FOPR 162
Basic
Forest
Geomatics
FOPR 162
Course Objectives
• You will learn the basics of “Plane Surveying”
which will provide the ability to perform
accurate surveys for most types of forestry
applications
• Your skills will include:
–
–
–
–
proper care and use of survey instruments
tests for adjustment of equipment
field techniques for measurement and layout
understanding different types of errors and how
they affect surveys
– knowing calculations and methods of error
adjustment for different types of surveys
Introduction
Definition of Surveying
Major Surveying Types:
Geodetic & Plane
The science, art, and technology of determining the
relative positions of points above, on, or beneath the
earth’s surface, or of establishing such points
• Geodetic Surveying
The Purpose of Surveying
• To establish property
boundaries
• To establish road
locations
• To determine deflection
• To establish operational
boundaries
– Accounts for the curvature of the earth
– More accurate for large areas
– Assumes that the earth is an ellipsiod
• To locate inventory
plots
• To collect mapping
information
• To lay out camps / DLS
• Site mapping for
bridges and structures
• Navigating in the woods
• Plane Surveying
– Assumes that the earth is flat and horizontal
Plane
Geodetic
1
FOPR 162
Units of Measure
Causes of Errors in Measurements
– this
course uses SI (metric)
Î Distance: meters (m)
Î Angles: degrees + minutes + seconds (° ‘ “)
or percent (%)
3
Î Area: square meters (m ) or hectares (ha)
9 Natural: wind, temperature, gravity,
magnetism
9 Instrument: poor construction or
adjustment of instrument
9 Personal: limitations of human
perception, eye sight, rod out of
plumb
Significant Figures
–all recording must be to correct # significant
figures
# of certain figures + 1 questionable figure
–153.65, 24, 2.4, 0.0024, .020?
–6100?
Types of Errors
„
„
„
Systematic errors
– accumulate during survey
Î incorrect chain length
Î rod out of plumb
Random errors
¾ tend to cancel
¾ what's left after mistakes and systematic
errors are eliminated
Mistakes / Blunders
– recording errors, inexperience,
misunderstanding, communication errors
2
FOPR 162
Boundary Systems
Accuracy and Precision
– Metes and Bounds System:
Precision
Î degree of consistency of group of
measurements
Î depends on instrument sensitivity &
operator skill
„ Accuracy
Î agreement with true value
• Original measurements were in “chains”, “rods”,
etcetera. This has often been updated to feet and
inches, and is slowly being converted to meters.
Each survey has a POB (point of beginning) [or POC
(point of commencement)], defined corners, lengths
and directions of the property sides, names of
adjoining property owners (e.g. TFL boundary), and
areas.
„
– Coordinate System:
• This system uses x and y measurements (departure
and latitude ) from a known or established point.
Tie Points
Boundary Systems - continued
– At the beginning and end of any survey you need
a permanent “benchmark” to tie in to. Such a tie
will provide a means of evaluating the accuracy of
your survey.
• Block and Lot
• Surveyed parcels of land are described in terms of
Land District, section, parcel, lot
3
FOPR 162
Distance Measurement
„
Lesson 1
Distance & Slope Measurement:
Trigonometric Leveling
OBJECTIVES
– To learn:
– Correct use of measuring tapes
– Correction of tape errors
– Break chaining and slope chaining
procedures
– Calculation of horizontal and vertical
distances from slope distance
– Characteristics of repeated measurements
– Trigonometric leveling procedures
Vertical Distance Measurement - VD
Horizontal Distance Measurement - HD
Normally want to determine horizontal distance
Standard Methods:
– taping
– Electronic Distance Measurement (EDM)
– pacing
– range finders
– stadia
– subtense bars
– odometers
– indirect
– GPS & LORAN
„ We will concentrate on taping & stadia
„
Objectives
– To measure the slope of the ground between
two field stations
– To relate the effect of slope on distance
measurements and allow for the calculation
of HD from SD measurements
– To determine the change in elevation
between two adjacent field stations by
calculating the VD
„ Devices - Dumpy Level, Abney level, Clinometer,
Altimeter
„
„
4
FOPR 162
Vertical Measurement Clinometer
Vertical Measurement Clinometer
Measures slope in percent or degrees from one
station to another. You also need slope distance
to determine HD or DE
„ Most common type - liquid filled Suunto
clinometer
„ Requires ability to view depth - View crosshair
while sighting on crewmate or control point at
specified height
„ Used in leveling (ground profiles) and measuring
heights (ie; trees)
„ Use slope tables to quickly determine HD and DE
- or compute these values...
