

    This instrument uses electronic digital image processing to evaluate a bar coded survey staff. The process is very similar to the commonly used bar coding systems used in industry. The system was first introduced in 1990 by the Wild company (now Leica Geosystems) and has since been adopted by others. Digital image processing allows height and distance data to be read and recorded electronically- thus eliminating manual booking errors and also reducing the time taken for the manual staff reading and booking process.

    Figure 9: Digital level

  • 9

    The Levelling staff

    There are a number of different types of staffs that are used for levelling. They are usually made of rigid fibreglass or aluminium box sections of about 1.5m in length, that telescope or socket together. Other varieties are made from wood and hinge together. The total height for a staff is generally no higher than 5m.

    The most common staffs are graduated in metres, decimetres and centimetres using alternating colours of black and red (see picture at right). There are a variety of ways in which they are graduated. The two most common types of staff faces are the E type face or the 5mm graduated face.

    The staff should be held vertical over the point to be measured with the face of the staff pointing towards the level. The staff can be held vertical with the aid of a bubble which has a convenient right angle handle which allows it to be held against the staff. If a precise level is to be taken at a point then care must be taken too ensure the staff is correctly positioned over the point and it does not subside under the weight of the staff. For points that are not clearly defined or on soft ground a change plate is often used.

    Figure 10 Setting up the Automatic level

    1. Set up the legs of the tripod so that they are firmly secured and the plate is approximately horizontal.

    2. Take the level out of the box, and while still holding firmly, attach it to the head of the legs using the plate nut.

    3. Centralise the circular bubble using the foot screws on the tribrach of the level.

    4. Eliminate parallax - This will occur when the crosshairs are not brought into sharp focus. Parallax is present when the crosshairs appear to move when the position of your eye changes. To eliminate parallax it is necessary to bring the cross hairs into sharp focus using the eyepiece focusing screw.

    5. Centre the vertical cross-hair on the levelling staff and clamp the telescope. 6. Focus onto the staff. 7. With the staff in the field of view as shown above in figure 10, record the staff

    reading as shown by the horizontal cross-hair.

  • 10

    Figure 11

    Booking and Reduction - Collimation method

    Levelling observations are recorded either in specially ruled levelling books or levelling sheets that are later stapled into a book. Information that should be included on the sheet is the following: Heading statement describing the job, date, personnel and Bench Mark data used. The reduction of the observations should be done by either the Collimation method or the Rise and fall method. The Rise and Fall method of booking is discussed later. Spaces should be left for the relevant checks to be carried out. Column left to enable to sets of observations to be compared or adjusted. The steps for the reduction of the observations are as follows: Determining the height of collimation and enter it into the appropriate column. Collimation Height (CH) = RL BM+ BS Determine the RL for each intermediate point by subtracting the intermediate sight staff reading from the collimation height. Results are written into the 'OUT' column. RL Point = CH InterSight The Foresight reading is subtracted from the collimation height and compared to the known RL of the Bench mark (BM). RL BM = CH - FS Reductions must be checked by:

  • 11

    CH = (Sum of BS + Sum of IS + Sum of FS + RLout )/n n = number of staff readings If a second set of levelling observations are taken from a new point, the staff observations are booked and reduced in a similar manner to the first set. The reduced RL for each point is placed in the REDUCEDBACK column. The seconded set of reduced RL's are transcribed onto the first levelling sheet in the REDUCEDBACK column. If no gross errors are evident the mean RL for each point is determined and written into the adjusted column.

    Figure 12: Booking sheet following the height of Collimation method Series Levelling (Rise and Fall)- Field Procedure

    What is Series Levelling?

    Series Levelling can be described as the procedure of moving the instrument through a succession of points. At each level position a BS and FS is read and booked. See figure 13. This procedure becomes necessary when heights are needed at a number of points or the BM is some distance from the site of interest.

    Series levelling should commence and terminate on points of known RL (Bench Marks).

    Referring to figure 13 the field procedure is a follows:

  • 12

    Figure 13

    1. The instrument is set up approximately 50m to 100m away from the BM (point A) and a staff reading to the nearest 0.001m is observed and booked.

    2. The staff is moved to the FS position (point B) and again the observation is observed to 0.001m and booked. (An example of the booking sheet is shown in figure 14)

    Care should be taken that the BS distance is equal or nearly equal to the FS distance.

    This point (A) is referred to as a change point and must be a firm physical object.

    3. The level is then moved to point 2 approximately 50m to 100m from staff, as shown in the diagram above. A BS reading is then observed to the staff at B to the nearest 0.001m.

    4. If any levels are required to objects in the immediate area of the level, then intermediate levels are taken and booked in the intermediate sight column.

    5. The staff is then moved to the next change point (C) and a FS reading taken. 6. These steps are repeated until the end of the levelling run is reached - the

    last point (FS) should always be a point of known RL. When the observations are complete:-

    7. The observations are then reduced as shown below.

    Series Reduction Steps

    Sum the BS Column and sum the FS column Sum of BS minus sum of FS should equal the first RL minus last RL Calculate the rise and fall for each line Sum the rise and fall columns. The difference between the two columns must

    equal the difference between BS and FS column. Using the rise or fall value for each line calculate the RL for each change

    point. Subtract the last RL from the first RL. The result should be the same as BS -

    FS

  • 13

    Figure 14: Rise and Fall booking sheet Visit the following website to view an online demonstration of levelling: http://www.cage.curtin.edu.au/leap/virtu-o-modules/virtu-o-levelling/1_LevellingModuleStart/1_LevellingModuleStart.html The following pages are taken from Schofield W (2001) Engineering Surveying: Theory and examination problems for students, 5th ed, Butterworth-Heinemann, Oxford, ISBN 0 7506 4987 9 and demonstrate the methods of undertaking a levelling exercise and booking.

  • 14

  • 15

  • 16

  • 17

    Trigonometrical Levelling An alternative to using a level is to use a theodolite and measure angles and distances to determine changes in level. The following pages sourced from Schofield W (2001) Engineering Surveying: Theory and examination problems for students, 5th ed, Butterworth-Heinemann, Oxford, ISBN 0 7506 4987 9 give details of this method.

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  • 1

    Angle Measurement

    Theodolites

    A theodolite is an instrument specifically designed for measuring angles. A horizontal angle is the angle in Plan subtended by two distant points at the station where the theodolite is set up. A vertical angle is one between a distant point and a vertical axis (zenith angle) or a horizontal axis (angle of elevation or depression) at the geometric centre of the theodolite. Note: the geometric centre is the intersection point of the vertical and the horizontal axes of rotation and the line of collimation of the telescope, as illustrated in figure 1.

