Contouring

A contour is an imaginary line joining points of equal elevation on the ground. Every point on a contour has the same RL (reduced level), and a contour is the line in which a level surface cuts the ground.

Characteristics of contours

1. Contours are closed curves. Every contour must close on itself, either inside the map or outside it. It cannot start or end abruptly.
2. All points on a contour have the same elevation.
3. Contours never cross each other, except at an overhanging cliff or a cave. Two contours of different elevation cannot meet or cross because one point cannot have two elevations.
4. Contours of different elevations do not unite to form one line, except at a vertical cliff, where they coincide and appear as one line.
5. Closely spaced contours indicate a steep slope; widely spaced contours indicate a gentle slope. Equally spaced contours indicate a uniform slope, and straight, equal and parallel contours indicate an inclined plane.
6. Contours cross ridge lines and valley lines at right angles. The direction of the steepest slope is at right angles to the contour.
7. A set of concentric closed contours with the higher value inside represents a hill; with the lower value inside it represents a depression (pond or pit).
8. Contours bend as V or U shapes across a ridge and a valley. For a ridge (spur) the bend of the contour points towards the lower ground; for a valley the bend points towards the higher ground.
9. Contour of the same elevation may reappear on the other side of a ridge or a valley, so one contour can appear twice in a map.
10. A contour cannot lie between two contours of higher (or lower) values unless it closes on itself (hill or depression).
11. Contours of a saddle (col) show a pair of hills (high) and a pair of depressions (low) opposite to each other, with the saddle point at the neck.

        HILL                    DEPRESSION
      .--100--.                .--100--.
     / .-110-. \              / .--90-. \
    | / (120) \ |            | /  (80)  \ |
     \ '-----' /              \ '-----' /
      '-------'                '-------'
   RL rises to the centre     RL falls to the centre

   RIDGE (v points downhill)     VALLEY (^ points uphill)
      130  \   /                    120     /\
            \ /                           /  \
      120  \   /                    110   /    \
            \_/                          /      \
      110  \     /                  100 /        \
            \___/
   high ground at top            high ground at top

   Steep slope      Gentle slope      Uniform slope
   (close)          (wide apart)      (equal spacing)
   ----100          ----100           ----100
   ----101                            ----101
   ----102                            ----102
   ----103          ----101           ----103

Definitions

• Contour interval (CI) is the constant vertical distance between two consecutive contours, for example 1 m or 5 m. It is the same over one map.
• Contour gradient is the line of constant slope (uniform gradient) that cuts the contours at the same horizontal equivalent. It is laid on the ground to fix the route of a road, railway, canal or pipe line. The gradient = CI / HE = 1 in (HE/CI).
• Index contour is every fifth contour (for example every 5th line when CI = 2 m, giving 10 m, 20 m, 30 m ...) drawn as a thicker line and labelled with its RL, so that the map can be read quickly. The other contours are called intermediate (minor) contours.

Main characteristics of contours

1. Contours are closed curves. Every contour must close on itself, either inside the map or outside it. It cannot start or end abruptly.
2. All points on a contour have the same elevation.
3. Contours never cross each other, except at an overhanging cliff or a cave. Two contours of different elevation cannot meet or cross because one point cannot have two elevations.
4. Contours of different elevations do not unite to form one line, except at a vertical cliff, where they coincide and appear as one line.
5. Closely spaced contours indicate a steep slope; widely spaced contours indicate a gentle slope. Equally spaced contours indicate a uniform slope, and straight, equal and parallel contours indicate an inclined plane.
6. Contours cross ridge lines and valley lines at right angles. The direction of the steepest slope is at right angles to the contour.
7. A set of concentric closed contours with the higher value inside represents a hill; with the lower value inside it represents a depression (pond or pit).
8. Contours bend as V or U shapes across a ridge and a valley. For a ridge (spur) the bend of the contour points towards the lower ground; for a valley the bend points towards the higher ground.
9. Contour of the same elevation may reappear on the other side of a ridge or a valley, so one contour can appear twice in a map.
10. A contour cannot lie between two contours of higher (or lower) values unless it closes on itself (hill or depression).

        HILL                    DEPRESSION
      .--100--.                .--100--.
     / .-110-. \              / .--90-. \
    | / (120) \ |            | /  (80)  \ |
     \ '-----' /              \ '-----' /
      '-------'                '-------'
   RL rises to the centre     RL falls to the centre

   RIDGE (v points downhill)     VALLEY (^ points uphill)
      130  \   /                    120     /\
            \ /                           /  \
      120  \   /                    110   /    \
            \_/                          /      \
      110  \     /                  100 /        \
            \___/
   high ground at top            high ground at top

   Steep slope      Gentle slope      Uniform slope
   (close)          (wide apart)      (equal spacing)
   ----100          ----100           ----100
   ----101                            ----101
   ----102                            ----102
   ----103          ----101           ----103

Definitions

• Contour interval (CI) is the constant vertical distance between two consecutive contours, for example 1 m or 5 m. It is the same over one map.
• Horizontal equivalent (HE) is the horizontal distance between two consecutive contours. It varies with the slope of the ground: small on steep ground and large on gentle ground. $HE = \dfrac{CI}{\tan\alpha}$, where $\alpha$ is the ground slope angle.
• Index contour is every fifth contour (for example every 5th line when CI = 2 m, giving 10 m, 20 m, 30 m ...) drawn as a thicker line and labelled with its RL, so that the map can be read quickly. The other contours are called intermediate (minor) contours.

