CIEN 30033
ENGR. EDNA P. ARROJADO
ENGR. JOHN PATRICK B. CID
ENGR. KENNETH BRYAN M. TANA
ENGR. MARC ERICK VON A. TIOSING
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FUNDAMENTALS OF SURVEYING 2
The Overview:
This course focuses to provide students with general introduction to omitted
measurements of a closed traverse, subdivision of lots, principle of the stadia, contouring,
hydrographic surveys with area and volume computations, and astronomical observation.
Module 1: Omitted Measurements
Learning Objectives:
At the end of this lesson, the learner will be able to:
•
•
•
Recall the parameters and characteristics of a Closed Traverse
Comprehend and compute for the omitted information of a closed traverse.
Embody the principles in computing omitted measurements in any given case.
Course Materials:
Introduction:
A traverse is a form of control survey used in a wide variety of engineering and property surveys.
Essentially, traverses are a series of established stations tied together by angle and distance.
Traverses can be an Open Traverse or Closed Traverse. In this chapter we will focus more on
closed traverse.
Discussion:
A closed traverse is one that either begins and ends at the same point or begins and ends at
points whose positions have been previously determined. Used often in computing land areas.
The principle of a closed traverse is that the sum of latitude must be equal to zero as well as the
sum of the departures. The traverse shall be adjusted when this condition isn't satisfied.
Recall that the important components of a traverse computations are Latitude and Departure.
The latitude of a line is its projection onto the reference meridian or a north-south line. Latitude
are sometimes referred to as northings or southings. Latitude of lines with northerly bearings are
designated as being north (N) or positive (+); those in a southerly direction are designated as
south (S) or negative (-). On the other hand, the departure of a line is its projection onto the
reference parallel or an east-west line. Departures are east (E) or positive (+) for lines having
easterly bearings and west (W) or negative (-) for lines having westerly bearings.
In a closed traverse if lengths and bearings of all lines could not be measured due to certain
reasons, the techniques developed in the computation of latitudes and departures can be used to
solve for missing course information. The omitted or the missing measurements can be computed
provided the number of missing quantities shall not exceed two in number.
There are several other reasons why measurements are omitted in the field and are computed
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later in the office. Conditions or problems encountered may also be due to the presence of
obstacles, rugged terrain, lack of time, unfriendly or hostile landowners, etc. The following are
some of the common types of omitted measurements:
1. Length and Bearing of One Side is unknown.
2. Missing Data on two adjoining sides.
3. Missing Data on two non-adjoining sides.
The main technique in solving these types of omitted measurements is simply understanding the
principle of the closed traverse and the closing line.
Closing line is the side of the traverse that closes a loop of traverse which directly implies that
there's no error of closure (known sides are presumed to be correct without error). With that given,
we know that in a given closed traverse, ∑ 𝑙𝑎𝑡𝑖𝑡𝑢𝑑𝑒 = 0 & ∑ 𝑑𝑒𝑝𝑎𝑟𝑡𝑢𝑟𝑒 = 0, therefore, if we
assume that the omitted measurement in a closed traverse is the closing line itself. We can get
the algebraic sum of Latitude and Departures of known sides of the traverse to be the value
of latitude and departure of the closing line but shall be in opposite sign to satisfy the principle of
closed traverse.
If the algebraic sum of the latitudes and the algebraic sum of departures of the known sides are
designated by ‘CL’ and ‘CD’, respectively, then the length ‘L’ of the unknown side (assumed as
closing line) is:
𝑳 = √ 𝑪𝑫 𝟐 + 𝑪𝑳 𝟐
and the tangent of the bearing angle (𝛼), which is taken with due regard to sign, is:
𝐭𝐚𝐧(𝜶) =
−𝑪𝑫
−𝑪𝑳
For a closed traverse with omitted measurements on two adjoining sides, establishment of closing
line using the known sides of the traverse shall be executed. Thereafter, the omitted
measurements shall be obtained using basic principles of Trigonometry and Geometry.
With an added graphical solution, the determination of missing data as explained above are also
applicable even though the sides with two unknown quantities are non-adjoining; for different
cases of omitted measurements involving non-adjoining sides, the following principles are
adapted:
1st Principle: A line may be moved from one location to a second location parallel with
the first, and its latitude and departure will remain unchanged.
2nd Principle: The algebraic sum of the latitudes and the algebraic sum of the departures
of any system of lines forming a closed figure must be zero, regardless of the order in
which the lines are placed.
A solution by simultaneous equation will give the missing parts. In practice, however, this method
is seldom applied since geometric solutions are preferred by most engineers and surveyors.
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Activities/Assessment:
1. A traverse forming a close loop characterizes the lot of Mr. Romulo. Survey data are tabulated
below with a missing measurement due to a mad dog that has bitten a surveyor. Determine the
length and bearing of the missing line.
Solution. We assume line BC as closing line of the traverse A-B-C-D (lot of Mr. Romulo). Following
the principles of closed traverse, we need to satisfy the algebraic sum of Latitudes and Departures
must be zero; then, algebraic sum of Latitude and Departures of known sides of the traverse to
be the value of latitude and departure of the closing line but shall be in opposite sign.
Therefore, BC shall have a Latitude of -27.61 & Departure of 64.73. Length of course BC, 𝐿𝐵𝐶 =
64.73
√(−27.61)2 + 64.732 = 𝟕𝟎. 𝟑𝟕𝒎 , and Bearing BC, 𝛼 = tan−1 (−27.61) , 𝛼 = −𝟔𝟔. 𝟗°. Following the
signs of Lat and Dep of BC, Course BC is 70.37 m S 66°54’E.
2. From the given closed traverse shown below. Determine the bearing of course DE and EA.
COURSE
AB
BC
CD
DE
EA
BEARING
S 35°30’ W
N 57°15’ W
N 1°45’ E
?
?
LENGTH (m)
44.37
137.84
12.83
64.86
106.72
Solution. Let us establish first a closing line with an unknown length and bearing using readily
available or known lines (AB, BC & CD), say AD will be the closing line.
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COURSE
AB
BC
CD
AD
BEARING
S 35°30’ W
N 57°15’ W
N 1°45’ E
?
Tract A-B-C-D
LENGTH (m)
44.37
137.84
12.83
?
LAT
-36.122
74.568
12.824
?
DEP
-25.766
-115.929
0.392
?
Following the same manner of solving in problem number 1, we get the course AD as 150.317 m
S 70°3’26.8” W.
Realizing ∆𝐴𝐸𝐷 and by Cosine Law, we get ∠𝐸𝐴𝐷 = 21°51’22.03”, we know that Bearing of line
EA is computed as Bearing AD - ∠𝐸𝐴𝐷 (basing from North of Point A) hence, 70°3’26.8” 21°51’22.03” = 48°12’4.77” thus, Line EA is 106.72 m S 𝟒𝟖°𝟏𝟐’𝟒. 𝟕𝟕” E.
Therefore, bearing of line DE can be computed as ∠𝐸 – bearing EA. Using ∆𝐴𝐸𝐷, by sine law,
∠𝐸 = {59.63°, 120.37°}. As the drawing above suggests, we take ∠𝐸 = 120°22′ 49" and proceed
to computing Bearing DE; yielding to Line DE is 64.86 N 𝟕𝟐°𝟗’𝟓𝟖. 𝟕𝟐” E
Module 2: Subdivision (Land-Partitioning)
Learning Objectives:
At the end of this lesson, the learner will be able to:
•
•
•
Discuss the idea of land-partitioning.
Recall the trigonometric and geometric principles and formula in computing areas.
Define the dividing line and compute for its length and bearing for land partition in any
given case
Course Materials:
Introduction:
In this lesson and the succeeding lesson, four of the most common cases encountered in the
subdivision of land will be explained. These are: dividing an area into two parts by a line between
two points, dividing an area by a line running through a point in a given direction, to cut off a
required area by a line through a given point, and to cut off a required area by a line running in a
given direction.
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Discussion:
Dividing an Area into Two Parts by a Line between Two Points, in Fig. 2.1, ABCDEF represents
an irregular parcel of land to be divided into two parts (Tract ABCD and Tract DEFA) by a cut off
line extending from D to A.
It is assumed that the length and direction of each course has been earlier determined, the
latitudes and departures computed and adjusted, and the area of the whole tract computed.
The solution here is to determine the length and direction of the dividing line AD by trigonometric
and geometric computations then calculate the area of each parcel of land.
Fig 2.1 Tract A-B-C-D-E-F
Dividing an Area by a Line Running through a Point in a given Direction, in Fig. 2.2, ABCDEF
represents an irregular parcel of land to be subdivided into two parts by a cut off line (BP) running
in a given direction (indicated by Ɵ) which passes through point B. Out of the desired division,
tract BCDEP is formed on one side of the dividing line and tract FABD on the other side.