„
Vertical Measurement
Abney Level
Vertical Measurement
Abney Level
Used to measure slope in percent or degrees Uses different scales that can be exchanged for
measurement
„ equipped with leveling bubble and is more
accurate than the clinometer, but requires more
time for measurement
„ Uses a sighting tube that allows user to view next
station, leveling bubble, and scale
„ User can fix specific slope value on scale and remeasure for that value very quickly
„
5
FOPR 162
Chaining Equipment
Measuring Elevation Changes - Clinometer
Early surveyors used link chains
„ Hence tape and chain used interchangeably
today
„ Various types of tapes used
– steel, eslon, nylon, cloth, string
– adding tapes and subtracting tapes
„ Other taping equipment
– chaining pins
– hand level or clinometer
– tension & clamp handles
– thermometer
– range poles
– plumb bobs
– flagging
Sight in on crewmember located at next station,
sighting on a point equal to height of clinometer
„ Measure slope in degrees (α) or
percent (p) and SD
„ Compute HD and VD
„
„
HD
α or p
L
VD
SD
SD
α
HD=SD*cos(α)
L
VD=SD*sin(α)
α = arctan(p/100)
Break Chaining
Chaining Procedure - Level Ground
Used to obtain horizontal distance directly
on sloping ground
„ Procedure
– chain down hill if possible
– keep chain level
– use plumb bob from chest height
„
Lining in
„ Applying tension
„ Plumbing
„ Marking distance
„ Reading the tape
„ Recording distances, e.g.; 18.4 m
„
Adding
20
18
1
0
+1
15
Subtracting
19
2
1
Survey
Direction
0
6
3
.81
FOPR 162
Slope Chaining
„
Chaining Errors
Three categories
– Instrument errors
– Natural Errors
– Human Errors
„ Specific Errors (Systematic or Random?)
– Tape length errors
– Non standard temperature
– Inconsistent tape tension
– Tape sag
– Poor alignment
– Tape not horizontal
– Sloppy plumbing or marking
– Sloppy reading, interp, recording
„
Procedure
–
–
–
Using proper chaining methods
Measure slope distance and slope
Calculate horizontal distance (and vertical distance if
required)
α
HD
α
SD
α
HD
SD
HD=SD*cos(α)
α = arctan(p/100)
Correction of tape errors
„
Trigonometric Leveling
Incorrect length of tape
– one of most significant errors
l − l'
= (
Cl
) L
l'
LC = L + Cl
Measure Slope distance & slope
Calculate HD & VD
„ Calculate Station and Elevation of TP
„
„
α
Where:
(read as p%)
HD
α
VD
A
Cl = Correction value applied to measured
length to obtain true length
l = Actual tape length
l’ = Nominal tape length
L = Measured length of line
LC = Corrected or true length of line
SD
α
HD=SD*cos(α)
α = arctan(p (%)/100)
7
SD
VD=SD*sin(α)
B
FOPR 162
Basic Forest Geomatics
Angles, Directions & Compass Traverse
„
Lesson 2
Angles, Directions &
Compass Traverse
OBJECTIVES
To learn:
– The theory of the compass
– Conversion of azimuths to bearings,
computation of directions from angles
– Measurement of horizontal angles and direction
with the hand compass
– To run a closed traverse with hand compass
and chain
– To compute errors of closure and the
adjustment of latitudes (northing) & departures
(easting) with the compass rule
Angles & Directions
Vertical angles (measured in vertical plane)
Horizontal angles (measured in horizontal plane)
„ Components of an angle
– starting line
– direction of turn
– amount of turn ( angular distance)
„ Units of angular measurement
– Degrees, minutes, seconds
Î 360 degrees in circle,
60 min / deg,
60 sec / min
Direction
„
„
–
Radians 2*π radians in 360 deg.