    Figure 1 Theodolite Geometry

  • 2

    Figure 2 Theodolite Angle measurement is undertaken as part of control and detail surveys. It can also be used to set out angles on construction sites; it can also be used to extend straight lines and to check verticality (also called plumbing). In modern surveying theodolites are frequently used in combination with electromagnetic distance measuring equipment. 1 Theodolite construction The main component of any theodolite is a telescope equipped with cross hairs enabling precise sighting of distant points. This telescope is able to rotate about vertical and horizontal axes, with the angles of rotation measured on graduated circles. The theodolite is mounted on a tripod enabling it to be set with its axes correctly aligned and its geometric centre over a station. Theodolite precision depends on quality of construction, but more importantly on the magnification of the telescope, the lowest direct reading it can take (known as the least count) and the plate bubble sensitivity. The sensitivity is quoted as the number of seconds corresponding to one division (2mm) on the bubble tube of dislevelment 1.1 Telescope A telescope provides approximately 30x magnification with internal focussing producing a real erect image, which can be made coincident with the crosshairs. A gun sight allows for an initial pointing to be made. Horizontal and vertical clamps are provided to lock the telescope. With these locked tangent screws can be used to

  • 3

    allow small limited movement for exact sighting of a point. The circle-viewing eyepiece is generally located next to the telescope eyepiece. The barrels of both can be rotated to provide clear images of the circle graduations and of the crosshairs. 1.2 Circles 1.2.1 Optical Theodolite Graduated glass circles are enclosed and are viewed via an auxiliary eyepiece. A small mirror must be adjusted to provide adequate circle illumination by natural light or an artificial source. Many instruments have an optical micrometer controlled by a screw to enable angles to be read with sufficient precision. 1.2.2 Electronic Theodolite Glass circles are used for stability, electronic codings are scanned and angles displayed on an LCD screen. 1.3 Standards The telescope is supported by two standards and can rotate about a horizontal trunnion axis running between them. The clamp and tangent screws controlling this motion are located on one of the standards, as are the mirror for circle illumination and the micrometer screw. The vertical circle is located in one of the standards, on older instruments an attitude bubble for indexing the vertical circle was mounted on top of this standard. A clipping screw is fitted to the same standard to allow centring of this bubble. In modern instruments automatic indexing is achieved using a pendulum and damper inside the standard. 1.4 Plate assembly The standards project upwards from the plate assembly. This contains the horizontal clamp and tangent screw, the horizontal circle with a means of setting it (often a second clamp and tangent screw), an optical plummet and the plate bubble. The plate assembly above the circle, the standards and the plate bubble are called the upper motion or alidade. 1.5 Tribrach This supports the plate assembly, which is allowed to rotate horizontally above it. Three footscrews allow the instrument to be levelled so that the vertical axis is truly vertical. The plate bubble is used to check this levelling, a small bullseye bubble is often also provided to enable a quick approximation of levelling to be made. The tribrach is the lowest part of the theodolite and is bolted on top of a tripod prior to use. 1.6 Tripods These have pointed feet and telescopic legs to allow them to be set firmly over stations enabling the theodolite to be easily and precisely levelled and centred. The flat head and securing bolt allow theodolites to be moved by a small amount

  • 4

    sidewards to allow precise centring over a station as checked via the optical plummet in the plate assembly. Provision also exits for the hanging of a plumb bob. 2 Setting up For angle readings to be valid a theodolite must be set up with the vertical axis truly vertical (shown by the plate bubble) and with this axis over the station. Two methods of setting up exist, the first does not require a plumb bob, this works best when the instrument is set up over a point that is at the same level as the tripod feet. The second involves the use of a plumb bob. The first two steps are the same regardless which method is used. 1. Open the tripod legs; adjust the length of the legs so that the instrument when

    attached is at a convenient height. Clamp the legs and by estimation set the tripod over the station.

    2. Remove the theodolite from its box, bolt it centrally on the tripod and set the foot screws to the middle of their travel.

    Method 1 3. Tread one tripod foot into the ground, and looking through the optical plummet

    position the other two legs so that the station appears central. Tread in the other two feet. If on a hard smooth surface this cannot be done, the tripod legs must be prevented from slipping sidewards.

    4. If the station appears to be no longer central in the plummet, turn the foot screws until it appears central.

    5. Make the instrument level by raising or lowering the tripod legs. If a bullseye bubble is fitted use this, if not set the plate bubble parallel to the line between two legs, shorten or lengthen the legs to centre the bubble. Turn the alidade through 90 and recentre the bubble by adjusting the third leg. Repeat this until the bubble is to within two divisions of central. Sight through the optical plummet, if the instrument is not centred, recentre it by slackening the securing bolt, slide the theodolite horizontally and tighten the bolt.

    6. Finely level the theodolite using the foot screws. Set the plate bubble parallel to the line of two foot screws and turn these screws in opposite directions until the bubble is central. Turn the alidade trough 90 and recentre the bubble using the third foot screw. Return the bubble to its original position and repeat the procedure until the bubble remains central in both positions. The bubble will follow the direction of the left thumb. If the bubble is correctly set in its mounting, it will remain central whatever its disposition. This can be checked by turning the alidade through 180 from its first position. If the bubble moves off centre it should be brought halfway back with the foot screws with further adjustments made with all foot screws so that the bubble remains in this off-centre position. Figure 3 shows the plate with the bubble in its various positions.

    7. Precisely centre the theodolite. Loosen the securing bolt and carefully move the instrument sideways till the station appears central in the optical plummet.

    8. Repeat steps 6 and 7 until the theodolite remains centred and level. 9. Sighting on a plain background rotate the eyepiece to focus the crosshairs.

  • 5

    Figure 3 levelling the theodolite Method 2 3. Hang the plumb bob under the theodolite. Move the tripod legs and bring the bob

    over the station and level the tripod legs. The legs should be moved circumferentially for levelling and radially for centring. Tread the legs into the ground. If this moves the bob off centre shorten or lengthen the tripod legs to bring it back.

    4. Finely level the theodolite using the foot screws. Set the plate bubble parallel to the line of two foot screws and turn these screws in opposite directions until the bubble is central. Turn the alidade trough 90 and recentre the bubble using the third foot screw. Return the bubble to its original position and repeat the procedure until the bubble remains central in both positions. The bubble will follow the direction of the left thumb. If the bubble is correctly set in its mounting, it will remain central whatever its disposition. This can be checked by turning the alidade through 180 from its first position. If the bubble moves off centre it should be brought halfway back with the foot screws with further adjustments made with all foot screws so that the bubble remains in this off-centre position. Figure 3 shows the plate with the bubble in its various positions.