It is assumed that the length and direction of each course are known, the latitudes and departures
computed and adjusted, and the area of the whole tract computed. The solution will require the
calculation of the lengths BP and FP and the area of each of the two tracts.
Fig 2.2 Tract A-B-C-D-E-F with dividing lines
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The computations are further checked by determining if the algebraic sums of the latitudes and
of the departures of AB, BP, PF and FA are equal to zero.
Activities/Assessment:
1. A triangular lot has the following Azimuths and distances. Technical description of the lot is
tabulated below:
COURSE
1-2
2-3
3-1
AZIMUTH
180°
300°
40°
LENGTH (m)
?
?
960.22
The lot is to be divided such that the area of the southern portion would be 210,000 m 2. Compute
the position of the other end of the dividing line if the line starts at corner 3 of the lot. Express the
distance from corner 1. Determine the technical details of the said dividing line.
Solution:
We start of by plotting the tabulated technical description
of the lot. Assume length of the dividing line as Y, and the
distance of corner 1 to the dividing line along course 1-2
be X: let this intersection be point “4”
From the figure, we can get ∠3 by using intercepted ∠2
and azimuth of course 3-1. Such that, ∠3 = 180° − 40° −
60° = 80°. Let ∠134 be unknown angle ‘𝜃’.
Solving for X,Y & 𝜃.
1
Using Area of ∆134 = 210,000 m2 = 2 𝑋(960.22)sin (40°), X
= 680.47 m; by cosine law, we can get Y such that, 𝑌 2 =
𝑋 2 + 960.222 − 2𝑋(960.22)cos (40°), Y = 619.672 m.
𝑋
𝑌
For 𝜃, using sine law, sin (𝜃) = sin (40°), having X and Y
solved earlier. 𝜽 = 𝟒𝟒°𝟓𝟑′ 𝟕𝟕"
2. A parcel of land has technical description as tabulated below. The area of the lot is more or
less 1000 m2. Assume the lines forming the lot has negligible error of closure. If it is subdivided
into two portions such that the dividing line starts at the mid-point of Line 4-1 and must be parallel
to Line 1-2 of the boundary. Determine the length of the subdividing line, area of the eastern
subdivided lot and the distance of the nearest corner of subdividing line to corner 2.
COURSE
BEARING
LENGTH (m)
1-2
27.72
S 1°27’ E
2-3
38.21
S 88°57’ W
3-4
27.90
N 7° E
4-1
34.12
N 88°47’ E
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Solution. Technical Description was plotted and resulting intercepted angles on the corner points
were computed.
Another option in computing subdivision of lands is the mathematical method by trigonometric
formula (sine and cosine laws). Take a look at quadrilateral 1-2-5-6, it is the eastern lot subdivided
with respect to the given subdividing line.
Solving for unknown length 5-6, and 2-5. Draw line connecting corners 2 and 6, which divides the
shaded lot into two (2) triangles. Trial line 2-6 is chosen instead of line 1-5, to maximize the known
courses. Before we push through, we must get all the angles formed by these new boundary lines
forming the subdivided eastern area: ∠1 = 90°14′ , ∠2 = 89°36′ , ∠5 = 90°24′ & ∠6 = 89°46′.
Using ∆126, by cosine law, line 2-6 = 32.608 m. Then, angles ∠162 =
58°13′ 15.79, ∠126=∠265=31°32' 44.21", ∠625=58°3' 15.79 can be computed using known
trigonometric methods. Unknown lines 2-5 and 5-6 can be computed using sine law in ∆256. Line
2-5 = 17.06 meters, Line 5-6 (subdividing line) = 27.67 meters.
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By adding the areas of ∆126 + ∆256, (same procedure used in no.1 area of triangle), we get
total area of the eastern lot, 236.45 m2 + 236.02 m2 = 472.47 m2.
Module 3: Tachymetric Surveys
Learning Objectives:
At the end of this lesson, the learner will be able to:
•
•
•
•
•
Learn the definition and uses of tachymetric surveys.
Discuss the principle of the stadia method.
Familiarize with stadia hairs, stadia rods, stadia constants and stadia interval factor.
Discuss inclined stadia sights.
Determine stadia constant, stadia interval factor, horizontal and inclined measurements
for level and inclined line of sights respectively.
Course Materials:
Introduction:
In surveying, tachymetry is defined as a procedure of obtaining horizontal & vertical distances
and differences in elevation based on the optical geometry of the instrument employed. The word
tachymetry is derived from the Greek words “takhus metron” meaning ‘swift measurement’. It is a
branch of surveying where height and distances between ground marks are obtained by optical
means only. The distance between marks can be obtained without using a tape. The tachymeter
is any theodolite adapted, or fitted with an optical device to enable measurement to be made
optically. The Stadia System has various methods such as Fixed Hair Method and Movable Hair
Method, or Subtense Method. In this chapter, we will focus on the fixed hair method of stadia
measurements.
It is used to measure and produce topographic maps contain detailed information and contour
lines for the purpose of planning a construction project such as roads, buildings, manhole, culvert,
road, etc. It is also used in obtaining reduced levels between points on the surface of the earth.
Principle of Stadia Method:
As in the field of tacheometric surveying ‘Stadia Method’ is the most widely used procedure so
we will discuss the principle behind it. The stadia method follows the principle that in similar
isosceles triangles the ratio of the perpendicular to the base is constant. But before we go onto
detail in discussing the distance-elevation formula for Horizontal Sight in stadia method using the
principles of similar triangles in geometry. We must first be familiar with the components and
terminologies in this method.
The telescope of most surveying instruments are equipped with stadia hairs in addition to the
regular vertical and horizontal hairs (shown in the Fig. 3.1). Stadia hairs are equidistant from the
horizontal cross-hair and so positioned such that one is above and the other below. When stadia
hairs and the cross-hairs are simultaneously visible and in focus, they are called fixed stadia hairs.
In this type of arrangement, the stadia hairs are mounted on the same ring (or reticule) and in the
same plane as the cross-hairs, they are not adjustable with respect to each other.
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The stadia rods (shown in the Fig. 3.2) are main instrument on a surveying activity to observe
the measurements intercepted by the stadia hairs. There are various types of markings used in
stadia rods but all have geometric figures designed to be legible when used for measuring long
distances. Stadia rods are usually graduated in decimals of a foot but may be graduated in
decimals of a meter or a yard. As an aid in distinguishing the numbers and graduations, different
colors are used.
Fig.3.1 Tachymetric Survey Set-up
Fig.3.2 Graduated Rods
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Stadia Constants and Stadia Interval Factor
For this particular set-up (shown in the Fig. 3.3), the line of sight the telescope is horizontal and
the stadia rod is held vertical at a distant point. The apparent locations of the stadia hairs on the
rod are points A and B and the stadia intercept is S.
From similar triangles, we know that f:i = d:S which yields to d = (f/i)S, the horizontal distance
between the axis and the rod is given by the following equation D = d + (f+c), this is the distance
equation. Stadia intercept, S, is found by subtracting the reading of the upper and lower stadia
reading.
Fig. 3.3 Stadia Constants and Interval Factor
The constant K = f/i is called the multiplying constant or stadia interval factor, and is designated
by the letter K. For any given instrument, this value remains constant and depends only on the
spacing between the stadia hairs. The manufacturer of the instrument can space the stadia hairs
in relation to the focal length so as to obtain any convenient value of K to be desired. Most
common value of K is 100.
The constant C = (f + c) is known as the additive constant of the tacheometer but the latter one is
made zero by using an anallatic lens in the instrument. It is termed as stadia constant and
designated by the letter C referring to the distance from the center of the instrument to the principal
focus.
For computing horizontal distances from stadia readings with horizontal line of sight, we can use
the equation:
𝑫 = 𝑲𝑺 + 𝑪
Where D – distance from the instrument center to the face of the rod, K – stadia interval factor, S
– Stadia Intercept (difference of upper and lower stadia hair reading), and C – stadia constant.
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Inclined Stadia Sights
In actual field practice, most stadia measurements are inclined because of varying topography,
but the interval is still read on a vertical held rod. The inclined measurement, which is also
dependent on the observed vertical angle, is reduced to horizontal and vertical components of
the inclined line of sight. Looking at a standard set-up of inclined measurements (shown in the
fig. 3.4), this figure illustrates an inclined line of sight for an instrument set-up at point M with the
rod held vertically at N. The horizontal distance between the instrument and the rod is shown as
̅̅̅̅), and the vertical distance is VD (𝑃𝐷
̅̅̅̅).