Methods of contour interpolation

Contour interpolation is the process of locating the positions of contours of required values between two plotted points of known RL. It assumes the ground slope is uniform between the two points.

Methods of interpolation

1. Estimation: The positions of contours are estimated by eye between the two points. It is quick but rough.
2. Arithmetical calculation: The distance from the lower point to the contour is found by proportion:

$$x = \frac{h_c - h_1}{h_2 - h_1}\times d$$

where $h_1, h_2$ are the RLs of the two points ($h_2 > h_1$), $h_c$ is the contour value and $d$ is the plotted distance between the points. It is accurate but slow. 3. Graphical methods

• Tracing paper (transparent sheet) method: Parallel equidistant lines are drawn on tracing paper and numbered with the contour values. The paper is turned and moved until the lines for the two end RLs pass through the two points. The crossing of line AB with the lines of the paper marks the contours. It is quick and common.
• Similar triangles method: A line is drawn from each end point at right angles to AB, with lengths in proportion to the RLs (scale of levels). The ends are joined and the equal divisions are projected on AB by parallel lines.

The tracing paper method is quick, simple and accurate enough, so it is recommended for a large volume of work.

Contour interval

Contour interval (CI) is the constant vertical distance between two consecutive contours, for example 1 m or 5 m. It is the same over one map. It depends on the scale, relief of the ground and the purpose of the map.

Contour index

Index contour is every fifth contour (for example every 5th line when CI = 2 m, giving 10 m, 20 m, 30 m ...) drawn as a thicker line and labelled with its RL, so that the map can be read quickly. The other contours are called intermediate (minor) contours.

Plotting index and minor contours

The contours are drawn from the spot levels of surveyed points by interpolation. Example: two surveyed points A (RL 52.4 m) and B (RL 61.8 m) plotted 9.4 cm apart; CI = 2 m, so the index contour is every 5th contour (multiples of 10 m).

1. List the contour values between the two RLs: 54, 56, 58, 60 m. Of these, 60 m is an index contour (multiple of 10 m); 54, 56 and 58 are minor contours.
2. Assuming a uniform slope, find the distance of each contour from A: $x = \dfrac{h_c - 52.4}{61.8 - 52.4}\times 9.4$.

3. Mark these points on AB, repeat for all the pairs of points, and join the points of equal value with smooth curves.
4. Draw the index contour (60 m) thicker, with its value written in a break in the line; the minor ones are thin lines.

   A(52.4)  54   56   58  60(index)  B(61.8)
   o--------|----|----|----||--------o

Why contours never touch or cross

A contour joins points of one fixed elevation. If two contours of different elevations touched or crossed, the common point would have two different elevations at the same place, which is impossible. So contour lines never meet. The only exceptions are:

• a vertical cliff, where the contours of different RL fall on the same plan position and appear to merge into one line, and
• an overhanging cliff or cave, where the upper contour projects over the lower, so the lines appear to cross (the hidden part is drawn dotted).

Contour interval

Contour interval (CI) is the constant vertical distance between two consecutive contours, for example 1 m or 5 m. It is the same over one map. Small CI is used for flat ground and large-scale maps; large CI for hilly ground and small-scale maps.

Index contour

Index contour is every fifth contour (for example every 5th line when CI = 2 m, giving 10 m, 20 m, 30 m ...) drawn as a thicker line and labelled with its RL, so that the map can be read quickly. The other contours are called intermediate (minor) contours.

Direct method

In the direct method the contours are located and marked directly on the ground, and the points are then surveyed and plotted. It is accurate but slow and costly, so it is used for small areas and large-scale maps.

Steps

1. Establish control (traverse stations) and a benchmark in the area.
2. Set up a level at a convenient point, take a back-sight on the BM and find the height of instrument: $HI = RL_{BM} + BS$.
3. For a contour of RL $h$, the staff reading required is $r = HI - h$. A staff man moves up or down the slope until the staff reading equals $r$; that point lies on the contour. Mark it with a peg.
4. Find several points of the same contour, then move to the next contour (staff reading changed by the CI).
5. When the level is shifted, a new HI is found by sighting a fixed point (change point).
6. The pegs are then located by the plane table, theodolite (tacheometer), chain or compass from the control stations and plotted on the sheet. The points of equal RL are joined by a smooth curve.

For steep ground a clinometer or a hand level may be used with a ranging rod.

Indirect method

In the indirect method only selected points (spot levels) are surveyed and plotted, and the contours are drawn later by interpolation. It is cheaper, quicker and suitable for large areas and small-scale maps.