„
„
The DIRECTION of a line is its horizontal angle
from a reference meridian.
The reference meridian CAN be any straight line.
Me
ri
30° left of the meridian
d ia
n
Horizontal Angles – Type
Horizontal Angles - Direction
(Closed Traverse)
External Angles
G
+ or right or
clockwise
- or left or
counter clockwise
E
A
F
D
Internal Angles
B
Arrows indicate the direction of travel
Deflection Angles
Direction: Bearings & Azimuths
Direction of a line is its horizontal angle from a
reference meridian
„ Bearings
– measured from N or S towards E or W
„
N25W
N
S30W
Arrows indicate the
direction of travel
Bearings & Azimuths
„
Azimuths are measured clockwise from North
N25W
= 335°
N60E
W
C
N
N60E
= 60°
E
S55E
S
S30W
= 210°
S55E
= 125°
Conversion of bearings to azimuths and vice versa...
Directions from Angles
„
Calculation of directions from angles & visa versa
W
5° 5°
N4 r 31
o
N
Theory of the Compass
The compass needle shows the direction of
the magnetic meridian
„ The difference between true north and magnetic
north is the declination
„
-90°
N = 0°
+ 270°
E
5°
N4 45°
r
o
45°
Start
Declination
True direction vs. magnetic direction
„ The declination at Vancouver is 18° 51’ east (2005)
decreasing (shifting west) approximately 6’ annually
„
N
W
C
pa
om
N
ss
Declination
E
S
h
o rt
The magnetic north
pole is constantly
moving.
The current impact of
this in Vancouver is a
change of
approximately 1/10 of a
degree (6 minutes)
each year.
Finding Declination
Hand Compass
To find the declination of any point in Canada,
go to:
http://www.geolab.nrcan.gc.ca/geomag/home_e.shtml
You need to enter:
– Year
– Latitude in degrees and minutes
– Longitude in degrees and minutes
For example, UBC is 49° 15’ latitude / 123° 08’ longitude
Coordinates can be obtained from many maps or from numerous web
sites such as:
http://www.astro.com/cgi/aq.cgi?lang=e
Hand Compass
–
–
–
–
–
–
–
Common brands - Silva and Suunto
Available with - azimuth or bearing readings; check
bezel markings to determine which
For basic surveys use a model with a flip-up mirror,
rifle-type sight, & rotating bezel
Make sure declination is set properly
The compass should be level when taking a
bearing. See detailed instructions at
“Compass.pdf” in the course notes
Instructions are on a pdf file
There are many web sites that provide additional
information on the use of a compass. For example:
http://www.learn-orienteering.org/old/
Staff Compass
Staff Compass
Traverse - Open
Larger, more precise instrument than the hand
compass – 30” divisions vs 2° for hand compass
„ Mounted on a staff (Jacob's Staff) to provide
stability during use
„ Compass must be leveled prior to use
„ Compass is not liquid filled, and will become
damaged easily if the needle is not secured after
use
„ Compass is read using a front and rear sight
mounted opposite each other
„ Declination must be set
„
Traverse: a series of lines whose lengths &
directions are known
„ Open traverses do not have both PoC & PoT at
known points
„
N
0+00
PoC
1+37
0+75
1+25
PoT at
known
point
1+62
A traverse where the PoT is back at the PoT is a closed
traverse even if the PoC/PoT is not at a known point.
PoC
PoT
J
„
N
F
B
C
D
PoT
Most common in forestry
– cut block boundaries, road centerline (C/L),
deflection lines, location of inventory plots
– For each leg:
– measure slope (Suunto clinometer)
– slope distance (nylon chain)
– direction (hand compass)
G
H
I
A
Open Traverse
1+25
Compass & Chain Traverse
If the PoC and PoT are both at known points, or if they can be tied
to known points, then the traverse IS closed.
