credits: 10 learning hours: 100 sector: construction and

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TVET CERTIFICATE V in LAND SURVEYING

T T T S R 1 0 3 SURVEYING MEASUREMENT ADJUSTMENT

LSVSA501 PERFORM SURVEYING MEASUREMENT ADJUSTMENT

Credits: 10 Learning hours: 100

Sector: Construction and Building Services

Sub-sector: Land Surveying

Module Note Issue date: June, 2020

Purpose statement

This module describes the skills, knowledge and attitudes required to perform surveying measurement

adjustment. At the end of this module, participants will be able to identify principles of surveying

measurement, apply methods of surveying adjustment, and perform accuracy and precision of

measurements.

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Table of Contents

Elements of competence and performance criteria Page No.

Learning Unit Performance Criteria

1. Identify principles of

surveying measurement

1.1. Identify types of measurement 3

1.2. Identify sources of errors

1.3. Identify types of errors

2. Apply methods of

surveying adjustment

2.1. Identify methods of survey adjustment 9

2.2. Determine error propagation in surveying

measurements

2.3. Adjust errors in linear measurement

3. Perform Accuracy and

Precision of measurements

3.1. Determine indices of precision 34

3.2. Apply least squares adjustment in survey

measurement

3.3. Adjust errors in traverse surveys network

Total Number of Pages: 63

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Learning Unit 1 – Identify Surveying Measurement principles

1. Introduction to surveying measurement and adjustment

We currently live in what is often termed the information age. Aided by new and emerging technologies,

data are being collected at unprecedented rates in all walks of life. For example, in the field of surveying,

total station instruments, global positioning system (GPS) equipment, digital metric cameras, and satellite

imaging systems are only some of the new instruments that are now available for rapid generation of vast

quantities of measured data.

Geographic Information Systems (GISs) have evolved concurrently with the development of these new

data acquisition instruments. GISs are now used extensively for management, planning, and design. They

are being applied worldwide at all levels of government, in business and industry, by public utilities, and in

private engineering and surveying offices. Implementation of a GIS depends upon large quantities of data

from a variety of sources, many of them consisting of observations made with the new instruments, such

as those noted above.

Before data can be utilized, however, whether for surveying and mapping projects, for engineering design,

or for use in a geographic information system, they must be processed. One of the most important aspects

of this is to account for the fact that no measurements are exact. That is, they always contain errors.

The steps involved in accounting for the existence of errors in measurements consist of (1) performing

statistical analyses of the observations to assess the magnitudes of their errors and to study their

distributions to determine whether or not they are within acceptable tolerances; and if the observations

are acceptable, (2) adjusting them so that they conform to exact geometric conditions or other required

constraints. Procedures for performing these two steps in processing measured data are principal subjects

of this book.

LO 1.1 – Identify Measurement Types

● Content/Topic 1 :Differentiate types of measurement

Measurements are defined as observations made to determine unknown quantities. They may be

classified as either direct or indirect.

Direct measurements are made by applying an instrument directly to the unknown quantity and

observing its value, usually by reading it directly from graduated scales on the device. Determining

the distance between two points by making a direct measurement using a graduated tape, or

measuring an angle by making a direct observation from the graduated circle of a theodolite or

total station instrument, are examples of direct measurements.

Indirect measurements are obtained when it is not possible or practical to make direct

measurements. In such cases the quantity desired is determined from its mathematical

relationship to direct measurements. Surveyors may, for example, measure angles and lengths of

lines between points directly and use these measurements to compute station coordinates. From

these coordinate values, other distances and angles that were not measured directly may be

derived indirectly by computation. During this procedure, the errors that were present in the

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original direct observations are propagated (distributed) by the computational process into the

indirect values. Thus, the indirect measurements (computed station coordinates, distances, and

angles) contain errors that are functions of the original errors. This distribution of errors is known

as error propagation. The analysis of how errors propagate is also a principal topic of this book.

L O 1.2. Identify Sources of Errors

Content/Topic 1 - Identification Of Errors In Measurements

By definition, an error is the difference between an observed value for a quantity and its true value, or

where E is the error in an observation, X the observed value, and its true value. It can be unconditionally

stated that (1) no observation is exact, (2) every observation contains errors, (3) the true value of an

observation is never known, and, therefore, (4) the exact error present is always unknown. These facts are

demonstrated by the following.

Errors in observations stem from three sources, and are classified accordingly.

Natural error: are caused by variations in wind, temperature, humidity, atmospheric pressure,

atmospheric refraction, gravity, and magnetic declination. An example is a steel tape whose length varies

with changes in temperature.

Instrumental errors: result from any imperfection in the construction or adjustment of instruments and

from the movement of individual parts. For example, the graduations

on a scale may not be perfectly spaced, or the scale may be warped. The effect of many instrumental

errors can be reduced, or even eliminated, by adopting proper surveying procedures or applying computed

corrections.

Personal errors: arise principally from limitations of the human senses of sight and touch. As an example, a

small error occurs in the observed value of a horizontal angle if the vertical crosshair in a total station

instrument is not aligned perfectly on the target, or if the target is the top of a rod that is being held

slightly out of plumb.

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L O 1.3 Identify Types of Errors

Content/Topic 1 – Identification Error Types

Errors in observations are of two types: systematic and random.

Systematic errors:

They arise from sources which act in a similar manner on observations.

The method of measurement, the instruments used and the physical conditions at the time of

measurements must all be considered in this respect.

So long as system conditions remain constant, the systematic errors will likewise remain constant. If

conditions change, the magnitudes of systematic errors also change. Because systematic errors

tend to accumulate, they are sometimes called cumulative errors.

Few examples of these errors include: Expansion of steel tapes, distance measuring (EDM)

instruments and collimation in a level.

Systematic errors are not revealed by taking the same measurement again with the same

instruments. The only way to check adequately for systematic error is to re-measure the quantity

by an entirely different method using different instruments.

Random errors:

Are those discrepancies remaining once blunders and systematic errors have been eliminated.

Even if a quantity is measured any times with the same instrument in the same way and if all

sources of systematic error have been removed, it is still highly unlikely that all results will be

identical. The differences caused mainly by limitations of instruments and observers are random

errors.

They are caused by factors beyond the control of the observer, obey the laws of probability, and

are sometimes called accidental errors. They are present in all surveying observations.

