lecture 1a - horizontal and vertical positioning

7/23/2019 Lecture 1A - Horizontal and Vertical Positioning

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Review of Horizontal and Vertical

PositioningLecture 1

GE 12 – General Surveying II

Department of Geodetic EngineeringTraining Center for Applied Geodesy and Photogrammetry GE 12

–General Surveying II



Objectives

By the end of the class, the students must be

able to:

– Recall survey operations for determining

horizontal positions.

– Recall the various levelling techniques for

determining vertical positions





HORIZONTAL POSITIONING



Survey Operations for Horizontal

Positioning

• Intersection

• Resection

• Traverse• Triangulation

• Trilateration





Survey Operations –

LOCATION BY INTERSECTION

• Operation employed if the

coordinates are given for

two ends of a line and

directions are observedfrom each end of this line

to a third point not on the

line in order to calculate

the coordinates of thethird point.

5Department of Geodetic EngineeringTraining Center for Applied Geodesy and Photogrammetry GE 12




Survey Operations – RESECTION

• Operation employed when angles between lines to threepoints of known position are observed from a point of

unknown position in order to calculate the coordinates of

the unknown point.





Survey Operations – TRIANGULATION

• Triangulation System - consists of a series of joined oroverlapping triangles in which an occasional line is measured

and the balance of the sides are calculated from angles

measured at the vertices of the triangles.





Survey Operations – TRILATERATION

• Trilateration System - also

consists also of a series of

joined or overlapping triangles

but the lengths of the triangle’s

sides are measured and few

directions or angles are

observed (only those required

to establish azimuth); same

idea as in triangulation.





Survey Operations - TRAVERSE

• Consists of a series of straight lines connecting successive

points whose lengths and directions have been determined

from field observations.

• Currently the most common of several possible methods for

establishing a series or network of monuments with knownpositions on the ground.

• Procedures will vary depending on the type of field angles

measured and whether bearings or azimuths are used to

describe directions.





Closed Traverse



Closed Loop Traverse



Survey Operations - TRAVERSE

Types of Traverse According to Method of Turning the Angles1. Interior-angle Traverse

2. Deflection-angle Traverse

3. Traverse by Angle to the Right

4. Azimuth Traverse





In dealing with a closed traverse, we have computations in:

1) Determining latitudes and departures

2) Calculating total error of closure

3) Balancing the survey

4) Determining adjusted positions of traverse stations

5) Area computation

6) Area subdivision





Projection of a line onto a reference

meridian or North-South line

Lines with Northerly bearings (+) LAT

Lines with Southerly bearings (-) LAT

Equal to distance*cosine of bearing angle

Latitude = d*Cos b





Projection of a line onto a reference parallel

or East-West line Lines with Easterly bearings (+) DEP

Lines with Westerly bearings (-) DEP

Equal to distance*sine of bearing angle

Departure = d*Sine b





Is usually a short line of unknown length and direction connecting the initial

and final traverse stations

22

)()( Lat Dep LEC

Lat

DepTan

Note:

In computing for , use the absolute values for Dep and Lat. Determinethe quadrant where the line lies using corresponding signs of the 2 sums.





Ratio of the linear error of closure to the perimeteror total length of the traverse

D LEC REC

REC = Relative Error of Closure

LEC = Linear Error of Closure

D = Total Length or perimeter of the traverse

Department of Geodetic EngineeringTraining Center for Applied Geodesy and Photogrammetry GE 12–

General Surveying II



D

d

C c Llat

Dd C c Ddep

clat = correction to latitude

cdep= correction to departure

CL= total closure in lat = Lat

CD= total closure in dep= Dep

d = length of any course

D = total length of the traverse





Lat

C Lat c L

lat

)(

Dep

C Depc Ddep

)(

Where:

clat = correction to latitude

cdep= correction to departure

CL= total closure in lat = Lat

CD= total closure in dep= Dep





Line Length(m)Azimuth

(from South)Line

Length

(m)

Azimuth from

(South)

AB 495.85 185o30’ DE 1020.87 347o35’

BC 850.62 226o02’ EF 1117.26 83o44’

CD 855.45 292o22’ FA 660.08 124o51’

Compute for:

1. Latitude and Departure of each line2. Bearing of the side error, LEC, REC

3. Adjust the traverse using Compass Rule





1. Latitude and Departure of each line

Line Distance (m) BearingLat

(N+, S-)