1. Method of squares (grid method): The area is divided into squares of 5 m to 20 m side (smaller for a rough ground). The corners are pegged and their levels are taken with a level. The grid is plotted, RLs are written at the corners and the contours are interpolated along the grid lines. It is suitable for small flat areas.
2. Method of cross-sections: Cross-sections are taken at right angles to a base line or a main line at fixed intervals (10 m to 30 m). Levels are taken at regular intervals and at the break points along each section. Used for route survey (roads, railways, canals).
3. Radial line method: Radial lines are set out from a central station at known angles; points on them are located by distance and their levels are taken (by a level or by tacheometry). It is suitable for a hilly area where a hill can be observed from the summit.
4. Tacheometric method: Spot heights and positions are obtained together with a theodolite and staff; used for hilly ground.

Contour gradient

Contour gradient is the line of constant slope (uniform gradient) that cuts the contours at the same horizontal equivalent. It is laid on the ground to fix the route of a road, railway, canal or pipe line. The gradient = CI / HE = 1 in (HE/CI).

Example: for a road of 1 in 20 gradient and CI = 1 m, the horizontal equivalent is 20 m, so a divider opened to 20 m (at map scale) is stepped from contour to contour to trace the route.

Index contour

Index contour is every fifth contour (for example every 5th line when CI = 2 m, giving 10 m, 20 m, 30 m ...) drawn as a thicker line and labelled with its RL, so that the map can be read quickly. The other contours are called intermediate (minor) contours.

Factors determining contour interval

1. Purpose of the map: a small CI is used for detailed engineering design (dam, building site); a large CI for a general reconnaissance map or a route survey.
2. Scale of the map: a large-scale map (1:500) has a small CI (0.25 m to 0.5 m); a small-scale map has a large CI (5 m to 20 m). The usual rule is CI (m) = (map scale denominator) / (some factor) so that lines do not crowd.
3. Nature of the ground: flat ground needs a small CI (0.25 to 1 m) to show the relief; steep or hilly ground needs a large CI (5 to 10 m or more) or the contours crowd together.
4. Time and money available: a small CI needs more field points and more time, so cost increases.
5. Accuracy required: if the work needs high accuracy (earthwork volume) a small CI is chosen.
6. Size of the area: for a large area the CI is large; for a small area the CI is small.

The contour interval should be kept constant for one map so that the slope can be seen from the spacing of the contours.

The contour interval is the constant vertical distance between two successive contours. It is chosen from the following factors.

1. Purpose of the map: a small CI is used for detailed engineering design (dam, building site); a large CI for a general reconnaissance map or a route survey.
2. Scale of the map: a large-scale map (1:500) has a small CI (0.25 m to 0.5 m); a small-scale map has a large CI (5 m to 20 m). The usual rule is CI (m) = (map scale denominator) / (some factor) so that lines do not crowd.
3. Nature of the ground: flat ground needs a small CI (0.25 to 1 m) to show the relief; steep or hilly ground needs a large CI (5 to 10 m or more) or the contours crowd together.
4. Time and money available: a small CI needs more field points and more time, so cost increases.
5. Accuracy required: if the work needs high accuracy (earthwork volume) a small CI is chosen.
6. Size of the area: for a large area the CI is large; for a small area the CI is small.

The contour interval should be kept constant for one map so that the slope can be seen from the spacing of the contours.

Index contour

Index contour is every fifth contour (for example every 5th line when CI = 2 m, giving 10 m, 20 m, 30 m ...) drawn as a thicker line and labelled with its RL, so that the map can be read quickly. The other contours are called intermediate (minor) contours.

Indirect method of contouring

In the indirect method only selected points (spot levels) are surveyed and plotted, and the contours are drawn later by interpolation. It is cheaper, quicker and suitable for large areas and small-scale maps.

1. Method of squares (grid method): The area is divided into squares of 5 m to 20 m side (smaller for a rough ground). The corners are pegged and their levels are taken with a level. The grid is plotted, RLs are written at the corners and the contours are interpolated along the grid lines. It is suitable for small flat areas.
2. Method of cross-sections: Cross-sections are taken at right angles to a base line or a main line at fixed intervals (10 m to 30 m). Levels are taken at regular intervals and at the break points along each section. Used for route survey (roads, railways, canals).
3. Radial line method: Radial lines are set out from a central station at known angles; points on them are located by distance and their levels are taken (by a level or by tacheometry). It is suitable for a hilly area where a hill can be observed from the summit.
4. Tacheometric method: Spot heights and positions are obtained together with a theodolite and staff; used for hilly ground.

Procedure for control: the area is first controlled by a traverse and levelled BMs. Spot levels of the points (grid corners, cross-section points or radial points) are taken from these. The points are plotted to scale, their RLs are written, and the contours are drawn by interpolation, with every fifth contour as an index contour.

Direct method of contouring

In the direct method the contours are located and marked directly on the ground, and the points are then surveyed and plotted. It is accurate but slow and costly, so it is used for small areas and large-scale maps.