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The magnitudes and algebraic signs of random errors are matters of chance. There is no absolute

way to compute or eliminate them, but they can be estimated using adjustment procedures known

as least squares

Characteristics of random errors:

they are small errors and may occur more frequently than large ones

they can be positive or negative

Gross errors or mistakes:

These are observer blunders and are usually caused by misunderstanding the problem,

carelessness, fatigue, missed communication, or poor judgment...

These types of mistakes can occur at any stage of a survey when observing, booking,

computing or plotting and they would obviously have a very damaging effect on the results

if left uncorrected.

By following strictly a well-planned observing procedure it is possible to reduce the number

that occurs and then independent checks at each stage should show up those that have

been made.

Precision, accuracy, Reliability, Uncertainty

A discrepancy is the difference between two observed values of the same quantity. A small discrepancy

indicates there are probably no mistakes and random errors are small. However, small discrepancies do

not preclude the presence of systematic errors.

Precision:

Refers to the degree of refinement or consistency of a group of observations and is evaluated on

the basis of discrepancy size. If multiple observations are made of the same quantity and small

discrepancies result, this indicates high precision. The degree of precision attainable is dependent

on equipment sensitivity and observer skill.

The precision is expressed in terms of standard deviation of the error.

Accuracy:

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Denotes the absolute nearness of observed quantities to their true values. As the true value never

known the accuracy also is never known.

The difference between precision and accuracy is perhaps best illustrated with reference to target

shooting.

Accuracy is defined as the mean error i.e.:

Accuracy and Precision.

Case(a): ideal case, observations (shootings) are precise

and accurate.

Case (b): illustrates the case where data have an accurate

mean value with unprecise measurements.

Typically, this represents the experiment where the unrefined measurement technique is used. For

example: measure of distance through step-pacing by an experienced surveyor.

Case (c): shooting (measurements) are precise but not accurate. This occurs when for example

measurements were made by an experienced observer (shooter) with uncalibrated instrument

(which was affected by a systematic error).

Case (d): worst case: measurements are neither precise nor accurate. It happens for example when

measurements are made by careless, inexperienced user or observer using uncalibrated instrument.

Reliability:

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Is a way of ensuring that any instrument used for measuring observations gives the same results

every time.

Uncertainty:

is a non-negative parameter characterizing the dispersion of the values attributed to a measured

quantity. It integrates both imprecision and inaccuracy. Uncertainty is expressed as root mean square error

(RMSE).

Assess Error Distribution During the Field Measurement

Then two basic rules are being used to distribute the error:

1—compass rule: error is distributed to the courses based on their length. Since course BM1-A is longer,

most of the correction belong to that course

2—transit rule: error is distributed to the courses based on latitude and longitude components of each

course; the course BM1-A runs from West to East, thus contains only longitude component and vice versa;

note that the courses do not change direction: transit is an equivalent to theodolite; assuming that

accuracy of angles is higher than accuracy of distances, then we prefer not to change the directions

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Learning Unit 2. Apply Methods of Surveying Adjustment

LO 2.1: Identify Methods of Surveying Adjustment

The methods of surveying adjustment are follow:

Chain and tape correction

Compass(Bowditch) rule

Transit rule

Adjustment by least squares

Content /Topic1 – Correction of chain and tape

In this method, steel tapes or wires are used to measure distance very accurately. Nowadays, EDM is being

used exclusively for accurate measurements but the steel tape still is of value for measuring limited lengths

for setting out purposes. Tape measurements require certain corrections to be applied to the measured

distance depending upon the conditions under which the measurements have been made. These

corrections are discussed below.

Systematic error correction in taping measurements

As like other observations there are three fundamental sources of errors in taping

1. Instrumental 2. Natural 3. Personal errors

1.1 Correction for Absolute Length

Due to manufacturing defects the absolute length of the tape may be different from its designated or

nominal length. Also with use the tape may stretch causing change in the length and it is imperative that

the tape is regularly checked under standard conditions to determine its absolute length. The correction

for absolute length or standardization is given by

Where

c = the correction per tape length,

l = the designated or nominal length of the tape, and

L= the measured length of the line.

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If the absolute length is more than the nominal length the sign of the correction is positive and vice versa.

1. Standard

a. Error occur when the length of the tape used is incorrect i.e. If the actual length of the tape

differs from its nominal graduated length due to defect in manufacture or repair.

b. Such errors are eliminated by checking the tape against a standard such as two marks

measured for the purpose so that the error per tape length is known.

1.2 Correction of systematic errors (length tape correction)

◦ If the error cannot be eliminated then a correction can be applied to remove the error.

◦ The correct lenth can be given by;

1.3 Slope correction

◦ It is already indicated that whenever slope distances are measured, they must be reduced to

horizontal plane.

◦ This can be done by computation from measured slope angle, or measured difference in

elevation, h of the two points in consideration.

standard

of Length

band usedLength of lengthmeasuredlengthCorrect

length recordedor measured

length tapenominal

length tapeactual

length measured toapplied be tocorrection

:

)'

'(

L

l'

l

C

Where

Ll

llC

l

l

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• H=L cos Or H=

The correction can be given by;

1.4 Tension Correction

Correct tension should be applied using a spring balance

If the standard tension is not applied a correection should be applied because the

lenght of the tape will have changed.

This correction is given by;

measuredlength

band for the elasticity of modulus sYoung'

band theof area sectional-cross

tensionstandard

tensionfield

;

)(

s

L

E

A

P

P

Where

LEA

PPCorrection

s

L

B

A

levelin difference

length measuredL

Where;

2

or ) cos1(

2

h

L

h-

LCorrection

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1.5 Temprature Correction

Correction is required if the tapes temprature, t at the time of measurement is not not equal to the

standard temprature, ts.

The temprature correction is given by;

1.6 Sag correction

Sag correction is necessary when the band (tape) is suspended between tripod heads i.e hung on

the catenary rather than being laid on the ground.

The correction for the sag must be applied if the tape has been standardized on a flat.