HD (𝑂𝐷
Fig. 3.4 Inclined Stadia Sights
From the figure where, K – stadia interval factor, a – upper stadia reading, b – lower stadia
reading, P – horizontal cross-hair reading or rod reading (RR), s – stadia intercept, 𝛼 – observed
vertical angle of elevation/depression, C – stadia constant, ID – inclined distance, HD – horizontal
distance, VD – vertical distance, HI – height of instrument, and DE – difference in elevation.
Described above, we can deduce the following equations essential for inclined stadia sights, as
follows:
𝑰𝑫 = 𝑲𝒔𝒄𝒐𝒔(𝜶) + 𝑪
𝑯𝑫 = 𝑲𝒔𝒄𝒐𝒔𝟐 (𝜶) + 𝑪𝒄𝒐𝒔(𝜶), 𝒐𝒓 𝑯𝑫 = 𝑰𝑫𝒄𝒐𝒔(𝜶)
𝑽𝑫 = 𝑲𝒔𝒄𝒐𝒔(𝜶)𝒔𝒊𝒏(𝜶) + 𝑪𝒔𝒊𝒏(𝜶), 𝒐𝒓 𝑽𝑫 = 𝑰𝑫𝒔𝒊𝒏(𝜶)
𝑫𝑬 = 𝑯𝑰 + 𝑽𝑫 − 𝑹𝑹
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Activities/Assessment:
1. An instrument was used in a level stadia sight with a constant of 0.30m was set-up on a line
between two points A & B, and the following hair readings was observed. Assuming stadia interval
factor of 99.5, find the length of the Course AB.
POSITION
ROD @ A
ROD @ B
UPPER
1.33m
1.972m
HAIR READINGS
MIDDLE
1.175m
1.854m
LOWER
1.02m
1.736m
Solution.
POSITION
ROD @ A
ROD @ B
‘S’
1.33-1.02 = 0.31m
1.972-1.736 = 0.236m
‘K’
99.5
99.5
‘C’
0.30 m
0.30 m
KS + C
31.145m
23.782m
Summing up the length we get from points A & B, LAB = 31.145 + 23.782 = 54.93m.
2. The following data were obtained by stadia measurement. vertical angle 18°23', and observed
stadia intercept = 2.20m. The stadia interval factor of the instrument used is 95.5 and C = 0.30m.
If the height of the instrument is 1.62m, and the rod reading is taken at 1.95m, determine HD, VD,
ID and DE.
Solution. We will use the readily available derived formulas for stadia with inclined sights.
Solving for ID, using 𝑰𝑫 = 𝑲𝒔𝒄𝒐𝒔(𝜶) + 𝑪, 𝑰𝑫 = (𝟗𝟓. 𝟓)(𝟐. 𝟐𝟎)𝒄𝒐𝒔(𝟏𝟖°𝟐𝟑′) + 𝟎. 𝟑𝟎, ID = 199.68m.
From here, we can get horizontal distance (HD) and vertical distance (VD), using 𝑯𝑫 = 𝑰𝑫𝒄𝒐𝒔(𝜶)
&
𝑽𝑫 = 𝑰𝑫𝒔𝒊𝒏(𝜶),
𝑯𝑫 = 𝟏𝟗𝟗. 𝟔𝟕𝟖𝟏𝟑𝒄𝒐𝒔(𝟏𝟖°𝟐𝟑′),
HD
=
189.49m;
𝑽𝑫 =
′
(
)
𝟏𝟗𝟗. 𝟔𝟕𝟖𝟏𝟑𝒔𝒊𝒏 𝟏𝟖°𝟐𝟑 , VD = 62.97m.
We can get the difference in elevation (DE) of point A and point B using 𝑫𝑬 = 𝑯𝑰 + 𝑽𝑫 − 𝑹𝑹,
𝑫𝑬 = 𝟏. 𝟔𝟐 + 𝟔𝟐. 𝟗𝟕𝟑 − 𝟏. 𝟗𝟓, DEAB = 62.64m.
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Module 4: Contour and Mapping
Learning Objectives:
At the end of this lesson, the learner will be able to:
•
•
•
•
•
•
•
Define the terms used and objectives in Contouring.
Familiarize with the characteristics of contour lines.
Discuss Direct & indirect method of locating Contour.
Learn the Interpolation of contour process
Discuss the idea of survey mapping
Familiarize with the types and uses of Maps
Solve surveying problems relating map scales and contours
Course Materials:
Fig 4.0 Visualization of Contour Maps
Five different types of contours have been designed by cartographers to portray the relief of the
ground surface and to make map reading easier. Most contours are shown in brown which can
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be taken in a general way to be representative of soil. It is however, also possible to include
contours in other colors.
In general, all the types used to conform to the characteristics of contours, but their symbolization
has been varied to make the relief picture more readily apparent. These contours are classified
into the following:
1.
2.
3.
4.
5.
Index contours
Intermediate contours
Supplemental contours
Depression contours
Approximate contours
Index Contours
As a convenience in scaling elevations, and to provide ease and speed in reading contours, a
contour is shown by a heavier line at regular intervals on a topographic map. These heavier lines
which are normally twice the gauge of the standard contours are called the index contours. They
are usually drawn every fifth contour and carry the contour number or elevation designation. Index
contours also serve as a visual reinforcement in the contour image. By representing the
information at two visual levels, it makes it easier to identify the major forms from a detail.
Intermediate Contours
In Fig. 4.1 and 4.2, the four lighter weight contours found between the index contours are the
intermediate contours. These lines are not usually labelled except where the terrain is relatively
flat and their elevations are not readily obvious. They conform to the contour interval specified for
the map.
In certain portions of the map where the intermediate contours are so closely spaced as to nearly
unite or merge into a single line, it is standard practice for readability not to portray the lines for
short distances. This technique is called feathering (see Fig. 4.2).
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Fig. 4.1 Contour Line Representations
Fig. 4.2 Feathering of Contours
Depression Contours
Depression contours are drawn to show low spots such as excavations around which contours
close. The symbol used is the index of intermediate contour to which ticks are drawn
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perpendicular to the contour line on the downhill side. An example of depression contours is given
in Figure 4.3.
Fig. 4.3 Depression Contours
Supplemental Contours
Certain portions of the map area sometimes are do flat or level that the contours are too far apart
to show properly important breaks in the terrain. To better depict the relief and remedy such
situations, supplemental or auxiliary contours are used. They are drawn as dashed lines (see Fig.
4.4) or lines of dots that begin and end when they approached the areas where the regular
contours close in on each other. Supplemental contours are usually drawn at one-half the
specified contour interval of the map. The conformation of the flatter ground as well as the rise
and fall of the more rugged areas can be more easily assessed by using this technique.
Fig. 4.4 Supplemental Contours
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Approximate Contours
In some instances, contour accuracy cannot be definitely determined. The area may be
inaccessible on the ground or it may be difficult to interpret contours from aerial photographs when
heavy cloud or shadows occur. To portray a reasonable idea of relative elevations, the map maker
has to make an educated guess rather than have a blank area in the map. In these cases, the
approximate contour (see Fig. 4.5) is used.
Fig 4.5 Approximate Contours
The Contour Interval
The constant vertical distance between two adjacent contour lines is termed the contour interval.
It must always be consistent within the limits of the map but may be varied between map sheets
to better portray the terrain. Neighboring contour lines are drawn either close together or far apart
to show changes in slope and relief variations. Whatever the contour interval, areas which change
rapidly in elevation will have more contour lines within a given plan distance than areas which
change slowly in elevation.
In design of topographic maps, the contour interval is commonly fixed at multiples of 0.5, 1, 2, 5,
10, 20, 50, and 100 meters. It is always stated within the margins of the map. For general types
of terrain, the contour interval used may be as follows: Flat to gently rolling, 1 to 5 m; hilly, 5 to 20
m; and mountainous, 25 to 100 m.
According to map scale, the following contour intervals are used.
Table 4.1 contour interval for different map scales
Scales
1/500
1/2,000
1/5,000
1/10,000
Interval
0.5m
1
2
5 or 10
Scale
1/25,000
1/50,000
1/100,000
1/250,000
Interval
10m
20
25
50
The contour interval and map scale are interrelated. In general, the smaller the scale, the larger
the contour intervals.
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Selection of Contour Intervals
In planning and designing the contour interval to be used for a topographic map, the following
factors should be considered.
1.
Relative Cost. The time and expense of field and office work is given important
consideration. The contour interval is a matter of economy with the smallest interval that
can be afforded desired. The smaller the interval, the greater the amount of field work,
reduction, and plotting required in the preparation of the map. The cost of the map will be
higher as the contour interval is reduced.
2.