Dep

(E+, W-)

AB 495.85 N 05

o

30' E +493.57 +47.53BC 850.62 N 46o02' E +590.53 +612.23

CD 855.45 S 67o38' E -325.53 +791.09

DE 1020.87 S 12o25' E -996.99 +219.51

EF 1117.26 S 83o

44' W -121.96 -1110.58FA 660.08 N 55o09' W +377.19 -541.70

=5000.13

Lat=+16.81

Dep=+18.08





Bearing of the side error:

'0547

075550268.1

81.1608.18tan

0

b

b

Bearing of the side error is S 47o05’ W

2. Bearing of the side error, LEC, REC





Linear Error of Closure (LEC):

687.24

)08.18()81.16( 22

LEC = 24.69 m

Relative Error of Closure (REC):

200

1

52.202

1

13.5000

69.24

say

REC = 1/200

2. Bearing of the side error, LEC, REC





LineDistance

(m)Latitude Departure

Correction

(by Compass Rule) Lat_adj Dep_lat

Lat

Dep

AB 495.85 493.57 47.53 -1.667 -1.793 491.903 45.737

BC 850.62 590.53 612.23 -2.860 -3.076 587.670 609.154

CD 855.45 -325.53 791.09 -2.876 -3.093 -328.406 787.997

DE 1020.87 -996.99 219.51 -3.432 -3.691 -1000.422 215.819

EF 1117.26 -121.96 -1110.58 -3.756 -4.040 -125.716 -1114.620

FA 660.08 377.19 -541.7 -2.219 -2.387 374.971 -544.087

Sum: 5000.13 16.81 18.08 -16.810 -18.080 0.000 0.000

3. Traverse Adjustment by Compass Rule



3. Traverse Adjustment by Compass Rule

Line

Adjusted Values (By Compass Rule)

Latitude DepartureDistance

(m)Bearing

Azimuth (from

South)

AB 491.903 45.737 494.025 N 5

o

19' E 185

o

19'

BC 587.670 609.154 846.419 N 46o02' E 226o02'

CD -328.406 787.997 853.692 S 67o23' E 292o37'

DE -1000.422 215.819 1023.436 S 12o

10' E 347o

50'

EF -125.716 -1114.620 1121.687 S 83o34' W 83o34'

FA 374.971 -544.087 660.783 N 55o26' W 124o34'



Line Lat Dep |Lat| |Dep|

Correction by

Transit RuleAdjusted Lat/Dep

Lat

Dep Lat_adj Dep_adj

AB 493.57 47.53 493.57 47.53 -2.855 -0.259 490.715 47.271

BC 590.53 612.23 590.53 612.23 -3.416 -3.331 587.114 608.899

CD -325.53 791.09 325.53 791.09 -1.883 -4.305 -327.413 786.785

DE -996.99 219.51 996.99 219.51 -5.768 -1.194 -1002.758 218.316

EF -121.96 -1110.58 121.96 1110.58 -0.706 -6.043 -122.666 -1116.623

FA 377.19 -541.7 377.19 541.70 -2.182 -2.948 375.008 -544.648

Sum: 16.81 18.08 2905.77 3322.64 -16.810 -18.080 0.000 0.000

4. Traverse Adjustment by Transit Rule



Line

Adjusted Values (By Transit Rule)

Latitude Departure Distance

(m)

Bearing Azimuth (from

South)

AB 490.715 47.271 492.987 N 5o30' E 185o30'

BC 587.114 608.899 845.849 N 46o03' E 226o03'

CD -327.413 786.785 852.191 S 67o24' E 292o36'

DE -1002.758 218.316 1026.248 S 12o17' E 347o43'

EF -122.666 -1116.623 1123.340 S 83o44' W 83o44'

FA 375.008 -544.648 661.266 N 55o27' W 124o33'

4. Traverse Adjustment by Transit Rule



VERTICAL POSITIONING



Leveling

Operation of measuring vertical distances,either directly or indirectly, to determine theelevation of points or their differences in

elevation.

Used for topographic mapping, suitabilityanalysis, and design, layout and construction of

structures to best conform to the configurationof the ground.





Level Surface

Vertical Line

Horizontal Line

Elevation

Mean Sea Level (MSL)

(Vertical Datum)

Diff in Elev.