Sag correction

No sag

e tempraturstandard

e tempraturtape

length measured

expansionlinear oft coefficien

;

)(

s

s

t

t

L

Where

ttLCorrection

tensionfield

length measured the

tape theoflength unit per weight

;

24 2

32

P

l

w

Where

P

lwCs

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Example 2.1. A line AB between the stations A and B was measured as 348.28 using a 20 m tape, too short

by 0.05 m. Determine the correct length of AB, the reduced horizontal length of AB if AB lay on a slope of 1

in 25, and the reading required to produce a horizontal distance of 22.86 m between two pegs, one being

0.56 m above the other.

Solution:

(a) Since the tape is too short by 0.05 m, actual length of AB will be less than the measured length. The

correction required to the measured length is

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Content /Topic2 – Correction by compass (Bowditch) Rule

The compass, or Bowditch rule adjusts the departures and latitudes of traverse courses in proportion to

their lengths. Although not as rigorous as the least-squares method; it does result in a logical distribution

of misclosures. Corrections by this method are made according to the following rules:

Note that the algebraic signs of the corrections are opposite those of the respective misclosures.

Example 10.4

Using the preliminary azimuths from Table 10.2 and lengths from Figure 10.1, compute departures and

latitudes, linear misclosure, and relative precision. Balance the departures and latitudes using the compass

rule.

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Solution

A tabular solution, which is somewhat different than that used in Example 10.3, is employed for computing

departures and latitudes (see Table 10.4).To compute departure and latitude corrections by the compass

rule, Equations (10.5) and (10.6) are used as demonstrated. By Equation (10.5) the correction in departure

for AB is

Content /Topic3 – Correction by transit Rule

CL=∑L x L /LT

CD=∑D x D/DT

CL=correction to latitude of any line

CD =correction to departure of any line

L =latitude of any line

D =departure of any line

LT=arithmetic sum of latitudes (ignoring the signs)

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DT =arithmetic sum of departures (ignoring the signs)

∑L = total error (algebraic sum) in latitude,

∑D = total error (algebraic sum) in departure

Content /Topic4 – Correction by least square methods

It is a general practice in surveying to always have redundant observations as they help in detection of

mistakes or blunders. Redundant observations require a method which can yield a unique solution of the

model for which the observations have been made. The least squares method provides a general and

systematic procedure which yields a unique solution in all situations.

Assuming that all the observations are uncorrelated then the least squares method of adjustment is based

upon the following criterion: “The sum of the weighted squares of the residuals must be a minimum”. If υ

1, υ 2, υ 3, etc., are the residuals and ω 1, ω 2, ω 3, etc., are the weights then φ = ω 1 υ 1 2 + ω 2 υ 2 2 + ω

3 υ 3 2 +……+ ω n υ n 2 = a minimum

= Σω ν ii i n = ∑ = 1

a minimum.

The above condition which the residuals have to satisfy is in addition to the conditions which the adjusted

values have to satisfy for a given model

LO 2.2. Error Propagation in Surveying Measurements

Content/Topic 1 : Propagation Of Error In Surveying Measurement

Error propagation in surveying measurements

Error in a series

Error in a sum

Error in redundant measurement

Error in a Series Describes the error of multiple measurements with identical standard deviations, such as

measuring a 1000’ line with using a 100’ chain. E series En

E sum is the square root of the sum of each of the individual measurements squared It is used when there

are several measurements with differing standard errors 2222 E 2 E 3 ... E n E sum E 1

Error in Redundant Measurements If a measurement is repeated multiple times, the accuracy increases,

even if the measurements have the same value n E red. meas.

https://images.slideplayer.com/26/8526622/slides/slide_4.jpg

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If you learn one thing… With Errors of a Sum (or Series), each additional variable increases the total error of the network With Errors of Redundant Measurement, each redundant measurement decreases the error of the network. As the network becomes more complicated, accuracy can be maintained by increasing the number of redundant measurements

Introduction to Adjustments Adjustment - “A process designed to remove inconsistencies in measured or computed quantities by applying derived corrections to compensate for random, or accidental errors, such errors not being subject to systematic corrections

LO 2.3- Identify Methods of Surveying Adjustment by application

Content/Topic 1 : Correction for tape measurement

Tape measurements require certain corrections to be applied to the measured distance

depending upon the conditions under which the measurements have been made. These

corrections are discussed below

Correction for Absolute Length Due to manufacturing defects the absolute length of the

tape may be different from its designated or nominal length. Also with use the tape may

stretch causing change in the length and it is imperative that the tape is regularly checked

under standard conditions to determine its absolute length. The correction for absolute

length or standardization is given by

Where

c = the correction per tape length,

l = the designated or nominal length of the tape, and

L= the measured length of the line.

If the absolute length is more than the nominal length the sign of the correction is positive

and vice versa

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Example. A distance of 220.450 m was measured with a steel band of nominal length 30 m.

On standardization the tape was found to be 30.003 m. Calculate the correct measured

distance, assuming the error is evenly distributed throughout the tape.

Solution:

Correction for Temperature If the tape is used at a field temperature different from the

standardization temperature then the temperature correction to the measured length is

where

α = the coefficient of thermal expansion of the tape material

tm = the mean field temperature, and

t0 = the standardization temperature.

The sign of the correction takes the sign of (tm- t0 ).

Correction for Pull or Tension If the pull applied to the tape in the field is different from the

standardization pull, the pull correction is to be applied to the measured length. This

correction is

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where

P = the pull applied during the measurement,

P0 = the standardization pull,

A = the area of cross-section of the tape, and

E = the Young‘s modulus for the tape material.

The sign of the correction is same as that of (P – P0).

Correction for Sag

For very accurate measurements the tape can be allowed to hang in catenary between two

supports (Fig. 2.1a). In the case of long tape, intermediate supports as shown in Fig. 2.1b,

can be used to reduce the magnitude of correction.

The tape hanging between two supports, free of ground, sags under its own weight, with

maximum dip occurring at the middle of the tape. This necessitates a correction for sag if

the tape has been standardized on the flat, to reduce the curved length to the chord length.

The correction for the sag is

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Where

W = the weight of the tape per span length.

The sign of this correction is always negative.

If both the ends of the tape are not at the same level, a further correction due to slope is

required. It is given by

Where

α = the angle of slope between the end supports

Correction for Slope If the length L is measured on the slope as shown in Fig. 2.2, it must be

reduced to its horizontal equivalent L cos θ. The required slope correction is

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Where

θ = the angle of the slope, and

h = the difference in elevation of the ends of the tape.