    5. Precisely centre the theodolite 6. Repeat steps 4 and 5 until the theodolite remains levelled and centred 7. Sighting on a plain background rotate the eyepiece to focus the crosshairs. 3 Telescope positions The telescope can be in two positions for observations. The vertical circle can be either to the left of the telescope as seen by the surveyor (face left) or to the right (face right). Usually one starts with face left readings, with repeat readings taken face right. 4 Horizontal angle readings Horizontal angles are those subtended by two distant stations. Horizontal circles of theodolites are graduated in a clockwise sense in plan and measured angles are generally quoted clockwise (ABC is the clockwise angle at B from A to C). It is also usually convenient to swing the alidade clockwise from the first to the second station for the first measurement of the angle. The value of the clockwise angle is the first reading subtracted from the second. If the second reading is lower than the first add 360 to it.

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    The first station sighted is taken as the reference object (RO). Ideally it is the most permanent, most distant and most sharply defined station.

    Figure 4 Horizontal angles Procedure 1. Ensure that the horizontal circle is clamped to the tribrach. 2. Release the horizontal and vertical clamps and swing the telescope to sight the

    target. 3. Look through the telescope and focus it on the target. Lock the clamps and use

    the tangent screws to bring the target into coincidence with the vertical cross hair. Check that parallax is eliminated.

    4. Read the horizontal circle and book the reading. 5. Release the clamps (be sure to keep the circle clamped to the tribrach) turn the

    alidade and move the telescope in elevation to sight the second target. Repeat the sighting, reading and booking procedure.

    Target sighting The target should be sighted as precisely as possible. Focus the telescope and crosshairs so that there is no parallax between target and crosshair images. Lines of sight should be at least 1m above ground level. If the target cannot be sighted directly: 1. Get the chainman to hold a spirit level over it indicating which side is to be

    sighted 2. Get the chainman to hold a plumb bob over the station 3. Set a tripod and plumb bob over it 4. Set a tripod and a surveying target over it Angle reading The smallest figured division on the circle of a theodolite is its least count, 20 seconds, 10 seconds, 5 seconds or 1 second in increasing order of precision. Three systems of circle reading exist: 1. Direct: the circle is marked in degrees and the minute and 20 second intervals

    are read off the index scale against the degree division. 2. Electronic: readings to the least count are automatically displayed. 3. Optical micrometer: the glass micrometer allows the reading of intervals smaller

    than can be scribed in the circle or index. The ray of light from the circle divisions can be moved sideways by passing through a plane sheet of glass which can be rotated. The surveyor turns the micrometer screw until a division on the circle

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    coincides with the fixed index mark. The amount of rotation is proportional to the sideways movement in minutes and seconds and is displayed on a micrometer scale viewed trough the circle telescope.

    Micrometer Operation 1. With reference to figure 5 turn the micrometer screw until the micrometer reads

    zero. 2. Turn the screw until the circle scale shows coincidence of a division and the

    index marker. On a one second instrument obtain coincidence of diametrically opposite divisions (see figure 6). Read the circle and micrometer scales adding values to get the angle values.

    The horizontal and vertical scales usually appear close to each other so Take care to read the correct value Where the micrometer serves both circles reset it to read vertical angle after the

    horizontal one

    Figure 5 and Figure 6 Rounds of angles procedure and booking Angles should be observed more than once so that:

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    1. Inconsistent values can be identified and rejected 2. A mean of several consistent values can be taken 3. For optical reading instruments both faces and different parts of the circle can be

    used so providing compensation for some instrumental errors. For an optical reading theodolite take a round of angles by commencing sighting the RO on face left, swinging right (clockwise in plan) and sighting the other stations in turn. At the final station transit the telescope to face right position without disturbing the circle, observe the stations in reverse order swinging left finally sighting the RO. Each subsequent round should be commenced at 180/n further clockwise around the circle for a total of n rounds. Table 1 shows booking and reduction Table 1

    Readings Calculations Face Mean

    direction Reduced Angle Inst at Sighting Left Right

    I Round 1

    A 000342 1800344 000343 000000 B 672437 2472433 672435 672052 C 1162103 2962055 1162059 1161716

    I Round 2

    A 904118 2704121 904120 000000 B 1580212 3380214 1580213 672053 C 2065830 265835 2065832 1161712

    5 Angle readings vertical Modern instruments have 360vertical circles with zero representing on face left a vertical upward, a horizontal or a vertical downward sight depending on manufacture. On modern instruments the first type known as zenith zero is the most common. Electronic instruments generally allow switching between zenith angles, horizontal angles and percentage grades. Figure 7 shows the most common figuring of vertical angles.

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    Figure 7 vertical circle figuring Indexing Older instruments are fitted with a altitude bubble connected to the vertical circle. It is located on one of the standards and is levelled by a clipping screw, this screw is distinguished from the other screws by being fluted rather than knurled. Before vertical angles are measured circle indexing must be undertaken by centring the altitude bubble with the clipping screw, this gives greater precision than obtainable with the plate bubble. Modern optical instruments have automatic vertical indexing. Most electronic instruments have automatic indexing. Angle measurement 1. Note circle zero position and figuring 2. Index the vertical circle (if not automatic) 3. Accurately sight the target, read and book the angle. Note 1. Micrometer is used as for horizontal circle readings 2. One second instruments give a mean of diametrically opposite readings 3. Take a mean of readings on both faces 5.1 Trigonometric heighting Trigonometrical heighting or levelling can be undertaken using a theodolite. Over long distances there will be error from the effects of earth curvature and refraction. This is a convenient method if an EDM is attached to the theodolite to enable distances to be measured.

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    Corrections can be evaluated as linear amounts as in precise levelling or as angular amounts to be applied to vertical angles.

    Figure 8 Trigonometrical heighting Figure 8 shows such an exercise being conducted between station A with an instrument height hI and station B with an instrument height hT. The zenith angle has been measured as and the slope distance been determined as D. Angular corrections of c (curvature) and kc (refraction taken as opposite in sign to c) are to be applied, note k is the coefficient of refraction and R is the earths radius. Rise AB = hI + Dsin[90 - + c(1 k)] hT Rise AB = hI + Dcos[ - c(1 k)] hT The curvature correction will be half the angle subtended at the earths centre by the distance AB. The slope distance D may be approximated to an arc, so: c (radians) = D/2R or c in seconds = 206265D/2R Combined correction (seconds) = 206265D(1-k)/2R A typical value for k is 0.14. Reciprocal trigonometrical heighting can also be undertaken (see reciprocal levelling) 6 Instrument errors and permanent adjustments For a discussion of these please refer to a standard surveying text book. The following pages are taken from Schofield W (2001) Engineering Surveying: Theory and examination problems for students, 5th ed, Butterworth-Heinemann, Oxford, ISBN 0 7506 4987 9.