0+40
1+37
0+75
1+62
Traverse - Closed
0+00
PoC at
known point
0+40
E
0+00
PoC
0+40
1+37
0+75
1+25
1+62
PoT
Compass & Chain Traverse
Traverse Computations
Types of Errors
– Local attraction: jewelry, glasses, machinery,
old cables, power lines
– incorrect declination
– visibility or target not on turning point (TP)
– chain not straight
– recording errors
„ To detecting local attraction use fore-shot (FS) &
back-shot (BS) - should be 180° different
„
• Three basic measurements are taken during a
traverse:
– Slope distance (with a chain)
– Slope (with a clinometer or abney)
– Direction (with a compass)
• Slope Distance is converted to
horizontal distance (HD) and
difference in elevation (DE or ∆E)
– This is usually done in the field using slope tables
20°
200°
SD
∆E = SD sin θ
θ
HD = SD cos θ
Slope in % (and equivalent degrees)
Slope
1
Distance 0.57
10
10.00
0.10
11
11.00
0.11
12
12.00
0.12
13
13.00
0.13
14
14.00
0.14
15
15.00
0.15
16
16.00
0.16
17
17.00
0.17
18
18.00
0.18
19
19.00
0.19
20
20.00
0.20
21
21.00
0.21
22
22.00
0.22
23
23 00
2
1.15
10.00
0.20
11.00
0.22
12.00
0.24
13.00
0.26
14.00
0.28
15.00
0.30
16.00
0.32
17.00
0.34
18.00
0.36
19.00
0.38
20.00
0.40
21.00
0.42
22.00
0.44
23 00
3
1.72
10.00
0.30
11.00
0.33
11.99
0.36
12.99
0.39
13.99
0.42
14.99
0.45
15.99
0.48
16.99
0.51
17.99
0.54
18.99
0.57
19.99
0.60
20.99
0.63
21.99
0.66
22 99
4
2.29
9.99
0.40
10.99
0.44
11.99
0.48
12.99
0.52
13.99
0.56
14.99
0.60
15.99
0.64
16.99
0.68
17.99
0.72
18.98
0.76
19.98
0.80
20.98
0.84
21.98
0.88
22 98
5
2.86
9.99
0.50
10.99
0.55
11.99
0.60
12.98
0.65
13.98
0.70
14.98
0.75
15.98
0.80
16.98
0.85
17.98
0.90
18.98
0.95
19.98
1.00
20.97
1.05
21.97
1.10
22 97
6
3.43
9.98
0.60
10.98
0.66
11.98
0.72
12.98
0.78
13.97
0.84
14.97
0.90
15.97
0.96
16.97
1.02
17.97
1.08
18.97
1.14
19.96
1.20
20.96
1.26
21.96
1.32
22 96
7
4.00
9.98
0.70
10.97
0.77
11.97
0.84
12.97
0.91
13.97
0.98
14.96
1.05
15.96
1.12
16.96
1.19
17.96
1.26
18.95
1.33
19.95
1.40
20.95
1.47
21.95
1.54
22 94
8
4.57
9.97
0.80
10.96
0.88
11.96
0.96
12.96
1.04
13.96
1.12
14.95
1.20
15.95
1.28
16.95
1.36
17.94
1.44
18.94
1.52
19.94
1.59
20.93
1.67
21.93
1.75
22 93
9
5.14
9.96
0.90
10.96
0.99
11.95
1.08
12.95
1.17
13.94
1.25
14.94
1.34
15.94
1.43
16.93
1.52
17.93
1.61
18.92
1.70
19.92
1.79
20.92
1.88
21.91
1.97
22 91
10
5.71
9.95
1.00
10.95
1.09
11.94
1.19
12.94
1.29
13.93
1.39
14.93
1.49
15.92
1.59
16.92
1.69
17.91
1.79
18.91
1.89
19.90
1.99
20.90
2.09
21.89
2.19
22 89
11
6.28
9.94
1.09
10.93
1.20
11.93
1.31
12.92
1.42
13.92
1.53
14.91
1.64
15.90
1.75
16.90
1.86
17.89
1.97
18.89
2.08
19.88
2.19
20.87
2.30
21.87
2.41
22 86
12
6.84
9.93
1.19
10.92
1.31
11.91
1.43
12.91
1.55
13.90
1.67
14.89
1.79
15.89
1.91
16.88
2.03
17.87
2.14
18.86
2.26
19.86
2.38
20.85
2.50
21.84
2.62
22 84
13
7.41
9.92
1.29
10.91
1.42
11.90
1.55
12.89
1.68
13.88
1.80
14.87
1.93
15.87
2.06
16.86
2.19
17.85
2.32
18.84
2.45
19.83
2.58
20.82
2.71
21.82
2.84
22 81
14
7.97
9.90
1.39
10.89
1.53
11.88
1.66
12.87
1.80
13.86
1.94
14.86
2.08
15.85
2.22
16.84
2.36
17.83
2.50
18.82
2.63
19.81
2.77
20.80
2.91
21.79
3.05
22 78
Traverse Computations - Continued
9
1
1
1
1
1
1
1
1
1
1
2
2
2
• Traverse computations require the determination of