Purpose of the Map. When the map is to be utilized for the detailed design of
engineering constructions or for the measurement earthwork quantities, close contouring
will be required. In general, the area included in such a map will be comparatively small,
so that it may be quite practicable to locate contours with an interval as small as ½ meter.
A wider interval will be required for surveys of reservoirs, drainage areas, and lines of
communication.
3.
Nature of Terrain. The type of terrain and map scale with invariably define the
contour interval needed to produce a suitable density of contours. An interval which would
be sufficient to show the configuration of mountainous terrain would be inadequate to
portray the undulations of comparatively flat ground. Rugged terrain will require a larger
interval than gentle and rolling country. To portray adequately flat ground a relatively small
interval must be specified.
4.
Scale of the Map. The contour interval should be in inverse ratio to the scale of
the map. If the map scale is reduced, the interval must be increased; otherwise lines are
crowded, confuse the map user, and may possibly obscure some important map details It
should also be noted that when the map scale is reflected, the more refined should be the
measurements of the elevations of chosen points since a smaller interval would be used.
Characteristics of Contours
The following are some of the common characteristics of contours (see Fig. 4.6):
•
•
•
•
•
•
•
•
•
•
All points in any one contour have the same elevation.
Every contour closes on itself, either within or beyond the limits of the map. The closure
may occur within the mapped area, but often happens outside the area and hence will not
appear on the map sheet.
A contour which closes on itself, either within or beyond the limits of the map indicates
either a summit or a depression. Contours which increase in elevation represent hills;
those which decrease in elevation portray valleys or excavations.
Contours on the ground cannot cross one another except where an overhanging cliff, a
vertical ledge or wall is represented on a map.
Contours are spaced evenly on a uniform slope.
Contours are straight and parallel to each other on a plane surface.
Irregular contours signify rough, rugged terrain.
The horizontal distance adjacent contours indicate the steepness of the slope of the
ground. Where the contours are relatively close together, the slope is comparatively steep;
where the contours are far apart, the slope is gentle.
Contours across curbs and a crowned sloping street in typical U-shaped curves.
As a contour approaches a stream, the contour turns upstream until it intersects the
shoreline. It then crosses the stream at right angles to the center of the bed and turns back
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along the opposite bank of the stream. If the stream has an appreciable width, the contours
are not drawn across the stream but are discontinued at the store, with which it merges.
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Fig. 4.6 Characteristics of Contours
Methods of Obtaining Contours
Both horizontal and vertical measurements are involved I the location and plotting of contours.
Since different instruments may be used, the field work is executed in various ways. However,
whatever method is employed, the main objective is always to locate details with minimum time
and effort. The accuracy of the resulting map will greatly depend upon the number and disposition
of the selected points.
There are several methods employed in determining the location and elevation of ground points
for purposes of plotting a contour map. The most common methods used are, however, divided
only onto the following two major classifications:
•
•
Direct Methods- are those in which the contours to the plotted are actually traced out in
the field. The procedure involves the locations and marking of a series of points on each
contour line. These points are surveyed and plotted in the field, and the appropriate
contours are drawn through them. The trace-contour method is one such example that is
used.
Indirect Methods- comprise in those which the points located as regards position and
elevating are not necessarily situated on the contours to be shown, but serve, on being
plotted, as basis for the interpolation of the required contours. The following field methods
for obtaining topographic detail fall under this classification: Coordinate method, controlling
point method, cross profile method, and the photogrammetric method.
Trace-Contour Method
One of the most accurate and direct procedure of locating contours is by trace-contour method.
Although this method is quite accurate, it tends to be slow and costly to undertake. This procedure
is thus only being used when it is absolutely necessary to meet rigid accuracy requirements. The
method is used advantageously in rolling country where the slopes are generally gentle.
The procedure consists of a series of rod readings taken along the same contour line from
successive setups of the instrument. When the transit and stadia rod are used, the instrument is
setup I one location near the elevation of the contour to be traced. The height of instrument is
obtained by back sighting on a point of known elevation, and the rod reading is computed when
the rod is on the contour to be traced. The rodman is then directed towards the expected path
that the contour will take. Points on the contour are found by trial and are located by the angle
and stadia distance. By this procedure, the entire length of the contour line is traced out on the
ground as far as the position of the instrument lines possible are taken from the instrument station.
The instrument is then moved ahead and another section is worked out.
Coordinate Method
One way of locating and plotting the contours of a given area is to utilize a grid or
coordinate system. The general procedure is summarized as follows:
1.
2.
3.
4.
On a grid system lay out the areas by establishing corner and perimeter stakes.
At the intersections of the grid lines determine the elevations.
Plot the points of known elevation to the desired scale in plan.
Draw the contour lines by interpolation.
Controlling-Point Method
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The controlling-point method is obtaining contours is considered to be one with the most universal
application. This method is applicable to practically every type of terrain and condition
encountered in topographic mapping. Its basic principle is that the readily visible and well defined
points on the are surveyed will form a framework on which other map details may be indicated or
upon which the interpolation of contours may be made much easier and faster.
The method can be advantageously performed in rough and rugged terrain. It is suitable for
surveys which serve the design and estimates of many engineering projects, such as reservoirs,
irrigation canals, hydro-electric developments, and drainage systems. One distinct advantage of
this system is that the amount of detail gathered can easily be modified to provide the desired
map.
Cross-Profile Method
The cross-profile method is principally used in locating contours along a route or other narrow
area of terrain. It is a modification of the method of cross-sectioning as used in route surveys. The
method is particularly appropriate for surveys required in the construction of roadways, railways,
canals, irrigation ditches, and in the installation of pipelines or sewer lines. The cross-profile
method of locating contours does not fall far short of the direct method in terms of accuracy,
provided that additional sections are run where required, and hat the ground is fairly uniform in
slope between the points located.
Photogrammetric Method
The availability of aerial photographs and the development of photogrammetric methods
have expanded rapidly our know-how in the construction of topographic maps. Using
photogrammetric methods, contours may be plotted from aerial photographs of the terrain with a
minimum of ground survey for control.
The main advantages of the compilation of topographic maps by aerial photographs over
ground methods are:
•
•
•
•
•
•
Speed of compilation.
High accuracy of the locations of planimetric features.
Faithful reproduction of the configuration of the ground by continuously traced contour
lines.
Reduction in the amount of control surveying required to control the mapping.
Freedom from interference by adverse weather and inaccessible terrain.
Photographic representation of the details of the area is obtained.
Among the disadvantages of mapping by aerial photographs are:
•
•
•
•
•
•
The difficulty of plotting in areas with heavy ground cover.
The high cost of mapping.
The difficulty of locating positions of contour lines in flat terrain.
The necessity for field editing and field completion.
Expensive cost of photogrammetric equipment.
The need to train equipment operators to use complicated photogrammetric equipment.
MAPPING
Map Titles
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Titles on engineering drawings usually appear in the lower right-hand corner of the sheet.
On maps the same convention is sometimes followed, although on many maps the shape
of the area portrayed may be such that the title is positioned elsewhere on the sheet to
give the drawing on a more balanced appearance.
Titles should be so constructed that they will readily catch the eye. The space occupied
by the title should be in proportion to the size of the map. Emphasis is placed on the most
important parts of the title by increasing their height and using upper-case letters.
Map Scales
Scale refers to the relationship which the distance between any two points on the map
bears to the corresponding distance on the ground. When a map is to be compiled, the
scale to be used is one of the important factors to be considered since the scale of the
map determines the type and precision of field surveys to be undertaken.
Map scales are portrayed in three different ways: (a) by words and figures or an
equivalence, (b) by scale ratio or representative fraction, and (c) graphically.
Equivalence Scale
Scales be may expressed as an equivalence or by words and figures. Such expressions
as 1 inch = 1 mile, 1 centimeter = 1 kilometer, and 3 inches = 200 feet, are expressions
of equivalence scales. Maps intended for the design of engineering constructions are
commonly plotted to this type of scales. Such scales usually vary between 1 cm = 1 meter
and 1 cm = 100 meters, the selected scale depending on the detail to be shown and the
extent of areas covered.
Scale Ratio or Representative Fraction
The topographic maps prepared by most mapping agencies are plotted to so-called
“natural scales” which are expressed as ratios. A fraction indicating a scale is termed the
representative fraction or scale ratio. It is usually referred to as the “RF” or “SR” for short.
The scale of a map can be expressed as a ratio, such as 1:5000, or as a fraction, as
1/5000. In either case, 1 unit on the map corresponds to 5,000 units to the ground. The
units of measurement may be feet, yards, meters, or any other convenient unit; however,
the units of both the map distance and ground distance must be the same. Thus 1 inch
on the map represents 5,000 inches on the ground, 1 foot represents 5,000 ft, and 1 meter
represents 5,000 meters. The RF or SR of a map is determined by the following formula.