(Level Line)





Orthometric

Height (H)

Ellipsoidal

Height (h)


General Surveying II 30



Terms

Mean Sea Level - arithmetic mean of water elevations over a

specific 19-year cycle; close approximation of the Geoid;

conforms to the earth’s gravitational field





Bench Mark (B.M.) - a definite point on an object, the elevation

and location of which are known; serves as point of

reference for levels; may be permanent (P.B.M.) or

temporary (T.B.M.)

Terms





Terms

Set-up - consists of a point supporting a backsight rod, a point supporting the

foresight rod, and a leveling instrument positioned between them

Section - an unbroken series of set-ups, made between two (2) permanent

control points.

Level Loop / Circuit - a line of levels that ends at the same point of its beginning

Level Network - consists of a number of intersecting lines of levels that are tied

into known benchmarks


–General Surveying II 33

e c t i o n



B.M.

T.P. 1

T.P. 2 B .

S . B

. S .

F.S.

F.S.

Terms

Turning point - a fixed point or object, often temporary in character,

between two bench marks upon which foresight and backsight rod

readings are taken; a B.M. may be used as a T.P.





Backsight (B.S.) - a rod reading taken on a point of known elevation (BM or TP)usually taken with the level sighting back along the line

Backsight distance - horizontal distance from level to rod on a B.S.

Terms

B.M.

T.P. 1

T.P. 2 B .

S .

B . S .

F.S.

F.S.

Backsight

Distance

Backsight

Distance





Foresight (F.S.) - a rod reading taken on a point of unknown elevationForesight distance - horizontal distance from level to rod on a F.S.

Intermediate Foresight (I.F.S.) - other staff reading between the backsight and

foresight in the same set-up of the instrument.

Terms

B.M.

T.P. 1

T.P. 2 B .

S .

B . S .

F.S.

F.S.

Foresight

Distance

Foresight

Distance





B.M.

T.P. 1T.P. 2 B

. S .

B .

S .

F.S.

F.S.

Datum

H.I.1

H.I.2

Terms



Height of the instrument (H.I.) - elevation of the line of sight of the telescope

above the datum when the instrument is leveled



Basic Equation in Leveling

Elev B = Elev A + BS - FS

STA. B.S. H.I. F.S ELEV.

BM1 8.46 755.11

763.57

TP1 1.23 762.34


–General Surveying II 38

Known Elev + BS = HI

HI – FS = New Elev



Sample Computation – Differential Leveling

PROCEDURES:

1. A line of levels is run from BM A to BMB

2. Leveling instrument is set-up at any convenient

location along the level route and backsight is taken on

a rod held vertically on BM A.

3. The rodman moves forward along the general direction

of BMB and hold the rod at a convenient turning point

(TP1). (Making sure that the backsight distance is

approximately equal to the foresight distance.)

4. Then, the level is transferred again and a backsight is

taken at (TP1) and foresight at (TP2). Repeat

procedures until foresight is taken on BMB.Department of Geodetic EngineeringTraining Center for Applied Geodesy and Photogrammetry GE 12





MSL

Line of Sight

H.I.

BS

FS

ELEV BM1

BM1

TP1

H.I. = ELEVBM1 + BS

Elevi = H.I. - F.S.





Sta. B.S. H.I F.S. Elev

B.M. 1 3.251 ________ 72.105

T.P.1 2.539 ________ 0.012 ______

T.P.2 3.572 ________ 0.338 ______

B.M.2 0.933 ________ 3.112 ______

T.P.3 0.317 ________ 3.306 ______

T.P.4 0.835 ________ 2.716 ______

T.P.5 0.247 ________ 3.542 ______

B.M.3 3.786 ______

75.356

H.I. = B.S. + ElevBM







B.M. 1 3.251 ________ 72.105

T.P.1 2.539 ________ 0.012 ______

T.P.2 3.572 ________ 0.338 ______

B.M.2 0.933 ________ 3.112 ______

T.P.3 0.317 ________ 3.306 ______

T.P.4 0.835 ________ 2.716 ______

T.P.5 0.247 ________ 3.542 ______

B.M.3 3.786 ______

75.356

Elevi= H.I. - F.S.