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    Linear Measurement

    1 Taping Engineering surveys require steel tapes, the most popular being 30 metre tapes made of mild steel with a protective covering. Two types of winder are in common usage:

    1. Open frame: reduces the build up of dirt on the tape 2. Closed or encased winder: more compact

    Markings on tapes are every 5mm with 1mm divisions for the first and last metre or 1mm divisions throughout. Additionally a 3 m or 5m long spring loaded pocket tape is also useful, especially in terms of setting out. Fifty and 100 m tapes are also used, for longer measurements it is more convenient to employ electromagnetic forms of measurements. 1.1 Using a Tape a) Keep the tape taut so that it is straight in plan and elevation b) Align the tape precisely with the points being measured. At times it can be difficult

    to measure from the start point of the tape, this is especially so when tension is to be applied. In these cases an additional handle with a spring clip can be used on the tape to give a firm grip, or the 1m mark can be used as the zero mark (in this case ensure 1m is deducted from the tape reading).

    c) Apply the correct tension to the tape. d) Distances of a greater length than the length of the tape need to be measured in

    two or more bays. At the end of each bay a precise stable mark must be made, examples include a peg with a nail or a sharp mark on concrete)

    1.2 Tape care Tapes need to be well looked after. In unwinding a tape care should be taken to ensure that the tape is not pulled off the winder. When using a tape ensure that it does not become kinked, scratched, stretched or broken. Winding in a tape should be undertaken with care to avoid kinking, twisting and the ingress of dirt and moisture into the winder. Periodic maintenance includes removing the tape from the winder and cleaning the tape. Tapes and winders should also be lightly oiled. Broken tapes can be repaired by riveting connected strips across the break, however this is only a temporary repair and broken tapes should be replaced. 1.3 Tape Corrections 1.3.1 Slope Reduction A measurement can be taken on a slope where a horizontal distance is required, to do this either the zenith angle (), or the difference in elevation V at the ends of the tape need to be determined as shown in figure 1. The zenith angle can be determined with a theodolite and the latter by leveling.

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    With reference to figure 1:

    sin)(D (H) Distance Horizontal 22 DV ==

    Figure 1: Slope reduction 1.3.2 Standardization Correction In some cases tapes can have a long enough lifetime to become permanently stretched. If such stretching is suspected the tape needs to be compared with a known length, e.g. a new tape. Then all measurements taken with the suspect tape need to be corrected by:

    length nominal Tapelength actual Tape

    1.3.3 Tension Correction The corrections that are applied for tension effects, temperature, sag and altitude are small and can be neglected if the tape is used at a tension close to the standard value and on the flat. A spring balance connected to the tape is used to measure its tension. From a measurement viewpoint it is convenient to attach the balance to the measuring end to enable measurement of tension and distance at the same time. The balance can be connected by roller grips. The simplest measurement is to set a standard tension, for example 45 or 50 Newtons; this value should be marked close to the zero of the tape. If non-standard tensions are used the thinning or thickening of the tape,

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    Youngs modulus of the tape and thickness will be required to calculate the true value. 1.3.4 Sag correction Tapes are standardised on the flat, thus, where possible measurements should be made on flat or nearly flat surfaces. Where this is not possible tapes are suspended. When suspended tapes take the form of a catenary and so the measurement will be too long. To calculate the effect of the sag or catenary the tension and weight of the tape per unit length need to be known, either from weighing the tape or from manufacturers specifications. 1.3.5 Temperature Tapes will expand or contract depending on the temperature at which the measurements are taken compared to that at which they were standardised. To determine this effect the temperature of the measurement needs to be determined and the coefficient of linear expansion known. 1.3.6 Altitude Correction Measurements need to be corrected to give the equivalent distance at sea level or other datum level used. Table 1 shows correction factors and formulae and typical values of tension, sag, temperature and altitude corrections. 1.3.7 Compromise tension Where a suspended steel tape is used a compromise tension can be applied such that the effects of sag are balanced by a tension higher than the standard one. Values are: 70 N for a 10mm wide tape 105 N for a 13mm wide tape 1.3.8 Accuracy of taping Manufacturers quote a standard error of 3mm, in a 30m long tape for steel tapes.

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    Table 1

    Tape Corrections Typical physical data (note values should be checked for each tape) Ps = 50N, ts = 20C, = 0.0000112/C, E = 200000 N/mm2, R = 6370 km Source of error Tension

    variation Sag Temperature

    Variation Altitude Effect

    Correction AE

    DPP s )( 2

    32

    24PDw

    Dtt s )(

    RDh

    Typical correction values for 30m length with data above

    P = 70 A = 3.25 Correction = + 0.0009

    P = 70 w = 0.240 Correction = - 0.0132

    t = 10C Correction = -0.0034

    h = 1000m Correction = -0.0047

    Symbols P = Field tension (N) Ps=Standard tension (N) D = measured distance (m) A = cross sectional area (mm2) E = Youngs Modulus (N/mm2) w =weight/unit length (N/m) t = field temperature (C) ts = standard temperature (C) = coefficient of linear expansion (/C) h = height above sea level (km) R = radius of earth (km) The following pages are sourced from Schofield W (2001) Engineering Surveying: Theory and examination problems for students, 5th ed, Butterworth-Heinemann, Oxford, ISBN 0 7506 4987 9 and deal with the subject of electromagnetic distance measuring (edm).

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    Map Projection Overview Peter H. Dana These materials were developed by Peter H. Dana, Department of Geography, University of Texas at Austin, 1995. These materials may be used for study, research, and education in not-for-profit applications. If you link to or cite these materials, please credit the author, Peter H. Dana, The Geographer's Craft Project, Department of Geography, The University of Colorado at Boulder. These materials may not be copied to or issued from another Web server without the author's express permission. Copyright 1999 Peter H. Dana. All commercial rights are reserved. If you have comments or suggestions, please contact the author or Kenneth E. Foote at [email protected]. Revised: 2000.10.3. KEF. Introduction Map projections are attempts to portray the surface of the earth or a portion of the earth on a flat surface. Some distortions of conformality, distance, direction, scale, and area always result from this process. Some projections minimize distortions in some of these properties at the expense of maximizing errors in others. Some projections are attempts to only moderately distort all of these properties. Conformality When the scale of a map at any point on the map is the same in any direction, the projection is conformal. Meridians (lines of longitude) and parallels (lines of latitude) intersect at right angles. Shape is preserved locally on conformal maps. Distance A map is equidistant when it portrays distances from the center of the projection to any other place on the map. Direction A map preserves direction when azimuths (angles from a point on a line to another point) are portrayed correctly in all directions. Scale Scale is the relationship between a distance portrayed on a map and the same distance on the Earth. Area 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. Different map projections result in different spatial relationships between regions.

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    Three Different Map Projections of the United States

    Albers Equal Area and Lambert Conformal Conic Projections of North America Map projections fall into four general classes.