SR= MD/GD
Where: MD – the map distance or the scaled distance between any two selected points
on the map, GD – the corresponding distance on the ground.
Graphic Scales
The scale consists of two parts, a primary scale on the right and an extension scale on
the left. The primary scale is divided into major divisions of ground distance while the
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extension scale shows the subdivisions of a division on the primary scale into convenient
fractions, usually tenths. The extension scale is portrayed to enable the map user to
measure shorter or fractional distances more precisely. An index mark or zero point
separates to two scales.
Classification of Map Scales
Map scales are generally classified as large, medium, and small. Their respective
scale ranges are as follows:
•
•
•
Large-Scale Maps- are those having scales of 1:2,000 or larger and with
contour intervals ranging from 0.10 to 2.0 meters.
Medium-Scale Maps- are maps having scales ranging from 1:2,000 to 1:
10,000 and with contour intervals ranging from 1.0 to 5.0 meters. They are
also referred to as intermediate scales.
Small-Scale Maps- are maps having scales of 1:10,000 or smaller and with
contour intervals ranging from 5 to 2,000 meters.
Linear Interpolation of Contour by Analytical Method
Arithmetical computations are employed where high accuracy is desired in locating
contour lines. This method is well suited for drawing large-scale maps. Distances between
points of known elevations are measured and the location on contour points are
determined by proportions.
Activities/Assessment:
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1. Determine the scale of a sketch wherein one centimeter represents 100 meters on the ground
– Using the formula SR = MD/GD, and the conversion 1cm = 100m. SR = 1/(100*100), therefore
the scale was used is 1:10000
2. The distance on a map is 8 cm for two specific points. The map was drawn with an unknown
scale, to verify the actual distance on ground the line was paced and the paced distance was 4
kilometers. What is the map scale? – Solution. Using the formula SR = MD/GD, SR =
8/(4*1000*100), therefore the scale was used is 1:50000.
3. On a map with a scale of 1cm = 3000m, 5.25cm power line was measured. What is the actual
ground length of the power cable? – using ratio and proportion we know that per 1cm of map
length corresponds to 3000m actual length thus, 5.25*3000 = 15,750m was the ground
distance.
4. For the contour map shown below. What is the elevation of point Z and the direction of flow the
river?
Answer: 220 meters & North-Eastern flow towards the ocean.
Module 5: Hydrographic Surveys
Learning Objectives:
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At the end of this lesson, the learner will be able to:
• Learn the definition and uses of Hydrographic surveys.
• Visualize and be familiar with Bathymetric Surveys and Sounding Methods
• Discuss Tides and its importance.
• Compute the capacity of bodies of water using End-Area Method and Simpson’s 1/3
Rule.
• Familiarize with Stream Gaging, Determination of River Discharge, and get the idea of
stream velocity measurement by floats
Course Materials:
Discussion:
Hydrographic surveying is the wet equivalent of topographic surveying. Its objective is to delineate
the shape of a portion of the earth’s surface concealed by water. The surface being mapped
cannot be observed directly or occupied, so it is necessary to infer topography from depth
measurements. Simply, it is the process of deducing underwater topography from numerous
discrete observations of depth at positions throughout the survey area. The quality of its product
depends on the accuracy and density of these observations. An important type of hydrographic
surveys is the bathymetric survey, which originates from the Greek terms bathus (deep) and
metron (measurement). This is the underwater equivalent to a land survey operation known as
hypsometry.
Unlike land surveys that use static or stationary equipment to conduct measurements, bathymetry
(shown in figure 5.1) involves a dynamic ship that is moving while carrying out observations. Thus,
special allowances have to be made for any error that arises out of this motion.
Fig. 5.1 Hydrographic Surveys
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Hydrographic surveys can be used for preparation of nautical charts, assessing silting in harbors,
for planning dredging, determination of contour, design of projects like dams, bridges, reservoirs,
marine structures, ports harbors, off-shore structures, flood control, hydroelectric power plant
planning, determining volume of water and navigation at seas.
Common terms used in Hydrographic Surveys are the following:
• Stream gaging – process of measurements in streams and rivers for the purpose of
predicting the rate of discharge at various water levels.
• Fathom – measurement of water depth equivalent to 6 feet
• Hydrographic Chart – a map or chart used in navigation at seas, it is similar to topographic
map.
• Tidal Datum – are specific tide levels which are used as surface of reference for the
purpose of measurement of depth as well as elevation.
• Discharge – volume of water flowing at a given section of water flow. Commonly expressed
as volume per unit of time.
• Stream – steady flow of water running along the surface of the Earth.
• Tidal Current – the horizontal movement of water accompanying tides and which is
produced by the combined action of astronomical, hydrological, and meteorological
factors.
• Flood Tide – the period within which water surface is rising and moving toward the shore.
• Ebb Tide – the period within which the water surface is falling and moving seaward.
• Slack Water – the instant at which the tidal current is changing direction and flows neither
in nor out.
• Set – direction of the current flow.
• Drift – refers to the speed of the current flow.
• High Water – refers to the maximum height to which the water surface rises above the
standard datum. (a.k.a. High Tide)
• Low Water – it is the greatest depression of the water surface below the standard datum.
• Tide Range – the range of the tides between high tide and low tide.
• Mean Sea Level – (msl) datum for the first-order level net of many countries and is
increasingly used as the base for general levelling operations. It is defined as the average
height of the sea for all stages of the tide.
• Mean Low Water – (mlw) is the mean of all low waters as observed over long periods.
• Mean Low Lower Low Water – (mllw) this datum is the average of all heights of the lower
of the two low waters that occur in each lunar day.
• Mean Low Water Springs – (mlws) defined as the mean of low waters of the spring tides
occurring a day or two after new or full moon.
Preliminary Steps in Hydrographic Surveying
The method starts by locating special control points along the shore line. The sounding method
is employed to determine the depth at various points by means of stationary boats. Sounding
locations can be either made from boat to the control points or by fixing a point in the boat and
taking sounding from the control point. Before this procedure certain preliminary steps have to be
made:
•
Reconnaissance. As every project require a start-up plan to complete it effectively and
economically, reconnaissance has to be undergone. A complete reconnaissance of whole
survey area to choose the best way of performing the survey. This would facilitate
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•
•
satisfactory completion of the survey in accordance with the requirements and
specifications governing such work. Aerial photographs would help this study.
Locate Horizontal Control. The horizontal control is necessary to locate all features of the
land and marine in true relative positions. Hence a series of lines whose lengths and
azimuths are determined by means of either triangulation or any other methods.
Tachymetric and plane table survey can be conducted in order to undergo rough works.
Locate vertical Control. Before sounding establishment of vertical control is essential to
determined. Numerous benchmarks are placed in order to serve as vertical control. Setting
and checking the levels of the gauges are uses of benchmarks
Sounding
The process of determining depth below water surface is called as sounding. The step before
undergoing sounding is determining the mean sea level. If the reduced level of any point of a
water body is determined by subtracting the sounding from mean sea level, hence it is analogous
to levelling.
In simpler terms, it is the measurement of the vertical depth from the level surface of the water to
the bed of the lake, river or sea – a series of soundings whether taken at random points or on a
grid can be used to prepare a plan showing the topographic features of the land covered by the
water.
The essential equipment used for undergoing sounding are:
•
•
Shore signals and buoys – These are required to mark the range lines. A line
perpendicular to shore line obtained by line joining 2 or 3 signals in a straight line constitute
the range line along which sounding has to be performed. Angular observations can also
be made from sounding boats by this method. To make it visible from considerable
distance in the sea it is made highly conspicuous. A float made of light wood or air tight
vessel which is weighted at bottom kept vertical by anchoring with guywires are called
buoys. In order to accommodate a flag a hole is drilled. Under water deep, the range lines
are marked by shore signals & the buoys
Sounding Equipment:
1. Sounding boat, a flat bottom of low draft is used to carry out sounding
operation. Large size boats with motor are used for sounding in sea. The
soundings are taken through wells provided in the boat.
2. Sounding pole or rod. Rod made of seasoned timber 5 to 10cm diameter
and 5 to 8m length. A lead shoe of sufficient weight is connected at bottom
to keep it vertical. Graduations are marked from bottom upwards. Hence
readings on the rod corresponding to water surface is water depth.
3. Lead line. A graduated rope made of chain connected to the lead or sinker
of 5 to 10kg, depending on current strength and water depth. Due to deep
and swift flowing water variation will be there from true depth hence a
correction is required.
4. Sounding Machine. It is used when the water depth is too great where lead
line cannot be used. It is an upgraded lead line.
5. Fathometer. It is a measuring device which computes and records the time
of sound wave from the moving vessel down to the bottom of lake/sea and
going back to the vessel.