The sign of this correction is always negative

Correction for Alignment If the intermediate points are not in correct alignment with ends

of the line, a correction for alignment given below, is applied to the measured length (Fig.

2.3).

Where

d = the distance by which the other end of the tape is out of alignment.

The correction for alignment is always negative

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Example 2.2. A line AB between the stations A and B was measured as 348.28 using a 20 m

tape, too short by 0.05 m. Determine the correct length of AB, the reduced horizontal

length of AB if AB lay on a slope of 1 in 25, and the reading required to produce a horizontal

distance of 22.86 m between two pegs, one being 0.56 m above the other. Solution:

(a) Since the tape is too short by 0.05 m, actual length of AB will be less than the measured

length. The correction required to the measured length is

It is given that

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Example 2.3. A tape of standard length 20 m at 85°F was used to measure a base line. The

measured distance was 882.50 m. The following being the slopes for the various segments

of the line:

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Calculate the true length of the line if the mean temperature during measurement was 63°F

and the coefficient of thermal expansion of the tape material is 6.5 × 10–6 per °F

Example 2.4. A base line was measured by tape suspended in catenary under a pull of 145

N, the mean temperature being 14°C. The lengths of various segments of the tape and the

difference in level of the two ends of a segment are given in Table

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If the tape was standardized on the flat under a pull of 95 N at 18°C determine the correct

length of the line. Take

Cross-sectional area of the tape = 3.35 mm2

Mass of the tape = 0.025 kg/m

Coefficient of linear expansion = 0.9 × 10–6 per °C

Young‘s modulus = 14.8 × 104 MN/m2

Solution:

It is given that

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Correct length = 119.631 – 0.0056 = 119.6254 m

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Example 2-5

A 30-m steel tape standardized at 20°C and supported throughout under a tension of 5.45

kg was found to be 30.012 m long. The tape had a cross-sectional area of 0.050 cm2 and a

weight of 0.03967 kg/m. This tape was held horizontal, supported at the ends only, with a

constant tension of 9.09 kg, to measure a line from A to B in three segments. The data

listed in the following table were recorded. Apply corrections for tape length, temperature,

pull, and sag to determine the correct length of the line.

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(e) Finally, corrected distance AB is obtained by adding all corrections to the measured

distance, or

AB = 81.151 + 0.0324 - 0.0048 + 0.0030 - 0.0504 = 81.131 m

Home work

1) A 100-ft steel tape standardized at 68°F and supported throughout under a tension of 20

lb was found to be 100.012 ft long. The tape had a cross-sectional area of0.0078 in.2 and a

weight of 0.0266 lb/ft. This tape is used to lay off a horizontal distance CD of exactly 175.00

ft. The ground is on a smooth 3% grade, thus the tape will be used fully supported.

Determine the correct slope distance to layoff if a pull of 15 lb is used and the temperature

is 87°F.

2) A tape of 30 m length suspended in catenary measured the length of a base line. After

applying all corrections the deduced length of the base line was 1462.36 m. Later on it was

found that the actual pull applied was 155 N and not the 165 N as recorded in the field

book. Correct the deduced length for the incorrect pull. The tape was standardized on the

flat under a pull of 85 N having a mass of 0.024 kg/m and cross-sectional area of 4.12 mm2.

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The Young‘s modulus of the tape material is 152000 MN/ m2 and the acceleration due to

gravity is 9.806 m/s2.

Content/Topic 2 - Identification Of Systematic And Random Error In Differencial Levelling

2.3.1 Systematic errors in differential levelling

1. Collimation error: occurs when the line of sight of an instrument is not truly horizontal and is

minimized by keeping sight distances short and balanced.

For an individual setup, the resulting error in an elevation difference due to collimation is:

For all setups, the formula becomes:

By combining all back sights together and foresights together the formula becomes:

The collimation error determined from equation above is treated as a correction and thus

subtracted from the observed elevation difference to obtain the corrected value.

Example1: a level that has a collimation error of 0.04mm/m is used on level line where the back

sight distances sum to 863m and foresight distance sum to 932m. If the observed elevation

distance for the line 22.865m what is the corrected elevation difference?

Solution:

ec=0.00004(863-932)= -0.0028m

Thus the corrected elevation difference is 22.865m-(-0.0028m) = 22.868m

2. Earth curvature and refraction

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As the line of sight extends from an instrument, the level surface curves down and away. This

condition always causes rod readings to bee to high. Curvature effect.

As the line of sight extends from instrument. Refraction bends it toward the Earth’s surface,

causing readings to be too low. Refraction effect.

The combined effect of earth curvature and refraction on an individual sight always causes a

rod reading to too high by an amount approximated as: Where hCR is the

error in reading (in feet or meters), CR is 0.0675 when D is in units of meters or 0.0206 when D

is in units of feet, and D is the individual sight distance.

The effect of this error on a single elevation difference is minimized by keeping backsight and

foresight distances short and equal.

For the unequal sight distances, the resulting error is expressed as:

Where eCR is the error due to curvature and refraction and D1 and D2 are the individual back

sight and foresight distances that occur in a line of levels.

The equation above is the correction and is subtracted from the observed elevation difference to get the

corrected value. The sum of the backsight and foresight

distances of the whole level line will yield incorrect results, these distances should be squared before

they are summed.

Example2: An elevation difference between two stations on a hillside is determined to be 1.256m.

What would be the error in the elevation difference and the corrected elevation difference if the back

sight distance were 100m and the foresight distance 20m only?

Solution: substituting the distance into equation above and using CR=0.0675 gives us

From this the corrected elevation difference is h= 1.256m-0.0006m=1.255m

For the line of differential leveling, the combined effect of this error is:

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Regrouping back sight and fore sight distances:

3. Combined effects of systematic errors on elevation differences for one instrument set up, a

corrected elevation difference h:

Where r1 is the back sight rod reading, r2 is the foresight reading, others terms are defined as equation

(3):

For a line of levels, the corrected elevation differences h become:

rBs

and rFs are respectively the sum of all back sight and fore sight, D2 FS are respectively the sum of

squares of back sights and foresights of whole leveling.

Content/Topic 2 – Random errors in differencial leveling

Reading rod errors

Rod reading errors for any individual sight distance D is: r= Dr/D

Where r/D is the estimated error in the rod reading per unit length of sight distance and D is the length of

sight distance.