75.344







B.M. 1 3.251 ________ 72.105

T.P.1 2.539 ________ 0.012 ______

T.P.2 3.572 ________ 0.338 ______

B.M.2 0.933 ________ 3.112 ______

T.P.3 0.317 ________ 3.306 ______

T.P.4 0.835 ________ 2.716 ______

T.P.5 0.247 ________ 3.542 ______

B.M.3 3.786 ______

75.356

77.883

81.117

78.938

75.949

74.068

70.773

75.344

77.545

78.005

75.632

73.233

70.526

66.987

Arithmetic Check:

BM1elev 72.105

+ +

∑ B.S. 11.694- -

∑ F.S. 16.812

= _______

BM3elev 66.987

H.I. = B.S. + ElevBM Elev

i= H.I. - F.S.

OK!∑ B.S. = 11.694 ∑ F.S. = 16.812






Personal Errors – Unequal Backsight & Foresight Distances

Sources of Errors in Leveling



Ghilani, et. al.



Errors due to the Curvature of the Earth

Sources of Errors in Leveling



http://www.oregon.gov/ODOT/HWY/GEOMETRONICS/docs/Leveling_Errors.pdf



• When a line of level makes a complete circuit, almost invariably the

BMelevinitial ≠ Bmelevfinal

• This difference is the error of running the circuit and is called the Error

of Closure.

• Result intermediate BMs are also in error

• Problems:

Determining the error for intermediate points

Adjusting their corresponding elevations

Error of Closure



• The appropriate correction to the observed elevation of a given bench

mark in the circuit is directly proportional to the distance of the BM

from the point of beginning.

C E L

d C

C = correction to be applied to a BM

d = distance of a BM from the point of beginning

L = length of the circuit

EC = error of closure of the level circuit = Elevobserved - Elevtheoretical

Corrections to the Observed Elevations



Point

Distance from

B.M.1

(km)

Observed

elevation

(m)

B.M.1 0 150.92

B.M.2 0.35 238.45

B.M.3 0.89 203.3

B.M.4 1.24 165.81

B.M.1 1.78 151.04

Solve for the adjusted

elevations of the intermediate

points based on the given

leveling data on the table.

Sample Problem:




Point

Distance from

B.M.1

(km)

Observed

elevation

(m)

Corrections

(m)

Adjusted

Elevation

(m)

B.M.1 0 150.92 0.00

B.M.2 0.35 238.45 -0.02 238.43

B.M.3 0.89 203.3 -0.06 203.24

B.M.4 1.24 165.81 -0.08 165.73

B.M.1 1.78 151.04 -0.12 150.92

C E L

d C

EC = Elevobs - Elevtheo = 151.04 – 150.92 = 0.12 m

L = 1.78 km

C = - d (in km) * 0.12 m

1.78 km




Methods in Leveling

1. Direct or Spirit Leveling (Differential, Double Rod, Three Wire)

2. Indirect or Trigonometric Leveling

3. Reciprocal Leveling

4. Stadia Leveling

5. Barometric Leveling6. Profile Leveling

7. Borrow-Pit Leveling

8. Gravimetric Leveling

9. Inertial Positioning System

10. GPS Survey

Department of Geodetic Engineering

Training Center for Applied Geodesy and Photogrammetry GE 12–




• The operation of determining the elevation ofpoints some distance apart by a series of set-ups of a leveling instrument along a designatedroute.

• Measure vertical distances directly

• Most precise method of determining elevations

• Forms of direct leveling include:

• Differential, Double Rod and Three Wire Leveling

Methods in Leveling – Direct/Spirit






Methods in Leveling – Differential

• Requires a series of set ups of the instrument along the general

route and, for each set up, a rod reading back to a point of known

elevation and forward to a point of unknown elevation.

• Determining the elevation of points some distance apart.

• CHECKS:

1. Rerunning levels on the same route

2. “Tying on” to a previously established BM near the end of the

level line3. Returning to the initial BM level loop










Note: effect of earth’s curvature and refraction is reduced to negligible amount

BM1

BM2







BM1

(known)

TP1

TP2 BM2

(known)








B.M. 1 3.251 ________ 72.105

T.P.1 2.539 ________ 0.012 ______

T.P.2 3.572 ________ 0.338 ______ B.M.2 0.933 ________ 3.112 ______

T.P.3 0.317 ________ 3.306 ______

T.P.4 0.835 ________ 2.716 ______

T.P.5 0.247 ________ 3.542 ______

B.M.3 3.786 ______



LEVELS FOR BENCHMARKS ALONG RIDGE ROAD


B.M. 1 3.251 75.356 72.105

T.P.1 2.539 77.883 0.012 75.344

T.P.2 3.572 81.117 0.338 77.545

B.M.2 0.933 78.938 3.112 78.005

T.P.3 0.317 75.949 3.306 75.632

T.P.4 0.835 74.068 2.716 73.233

T.P.5 0.247 70.773 3.542 70.526

B.M.3 3.786 66.987 [ 66.980]