6. Tide Gage. This is an instrument for measuring the height of the tide.
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•
Instruments for measuring angles
Tides
Tides are very long-period waves that move through the oceans in response to the forces exerted
by the moon and sun. Tides originate in the oceans and progress toward the coastlines where
they appear as the regular rise and fall of the sea surface caused by the combined gravitational
effects of the sun and moon, with the moon having the major effect influenced by:
• terrestrial gravity
• earth’s rotation
• land masses
• weather systems
Tides vary on timescales ranging from hours to years due to a number of factors, which determine
the lunitidal interval. To make accurate records, tide gauges at fixed stations measure water level
over time. Gauges ignore variations caused by waves with periods shorter than minutes. These
data are compared to the reference (or datum) level usually called mean sea level.
While tides are usually the largest source of short-term sea-level fluctuations, sea levels are also
subject to forces such as wind and barometric pressure changes, resulting in storm surges,
especially in shallow seas and near coasts.
Capacity of Lakes and Reservoirs
One application of hydrographic surveying is determining the volume and capacity of water bodies
engaged in engineering designs. In the design of water supply systems, irrigation projects,
structures for aquaculture development, and hydroelectric power generating stations it is
necessary to determine the volume of water which could be contained and generated by a
supporting reservoir or lake. Two methods in determining volume of bodies of water are Crosssection Method and Contour Method.
Trapezoidal Rule, Simpson’s 1/3 Rule, and End-Area Method
Primarily, the methods of determining the volume of water bodies depends the on the crosssection of water in a given length. This cross-sectional area will be multiplied to a length which
yields to a volume. Trapezoidal rule and Simpson’s 1/3 rule are methods used for approximating
these sectional areas.
The assumption made in using the trapezoidal rule is that the ends of the offset in the boundary
line are assumed to be connected by straight lines, thereby forming a series of trapezoids. When
offsets are taken fairly close together and when the curves are flat; no considerable error is
introduced by this assumption. Assuming offsets are spaced at regular intervals (shown in Fig.
5.2), the total area of the cross-section can be solved by the formula:
𝑨=(
𝒉𝟏 + 𝒉𝒏
+ ∑ 𝒉) 𝒙 (𝒅)
𝟐
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Where h1 & hn are first and last offset, ∑ 𝒉 – sum of the intermediate offsets, and d – common
interval. In our case here in Fig. 5.2, the first and last offsets are h 1 & h5, while its intermediate
offsets are h2, h3, & h4.
Fig. 5.2 Hydrographic Survey Cross-Section with Equal Perpendicular Offsets (5)
For specific cases having the spacing at irregular intervals (unequal spacing of offsets). The
formula which follows the same manner, can be used for ‘n’ number of offsets:
𝒏−𝟏
∑ 𝒅𝒊 (𝒉𝒊 + 𝒉𝒊+𝟏 )
𝒊=𝟏
On the other hand, Simpson’s One-Third Rule is based on the assumption that the curved
boundary consists of series of parabolic arcs, where each arc is continuous over three adjacent
offsets that are equally spaced, because of this assumption, the rule is only applicable when there
are odd number of offsets with equal spacing. The area can be computed by the formula:
𝑨=
𝒅
[(𝒉𝟏 + 𝒉𝒏 ) + 𝟐 ∑ 𝒉(𝑶𝑫𝑫) + 𝟒 ∑ 𝒉(𝑬𝑽𝑬𝑵) ]
𝟑
Where h1 & hn are first and last offset, ∑ ℎ(𝑂𝐷𝐷) – sum of the odd intermediate offsets, ∑ ℎ(𝐸𝑉𝐸𝑁) –
sum of the even intermediate offsets, and d – common interval. Looking again in Fig. 5.2, the first
and last offsets are h1 & h5, while its odd intermediate offset is h 3, & even intermediate offsets are
h2 & h4.
After determining the area of cross-section which is to be multiplied to the distance of the sections
to get the specific volume is the next step. This process of computation is referred as the End
Area Method, which also follows the trapezoidal rule in a way. Cross-section Method & Contour
Method used in computing the capacity of lakes and reservoirs employ the same principle as the
End-Area Method.
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Fig. 5.3 Body of Water taken with Cross-Sections
The formula used in Cross-Section Method is:
𝑽 = 𝑫(𝑨𝟏 + 𝑨𝟐 )
Where V – the volume of prismoid between cross-sections 1 & 2. D – distance between sections
1 & 2. A1 & A2 are the cross-sections (End Areas). For better visualization, take a look at Fig. 5.3.
The volume of the entire water body shall be computed by adding all of the individual computed
prismoids.
On the other hand, the Contour Method maximizes the uniform contour interval that simplifies
the formula for End-Area. The volume of water contained between consecutive contours is
determined by using the formula:
𝒗 = 𝒉(𝑨𝒍 + 𝑨𝒉 )
Where v – volume of prismoid between two-consecutive contours, h – contour interval, 𝑨𝒍 – area
enclosed by lower contour, and 𝑨𝒉 – area enclosed by higher contour. Observe uniformity of units
to get desired volume.
The total volume of water contained in the reservoir or lake is the sum of the volumes of the
individual prismoids. Therefore, the approximate total Volume can be computed using Contour
Method via:
𝑨𝟎 + 𝑨𝒏
𝑽=(
+ ∑ 𝑨𝒊 ) 𝒙 (𝒉)
𝟐
Where A0 – Area bounded at water surface, A n – Area bounded at lowest contour, ∑ 𝐴𝑖 – sum of
areas enclosed by the intermediate contours, and h – contour interval.
Stream Gaging
Stream gaging is a technique used to measure the discharge, or the volume of water moving
through a channel per unit time, of a stream. The height of water in the stream channel, known
as a stage or gage height, can be used to determine the discharge in a stream. When used in
conjunction with velocity and cross-sectional area measurements, stage height can be related to
discharge for a stream. If a weir or flume (devices, generally made of concrete, located in a stream
channel that have a constant, known shape and size) is used, mathematical equations based on
the weir or flume shape can be used in conjunction with stage height, negating the need for
velocity measurements
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Measurement of Stream Velocity & Discharge
Measurement of stream discharge are usually made in connection with the design of water supply
systems, flood protection works, hydroelectric power development, irrigation systems, and fish
farming structures. While stream velocities are commonly determined either by the use of floats
or current meters, various methods of determining discharge are Velocity-Area Method, SlopeArea, and Weir Method.
The velocity area method suggests that the total discharge Q is the sum all small discharges. (Q
= AV = a1v1 + a2v2 + …. + anvn, comprising the same section. Peak flows are the primary concern
of engineers when it comes to measuring flows; Slope-Area Method is fit in this regard. The results
obtained by this method are only approximate and are inferior in precision to those in which the
velocity is actually observed. The limitations of the method lie in the difficulty of selecting a correct
value of roughness coefficient (n). However, this method is useful in making rough estimates of
discharges.
The approximate value of discharge is computed by the formula:
𝑸 = 𝑨(𝑽) 𝒐𝒓 𝑸 = 𝑨(𝑪)√𝑹𝑺
Where Q – discharge (m3/s), V – mean velocity of the stream (m/s), A – mean cross-sectional
area in the reach (m 2), C – Kutter’s Variable Coefficient, R – hydraulic mean depth or hydraulic
radius (m), S – longitudinal slope of the water surface (m/m)
Kutter’s Coefficient is a variable basically depending upon the roughness of the bed and its slope,
also with the hydraulic radius. It is derived by the formula:
𝟏
𝟎. 𝟎𝟎𝟏𝟓𝟓
+ 𝟐𝟑 +
𝒏
𝑺
𝑪=
𝒏
𝟎. 𝟎𝟎𝟏𝟓𝟓
𝟏+
(𝟐𝟑 +
)
𝑺
√𝑹
The roughness coefficient (n) in the above formula is a retardation factor that will depend on the
character and slope of the stream bed.
Manning’s Velocity Equation
In determining velocities of stream flow, Manning’s formula showed a better fit, and is denoted as
follows:
𝑽=
𝑹𝟐/𝟑 𝑺𝟏/𝟐
𝒏
This version of the formula is in S.I. units, where V – velocity of uniform flow (m/s), R – hydraulic
mean depth or hydraulic radius (m), S – longitudinal slope of the water surface (m/m), and n –
roughness coefficient.
Activities/Assessment:
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1. A series of perpendicular offsets were taken from a hydrographic survey line. These offsets
were taken 2.5 meters apart and were measured in the following order, 0, 2.6, 4.2, 4.4, 3.8, 2.5,
4.5, 5.2, 1.6, and 5.0 meters. By trapezoidal rule find the cross-sectional area.