Instrument-leveling errors

The estimated error in leveling for an automatic compensator or level vial is generally given in the technical

data for each instrument.

For precise levels, this information is listed in arc second or an estimated elevation error for a given

distance: for example: the estimated error of 1.5mm/km correspond to: 1.5/1,000,000*p=0.3”. This

error ranges between 0.1”and 0.2” for a precise level and 10” for a less precise level. N.B:

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P=206264.8”/radian which comes from the fact that 206264.8”=1radian

Rod plumbing errors

While a level rod that is held nonvertical always causes the reading to be high, this error will appear

random in a leveling network due to its presence in all back sight and foresight distances of the network.

While a level rod that is held vertical always causes the reading to be too high, this error will appear

random in leveling network, due to its presence in all back sight and foresight distances of the network.

Where D is the linear amount that the rod is out of plumb at the location of the rod reading “r”. ”d”

depends on the rod level bubble centering error and the reading location.

If the rod bubble is out of level by , d is:

D=r sin

Subtitute (10) into (9):

Having difference in elevation the formula (11) becomes:

Example 3: assume that a rod level bubble is within 5’ of level and the rod reading is at 4m. What is

the estimated error in the rod reading?

Solution:

From the equation (11) we get:

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Learning Unit.3. Perform Accuracy and Precision of Measurements

LO 3.1. Adequate Determination Of Indices Of Precision

Content /Topic3 - Determination of Indices Of Precision

Measures of central tendency are computed statistical quantities that give an indication of the value within

a data set that tends to exist at the center. The arithmetic mean, the median, and the mode are three such

measures. They are described as follows:

ARITHMETIC MEAN. For a set of n observations, y1, y2,...,yn, this is the average of the observations. Its

value, is computed from the equation:

(2.1)

Median.

As mentioned previously, this is the midpoint of a sample set when arranged in ascending or descending

order. One-half of the data are above the median and one-half are below it. When there are an odd

number of quantities, only one such value satisfies this condition. For a data set with an even number of

quantities, the average of the two observations that straddle the midpoint is used to represent the

median.

Mode.

Within a sample of data, the mode is the most frequently occurring value. It is seldom used in surveying

because of the relatively small number of values observed in a typical set of observations. In small sample

sets, several different values may occur with the same frequency, and hence the mode can be meaningless

as a measure of central tendency.

True Value, µ:

A quantity’s theoretically correct or exact value

Error, ε:

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The difference between a measured quantity and its true value. The true value is simply the population’s

arithmetic mean. Since the true value of a measured quantity is indeterminate, errors are also

indeterminate and are therefore only theoretical quantities.

repeated for convenience here, errors are expressed as:

(2.2)

Where Yi is the individual observation associated with εi and is the true value for that quantity.

True value, µ: a quantity’s theoretically correct or exact value. As noted in Section 1.3, the true value can

never be determined.

Most Probable Value

Most probable value, that value for a measured quantity which, based y: on the observations, has the

highest probability of occurrence. It is derived from a sample set of data rather than the population and is

simply the mean if the repeated measurements have the same precision

Residual, :

The difference between any individual measured quantity and the most probable value for that quantity.

Residuals are the values that are used in adjustment computations since most probable values can be

determined. The term error is frequently used when residual is meant, and although they are very similar

and behave in the same manner, there is this theoretical distinction. The mathematical expression for a

residual is

(2.3)

Where vi is the residual in the ith observation, yi, and is the most y probable value for the unknown

Variance, : a value by which the precision for a set of data is given. Population variance applies to a

data set consisting of an entire population. It is the mean of the squares of the errors and is given by

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(2.4)

Sample variance applies to a sample set of data. It is an unbiased estimate for the population variance

given in Equation above and is calculated as

(2.5)

Standard error, :

the square root of the population variance. From Equation (2.4) and this definition, the following equation

is written for the standard error:

(2.6)

Standard Deviation, S: the square root of the sample variance. It is calculated using the expression

(2.7)

where S is the standard deviation, n -1 the degrees of freedom or number of redundancies, and

the sum of the squares of residuals. Standard deviation is an estimate for the

standard error of the population. Since the standard error cannot be determined, the standard deviation is

a practical expression for the precision of a sample set of data. Residuals are used rather than errors

because they can be calculated from most probable values, whereas errors cannot be determined. Again,

as discussed in Section 3.5, for a sample data set, 68.3% of the observations will theoretically lie between

the most probable value plus and minus the standard deviation, S. The meaning of this statement will be

clarified in an example that follows.

Standard Deviation Of The Mean: the error in the mean computed from a sample set of measured values

that results because all measured values contain errors. The standard deviation of the mean is computed

from the sample standard deviation according to the equation

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Notice that as n → ∞, then → 0. This illustrates that as the size of Sy the sample set approaches the total

population, the computed mean will approach the true mean µ

Alternative Formula For Determining Variance

From the definition of residuals Equation is rewritten as

©

Substituting Equation (2.1) for into Equation (c) and dropping the bounds y for the summation yields

(D)

Expanding Equation (d) gives us

Rearranging Equation (e) and recognizing that (∑ yi /n)2 occurs n times in Equation (e) yields

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Adding the summation symbol to Equation (ƒ) yields

Factoring and regrouping similar summations in Equation (g) produces

Multiplying the last term in Equation (h) by n/n yields

Finally, by substituting Equation (2.1) in Equation (i), the following expression for the variance results:

Using Equation (2.10), the variance of a sample data set can be computed by subtracting n times the

square of the data’s mean from the summation of the squared individual observations. With this equation,

the variance and the standard deviation can be computed directly from the data. However, it should be

stated that with large numerical values, Equation (2.10) may overwhelm a handheld calculator or a

computer working in single precision. If this problem should arise, the data should be centered or Equation

(2.5) used. Centering a data set involves subtracting a constant value (usually, the arithmetic mean) from

all values in a data set. By doing this, the values are modified to a smaller, more manageable size.

Example 2.1 Using the data from Table 2.2, determine the sample set’s mean, median, and mode and the

standard deviation using both Equations (2.7) and (2.10). Also plot its histogram. (Recall that the data of

Table 2.2 result from the seconds’ portion of 50 theodolite directions.)