Steps

1. Establish control (traverse stations) and a benchmark in the area.
2. Set up a level at a convenient point, take a back-sight on the BM and find the height of instrument: $HI = RL_{BM} + BS$.
3. For a contour of RL $h$, the staff reading required is $r = HI - h$. A staff man moves up or down the slope until the staff reading equals $r$; that point lies on the contour. Mark it with a peg.
4. Find several points of the same contour, then move to the next contour (staff reading changed by the CI).
5. When the level is shifted, a new HI is found by sighting a fixed point (change point).
6. The pegs are then located by the plane table, theodolite (tacheometer), chain or compass from the control stations and plotted on the sheet. The points of equal RL are joined by a smooth curve.

For steep ground a clinometer or a hand level may be used with a ranging rod.

Uses of a contour map

1. To find the elevation of any point and the slope between two points from the map (without going to the field).
2. To draw the longitudinal section and cross-sections of a road, railway, canal or pipe line on paper.
3. To fix the best alignment of a road, railway or canal with a given gradient (contour gradient).
4. To find the intervisibility of two points (to decide if a signal station can be seen from another).
5. To calculate the capacity of a reservoir, the catchment area and the volume of earthwork (area between contours).
6. To select suitable sites for a dam, building, bridge, airport or a water tank, and to mark the drainage lines (watershed).
7. To find the quantity of cutting and filling for levelling a site and to plan irrigation and drainage layout.
8. To study the general topography (hills, valleys, ridges) of a large area, and for the military and town planning.

Contour interpolation is the process of locating the positions of contours of required values between two plotted points of known RL. It assumes the ground slope is uniform between the two points.

Methods of interpolation

1. Estimation: The positions of contours are estimated by eye between the two points. It is quick but rough.
2. Arithmetical calculation: The distance from the lower point to the contour is found by proportion:

$$x = \frac{h_c - h_1}{h_2 - h_1}\times d$$

where $h_1, h_2$ are the RLs of the two points ($h_2 > h_1$), $h_c$ is the contour value and $d$ is the plotted distance between the points. It is accurate but slow. 3. Graphical methods

• Tracing paper (transparent sheet) method: Parallel equidistant lines are drawn on tracing paper and numbered with the contour values. The paper is turned and moved until the lines for the two end RLs pass through the two points. The crossing of line AB with the lines of the paper marks the contours. It is quick and common.
• Similar triangles method: A line is drawn from each end point at right angles to AB, with lengths in proportion to the RLs (scale of levels). The ends are joined and the equal divisions are projected on AB by parallel lines.

The tracing paper method is quick, simple and accurate enough, so it is recommended for a large volume of work.

Application of the tracing paper method for a large area

1. Plot all spot levels on the sheet, with their RLs written beside each point.
2. For the CI = 1 m (say), draw 10 equal-spaced parallel lines on tracing paper (or use a ruled transparent sheet) and number them 0 to 10 for the required levels.
3. To interpolate between A (RL 52.4) and B (RL 57.8), place the paper so that the line for 52.4 passes through A (estimate between lines 52 and 53) and the line 57.8 passes through B.
4. Prick the points where AB crosses the lines 53, 54, 55, 56, 57 on the sheet.
5. Repeat for all pairs of neighbouring points, then join the equal values by smooth curves and thicken every fifth contour.

Methods of contouring

Contouring is done by two methods: the direct method and the indirect method.

Direct method

In the direct method the contours are located and marked directly on the ground, and the points are then surveyed and plotted. It is accurate but slow and costly, so it is used for small areas and large-scale maps.

Steps

1. Establish control (traverse stations) and a benchmark in the area.
2. Set up a level at a convenient point, take a back-sight on the BM and find the height of instrument: $HI = RL_{BM} + BS$.
3. For a contour of RL $h$, the staff reading required is $r = HI - h$. A staff man moves up or down the slope until the staff reading equals $r$; that point lies on the contour. Mark it with a peg.
4. Find several points of the same contour, then move to the next contour (staff reading changed by the CI).
5. When the level is shifted, a new HI is found by sighting a fixed point (change point).
6. The pegs are then located by the plane table, theodolite (tacheometer), chain or compass from the control stations and plotted on the sheet. The points of equal RL are joined by a smooth curve.

For steep ground a clinometer or a hand level may be used with a ranging rod.

Indirect method

In the indirect method only selected points (spot levels) are surveyed and plotted, and the contours are drawn later by interpolation. It is cheaper, quicker and suitable for large areas and small-scale maps.

1. Method of squares (grid method): The area is divided into squares of 5 m to 20 m side (smaller for a rough ground). The corners are pegged and their levels are taken with a level. The grid is plotted, RLs are written at the corners and the contours are interpolated along the grid lines. It is suitable for small flat areas.
2. Method of cross-sections: Cross-sections are taken at right angles to a base line or a main line at fixed intervals (10 m to 30 m). Levels are taken at regular intervals and at the break points along each section. Used for route survey (roads, railways, canals).
3. Radial line method: Radial lines are set out from a central station at known angles; points on them are located by distance and their levels are taken (by a level or by tacheometry). It is suitable for a hilly area where a hill can be observed from the summit.
4. Tacheometric method: Spot heights and positions are obtained together with a theodolite and staff; used for hilly ground.