Solution. Taking h1 = 0m & h10 = 5.0m, intermediate offsets are: 2.6, 4.2, 4.4, 3.8, 2.5, 4.5, 5.2,
𝒉 +𝒉
0+5
1.6, and d = 2.5m. Using 𝑨 = ( 𝟏 𝟐 𝒏 + ∑ 𝒉) 𝒙 (𝒅), 𝐴 = ( 2 + 2.6 + 4.2 + 4.4 + 3.8 + 2.5 +
4.5 + 5.2 + 1.6 ) 𝑥 (2.5). Area = 78.25 m2
2. From a transit line to the edge of a river, a series of perpendicular offsets are taken. These
offsets are spaced at 4.0 meters apart and were measured in the following order: 0.5, 1.4, 2.5,
5.6, 8.5, 7.4, 3.8, 5.1, and 2.3. By Simpson's One-Third Rule compute the area.
Solution. Counting the number of offsets (9) – we don’t need to modify the formula since all the
𝒅
𝟒.𝟎
conditions were satisfied. Using, 𝑨 = 𝟑 [(𝒉𝟏 + 𝒉𝒏 ) + 𝟐 ∑ 𝒉(𝑶𝑫𝑫) + 𝟒 ∑ 𝒉(𝑬𝑽𝑬𝑵) ], 𝑨 = 𝟑 [(𝟎. 𝟓 +
𝟐. 𝟑) + 𝟐(𝟐. 𝟓 + 𝟖. 𝟓 + 𝟑. 𝟖) + 𝟒(𝟏. 𝟒 + 𝟓. 𝟔 + 𝟕. 𝟒 + 𝟓. 𝟏)], Area = 147.20 m2
3. In the accompanying plot, the boundary of the water surface of a reservoir is shown by the
irregular outline ABCDEFGH. Soundings were taken on parallel ranges BH, CG and DF for the
purpose of determining the volume of water in the reservoir. All units are in meters. Calculate the
sectional areas of ranges BH, CG and DF, assuming ranges A & E are having negligible areas;
determine the volume of water between ranges and the total volume at the reservoir.
Solution.
RANGE
A
BH
SECTIONAL AREA (m2)
0.00
0.5[3.0*1.8+4.6(1.8+3.6)+5.8(3.6+3.0)+2.5(3.0+1.2)+2.5*1.2] = 41.01 m2
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CG
DF
E
0.5[7.3*4.3+3.9(4.3+5.5)+3.9(5.5+5.5)+7.9(5.5+4.6)+5.5(4.6+4.3)+7.3*4.3]
= 136.32 m2
0.5[8.9*6.1+8.5(6.1+6.1)+7.3(6.1+5.5)+6.1(5.5+4.3)+6.4*4.3] = 164.985m2
0.00
PRISMOID
A – BH
BH – CG
CG – DF
DF – E
TOTAL
VOLUME
0.5(0.0+41.01)*11.0 = 225.555 m3
0.5(41.01+136.32)*13.0 = 1152.645 m3
0.5(136.32+164.985)*16.5 = 2485.76625 m3
0.5(164.985+0)*14.6 = 1204.3905 m3
5068.35675 m3
4. From the results of a hydrographic survey of a lake the following data were obtained.
CONTOUR
A
B
C
D
E
AREA BOUNDED BY CONTOUR (m 2)
1240.71
752.79
397.77
178.44
48.33
If the vertical distance (h) between the contour levels is 1.50 meters, determine the total volume
of the lake above the level of contour E.
𝑨 +𝑨
1240.71+48.33
Solution. Using 𝑽 = ( 𝟎 𝒏 + ∑ 𝑨𝒊 ) 𝒙 (𝒉), 𝑉 = (
𝟐
2
Total Volume of water above level E = 2960.28 m3
+ 752.79 + 397.77 + 178.44) 𝑥 (1.50).
5. Given the following data for a stream of uniform flow: A = 6.97 m 2, Pw = 9.76 m (wetted
perimeter), S = 0.007, and n = 0.025. Assuming constant stream bed slope and little variation in
the cross-section and condition of the bed, determine the hydraulic radius, velocity and discharge
of the stream using a. Manning’s Formula & b. Kutter’s Coefficient.
Solution.
Hydraulic Radius is obtained by A/P w = 6.97/9.76, R = 0.71414 m.
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𝑹𝟐/𝟑 𝑺𝟏/𝟐
By Manning’s Formula, 𝑽 =
, V = 0.714142/3(0.0071/2)/0.025, V = 2.67381 m/s, discharge
𝒏
can be computed as Q = AV, Q = 6.97(2.67381), Q = 18.64 m3
By Kutter’s Coefficient, 𝑪 =
𝟏
𝟎.𝟎𝟎𝟏𝟓𝟓
+𝟐𝟑+
𝒏
𝑺
𝒏
𝟎.𝟎𝟎𝟏𝟓𝟓
𝟏+ (𝟐𝟑+
)
𝑺
√𝑹
, 𝑪=
𝟏
𝟎.𝟎𝟎𝟏𝟓𝟓
+𝟐𝟑+
𝟎.𝟎𝟐𝟓
𝟎.𝟎𝟎𝟕
𝟎.𝟎𝟐𝟓
𝟎.𝟎𝟎𝟏𝟓𝟓
𝟏+
(𝟐𝟑+
)
𝟎.𝟎𝟎𝟕
√𝟎.𝟕𝟏𝟒𝟏𝟒
, C = 37.47633148, with C
solved, we can get the 𝑉 = 𝐶√𝑅𝑆, 𝑉 = 37.47633148√0.71414(0.007), V = 2.64971 m/s,
discharge can be computed as Q = AV, Q = 6.97(2.64971), Q = 18.47 m3
Module 6: Astronomical Observations
Learning Objectives:
At the end of this lesson, the learner will be able to:
•
•
•
•
•
•
Discuss Astronomical Observations and its purposes
Familiarize with numerous astronomical terms
Define Latitude, Parallels, Longtitude and Meridian
Learn and define the different types of celestial coordinate system
Discuss the PZS triangle
Apply knowledge in problems pertaining to time & arc measures, as well as PZS triangle
Course Materials:
Introduction:
Surveying has been defined as the science of determining positions of points on the earth’s
surface. The four components of surveying measurements are: (1) vertical (elevations), (2)
horizontal (distances), (3) relative direction (angles), and (4) absolute direction (azimuths). Due
to recent developments in technology, the accuracy and efficiency of measuring these first three
components have increased dramatically. This has resulted in accurate determination of the
size and shape of figures. Unfortunately, determination of the orientation of figures, the fourth
component, has not kept pace, even though inexpensive technology and equipment exist, such
as precise timepieces, portable time signal receivers, ephemerides, programmable calculators,
and computers. The purpose of this chapter is to provide sufficient theory, calculations, and
field procedures so surveys in both the northern and southern hemispheres can be accurately
oriented without significant increases in time and expense, in both the northern and southern
hemispheres.
Celestial Sphere and Definitions
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To better visualize positions and movements of the sun, stars, and celestial coordinate circles,
they are projected onto a sphere of infinite radius surrounding the earth. This sphere conforms
to all the various motions of the earth as the earth rotates on its axis and revolves around the
sun. As shown in the figure (shown in the fig. 6.1) the celestial sphere and principal circles
necessary to understand celestial observations and calculations. An assumption is made that all
heavenly bodies are all fixed within a gigantic sphere with Earth on the center, as explained.
Definitions of important points and circles on the sphere follow:
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Great circle. A circle described on the sphere's surface by a plane that includes the
sphere's center.
Zenith (Zn). The point directly overhead or where an observer's vertical line pierces the
celestial sphere. Opposite zenith is the nadir.
Equator. A great circle on the celestial sphere defined by a plane that is perpendicular to
the poles. Horizon. A great circle on the celestial sphere defined by a plane that is
perpendicular to an observer's vertical.
Hour circle. A great circle that includes the poles of the celestial sphere. It is analogous to
a line of longitude and perpendicular to the equator.
Vertical circle. A great circle that includes the zenith and nadir. It is perpendicular to the
horizon.
Meridian. The hour circle that includes an observer's zenith. It represents north-south at
an observer's location.
Altitude (h). The angle measured from the horizon upward along a vertical circle. It is the
vertical angle to a celestial body.
Declination (𝛿). The angular distance measured along an hour circle north (positive) or
south (negative) from the equator to a celestial body. It is analogous to latitude.
Prime meridian. Reference line (zero-degree longitude) from which longitude is measured.
It passes through the Royal Naval Observatory in Greenwich, England; hence, it is also
known as Greenwich meridian.