SOLUTION Mean: From Equation (2.1) and using the ∑yi value from Table 2.4, we have

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Median: Since there is an even number of observations, the data’s midpoint lies between the values that

are the 25th and 26th numerically from the beginning of the ordered set. These values are 23.4 and 23.5”,

respectively. Averaging these observations yields 23.45”

Mode: The mode, which is the most frequently occurring value, is 23.8”. It appears three times in the

sample. Range, class width, histogram: These data were developed in Section 2.4, with the histogram

plotted in Figure 2.1. Standard deviation. The standard deviation for the sample set is computed as

3.1.5. CONFIDENCE LIMITS

After establishing the sample mean as estimate of the true value of the quantity, the range of

values within which the true value should lie for a given probability is required. This range is called

the confidence interval, its bounds called the confidence limits. Confidence limits can be established

for that stated probability from the standard deviation for a set of observations. Statistical tables

are available for this purpose. A figure of 95% frequently chosen implies that nineteen times out of

twenty the true value will lie within the computed limits. The presence of a very large error in a set

of normally distributed errors, suggests an occurance to the contrary and such an observation can

be rejected if the residual error is larger than three times the standard deviation.

3.1. 6. WEIGHT

This quantity is known as weight of the measurement indicates the reliability of a quantity. It is

inversely proportional to the variance (2) of the observation, and can be expressed as

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Where k is a constant of proportionality. If the weights and the standard errors for observations x1,

x2, ,….., etc., are respectively 1 , 2 ω ,….., etc., and 1 , 2 ,….., etc., and u is the standard error

for the observation having unit weight then we have

The weights are applied to the individual measurements of unequal reliability to reduce them to one

standard. The most probable value is then the weighted mean xm of the measurements. Thus

and standard error of the wieghted mean

The standard deviation of an observation of unit weight is given by

and the standard deviation of an observation of weight n is given by

3.1.7. PROPAGATION OF ERROR

The calculation of quantities such as areas, volumes, difference in height, horizontal distance, etc., using

the measured quantities distances and angles, is done through mathematical relationships between the

computed quantities and the measured quantities. Since the measured quantities have errors, it is

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inevitable that the quantities computed from them will not have errors. Evaluation of the errors in the

computed quantities as the function of errors in the measurements is called error propagation.

3.1.8. Normal Distribution

Measured quantities in surveying (angles, distances,…) and in many surveying domain are random

variables that obey normal distribution. The corresponding probability density is:

being the true value of the measured quantity, the error of measurements equals:

=x-

In that manner, the probability density can be written as:

This is the probability density to make an error of measurements.

To that of making an error comprises between and 1

Particular case: P (-

Can be written as = i.e the probability to make an error of equals 0.68 or68%.

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Example 2.1:

Suppose that a line has been observed 10 times using the same equipment and procedures. The results are

shown in column (1) of the following table. It is assumed that no mistakes exist and that the observations

have been corrected for all systematic errors. Compute the most probable value for the line length, its

standard deviation and errors having 50% ,90% and 95% probability.

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Example 2.2

To clarify definitions and use the equations given in Sections see above, suppose that a line has been

observed 10 times using the same equipment and procedures. The results are shown in column (1) of the

following table. It is assumed that no mistakes exist, and that the observations have already been

corrected for all systematic errors. Compute the most probable value for the line length, its standard

deviation, and errors having 50%, 90%, and 95% probability

By Equation

the residuals are calculated. These are tabulated in column (2) and their squares listed in column (3). Note

that in column (2) the algebraic sum of residuals is zero. (For observations of equal reliability, except for

round off, this column should always total zero and thus provide a computational check.)

Example 2.3. The following are the observations made on the same angle:

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Determine

(a) the most probable value of the angle,

(b) The range,

(c) The standard deviation,

(d) The standard error of the mean, and

(e) The 95% confidence limits.

Solution:

For convenience in calculation of the required quantities let us tabulate the data as in Table 2.3.

The total number of observations n = 10.

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Table 2.3

It is a common practice in surveying to reject any observation that differs from the most probable value by

more than three times the standard deviation.

LO 3.2. Apply Least Squares Adjustment in Survey Measurement

● Content/Topic 1 –Application Of Least Squares adjustment In Surveying Measurement

In surveying (geomatics), after eliminating mistakes and making corrections for systematic errors, the

presence of the remaining random errors will be evident in the form of misclosures. Examples include

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sums of interior angles in closed polygons misclosures in closed leveling circuits, and traverse misclosures

in that do not total (n-2)1800, departures and latitudes. To account for these misclosures, adjustments are

applied to produce mathematically perfect geometric conditions. Although various techniques are used,

the most rigorous adjustments are made by the method of least squares, which is based on the laws of

probability. Now least-squares calculations are handled routinely and making adjustments by this method

is rapidly becoming indispensable in modern surveying (geomatics). The method of least squares is

currently being used to adjust all kinds of observations, including differences in elevation, horizontal

distances, and horizontal and vertical angles. It has become essential in the adjustment of GNSS

observations and is also widely used in adjusting photogrammetric data. Adjustments by the least-squares

method have taken on added importance with the most recent surveying accuracy standards. These

standards include the use of statistical quantities that result from least squares adjustment. Thus in order

to evaluate a survey for compliance with the standards, least-squares adjustments must first be

performed.

3.2.1 ADVANTAGES OF LEAST SQUARES

Least-squares adjustments provide several advantages over other arbitrary methods.

1. First of all, because the method is based upon the mathematical theory of probability, it is the

most rigorous of adjustment procedures.

2. It enables all observations to be simultaneously included in an adjustment, and each observation

can be weighted according to its estimated precision.

3. Furthermore, the least squares method is applicable to any observational problem regardless of its

nature or geometric configuration.

4. In addition to these advantages, the least-squares method enables rigorous statistical analyses to

be made of the results of the adjustment, that is, the precisions of all adjusted quantities can be

estimated, and other factors investigated.

5. The least-squares method even enables presurvey planning to be done so as to ensure that

required precisions of adjusted quantities are obtained in the most economical manner.

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3.2.2. FUNDAMENTAL CONDITION OF LEAST SQUARES

The method of least-squares adjustment is derived from the equation for the normal distribution curve. It

produces that unique set of residuals for a group of observations that have the highest probability of

occurrence.