Uses of contour maps

1. To find the elevation of any point and the slope between two points from the map (without going to the field).
2. To draw the longitudinal section and cross-sections of a road, railway, canal or pipe line on paper.
3. To fix the best alignment of a road, railway or canal with a given gradient (contour gradient).
4. To find the intervisibility of two points (to decide if a signal station can be seen from another).
5. To calculate the capacity of a reservoir, the catchment area and the volume of earthwork (area between contours).
6. To select suitable sites for a dam, building, bridge, airport or a water tank, and to mark the drainage lines (watershed).
7. To find the quantity of cutting and filling for levelling a site and to plan irrigation and drainage layout.
8. To study the general topography (hills, valleys, ridges) of a large area, and for the military and town planning.

Contour gradient

Contour gradient is the line of constant slope (uniform gradient) that cuts the contours at the same horizontal equivalent. It is laid on the ground to fix the route of a road, railway, canal or pipe line. The gradient = CI / HE = 1 in (HE/CI).

Characteristics of contours

1. Contours are closed curves. Every contour must close on itself, either inside the map or outside it. It cannot start or end abruptly.
2. All points on a contour have the same elevation.
3. Contours never cross each other, except at an overhanging cliff or a cave. Two contours of different elevation cannot meet or cross because one point cannot have two elevations.
4. Contours of different elevations do not unite to form one line, except at a vertical cliff, where they coincide and appear as one line.
5. Closely spaced contours indicate a steep slope; widely spaced contours indicate a gentle slope. Equally spaced contours indicate a uniform slope, and straight, equal and parallel contours indicate an inclined plane.
6. Contours cross ridge lines and valley lines at right angles. The direction of the steepest slope is at right angles to the contour.
7. A set of concentric closed contours with the higher value inside represents a hill; with the lower value inside it represents a depression (pond or pit).
8. Contours bend as V or U shapes across a ridge and a valley. For a ridge (spur) the bend of the contour points towards the lower ground; for a valley the bend points towards the higher ground.
9. Contour of the same elevation may reappear on the other side of a ridge or a valley, so one contour can appear twice in a map.
10. A contour cannot lie between two contours of higher (or lower) values unless it closes on itself (hill or depression).
11. Contours of a saddle (col) show a pair of hills (high) and a pair of depressions (low) opposite to each other, with the saddle point at the neck.

        HILL                    DEPRESSION
      .--100--.                .--100--.
     / .-110-. \              / .--90-. \
    | / (120) \ |            | /  (80)  \ |
     \ '-----' /              \ '-----' /
      '-------'                '-------'
   RL rises to the centre     RL falls to the centre

   RIDGE (v points downhill)     VALLEY (^ points uphill)
      130  \   /                    120     /\
            \ /                           /  \
      120  \   /                    110   /    \
            \_/                          /      \
      110  \     /                  100 /        \
            \___/
   high ground at top            high ground at top

   Steep slope      Gentle slope      Uniform slope
   (close)          (wide apart)      (equal spacing)
   ----100          ----100           ----100
   ----101                            ----101
   ----102                            ----102
   ----103          ----101           ----103

Principal characteristics of contours

1. Contours are closed curves. Every contour must close on itself, either inside the map or outside it. It cannot start or end abruptly.
2. All points on a contour have the same elevation.
3. Contours never cross each other, except at an overhanging cliff or a cave. Two contours of different elevation cannot meet or cross because one point cannot have two elevations.
4. Contours of different elevations do not unite to form one line, except at a vertical cliff, where they coincide and appear as one line.
5. Closely spaced contours indicate a steep slope; widely spaced contours indicate a gentle slope. Equally spaced contours indicate a uniform slope, and straight, equal and parallel contours indicate an inclined plane.
6. Contours cross ridge lines and valley lines at right angles. The direction of the steepest slope is at right angles to the contour.
7. A set of concentric closed contours with the higher value inside represents a hill; with the lower value inside it represents a depression (pond or pit).
8. Contours bend as V or U shapes across a ridge and a valley. For a ridge (spur) the bend of the contour points towards the lower ground; for a valley the bend points towards the higher ground.
9. Contour of the same elevation may reappear on the other side of a ridge or a valley, so one contour can appear twice in a map.
10. A contour cannot lie between two contours of higher (or lower) values unless it closes on itself (hill or depression).
11. Contours of a saddle (col) show a pair of hills (high) and a pair of depressions (low) opposite to each other, with the saddle point at the neck.

        HILL                    DEPRESSION
      .--100--.                .--100--.
     / .-110-. \              / .--90-. \
    | / (120) \ |            | /  (80)  \ |
     \ '-----' /              \ '-----' /
      '-------'                '-------'
   RL rises to the centre     RL falls to the centre

   RIDGE (v points downhill)     VALLEY (^ points uphill)
      130  \   /                    120     /\
            \ /                           /  \
      120  \   /                    110   /    \
            \_/                          /      \
      110  \     /                  100 /        \
            \___/
   high ground at top            high ground at top

   Steep slope      Gentle slope      Uniform slope
   (close)          (wide apart)      (equal spacing)
   ----100          ----100           ----100
   ----101                            ----101
   ----102                            ----102
   ----103          ----101           ----103

Uses of a contour map

1. To find the elevation of any point and the slope between two points from the map (without going to the field).
2. To draw the longitudinal section and cross-sections of a road, railway, canal or pipe line on paper.
3. To fix the best alignment of a road, railway or canal with a given gradient (contour gradient).
4. To find the intervisibility of two points (to decide if a signal station can be seen from another).
5. To calculate the capacity of a reservoir, the catchment area and the volume of earthwork (area between contours).
6. To select suitable sites for a dam, building, bridge, airport or a water tank, and to mark the drainage lines (watershed).
7. To find the quantity of cutting and filling for levelling a site and to plan irrigation and drainage layout.
8. To study the general topography (hills, valleys, ridges) of a large area, and for the military and town planning.