∑ B.S.= 11.694 ∑ F.S.= - 16.812

11.694

B.M. 1 72.105 - 5.118 66.987

Error of Closure = 66.987-66.980 = +0.007 m







Methods in Leveling – Double Rod

BM1

(known)

TP1 H

TP1 L

TP2 H

TP2 L

BM2

(unknown)

Methods in Leveling–

Double



TRAINING CENTER FOR APPLIED GEODESY AND PHOTOGRAMMETRY

COLLEGE OF ENGINEERING, UNIVERSITY OF THE PHILIPPINES, DILIMAN

Arithmetic Check:

Mean Elev BM = 149.246

(149.149 + 149.343)/2

∑BS = +19.838

∑FS = - 15.126

Sum = (+ 4.714)/2 = + 2.356

Elev BMA = + 146.890

Sum/2 = + 2.356

BMB elev = + 149.246

Methods in Leveling –

DoubleRod






Methods in Leveling – Three Wire

Ghilani, et. al.






Methods in Leveling – Three Wire

(Homework)

BACKSIGHT FORESIGHT

STA. HAIR RDGS MEAN RDG s H.I. F.S. MEAN RDG s ELEV.

BM1

1.152 ________ ________ ________

________ ________ 444.2420.935

0.718

TP1

2.784

________ ________ ________

1.117

________ ________ ________ 2.420 0.899

2.057 0.682

TP2

1.713

________ ________ ________

1.900

________ ________ ________ 1.440 1.537

1.166 1.172

BM2

1.450

________ ________ ________ 1.177

0.904



Methods in Leveling – Trigonometric




Ghilani, et. al.



Methods in Leveling – Reciprocal




Ghilani, et. al.



• Trigonometric + Direct leveling

• Vertical distances are determined by tacheometry by using transit

and level rod.

• Process of taking stadia measurements consists of observing,

through the telescope, the apparent locations of the two stadiahairs, which is held in a vertical position.

Methods in Leveling – Stadia




BACKSIGHT FORESIGHT

STA. INTERVAL

VERT.

ANGLE ROD RDG VD INTERVAL

VERT.

ANGLE

ROD

RDG VD ΔELEV ELEV



Methods in Leveling – Stadia




a

V

H = Ks cos2α + C cos α

V = ½ Ks sin2α + C sin α

K = (f/i) = stadia interval factor

C = f + c = stadia constant

s = stadia intervalα = vertical angle



• Operation of determining elevations of points at short measured

intervals along a definitely located line, such as the center line for a

highway, railroad, canal or a sewer. (usually by direct leveling)

• Stakes or other marks are placed at regular intervals along the line;

the intervals between stakes usually is 100m, 50m, 20m and 10m.

Methods in Leveling – Profile










Ghilani, et.al.







STA. B.S. H.I F.S. I.F.S ELEV.

BM 30 3.478 33.478 30.000

0 + 00 3.617

0 + 05 5.141

+ 10 1.720

TP 1 3.314 0.913

+ 20 2.860

+ 29.5 1.852

+ 30 1.805

TP 3 0.081 2.289

TP 4 0.333 3.661

BM 30 1.974



References

Davis, R.E., et. al (1981). Surveying: Theory and Practice. USA: McGraw-Hill, Inc.

Ghilani, C.D., et.al. (2008). Elementary Surveying: an Introduction to Geomatics. USA:Pearson Education, Inc.

La Putt, J.P. (2007). Elementary Surveying. Philippines: National Book Store.

Schofield, W.. Et. Al (2007). Engineering Surveying. UK: Elsevier Ltd.

Anderson, James & E. Mikhail (1998), Surveying: Theory and Practice 7th Edition,McGraw-Hill Companies, Inc.

• GE 10 Leveling Principles and Operations/ Leveling Errors and Adjustments LectureNotes prepared by Engr. Jeark A. Principe.

* GE 10 Triangulation and Trilateration Lecture Notes prepared by Engr. Jeark A.Principe.

lecture 1a - horizontal and vertical positioning

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