For a group of equally weighted observations, the fundamental condition enforced by the least-squares

method is that the sum of the squares of the residuals is a minimum. Suppose a group of m observations of

equal weight were taken having residuals v1, v2, v3, v4… vm. Then, in equation form, the fundamental

condition of least squares is

For any group of observed values, weights may be assigned to individual observations according to a priori

(before the adjustment) estimates of their relative worth or they may be obtained from the standard

deviations of the observations if available. An equation expressing the relationship between standard

deviations and weights, is

In Equation (16.2), wi is the weight of the ith observed quantity and the variance of that observation.

This equation states that weights are inversely proportional to variances. If observed values are to be

weighted in least-squares adjustment, then the fundamental condition to be enforced is that the sum of

the weights times their corresponding squared residuals is minimized or, in equation form

Some basic assumptions underlying least-squares theory are that (1) mistakes and systematic errors have

been eliminated, so only random errors remain in the set of observations; (2) the number of observations

being adjusted is large; and (3) as stated earlier, the frequency distribution of the errors is normal.

Although these basic assumptions are not always met, least-squares adjustments still provide the most

rigorous error treatment available.

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3.2.3. LEAST-SQUARES ADJUSTMENT BY THE OBSERVATION EQUATION METHOD

Two basic methods are employed in least-squares adjustments: (1) the observation equation method and

(2) the condition equation method. The former is most common and is the one discussed herein. In this

method, “observation equations” are written relating observed values to their residual errors and the

unknown parameters. One observation equation is written for each observation. For a unique solution, the

number of equations must equal the number of unknowns. If redundant observations are made, the least-

squares method can be applied. In that case, an expression for each residual error is obtained from every

observation equation. The residuals are squared and added to obtain the function expressed in either

Equation above

To minimize the function those equations, partial derivatives of the expression are taken with respect to

each unknown variable and set equal to zero. This yields a set of so-called normal equations, which are

equal in number to the number of unknowns. The normal equations are solved to obtain most probable

values for the unknowns. The following elementary examples illustrate the procedures.

Example

Using least squares, compute the most probable value for the equally weighted distance observations of

Example 3.1.

Solution

1. For this problem, as was done in Example 3.1, let be the most probable value of the observed

length. Then write the following observation equations that define the residual for any observed

quantity as the difference between the most probable value and any individual observation:

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Note that this answer agrees with the one given for Example for MPV. Note also that this procedure

verifies the statement given above that the most probable value for an unknown quantity, measured

repeatedly using the same equipment and procedures, is simply the mean of the observations.

In Figure below, the three horizontal angles observed around the horizon are x = 42°12’13”, y = 59°56’15”,

z = 257°51’35”. Adjust these angles by the least-squares method so that their sum equals the required

geometric total of 360º..

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Solution

1. Form the observation equations

2. Write an expression that enforces the condition that the sum of the three adjusted angles total 360°.

x + y + z = 360° (d)

3. Substitute Equations (a), (b), and (c) into Equation (d), and solve for v3

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Example 16.3

Adjust the three equally weighted distance observations taken (in feet) between points A, B, and C of

Figure below.

Solution

1. Let the unknown distances AB and BC be x and y, respectively. These two unknowns are related through

the observations as follows:

x + y = 393.65

x = 190.40

y = 203.16

2. Values for x and y could be obtained from any two of these equations so that the remaining equation is

redundant. However, notice that values

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2.5.4 MATRIX METHODS IN LEAST-SQUARES ADJUSTMENT

It has been noted that least-squares computations are quite lengthy, and therefore generally performed on

a computer. Their solution follows a systematic procedure hat is conveniently adapted to matrix methods.

In general, any group of observation equations may be represented in matrix form as

Where A is the matrix of coefficients for the unknowns, X the matrix of unknowns, L the matrix of

observations, and V the matrix of residuals. The detailed structures of these matrices are

The normal equations that result from a set of equally weighted observation equations are given in matrix

form by

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In Equations AT A is the matrix of normal equation coefficients for the unknowns. Premultiplying both sides

of Equation (AT A)-1 by and reducing yields

Equation (16.6) is the least-squares solution for equally weighted observations. The matrix X consists of

most probable values for unknowns for a system of weighted observations, the

following equation provides the X matrix:

In Equation (16.7) the matrices are identical to those of the equally weighted case, except that W is a

diagonal matrix of weights defined as follows

Solve Example 16.3 using matrix methods.

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Solution

LO 3.3 . Adjust Errors in Traverse Surveys Network

Content/Topic 1- Introduction Traverse Adjustment Method

A traverse is a series of consecutive lines whose ends have been marked in the field and whose lengths and

directions have been determined from observations.

In traditional surveying by ground methods, traversing, the act of marking the lines, that is, establishing

traverse stations and making the necessary observations is one of the most basic and widely practiced

means of determining the relative locations of points.

There are two kinds of traverses: closed and open.

Two categories of closed traverses exist: polygon and link. In the polygon traverse, the lines return to the

starting point, thus forming a closed figure that is both geometrically and mathematically closed. Link

traverses finish upon another station that should have a positional accuracy equal to or greater than that

of the starting point. The link type (geometrically open, mathematically closed), as illustrated in must have

a closing reference direction, for example, line Closed traverses provide checks on the observed angles and

distances, which is an extremely important consideration. They are used extensively in control,

construction, property, and topographic surveys.

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1.1. ANGLE MISCLOSURE

The angular misclosure for an interior-angle traverse is the difference between the sum of the observed

angles and the geometrically correct total for the polygon. The sum, of the interior angles of a closed

polygon should be

Where n is the number of sides, or angles, in the polygon. This formula is easily derived from known facts.

The sum of the angles in a triangle is 180°; in a rectangle, 360°; and in a pentagon, 540°.Thus, each side

added to the three required for a triangle increases the sum of the angles by 180°, if the direction about a

traverse is clockwise when observing angles to the right, exterior angles will be observed. In this case, the

sum of the exterior angles will be

Example 2.1

For the traverse of Figure below, the observed interior angles are given in Table. Compute the adjusted

angles.

Solution

The computations are best arranged as shown in Table. The first part of the adjustment consists of

summing the interior angles and determining the misclosure according to Equation above, which in this

instance, as shown beneath column 2, is +11”. The remaining calculations are tabulated, and the rationale

for the procedures follows.