Underwater mapping (hydrographic or bathymetric survey) is the survey that finds the depth and shape of the bed of a river, lake, reservoir, harbour or sea, and shows it on a map with the shore features. The relief of the bed is shown by lines of equal depth called isobaths (underwater contours).

Objectives and uses

• Planning of dams, reservoirs and barrages: storage capacity and silting (sedimentation) surveys.
• Design and maintenance of harbours, ports and navigation channels; dredging quantities.
• Location of bridge piers and scour studies; river training and flood control.
• Laying of pipelines, cables and intake structures; navigation charts.

Field work

1. Reconnaissance and shore control: A traverse or base line is established on the shore and the shore line is surveyed with level, theodolite or total station.
2. Tide or water level gauge: A gauge is fixed on the shore and read at regular intervals so that every sounding can be reduced to a common datum (RL of the water surface minus depth).
3. Sounding: Depths are measured along planned lines (parallel or radial) from a boat.
   • Sounding rod or pole for depths up to about 5 m.
   • Lead line (weight on a graduated line) for depths up to about 20 m.
   • Echo sounder: a transducer sends a sound pulse and the time to return gives the depth, $d = \frac{v\,t}{2}$ ($v \approx 1500$ m/s in water). It is fast and gives a continuous record.
   • Multibeam sonar for complete coverage of the bed.
4. Location of the sounding points: The position of the boat at each sounding is fixed by: angles from two shore stations (intersection), one angle and a distance along a range, two angles on a sextant to three known shore marks (resection), distance by EDM, or by GPS (DGPS/RTK) which is the usual method now.
5. Booking: Each depth is recorded with time, point number and gauge reading.

Reduction and plotting

• The reduced level of the bed at each point: $RL_{bed} = RL_{water} - d$, where $RL_{water}$ is taken from the gauge at the time of sounding (and the transducer depth is corrected).
• The sounding points are plotted to scale on the shore map and their reduced depths written.
• Isobaths are drawn by interpolation, as for ordinary contours, at a suitable interval.