the three-dimensional coordinate of each TP of a
traverse
– The horizontal “X” coordinate
is called the “departure” or “easting”
– The horizontal “Y” coordinate
is called the “latitude” or “northing”
– The vertical coordinate is the elevation
• Beginning with a known or “assigned” coordinate for
the PoC, a departure, latitude, and ∆E are calculated
for each “leg” of a traverse and are added to the
preceding TP to determine the subsequent TP
coordinate
Traverse Computations
Latitude or Northing (“y”)
Departures or Easting (“x”)
Your calculator will return the sine and cosine of any azimuth
from 0° to 360° as a positive or negative figure.
E
W
Lat
Dep
Lat = H cos α
S
α
W = 270°
Dep = H sin α
H
Where α = the Azimuth
sine sine -
cosine +
H
α
cosine +
360° = N = 0°
N
cosine -
„
cosine -
„
Traverse Computations
sine +
sine +
E = 90°
S = 180°
Caution: Be certain that your calculator is set to degrees!!!
Traverse Computations
„
The sum of individual departures & latitudes for a
closed traverse should total zero
J
I
H
G
Plot of closed traverse
A
B
C
F
D
E
ErrorClosure = ( ∑ Lats) + ( ∑ Deps) 2
2
Allowable Error = 1% of traverse length
0.4
23
FORESHORE PARK
2.5
18
9
27
.0
325.0
9
367.
13
3.
9
.9
91
40
7.6
46.0
0.
0
5.7
61
44
5.3
3
0.1
57
1
SCALE 1:1000
523.
491.
Traverse Computations
„
„
From Example
Adjustment of lats & deps
– sum lats = -0.084m,
– sum deps = -0.230m
– error = 0.244m
– allowable= 1% of 615.75m or 6.15m
– use compass rule to adjust
Î adjust each leg as its proportion of total
traverse length
PLOTTING TRAVERSES
The most accurate method of plotting a traverse is to determine the latitude (north – south displacement) and
departure (east – west displacement) of each station of the traverse. The displacement of each station is
calculated from the horizontal distance and the bearing of the course connecting the station from the previous
station. Normally, the Point of Commencement (PoC) is given a defined “x,y” (departure, latitude) coordinate
and the subsequent courses are added to produce a coordinate for each station.
For latitudes (y coordinate), courses toward the north are positive; toward the south is negative. For departure
(x coordinate), east is positive and west negative. Traverses in azimuths (direction stated as an angle between
0º and 360º) are very easy to calculate, because the direction sign is provided by the calculator.
For a course of known horizontal distance (H) and azimuth (Aº):
Latitude = H cos Aº
Departure = H sin Aº
Traverses in “Bearings” state the angle of the course to the east or west of the North or South direction (e.g.
N32W or S46E). Calculations using bearings can be more labor intensive, especially in assigning the positive
or negative value of the course. A traverse table (next page) is often useful in assigning latitudes and
departures. This table provides values for 0º to 45º reading from the top down, and for 45º to 90º reading from
the bottom up.
Example:
A course runs N28ºW and is 132 meters in length.
From the Table, the Latitude for 100 m is
88.3 m
Departure is
for 30 m is
26.5 m
for 2 m is
1.77 m
Total
116.57 m
Note that departure is negative because it runs westward.