Longitude (𝜆). Angle measured at the pole, east or west from the prime meridian. Varies
from zero degrees to 180°E and 180°W.
Latitude (𝜙). Angle measured along a meridian north (positive) and south (negative) from
the equator. It varies from zero to 90°.
Greenwich hour angle (GHA). Angle at the pole measured westward from the prime
(Greenwich) meridian to the hour circle through a celestial body. It is measured in a plane
parallel to the equator and varies from zero to 360°. The GHA of a celestial body is always
increasing-moving westward-with time.
Local hour angle (LHA). Angle at the pole measured westward from an observer's
meridian to the hour circle through a celestial body. It differs from GHA by the observer's
longitude.
Meridian angle (t). Equivalent to LHA, except it is measured either eastward or westward
and is always less than 180°.
Astronomical triangle (PZS triangle) Spherical triangle formed by the three points: (1)
celestial pole P, (2) observer's zenith Zn, and (3) celestial body S.
Astronomical refraction. As light from a celestial body penetrates the earth's atmosphere,
direction of the light ray is bent. Astronomical refraction is the angular difference between
the direction of a light ray when it enters the atmosphere and its direction at the point
Ephemeris. It is an astronomical almanac containing table giving the computed positions
of the sun, the planets, and various stars for any day of a given period.
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•
Horizon. The line where the earth seems to meet the sky, is the apparent junction of earth
and sky.
In surveying, our interest in astronomy is basically with respect to sides and angles of spherical
triangles on the celestial sphere. When observations are made for latitude, longitude, or azimuths,
surveyors and engineers are not concerned with the distances between celestial bodies. Since
any problem involving angular distances between points and angles between planes at the center
of the sphere may be readily determined by spherical trigonometry, the idea of celestial sphere is
adopted in field of surveying.
Fig. 6.1 Celestial Sphere
Parallels and Meridians
Latitude is used together with longitude to specify the precise location of features on the surface
of the Earth.
Latitude is a geographic coordinate that specifies the north–south position of a point on the
Earth's surface. Latitude is an angle which ranges from 0° at the Equator to 90° (North or South)
at the poles. Lines of constant latitude, or parallels, run east–west as circles parallel to the
equator.
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Longitude is a geographic coordinate that specifies the east–west position of a point on the
Earth's surface, or the surface of a celestial body. It is an angular measurement, usually
expressed in degrees. Meridian is an imaginary circle in a plane perpendicular to the planes of
the celestial equator and horizon.
Longitudes can be expressed in terms of arc and time measure. Arc measure is expressed in
degrees-minutes-seconds ( ˚ ‘ “ ) while time measure is expressed in hours minutes seconds ( h
m s
). A careful distinction between arc & time measure must be observed since 1-minute arc is
not equal to 1-minute time, however, the relationship between the arc & time measurement is
described by the ratio 360˚ (arc) = 24 h (time), we can deduce that 1’ (arc) = 4 s (time).
Celestial Coordinate System
The celestial coordinate system is best understood as a projection of the terrestrial coordinate
system, outward into space onto the celestial sphere.
Spherical coordinate systems are used to define the positions of heavenly bodies and points of
reference on the celestial sphere. The location of a body is usually expressed in terms of two
perpendicular components of curvilinear coordinates. Different celestial coordinate systems are
adopted and the most common are: Horizon System, Equatorial System, and Local Hour
Angle System. The origin of coordinates for all these reference systems may be at the point of
observation or at the center of the earth.
Horizon System (shown in the fig. 6.2), in this coordinate system the horizon is the primary
reference circle and secondary are vertical circles. To define the location of a heavenly body from
the horizon and the meridian of the celestial sphere, the required angular distances are its azimuth
and its altitude. The position of an observer’s horizon system on the celestial sphere with
reference to other systems is entirely dependent on the observer’s position on earth. The azimuth
of a heavenly body is a spherical angle and is defined as the angular distance measured along
the horizon from the observer’s meridian to the vertical circle through the body.
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Fig. 6.2 Horizon System
The Equatorial System. The equatorial coordinate system is a celestial coordinate system widely
used to specify the positions of celestial objects. It may be implemented in spherical or rectangular
coordinates, both defined by an origin at the center of Earth, a fundamental plane consisting of
the projection of Earth's equator onto the celestial sphere (forming the celestial equator), a primary
direction towards the vernal equinox, and a right-handed convention.
The origin at the center of Earth means the coordinates are geocentric, that is, as seen from the
center of Earth as if it were transparent. The fundamental plane and the primary direction mean
that the coordinate system, while aligned with Earth's equator and pole, does not rotate with the
Earth, but remains relatively fixed against the background stars. A right-handed convention
means that coordinates increase northward from and eastward around the fundamental plane.
The declination measures the angular distance of an object perpendicular to the celestial equator,
positive to the north, negative to the south. For example, the north celestial pole has a declination
of +90°. The origin for declination is the celestial equator, which is the projection of the Earth's
equator onto the celestial sphere. Declination is analogous to terrestrial latitude.
The right ascension, measures the angular distance of an object eastward along the celestial
equator from the vernal equinox to the hour circle passing through the object. The vernal equinox
point is one of the two where the ecliptic intersects the celestial equator. Analogous to terrestrial
longitude, right ascension is usually measured in sidereal hours, minutes and seconds instead of
degrees, a result of the method of measuring right ascensions by timing the passage of objects
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across the meridian as the Earth rotates. There are 360°/24h = 15° in one hour of right ascension,
and 24h of right ascension around the entire celestial equator.
Hour angle. Alternatively, to right ascension, hour angle, a left-handed system, measures the
angular distance of an object westward along the celestial equator from the observer's meridian
to the hour circle passing through the object. Unlike right ascension, hour angle is always
increasing with the rotation of Earth. Hour angle may be considered a means of measuring the
time since upper culmination, the moment when an object contacts the meridian overhead.
Astronomical PZS triangle
A spherical triangle is the figure formed by joining any three points on the surface of a sphere by
arcs of great circle. PZS spherical triangle (shown in the fig. 6.3) is called the astronomical
triangle, where: P – north pole of the celestial sphere, Z – observer’s zenith, S – heavenly body
observed (sun or stars). Since it is a true spherical triangle formed by great circles, many formulas
of spherical trigonometry are applicable in solving astronomical triangles.
The astronomical triangle consists of the following three sides and three angles:
• Polar distance (PS or p). Angular distance from the celestial pole P to a celestial body S.
Known as codeclination: p = 90° - 𝛿.
• Zenith distance (ZS or z). Angular distance along a vertical circle from an observer's zenith
Zn to a celestial body S. Known as co-altitude: z = 90° - h.
• Side PZ. Angular distance from the pole P to an observer's zenith Zn. Angle Z. Angle
measured at the zenith Zn, in a plane parallel with the observer's horizon, from the pole to
a celestial body.
• Angle Z is the true azimuth of the heavenly body.
• Angle S. Angle at a celestial body between the pole P and observer's zenith Zn. Known
as the parallactic angle.
• Angle t is known as the meridian angle
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Fig. 6.3 The Pole – Zenith – Star (PZS) Triangle
Activities/Assessment:
1. Convert the time measure of 8 h 40m 55s to equivalent arc measure.
Solution. We know that 360° is equivalent to 24h, 15° = 1h, 1° = 4m, 1’ = 4s. Therefore, 8h (15°/1h)
40m(1°/4m) 55s(1’/4s) arc time is deduced, solving, arc time is 120° + 10° + 13 ¾’, reducing ¾’
further ¾’(60”/1’) = 45”.
Final answer: 130° 13’ 45”
2. The longitude of New York is 73°57’30” West and Bontoc, Philippines is at 122°36’15” East.
Determine the difference in Solar Time between the two places.
Solution. Prime Meridian is located at 0° longitude, meridians will span at West and East from it,
making angles from 0° to 180° longitude. Longitudinal distance between Bontoc and New York is
122°36’15” + 73°57’30” (accounting for directions west and east). The difference in arc measure
is equal to 196°33’45”, converting to time measure using the proportion 15° = 1h, the difference
in solar time 13h 6m 15s.
3. During an instant of observation at latitude N 18° 10.1’, the sun’s apparent declination and true
altitude were recorded as 14° 10.5’ and 32° 50.2’, respectively assuming it was already corrected
due to error of refraction & parallax. Determine the length (in angular units) of the three sides of
the astronomical PZS triangle.
Solution. Polar Distance/Co-declination (Side PS = 90° - 14°10.5’), Pole to Zenith/Co-Latitude
(Side PZ = 90° - 18°10.1’), and Zenith Distance/Co-Altitude (Side ZS = 90° - 32°50.2’). Solving
values, PS = 75°49’30” , PZ = 71°49’54”, and ZS = 57°9’48”