   shore base
   A ---------- B
   | \         /
   |  \  .   . /    . = sounding points
   |   \  . . /       located by angles at A and B
   |    \ . /
   ~~~~~~~~~~~~ water

Accuracy

Sounding is corrected for the tide, the speed of sound, the draught of the boat and rolling. Fixing sounding positions with errors less than the plotting accuracy (0.5 mm on the map) is required.

Contour interval

Contour interval (CI) is the constant vertical distance between two consecutive contours, for example 1 m or 5 m. It is the same over one map.

Index contour

Index contour is every fifth contour (for example every 5th line when CI = 2 m, giving 10 m, 20 m, 30 m ...) drawn as a thicker line and labelled with its RL, so that the map can be read quickly. The other contours are called intermediate (minor) contours.

Procedure for contour plotting (applying the surveying principle)

The surveying principle of working from the whole to the part is followed: a strong framework of control is fixed first, and details and contours are added within it.

1. Control survey: A traverse or triangulation framework is made and plotted accurately. Benchmarks are fixed by levelling.
2. Field survey of contour points: The ground points are located (and their RLs found) by the direct or the indirect method (grid, cross-section or tacheometry) from the control stations.
3. Plotting the points: The control stations are plotted first by coordinates; the survey points are then plotted from them to scale. The RL of each point is written beside it.
4. Interpolation: Between each pair of neighbouring points the contours of required values are located by proportion, assuming a uniform slope between the points.
5. Drawing the contours: Points of equal RL are joined by smooth curves. Every fifth contour is drawn thick as an index contour and labelled. Contours are drawn without crossing each other.
6. Checking: The map is checked with the ground (shape of hills, valleys, ridge lines) and the title, scale, CI and north line are added.

Uses of contours

1. To find the elevation of any point and the slope between two points from the map (without going to the field).
2. To draw the longitudinal section and cross-sections of a road, railway, canal or pipe line on paper.
3. To fix the best alignment of a road, railway or canal with a given gradient (contour gradient).
4. To find the intervisibility of two points (to decide if a signal station can be seen from another).
5. To calculate the capacity of a reservoir, the catchment area and the volume of earthwork (area between contours).
6. To select suitable sites for a dam, building, bridge, airport or a water tank, and to mark the drainage lines (watershed).
7. To find the quantity of cutting and filling for levelling a site and to plan irrigation and drainage layout.
8. To study the general topography (hills, valleys, ridges) of a large area, and for the military and town planning.

Six important characteristics of contours

1. Contours are closed curves. Every contour must close on itself, either inside the map or outside it. It cannot start or end abruptly.
2. All points on a contour have the same elevation.
3. Contours never cross each other, except at an overhanging cliff or a cave. Two contours of different elevation cannot meet or cross because one point cannot have two elevations.
4. Contours of different elevations do not unite to form one line, except at a vertical cliff, where they coincide and appear as one line.
5. Closely spaced contours indicate a steep slope; widely spaced contours indicate a gentle slope. Equally spaced contours indicate a uniform slope, and straight, equal and parallel contours indicate an inclined plane.
6. Contours cross ridge lines and valley lines at right angles. The direction of the steepest slope is at right angles to the contour.

        HILL                    DEPRESSION
      .--100--.                .--100--.
     / .-110-. \              / .--90-. \
    | / (120) \ |            | /  (80)  \ |
     \ '-----' /              \ '-----' /
      '-------'                '-------'
   RL rises to the centre     RL falls to the centre

   RIDGE (v points downhill)     VALLEY (^ points uphill)
      130  \   /                    120     /\
            \ /                           /  \
      120  \   /                    110   /    \
            \_/                          /      \
      110  \     /                  100 /        \
            \___/
   high ground at top            high ground at top

   Steep slope      Gentle slope      Uniform slope
   (close)          (wide apart)      (equal spacing)
   ----100          ----100           ----100
   ----101                            ----101
   ----102                            ----102
   ----103          ----101           ----103

Distances of contours from guide point A

Given: $RL_A = 1000.25$ m, $RL_B = 1001.32$ m, distance on the map $AB = 5$ cm, CI = 0.5 m.

Difference of level $= 1001.32 - 1000.25 = 1.07$ m.

Contours lying between A and B (multiples of 0.5 m): 1000.50 m and 1001.00 m (the next one, 1001.50 m, is above B).

The slope is taken as uniform, so the distance from A is proportional to the height above A:

$$x = \frac{h_c - 1000.25}{1.07}\times 5\ \text{cm}$$

   A(1000.25)   1000.50      1001.00        B(1001.32)
   o-------------|-------------|-------------o
   0          1.17 cm      3.50 cm        5 cm

Answer: the 1000.50 m contour is 1.17 cm and the 1001.00 m contour is 3.50 cm from A.

Contour index

An index contour is every fifth contour, drawn as a thicker line and labelled with its RL so that contours can be identified easily. With CI = 3 m, the index contours are at every 15 m: 1245 m, 1260 m, ... The contours between them are the intermediate (minor) contours.

Fixing index and minor contours between 1240.12 m and 1264.32 m

The ground between the two points has a uniform slope, so contour positions are found by proportion.

1. Contour values are multiples of CI = 3 m. Between 1240.12 and 1264.32 these are 1242, 1245, 1248, 1251, 1254, 1257, 1260, 1263 m.
2. Of these, 1245 m and 1260 m are multiples of 15 m, so they are index contours; the other six are minor contours.
3. The difference in level between the points is $1264.32 - 1240.12 = 24.20$ m. If the points are $L$ cm apart on the map, the distance of a contour of value $h_c$ from the lower point is

$$x = \frac{h_c - 1240.12}{24.20}\times L$$

Taking the plotted distance as $L = 12.1$ cm (so that $x = (h_c - 1240.12)/2$ cm):

4. Mark these positions on the line joining the points, repeat for the neighbouring points, join equal values with smooth curves, and draw the 1245 m and 1260 m contours thick and label them.

 1240.12  1242 1245 1248 1251 1254 1257 1260 1263   1264.32
   o-------|----||---|----|----|----|----||---|-------o
                 idx                     idx

The spacing between neighbouring contours is equal ($3/24.2\times 12.1 = 1.50$ cm) because the slope is uniform.

Why contour lines never touch or cross

All points of a contour have the same RL. If two contours of different RL touched or crossed, the common point would have two different elevations, which is impossible. The only exceptions are a vertical cliff, where contours of different value fall on the same plan line and appear to merge, and an overhanging cliff or cave, where the upper contour is drawn over the lower one and appears to cross it (the hidden part is dotted).