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    Cylindrical projections result from projecting a spherical surface onto a cylinder. When the cylinder is tangent to the sphere contact is along a great circle (the circle formed on the surface of the Earth by a plane passing through the center of the Earth)..

    Projection of a Sphere onto a Cylinder (Tangent Case)

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    In the secant case, the cylinder touches the sphere along two lines, both small circles (a circle formed on the surface of the Earth by a plane not passing through the center of the Earth).

    Projection of a Sphere onto a Cylinder (Secant Case)

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    When the cylinder upon which the sphere is projected is at right angles to the poles, the cylinder and resulting projection are transverse.

    Transverse Projection of a Sphere onto a Cylinder (Tangent Case)

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    When the cylinder is at some other, non-orthogonal, angle with respect to the poles, the cylinder and resulting projection is oblique.

    Oblique Projection of a Sphere onto a Cylinder (Tangent Case)

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    Conic projections result from projecting a spherical surface onto a cone. When the cone is tangent to the sphere contact is along a small circle.

    Projection of a Sphere onto a Cone (Tangent Case)

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    In the secant case, the cone touches the sphere along two lines, one a great circle, the other a small circle.

    Projection of a Sphere onto a Cone (Secant Case)

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    Azimuthal projections result from projecting a spherical surface onto a plane. When the plane is tangent to the sphere contact is at a single point on the surface of the Earth.

    Projection of a Sphere onto a Plane (Tangent Case)

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    In the secant case, the plane touches the sphere along a small circle if the plane does not pass through the center of the earth, when it will touch along a great circle.

    Projection of a Sphere onto a Plane (Secant Case)

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    Miscellaneous projections include unprojected ones such as rectangular latitude and longitude grids and other examples of that do not fall into the cylindrical, conic, or azimuthal categories Selected Map Projections Cylindrical Projections Cylindrical Equal Area Cylindrical Equal-Area projections have straight meridians and parallels, the meridians are equally spaced, the parallels unequally spaced. There are normal, transverse, and oblique cylindrical equal-area projections. Scale is true along the central line (the equator for normal, the central meridian for transverse, and a selected line for oblique) and along two lines equidistant from the central line. Shape and scale distortions increase near points 90 degrees from the central line. Behrmann Cylindrical Equal-Area Behrmann's cylindrical equal-area projection uses 30:00 North as the parallel of no distortion.

    Behrmann Cylindrical Equal-Area Gall's Stereographic Cylindrical Gall's stereographic cylindrical projection results from projecting the earth's surface from the equator onto a secant cylinder intersected by the globe at 45 degrees north and 45 degrees south. This projection moderately distorts distance, shape, direction, and area.

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    Gall's Sterographic Cylindrical Peters The Peters projection is a cylindrical equal-area projection that de-emphasizes area exaggerations in high latitudes by shifting the standard parallels to 45 or 47 degrees.

    Peters Mercator The Mercator projection has straight meridians and parallels that intersect at right angles. Scale is true at the equator or at two standard parallels equidistant from

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    the equator. The projection is often used for marine navigation because all straight lines on the map are lines of constant azimuth.

    Mercator Miller Cylindrical The Miller projection has straight meridians and parallels that meet at right angles, but straight lines are not of constant azimuth. Shapes and areas are distorted. Directions are true only along the equator. The projection avoids the scale exaggerations of the Mercator map.

    Miller Cylindrical

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    Oblique Mercator Oblique Mercator projections are used to portray regions along great circles. Distances are true along a great circle defined by the tangent line formed by the sphere and the oblique cylinder, elsewhere distance, shape, and areas are distorted. Once used to map Landsat images (now replaced by the Space Oblique Mercator), this projection is used for areas that are long, thin zones at a diagonal with respect to north, such as Alaska State Plane Zone 5001.

    Oblique Mercator (Alaska State Plane Zone 5001) Transverse Mercator Transverse Mercator projections result from projecting the sphere onto a cylinder tangent to a central meridian. Transverse Mercator maps are often used to portray areas with larger north-south than east-west extent. Distortion of scale, distance, direction and area increase away from the central meridian. Many national grid systems are based on the Transverse Mercator projection The British National Grid (BNG) is based on the National Grid System of England, administered by the British Ordnance Survey. The true origin of the system is at 49 degrees north latitude and 2 degrees west longitude. The false origin is 400 km west and 100 km north. Scale at the central meridian is 0.9996. The first BNG designator defines a 500 km square. The second designator defines a 100 km square. The remaining numeric characters define 10 km, 1 km, 100 m, 10 m, or 1 m eastings and northings.

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    British National Grid 100 km Squares The Universal Transverse Mercator (UTM) projection is used to define horizontal, positions world-wide by dividing the surface of the Earth into 6 degree zones, each mapped by the Transverse Mercator projection with a central meridian in the center of the zone. UTM zone numbers designate 6 degree longitudinal strips extending from 80 degrees South latitude to 84 degrees North latitude. UTM zone characters designate 8 degree zones extending north and south from the equator.

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    UTM Zones 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).

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    UTM Zone 14 Pseudocylindrical Projections Pseudocylindrical projections resemble cylindrical projections, with straight and parallel latitude lines and equally spaced meridians, but the other meridians are curves. Mollweide The Mollweide projection, used for world maps, is pseudocylindrical and equal-area. The central meridian is straight. The 90th meridians are circular arcs. Parallels are straight, but unequally spaced. Scale is true only along the standard parallels of 40:44 N and 40:44 S.

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    Mollweide Projection Eckert Projections Eckert IV Equal Area The Eckert IV projection, used for world maps, is a pseudocylindrical and equal-area. The central meridian is straight, the 180th meridians are semi-circles, other meridians are elliptical. Scale is true along the parallel at 40:30 North and South.

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    Eckert IV Equal Area Eckert VI Equal Area The Eckert VI projection , used for maps of the world, is pseudocylindrical and equal area. The central meridian and all parallels are at right angles, all other meridians are sinusoidal curves. Shape distortion increases at the poles. Scale is correct at standard parallels of 49:16 North and South.

    Eckert VI Equal Area Robinson The Robinson projection is based on tables of coordinates, not mathematical formulas. The projection distorts shape, area, scale, and distance in an attempt to balance the errors of projection properties.

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    Robinson Sinusoidal Equal Area Sinusoidal equal-area maps have straight parallels at right angles to a central meridian. Other meridians are sinusoidal curves. Scale is true only on the central meridian and the parallels. Often used in countries with a larger north-south than east-west extent.

    Sinusoidal Equal Area

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    Conic Projections Albers Equal Area Conic A conic projection that distorts scale and distance except along standard parallels. Areas are proportional and directions are true in limited areas. Used in the United States and other large countries with a larger east-west than north-south extent.