-46.9 m
-14.1 m
-0.94 m
-61.94 m
Closure:
For a closed traverse, the sum of the latitudes and departures of all the courses should be zero. Closing error is
calculated using the Pythagorean Theorem:
2
E = E lats + E deps
The error
2
E
 Total Traverse Length 
is expressed as an accuracy ratio 1 : 

Total Traverse Length
E


Acceptable horizontal accuracy with basic hand held instruments ranges from 1:100 for field traverses to 1:300
for location surveys. For more critical surveys, more sophisticated equipment is used to obtain acceptable
horizontal accuracy ranges up to 1:5000.
TRAVERSE TABLE : LATITUDES AND DEPARTURES
ANGLE
(Read
Down)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
10.0
Lat. Dep.
10.0 0.0
10.0 0.2
10.0 0.3
10.0 0.5
10.0 0.7
10.0 0.9
9.9 1.0
9.9 1.2
9.9 1.4
9.9 1.6
9.8 1.7
9.8 1.9
9.8 2.1
9.7 2.2
9.7 2.4
9.7 2.6
9.6 2.8
9.6 2.9
9.5 3.1
9.5 3.3
9.4 3.4
9.3 3.6
9.3 3.7
9.2 3.9
9.1 4.1
9.1 4.2
9.0 4.4
8.9 4.5
8.8 4.7
8.7 4.8
8.7 5.0
8.6 5.2
8.5 5.3
8.4 5.4
8.3 5.6
8.2 5.7
8.1 5.9
8.0 6.0
7.9 6.2
7.8 6.3
7.7 6.4
7.5 6.6
7.4 6.7
7.3 6.8
7.2 6.9
7.1 7.1
Dep. Lat.
10.0
20.0
Lat. Dep.
20.0 0.0
20.0 0.3
20.0 0.7
20.0 1.0
20.0 1.4
19.9 1.7
19.9 2.1
19.9 2.4
19.8 2.8
19.8 3.1
19.7 3.5
19.6 3.8
19.6 4.2
19.5 4.5
19.4 4.8
19.3 5.2
19.2 5.5
19.1 5.8
19.0 6.2
18.9 6.5
18.8 6.8
18.7 7.2
18.5 7.5
18.4 7.8
18.3 8.1
18.1 8.5
18.0 8.8
17.8 9.1
17.7 9.4
17.5 9.7
17.3 10.0
17.1 10.3
17.0 10.6
16.8 10.9
16.6 11.2
16.4 11.5
16.2 11.8
16.0 12.0
15.8 12.3
15.5 12.6
15.3 12.9
15.1 13.1
14.9 13.4
14.6 13.6
14.4 13.9
14.1 14.1
Dep. Lat.
20.0
30.0
Lat. Dep.
30.0 0.0
30.0 0.5
30.0 1.0
30.0 1.6
29.9 2.1
29.9 2.6
29.8 3.1
29.8 3.7
29.7 4.2
29.6 4.7
29.5 5.2
29.4 5.7
29.3 6.2
29.2 6.7
29.1 7.3
29.0 7.8
28.8 8.3
28.7 8.8
28.5 9.3
28.4 9.8
28.2 10.3
28.0 10.8
27.8 11.2
27.6 11.7
27.4 12.2
27.2 12.7
27.0 13.2
26.7 13.6
26.5 14.1
26.2 14.5
26.0 15.0
25.7 15.5
25.4 15.9
25.2 16.3
24.9 16.8
24.6 17.2
24.3 17.6
24.0 18.1
23.6 18.5
23.3 18.9
23.0 19.3
22.6 19.7
22.3 20.1
21.9 20.5
21.6 20.8
21.2 21.2
Dep. Lat.
30.0
40.0
Lat. Dep.
40.0 0.0
40.0 0.7
40.0 1.4
39.9 2.1
39.9 2.8
39.8 3.5
39.8 4.2
39.7 4.9
39.6 5.6
39.5 6.3
39.4 6.9
39.3 7.6
39.1 8.3
39.0 9.0
38.8 9.7
38.6 10.4
38.5 11.0
38.3 11.7
38.0 12.4
37.8 13.0
37.6 13.7
37.3 14.3
37.1 15.0
36.8 15.6
36.5 16.3
36.3 16.9
36.0 17.5
35.6 18.2
35.3 18.8
35.0 19.4
34.6 20.0
34.3 20.6
33.9 21.2
33.5 21.8
33.2 22.4
32.8 22.9
32.4 23.5
31.9 24.1
31.5 24.6
31.1 25.2
30.6 25.7
30.2 26.2
29.7 26.8
29.3 27.3
28.8 27.8
28.3 28.3
Dep. Lat.