Interpolation of contours

Interpolation of contours is the process of locating the positions of contours of given values between plotted points of known RL, assuming that the ground slope between the points is uniform. The surveyed points are only a few spot levels; the contours are drawn through them by this process.

Methods of interpolation

1. Estimation: The positions of contours are estimated by eye between the two points. It is quick but rough.
2. Arithmetical calculation: The distance from the lower point to the contour is found by proportion:

$$x = \frac{h_c - h_1}{h_2 - h_1}\times d$$

where $h_1, h_2$ are the RLs of the two points ($h_2 > h_1$), $h_c$ is the contour value and $d$ is the plotted distance between the points. It is accurate but slow. 3. Graphical methods

• Tracing paper (transparent sheet) method: Parallel equidistant lines are drawn on tracing paper and numbered with the contour values. The paper is turned and moved until the lines for the two end RLs pass through the two points. The crossing of line AB with the lines of the paper marks the contours. It is quick and common.
• Similar triangles method: A line is drawn from each end point at right angles to AB, with lengths in proportion to the RLs (scale of levels). The ends are joined and the equal divisions are projected on AB by parallel lines.

The tracing paper method is quick, simple and accurate enough, so it is recommended for a large volume of work.

Uses of contour maps

1. To find the elevation of any point and the slope between two points from the map (without going to the field).
2. To draw the longitudinal section and cross-sections of a road, railway, canal or pipe line on paper.
3. To fix the best alignment of a road, railway or canal with a given gradient (contour gradient).
4. To find the intervisibility of two points (to decide if a signal station can be seen from another).
5. To calculate the capacity of a reservoir, the catchment area and the volume of earthwork (area between contours).
6. To select suitable sites for a dam, building, bridge, airport or a water tank, and to mark the drainage lines (watershed).
7. To find the quantity of cutting and filling for levelling a site and to plan irrigation and drainage layout.
8. To study the general topography (hills, valleys, ridges) of a large area, and for the military and town planning.

Factors affecting contour interval

1. Purpose of the map: a small CI is used for detailed engineering design (dam, building site); a large CI for a general reconnaissance map or a route survey.
2. Scale of the map: a large-scale map (1:500) has a small CI (0.25 m to 0.5 m); a small-scale map has a large CI (5 m to 20 m). The usual rule is CI (m) = (map scale denominator) / (some factor) so that lines do not crowd.
3. Nature of the ground: flat ground needs a small CI (0.25 to 1 m) to show the relief; steep or hilly ground needs a large CI (5 to 10 m or more) or the contours crowd together.
4. Time and money available: a small CI needs more field points and more time, so cost increases.
5. Accuracy required: if the work needs high accuracy (earthwork volume) a small CI is chosen.
6. Size of the area: for a large area the CI is large; for a small area the CI is small.

The contour interval should be kept constant for one map so that the slope can be seen from the spacing of the contours.

Characteristics of contours with sketches

1. Contours are closed curves. Every contour must close on itself, either inside the map or outside it. It cannot start or end abruptly.
2. All points on a contour have the same elevation.
3. Contours never cross each other, except at an overhanging cliff or a cave. Two contours of different elevation cannot meet or cross because one point cannot have two elevations.
4. Contours of different elevations do not unite to form one line, except at a vertical cliff, where they coincide and appear as one line.
5. Closely spaced contours indicate a steep slope; widely spaced contours indicate a gentle slope. Equally spaced contours indicate a uniform slope, and straight, equal and parallel contours indicate an inclined plane.
6. Contours cross ridge lines and valley lines at right angles. The direction of the steepest slope is at right angles to the contour.
7. A set of concentric closed contours with the higher value inside represents a hill; with the lower value inside it represents a depression (pond or pit).
8. Contours bend as V or U shapes across a ridge and a valley. For a ridge (spur) the bend of the contour points towards the lower ground; for a valley the bend points towards the higher ground.
9. Contour of the same elevation may reappear on the other side of a ridge or a valley, so one contour can appear twice in a map.
10. A contour cannot lie between two contours of higher (or lower) values unless it closes on itself (hill or depression).

        HILL                    DEPRESSION
      .--100--.                .--100--.
     / .-110-. \              / .--90-. \
    | / (120) \ |            | /  (80)  \ |
     \ '-----' /              \ '-----' /
      '-------'                '-------'
   RL rises to the centre     RL falls to the centre

   RIDGE (v points downhill)     VALLEY (^ points uphill)
      130  \   /                    120     /\
            \ /                           /  \
      120  \   /                    110   /    \
            \_/                          /      \
      110  \     /                  100 /        \
            \___/
   high ground at top            high ground at top

   Steep slope      Gentle slope      Uniform slope
   (close)          (wide apart)      (equal spacing)
   ----100          ----100           ----100
   ----101                            ----101
   ----102                            ----102
   ----103          ----101           ----103

Methods of contouring

There are two methods:

1. Direct method: contours are traced on the ground and located. Suitable for small areas and large-scale maps.
2. Indirect method: spot levels are taken at grid points, cross-sections or radial lines, and contours are interpolated. Suitable for large areas. Its types are the method of squares, the method of cross-sections and the radial line method (tacheometry also is used).

Field procedure of the direct method

1. Establish control (traverse stations) and a benchmark in the area.
2. Set up a level at a convenient point, take a back-sight on the BM and find the height of instrument: $HI = RL_{BM} + BS$.
3. For a contour of RL $h$, the staff reading required is $r = HI - h$. A staff man moves up or down the slope until the staff reading equals $r$; that point lies on the contour. Mark it with a peg.
4. Find several points of the same contour, then move to the next contour (staff reading changed by the CI).
5. When the level is shifted, a new HI is found by sighting a fixed point (change point).
6. The pegs are then located by the plane table, theodolite (tacheometer), chain or compass from the control stations and plotted on the sheet. The points of equal RL are joined by a smooth curve.

For steep ground a clinometer or a hand level may be used with a ranging rod.