    Albers Equal-Area Conic Equidistant Conic Direction, area, and shape are distorted away from standard parallels. Used for portrayals of areas near to, but on one side of, the equator.

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    Equidistant Conic Lambert Conformal Conic Area, and shape are distorted away from standard parallels. Directions are true in limited areas. Used for maps of North America.

    Lambert Conformal Conic (North America) Polyconic The polyconic projection was used for most of the earlier USGS topographic

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    quadrangles. The projection is based on an infinite number of cones tangent to an infinite number of parallels. The central meridian is straight. Other meridians are complex curves. The parallels are non-concentric circles. Scale is true along each parallel and along the central meridian.

    Polyconic (North America) Azimuthal Projections Azimuthal Equidistant Azimuthal equidistant projections are sometimes used to show air-route distances. Distances measured from the center are true. Distortion of other properties increases away from the center point.

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    Azimuthal Equidistant

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    Lambert Azimuthal Equal Area The Lambert azimuthal equal-area projection is sometimes used to map large ocean areas. The central meridian is a straight line, others are curved. A straight line drawn through the center point is on a great circle.

    Lambert Azimuthal Equal Area

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    Orthographic Orthographic projections are used for perspective views of hemispheres. Area and shape are distorted. Distances are true along the equator and other parallels.

    Oblique Aspect Orthographic Projection

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    Stereographic Stereographic projections are used for navigation in polar regions. Directions are true from the center point and scale increases away from the center point as does distortion in area and shape.

    North Polar Stereographic

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    Miscellaneous Projections Unprojected Maps Unprojected maps include those that are formed by considering longitude and latitude as a simple rectangular coordinate system. Scale, distance, area, and shape are all distorted with the distortion increasing toward the poles.

    World: Unprojected Latitude and Longitude

    North America: Unprojected Latitude and Longitude

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    Texas: Unprojected Latitude and Longitude Texas State-Wide Projection In 1992, the Cartographic Standards Working Group proposed a Texas State-Wide Map Projection Standard for the GIS Standards Committee of the GIS Planning Council for the Department of Information Sciences. Earlier maps had often used projections designed for the continental United States

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    Texas: Lambert Conformal Conic designed for the Continental United States The new projection was designed to allow state-wide mapping with a minimum of scale distortion. A Lambert Conformal Conic Projection was proposed with an origin at 31:10 North, 100:00 West and with standard parallels at 27:25 North and 34:55 North. For plane coordinate use a false Easting and Northing of 1,000,000 meters were defined for the origin.

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    Texas: Texas State-Wide Map Projection

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    Space Oblique Mercator The Space Oblique Mercator is a projection designed to show the curved ground-track of Landsat images. There is little distortion along the ground-track but only within the narrow band (about 15 degrees) of the Landsat image.

    Space Oblique Mercator References Bugayevskiy, Lev M. and John P. Snyder. 1995. Map Projections: A Reference Manual. London: Taylor and Francis. Muehrcke, Phillip C. and Juliana O. Muehrcke. 1998. Map Use: Reading-Analysis-Interpretation, 4th ed. Madison, WI: JP Publications. Snyder, John P. 1987. Map projections: a working manual. USGS Professional Paper 1395. Washington, DC: United States Government Printing Office. Many of the maps on this page were produced using MapInfo's MapInfo and Golden Software's MapViewer and Surfer for Windows. Notes sourced from: http://geography.about.com/science/geography/gi/dynamic/offsite.htm?site=http%3A%2F%2Fwww.colorado.Edu%2Fgeography%2Fgcraft%2Fnotes%2Fgps%2Fgps.html

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    Control Surveys

    1 Introduction In surveying and setting out, it is required to carry out a control survey of the area in the early stages of the project. The objective is to establish of permanent or semi-permanent points for plan control and levelling. This enables a unified, self checking survey of the required precision to be produced. It should be noted that whereas an error in detailing often has no effect on the location of subsequent points and even if small may be insignificant, an error at the control stage may affect the whole survey. Thus, most control surveys must be tied to existing known points. Traditionally control surveys have been carried out employing linear, angular and levelling measurements, which connect the stations. Control surveys should incorporate checks for closure at several stages. Large discrepancies indicate that the work, either in part or in full, must be repeated. Small discrepancies, once corrections of systematic errors have been accounted for, within specified tolerances may either be left or as is more common mat be removed by the adjustment of the observations to produce mathematical consistency. Prior to the development of computer programs for survey adjustment, field methods were employed allowing manual adjustment to be performed; these however provided few redundant measures. Computer programs are now available that are able to adjust highly complex networks, in three dimensions, with many redundant measures. Survey adjustment procedures are in no way designed to compensate for poor field measurement. Development of EDM techniques and total station theodolites has enabled the rapid taking of extra measurements, these can provide further checks and if consistent these can strengthen a survey. Satellite surveying, commonly termed GPS, is now in use to establish control stations. All surveying operations use two receivers. One must be at a point of known coordinates to enable corrections to be determined and applied in the location of new stations. Redundant measures can be obtained by receiving signals from more than the minimum required of four satellites, and by the taking of paired observations from opposite ends of lines forming a network. 1.1 Planning a control survey 1 A specification linking precision with permitted tolerances must be drawn up. 2 The whole area must be reconnoitred 3 Suitable survey methods must be selected 4 Equipment must be of the appropriate type and of adequate precision 5 Control stations must be suitably positioned 6 The survey must be self-checking so that its validity can be determined 7 Acceptably small errors of closure can be neglected or eliminated by adjustment,

    as appropriate. Large errors will require part or all of the work to be repeated. 8 For complex projects an error analysis will indicate if the specified precision has

    been achieved

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    9 Periodical checks must be made when the stations are in use over a long time 1.2 Station Selection (Plan Control) 1 Stations should define a control figure of a suitable shape 2 Stations should permit the convenient surveying of detail 3 Stations should be intervisible for linear and angular measurements 4 Stations should be accessible so that instruments can be set up over them 5 For GPS fixing stations should be in an open location (not next to a cliff face at

    the base for example) 6 Stations should be free from the risk of disturbance 7 Stations should be easily referenced 1.3 Establishing bench marks If no benchmark exists nearby, temporary benchmarks (TBM) need to be established to exert levelling control. These TBMs must

    Be centrally located Free from the risk of subsidence or ground heave Sharply defined so that a staff repeatedly held on one will be at the same level