40.0
Horizontal Distance
50.0
60.0
Lat. Dep. Lat. Dep.
50.0 0.0 60.0 0.0
50.0 0.9 60.0 1.0
50.0 1.7 60.0 2.1
49.9 2.6 59.9 3.1
49.9 3.5 59.9 4.2
49.8 4.4 59.8 5.2
49.7 5.2 59.7 6.3
49.6 6.1 59.6 7.3
49.5 7.0 59.4 8.4
49.4 7.8 59.3 9.4
49.2 8.7 59.1 10.4
49.1 9.5 58.9 11.4
48.9 10.4 58.7 12.5
48.7 11.2 58.5 13.5
48.5 12.1 58.2 14.5
48.3 12.9 58.0 15.5
48.1 13.8 57.7 16.5
47.8 14.6 57.4 17.5
47.6 15.5 57.1 18.5
47.3 16.3 56.7 19.5
47.0 17.1 56.4 20.5
46.7 17.9 56.0 21.5
46.4 18.7 55.6 22.5
46.0 19.5 55.2 23.4
45.7 20.3 54.8 24.4
45.3 21.1 54.4 25.4
44.9 21.9 53.9 26.3
44.6 22.7 53.5 27.2
44.1 23.5 53.0 28.2
43.7 24.2 52.5 29.1
43.3 25.0 52.0 30.0
42.9 25.8 51.4 30.9
42.4 26.5 50.9 31.8
41.9 27.2 50.3 32.7
41.5 28.0 49.7 33.6
41.0 28.7 49.1 34.4
40.5 29.4 48.5 35.3
39.9 30.1 47.9 36.1
39.4 30.8 47.3 36.9
38.9 31.5 46.6 37.8
38.3 32.1 46.0 38.6
37.7 32.8 45.3 39.4
37.2 33.5 44.6 40.1
36.6 34.1 43.9 40.9
36.0 34.7 43.2 41.7
35.4 35.4 42.4 42.4
Dep. Lat. Dep. Lat.
50.0
60.0
Horizontal Distance
70.0
Lat. Dep.
70.0 0.0
70.0 1.2
70.0 2.4
69.9 3.7
69.8 4.9
69.7 6.1
69.6 7.3
69.5 8.5
69.3 9.7
69.1 11.0
68.9 12.2
68.7 13.4
68.5 14.6
68.2 15.7
67.9 16.9
67.6 18.1
67.3 19.3
66.9 20.5
66.6 21.6
66.2 22.8
65.8 23.9
65.4 25.1
64.9 26.2
64.4 27.4
63.9 28.5
63.4 29.6
62.9 30.7
62.4 31.8
61.8 32.9
61.2 33.9
60.6 35.0
60.0 36.1
59.4 37.1
58.7 38.1
58.0 39.1
57.3 40.2
56.6 41.1
55.9 42.1
55.2 43.1
54.4 44.1
53.6 45.0
52.8 45.9
52.0 46.8
51.2 47.7
50.4 48.6
49.5 49.5
Dep. Lat.
70.0
80.0
Lat. Dep.
80.0 0.0
80.0 1.4
80.0 2.8
79.9 4.2
79.8 5.6
79.7 7.0
79.6 8.4
79.4 9.7
79.2 11.1
79.0 12.5
78.8 13.9
78.5 15.3
78.3 16.6
77.9 18.0
77.6 19.4
77.3 20.7
76.9 22.1
76.5 23.4
76.1 24.7
75.6 26.0
75.2 27.4
74.7 28.7
74.2 30.0
73.6 31.3
73.1 32.5
72.5 33.8
71.9 35.1
71.3 36.3
70.6 37.6
70.0 38.8
69.3 40.0
68.6 41.2
67.8 42.4
67.1 43.6
66.3 44.7
65.5 45.9
64.7 47.0
63.9 48.1
63.0 49.3
62.2 50.3
61.3 51.4
60.4 52.5
59.5 53.5
58.5 54.6
57.5 55.6
56.6 56.6
Dep. Lat.