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ADJUSTEMENT OF ANGLES

For work of ordinary precision, it is reasonable to adopt corrections that are even multiples of the smallest

recorded digit or decimal place for the angle readings. Thus in this example, corrections to the nearest 1”

will be made.

The method above consists of subtracting 11”/5=2.2” from each of the five angles. However, since the

angles were read in multiples of 1”, applying corrections to the nearest tenth of a second would give a false

impression of their precision. Therefore it is desirable to establish a pattern of corrections to the nearest

1”, as shown in Table 10.1. First multiples of the average correction of 2.2” is tabulated in column (3). In

column (4), each of these multiples has been rounded off to the nearest 1”. As shown in Table above. First

multiples of the average correction of 2.2”is tabulated in column (3). In column (4), each of these multiples

has been rounded off to the nearest 11”. The adjusted interior angles obtained by applying these

corrections are listed in column (6). As another check, they must total exactly the true geometric value of

(2n-4)1800, or 540000’00”in this case.

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1.2 COMPUTATION OF PRELIMINARY AZIMUTHS OR BEARINGS

If a line of known direction exists within the traverse, computation of preliminary azimuths (or bearings)

proceeds as discussed in Chapter 7. Angles adjusted to the

proper geometric total must be used; otherwise the azimuth or bearing of the first line, when recomputed

after using all angles and progressing around the traverse, will differ from its fixed value by the angular

misclosure.

Azimuths or bearings at this stage are called “preliminary” because they will change after the traverse is

adjusted, as explained in Section 10.11. It should also be noted that since the azimuth of the courses will

change, so will the angles, which were previously adjusted. Briefly by the equation

Example 2.2

Compute preliminary azimuths for the traverse courses of Figure above, based on a fixed azimuth of for

line AW, a measured angle to the right of 234°17’18”

51°52’24”for WAE, and the angle adjustment by method 1 of Table above

1.3 COMPUTATION OF PRERIMINARY AZIMUTH

1.4 DEPARTURES AND LATITUDES

After balancing the angles and calculating preliminary azimuths (or bearings), traverse closure is checked

by computing the departure and latitude of each line. As illustrated in Figure 10.3, the departure of a

course is its orthographic projection on the east-west axis of the survey and is equal to the length of the

course

The forward bearing= Back bearing of previous + angle between the line

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multiplied by the sine of its azimuth (or bearing) angle. Departures are sometimes called easting or

westings. Also as shown in Figure 10.3, the latitude of a course is its orthographic projection on the north-

south axis of the survey, and is equal to the course length multiplied by the cosine of its azimuth (or

bearing) angle. Latitude is also called northing or southing. In equation form, the departure and latitude of

a line are

where L is the horizontal length and the azimuth of the course. Departures and latitudes are merely

changes in the X and Y components of a line in a rectangular grid system, sometimes referred to as ΔX and

ΔY In traverse calculations, east departures and north latitudes are considered plus; west departures and

south latitudes, minus. Azimuths (from north) used in computing departures and latitudes range from 0 to

360°, and the algebraic signs of sine and cosine functions automatically produce the proper algebraic signs

of the departures and latitudes.

1.5 TRAVERSE LINEAR MISCLOSURE AND RELATIVE PRECISION

Because of errors in the observed traverse angles and distances, if one were to begin at point A of a closed-

polygon traverse like that of Figure 10.1, and progressively follow each course for its observed distance

along its preliminary bearing or azimuth, one would finally return not to point A, but to some other nearby

point A’. Point A ‘would be removed from A in an east-west direction by the departure

misclosure, and in a north-south direction by the latitude misclosure.The distance between A and A’ is

termed the linear misclosure of the traverse. It is calculated from the following formula:

The relative precision of a traverse is expressed by a fraction that has the linear misclosure as its numerator

and the traverse perimeter or total length as its denominator, or

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The fraction that results from Equation (10.4) is then reduced to reciprocal form, and the denominator

rounded to the same number of significant figures as the numerator.This is illustrated in the following

example.

Based on the preliminary azimuths from Table 2.2 and lengths shown in Figure 2.1, calculate the

departures and latitudes, linear misclosure, and relative precision of the traverse.

Solution

In computing departures and latitudes, the data and results are usually listed in a standard tabular form,

such as that shown in Table above.The column headings and rulings save time and simplify checking.

In Table above, taking the algebraic sum of east (+) and west (-) departures gives the misclosure, 0.026 ft.

Also, summing north ( +) and south (-) latitudes gives the misclosure in latitude, 0.077 ft. Linear misclosure

is the hypotenuse of a small triangle with sides of 0.026 ft and 0.077 ft, and in this example its value is, by

following equation

For any closed traverse, the linear misclosure must be adjusted (or distributed) throughout the traverse to

“close” or “balance” the figure. This is true even though the misclosure is negligible in plotting the traverse

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at map scale. There are several elementary methods available for traverse adjustment, but the one most

commonly used is the compass rule (Bowditch method). As noted earlier, adjustment by least squares is a

more advanced technique that can also be used. These two methods are discussed in the subsections that

follow.

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REFERENCES

1. W. Schofield, M. Breach, Engineering surveying 6th edition, Kingston University,2007

2. CHARLES D. GHILANI, PAUL R. WOLF, Elementary surveying 13th edition, Pennsylvania State 2012

3. Basic surveying -theory and practice, Oregon Department of Transportation Geometronics Unit,

Bend, Oregon 2000

4. Kanetkar, T. P., and Kulkarni, S. V. 1981. Surveying and Leveling, Vol I. Pune Vidyarthi Griha Praksam

, Pune.

5. Murthy, V. V. N. 1982. Land and Water Management Engineering. Kalyani publishers, New Delhi.

6. Michael, A. M. 1989. Irrigation Theory and Practice. Vikas Publishing House Pvt. Ltd, New Delhi.

7. Michael, A. M., and Ojha, T. P. 1993. Principles of Agricultural Engineering – Vol. II. Jain Brothers,

New Delhi.

8. Mal, B. C. 2005. Introduction to Soil and Water Conservation Engineering. Kalyani publishers, New

Delhi.

9. Dr A M Chandra. 2005. Problem solving with theory and objective type question New Delhi'

Bangalore ' Chennai

credits: 10 learning hours: 100 sector: construction and

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