    Where GPS is to be used a number of points must be surveyed by satellite and by spirit levelling to enable correspondence between the reference ellipsoid and geoid (see later) to be achieved. 1.4 Linear Measurement Follow the principles outlined in the distance measuring section of these notes. Note: Equipment should be regularly calibrated Apply all required corrections Accurate indexing of tapes and centring of tripods at all stations should be

    undertaken 1.5 Angular measurement Follow the principles outlined in the angular measurement section of these notes, particularly: Theodolite permanent adjustments should be regularly checked Instruments and targets should be accurately centred Several rounds of angles should be taken on both face left and face right and

    mean values calculated 1.6 Levelling Follow the principles outlined in the levelling section of the notes, particularly noting: Instrument should be free of collimation error Closure checks and cross runs should be undertaken Effects of errors from staff tilt, curvature and refraction should be eliminated

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    2 Coordinates Rectangular coordinates (Cartesian coordinates) are used for the plotting of control surveys for numerous reasons: 1 Coordinate plotting eliminates angle or bearing plotting, a source of error 2 Each point is plotted directly from the axes, so eliminating cumulative plotting

    errors 3 The arithmetic for survey adjustments can be undertaken more readily on

    coordinates than it can on field measurements 4 Information can be stored in coordinate form 5 Surveys can be tied into existing coordinate systems (and transposed between

    systems) 6 Land areas within traverses can be easily calculated The basis of the Cartesian coordinate system is illustrated in figures 1 and 2.

    Figure 1: Cartesian Coordinates in a Plane.

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    Figure 2: Cartesian coordinates in a Plane with two points defining a line In surveying the axes are taken north south and east west, the distances of a point from them being easting (east positive) and northing (north positive). Eastings are always quoted before the northing. A coordinate system may be purely local with axes arbitrarily chosen or the state or national system may be used. It is customary to position the origin south and west of the survey area to make all coordinates positive. An example the UK national Grid is shown in Figure 3. Surveys in a limited area can be taken assuming a plane, assuming a flat world. Surveys like these that are related to the national or state grid and surveys of larger areas will need to be corrected to allow for the earths curvature.

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    Figure 3: The UK National Grid, 100 km squares, 2.1 Bearings The coordinates of a point can be found by determining the differences in easting and northing from a point already coordinated. These differences are calculated from the polar coordinates of the joining line. While the length can be measured in the field the bearing must be deduced from angular measurements. Bearings are measured clockwise from a meridian or north line (note the contrast here compared to the mathematical notation). Bearings such as this were formerly called Whole Circle Bearings or WCB. A bearing refers to a line where a start and an end are specified. Alternatively it can refer to a point (corresponding to line end) relating to another point (= line start). Sometimes the term back bearing is used; this is the bearing of a line taking the clockwise angle measured from the meridian at the end of the line. When converting from polar to rectangular (Cartesian) coordinates it is useful to know the quadrant of the line, see Table 1. The following figures provide details of the polar coordinate system.

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    Figure 4: Polar Coordinates in a Plane

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    Figure 5: Polar coordinates in a plane and conversion from polar to Cartesian Coordinates

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    Figure 6: Three dimensional Cartesian coordinates

  • 9

    Figure 7: Three dimensional polar coordinates

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    Figure 8: Conversion of three-dimensional polar coordinates to three-dimensional Cartesian coordinates

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    Figure 9: Distance between three dimensional Cartesian coordinates 2.2 Orienting Bearing Before any bearings can be evaluated an initial orienting bearing must be determined. If a survey is to be tied into an existing coordinate system two known points must be observed. The simplest method is to use one of the known points as a control station and take observations of the other with a theodolite. The bearing of the line can then be found by a rectangular to polar conversion. If approximate orientation is sufficient the magnetic bearing of a line can be taken employing a compass. 2.3 Coordinate Conversions These can be simply undertaken; the basis is detailed in the preceding figures.

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    Table 1: Bearing Quadrant Easting

    Difference Northing Difference

    0 to 90 1 + + 90 to 180 2 + - 180 to 270 3 - - 270 to 360 4 - + 3 Traversing Traversing is one of the traditional methods of undertaking a control survey in plan. Stations are set out to define a series of traverse lines or legs, the plan lengths of which can be measured as can the angles between pairs of lines at each station. Three types of traverse survey exist, as illustrated in figure 10: 1 Closed loop traverse, where the legs form a closed polygon 2 Closed tied (also called connecting or linked) traverse, where the traverse runs

    between two stations of known position. 3 Open traverse, where the lines, although starting from a known position do not

    finish at one.

    Figure 10: Traverse types. Closed traverses provide a check on the validity and accuracy of field measurements. The loop traverse is suited to many engineering applications, however for road, railway and pipeline projects the tied traverse is often more

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    appropriate. Open traverses provide no check and are not recommended, however in underground surveying no other option may be available. Angles are measured by theodolite, distances by taping or EDM. The development of the EDM has contributed to the dominance of traversing in modern surveying. Coordinate plotting is used (less errors) with checks and adjustments made during the calculation process. 3.1 Station Selection In addition to the criteria listed previously the following additional requirements are needed: The stations should form a traverse of suitable shape Only neighbouring stations along traverse lines need to be intervisible Where traverse lines are to be taped the ground should be accessible Traverse legs should be approximately equal in length Existing stations and reference objects should be incorporated. Stations should be firmly and clearly marked out and strongly referenced. 3.2 Angle measurement Additional aspects to those covered previously include the following. Ideally angles should be measured in sequence progressing around the traverse from the start station. In a number of cases this may not be convenient and care must be taken to measure all angles, importantly not forgetting the one at the closing station. Angles should be properly recorded, so that one knows if the angle is internal or external. Three-tripod equipment is commonly employed. The theodolite can be detached from its tribrach and exchanged with a target that is similarly detachable. While the theodolite is set up at the instrument station B, at the back station A and forward station C a tripod, tribrach, optical plummet and target are set up. Following observation of angle ABC the complete target assembly at A is transferred to station D, and the theodolite at station B and target at station C are detached and exchanged. Angle BCD is then measured. Three-tripod equipment is always used when measuring distance with an EDM. A combined prism and target is set up at each station sighted. Three-tripod equipment eliminates centring errors to the extent that at each station measurements are taken to and from the same point in plan. Errors can still exist in station coordinates if the targets have not been accurately plumbed over the stations.

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    3.3 Included Angles For consistency and ease of bearing calculation the included angle should be measured at all stations. This is the clockwise angle swept out when turning from the back to the forward station, in other words at B sighting A then turning clockwise to C. In a loop traverse carried out in an anticlockwise sense the included angles will be internal; a clockwise loop will produce external angles. 3.4 Angle closure checks Once all the angles of a closed traverse have been measured, a check should be conducted. This can usually be performed in the field. Loop Traverse Check Anticlockwise traverse: Sum of internal angles should equal (n 2) x 180 (internal) Clockwise traverse: Sum of Internal angles should equal (n + 2) x 180 (external) For a traverse round of n stations. Tied traverse check The sum of the internal angles should equal the final orienting bearing minus the first plus (n + 1) x 180 usually (n 1) x 180 sometimes (n + 3) x 180 infrequently for a traverse of n stations. As an alternative the bearings of the traverse legs can be successively calculated from the unadjusted angles and the deduced value of the final orienting bearing compared with the known value. Acceptable closure Closing errors will be affected by the precision of the equipment as well as the skill and care of the surveyor. Acceptable errors will depend on the precision of the survey as shown in table 2, second order precision is appropriate for many engineering surveys, geodetic precision may be required in tunnelling work.