80.0
100.0
90.0
Lat. Dep. Lat. Dep.
90.0 0.0 100.0 0.0
90.0 1.6 100.0 1.7
89.9 3.1 99.9 3.5
89.9 4.7 99.9 5.2
89.8 6.3 99.8 7.0
89.7 7.8 99.6 8.7
89.5 9.4 99.5 10.5
89.3 11.0 99.3 12.2
89.1 12.5 99.0 13.9
88.9 14.1 98.8 15.6
88.6 15.6 98.5 17.4
88.3 17.2 98.2 19.1
88.0 18.7 97.8 20.8
87.7 20.2 97.4 22.5
87.3 21.8 97.0 24.2
86.9 23.3 96.6 25.9
86.5 24.8 96.1 27.6
86.1 26.3 95.6 29.2
85.6 27.8 95.1 30.9
85.1 29.3 94.6 32.6
84.6 30.8 94.0 34.2
84.0 32.3 93.4 35.8
83.4 33.7 92.7 37.5
82.8 35.2 92.1 39.1
82.2 36.6 91.4 40.7
81.6 38.0 90.6 42.3
80.9 39.5 89.9 43.8
80.2 40.9 89.1 45.4
79.5 42.3 88.3 46.9
78.7 43.6 87.5 48.5
77.9 45.0 86.6 50.0
77.1 46.4 85.7 51.5
76.3 47.7 84.8 53.0
75.5 49.0 83.9 54.5
74.6 50.3 82.9 55.9
73.7 51.6 81.9 57.4
72.8 52.9 80.9 58.8
71.9 54.2 79.9 60.2
70.9 55.4 78.8 61.6
69.9 56.6 77.7 62.9
68.9 57.9 76.6 64.3
67.9 59.0 75.5 65.6
66.9 60.2 74.3 66.9
65.8 61.4 73.1 68.2
64.7 62.5 71.9 69.5
63.6 63.6 70.7 70.7
Dep. Lat. Dep. Lat.
90.0
100.0
90
89
88
87
86
85
84
83
82
81
80
79
78
77
76
75
74
73
72
71
70
69
68
67
66
65
64
63
62
61
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
ANGLE
(Read Up)
Adjusting Closure:
If a traverse comes within an acceptable closing error, adjustments can be made to the coordinates of each
station to eliminate the error. A common way of doing this is to adjust the latitude and departure for each leg
using a prorate formula:
Length of leg
Correction to Lat or Dep = −(Elat or dep )
Total Traverse Length
(Note the negative sign!)
Adjustments of this type do not actually correct errors. The error is systematically distributed over the survey to
make it disappear on paper.
Calculating Areas by Coordinates:
Once coordinates of a closed traverse are known, area can be quickly and precisely calculated using the area by
coordinate method. The following example illustrates the setup of a calculation matrix that will work for any
polygon. To close the calculation, the first coordinate must be repeated at the end. The direction of the
traverse or the data entry does not matter, as long as it is in sequence.
C
(xC,yC)
D
B
(xD,yD)
E
(xB,yB)
A
(xE,yE)
COORDRINATES
xA
yA
xB
yB
xC
yC
xD
yD
xE
yE
xA
yA
xA yB
xB yC
xC yD
xD yE
xE yA
xB yA
xC yB
xD yC
xE yD
xA yE
TOTALS
Σ
Σ
AREA = 0.5 |Σ
(xA,yA)
CALCULATION
-Σ
|
The area is one half of the absolute value of the difference in the sums.
Note that it is extremely important to carry algebraic signs through all the calculations. Negative coordinates
must carry their sign!
Another way to express this calculation (using numbered subscripts) for any polygon is:
Area = 0.5 |(x1y2 + x2y3 + … + xny1) – (y1x2 + y2x3 + … + ynx1)|
Where “n” is the last station.