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    Table 2: Precision of survey Acceptable angular

    misclosure Acceptable linear misclosure

    Geodetic 2n1/2 1 in 20000 Secondary 10n1/2 1 in 10000 Tertiary 20n1/2 1 in 5000 Fourth order 60n1/2 1 in 2000 n = number of stations 3.5 Bearing Calculation The bearing of one line, preferably at the start section must be known. Taking the lines in sequence (AB, BC, CD, etc): Bearing of Line = Bearing of previous line + included angle + 180 if sum < 180

    -180 if 180< sum< 540 -540 if sum > 540

    In a closed traverse a check on the arithmetic should be undertaken. Bearings should be successively calculated around the traverse. In a loop traverse the bearing of the first line calculated successively should equal its initial value. In a tied traverse the known and deduced values of the final orienting sight should be equal. 3.6 Station Coordinates Station coordinates are calculated by successively adding the line coordinate differences to the coordinates of the start station. In closed traverses the sums of the easting differences and of the northing differences are first calculated to determine the linear closing error (at best both equal to zero) if these are acceptably small it is distributed by adjusting the coordinate differences. For an open traverse the station coordinates can be calculated as soon as the coordinate differences have been evaluated. The following example shows calculations for an open traverse, the subsequent examples deal with how to adjust closed traverses. 3.7 Misclosure and adjustments: Bowditchs method In a closed traverse a check is made on the linear misclosure when the coordinate differences are determined. The misclosure in both the eastings eE and northings eN can be calculated. The linear misclosure e is determined using the Pythagorean theorem as:

    2/122 )]()[( NE eee += The fractional linear misclosure is = 1/H/e where H is the sum of the line lengths. For a loop traverse E and N should equal zero, therefore:

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    ==

    Ne

    Ee

    N

    E

    for a tied traverse with start station coordinates (EA;NA) and end station coordinates (Ez;NZ)

    ==

    )(

    )(

    AZN

    AZE

    NNNe

    EEEe

    Provided that the linear misclosure is within acceptable limits (Table 2) adjustments can be applied to remove the error, making all derived coordinates consistent. The best known traverse adjustment method is the Bowditch method for engineering surveys. It is based on the assumption that the closing error has been generated at a constant rate around the traverse. Thus, at any point the direction of adjustment will have a bearing at 180 from the line of misclosure and the magnitude of the of the adjustment will be in proportion to the distance around the traverse from the start. The adjustment is applied to the coordinate differences in proportion to the respective line lengths, the sign being opposite to that of the error, so:

    =

    =

    HHe

    HHe

    ABN

    ABE

    AB

    AB

    N toadustment

    E toadustment

    where HAB is the horizontal length of the line AB The total adjustments are eE and eN removing all closing error. An integral part of the Bowditch method is the initial adjustment of angles as dealt with previously. 3.8 Other methods of Traverse adjustment The Bowditch method evolved in the days of compass and chain surveys. It makes the assumption that bearing and linear measurements have similar error effects. As a method the angles are adjusted twice and problems can occur. When closing errors are small the method of adjustment, so long as it has a logical foundation, is often immaterial. Large errors indicate a mistake in the field or faulty equipment. If a more rigorous adjustment is needed the method of least squares as described later in these notes should be employed. Gross errors If a large angular or linear misclosure is found in a traverse a gross error in the fieldwork or calculations has occurred. Initially one would check the calculations before retaking the measurements.

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    Gross angular errors A single gross angular error can be located by calculating the coordinates in anticlockwise and clockwise directions for a loop traverse and from each end of a tied traverse. The station where the error has occurred will have the same coordinates. Gross linear errors If a single gross linear error has occurred the faulty line will be parallel to the linear misclosure. Two or more gross errors These can be difficult to locate. 4 Intersection A point can be located in plan by two intersecting sights by theodolite from the end points of a baseline of accepted length. The triangle formed should be well conditioned. The unmeasured angle is deduced so that the sine rule can be used to determine one or both distances from the fixed stations at the end of the baseline. As long as the coordinates of one station and the baseline bearing are known, then the coordinates of the intersected point can be calculated, see figure 11 (a). Using this method there is no check on the fieldwork; this implies that the horizontal angles need to be measured carefully. A better method is illustrated in figure 11 (b). This involves measuring the angles of intersecting sights from three or more stations. Thus redundant measures are included; as such a variation of coordinate adjustment should be carried out. Vertical angles from two stations also provide a redundant measure.

    Figure 11

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    5 Resection An instrument station can be located by measuring angles subtended by three distant stations whose positions are known. The method has been useful for relating a station to triangulated points and for locating a station on an offshore structure.

    Figure 12 If the four stations involved are concyclic, there can be no solution; if they are nearly concyclic the fix of the instrument station will be weak. Therefore either the unknown station needs to lie within the fixed stations or the three fixed stations should present a convex formation to an unknown point. With reference to figure 12 a stations A, B and C are three known stations observed from point P. Lengths l1 and l2 and angle c will be known or can be deduced from coordinates. Angles and are measured at P. Denoting the unknown angle CAP as , then angle PBC = 360 - ( + + + c)

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    Then

    [ ]

    )cos(sinsin)sin(sintan

    Hencesin)cos(cos)sin(sin(sinsin

    sin)sin(

    sinsin

    sinc)----sin(360CP and

    sinsin

    21

    21

    21

    21

    1

    cllcl

    ccll

    cll

    CPlCP

    +++++=

    +++++=

    +++=

    ==

    Thus triangle APC can be solved and P related to A , B and C. 6 Networks A control network involves measurements of linear distance and angles between stations. No restrictions exist on which distances and angles should be measured in contrast with the methods of traversing, triangulation and trilateration. Provided measurements can be taken to sufficient precision a control network can establish the levels as well as the positions of all stations. Prior to the development of computer based least squares adjustment processes, control methods were geared to manual calculations and adjustments. Such adjustments were often undertaken sequentially to satisfy geometric conditions in turn and to adjust a small part of the whole survey at a time. Redundant measurements provide additional checks, however with manual calculation increased the complexity of the process. The adoption of EDM techniques for fieldwork and the use of computer programs to undertake calculations, adjustment and strength analysis have meant that a network process allows many more field observation