7.10. Determining areas from aerial photographs

Areas of land surfaces can be determined directly from aerial photographs. This requires, however, the delineation of the boundaries of the units whose areas are to be determined. The area of many regularly shaped surfaces can be calculated by using the appropriate formulas shown in Appendix, Table 1. Areas of non regular shaped surfaces require more advanced calculus and techniques discussed later in this chapter. A variety of methods for determining land areas of any unit shape from aerial photographs or maps have been developed. Instruments and techniques for determining areas range from simple to complex, archaic to high technology, cheap to very expensive. The principal methods and devices used for area determination are: integration, coordinates, weighing, transects, dot grid, and planimeters.

Whether the surface is regular or irregular, mathematical area measurements basically require knowledge of the linear magnitude of two dimensions. However, beside accurate mathematical formulas, several practical techniques and procedures are available to estimate surface areas.

Area determination of the earth's surface usually considers the area of the orthogonal projection of the irregular land surface to a datum-level plane. Area determination for portions of the earth's surface can be expressed as the square of any of the linear units. The most common area units are the international metric system units: square centimeters (cm2), square meters (m2), hectares (ha), and square kilometers (km2). However, in various countries, other units have become standard through common usage. For example, in the United States, the acre (43,560 square feet or 10 square chains) is the most common area unit (see Appendix, Table 2 for converting factors).

7.10.1. Area by coordinates

If we can establish an xy-axis system on a plane (e.g., aerial photos), the area of a polygon of any shape can be computed. Suppose we take the east-west direction of a photograph as the x-axis, the north-south direction as a y-axis, and the xy origin as (0,0). If a polygon of any shape is bounded by straight segments, we can determine the x and y coordinates of the vertices by measuring the distance between the origin and the orthogonal projection of that vertex to the x-axis (for x coordinates of the vertex) and to the y axis (for y coordinates of the vertex) (figure 7.18).
 
Figure 7.18. Area by coordinates; xi and yi are the photo coordinates of vertices.

If the coordinates are designated as x1, x2, x3, ....., xn and the y coordinates are y1, y2, y3, ...., yn, then the area of the polygon whose vertices are (y1, y1), (y2, y2), (y3, y3), ..., (yn, yn) is the absolute value of:

if the ground coordinates of vertices are known the ground area may be directly determined by substituting these coordinates in equation 7.25. Otherwise, the scale of the photograph may be used to determine the surface area on the ground as follows: after determining the polygon area using the photo coordinates, this surface is then multiplied by the square of the photo scale reciprocal:

Photo area ´ PSR2 = Ground area

Example 7.19

The photo coordinates of the 5 vertices of a polygon delineated on a 1:12,000 vertical aerial photograph were measured to be x1 = 5.12 cm, y1 = 1.54 cm, x2 = 5.61 cm, y2 = 4.37 cm, x3 = 7.13 cm, y3 = 6.20 cm, x4 = 9.83, y4 = 6.81 cm, x5 = 8.08 cm, y5 = 3.44 cm. Find the area of the polygon.

Solution:

Using equation 7.25, the photographic area of the polygon is:

½(|[(5.12)(4.37) + (5.61)(6.20) + (7.13)(6.81) + (9.83)(3.44) + (8.08)(1.54)] -

[(5.61)(1.54) + (7.13)(4.37) + (9.83)(6.20) + (8.08)(6.81) + (5.12)(3.44)]|)

= |151.97 - 173.38| = 21.41 cm2

The ground area of the polygon is then:

21.41 cm2 ´ (12000)2 = 308304 104 cm2 or 30.83 hectares

Example 7.20

Using the same photograph in example 7.19, another polygon was delineated and the ground coordinates of its 5 vertices were found to be X1 = 514852, Y1 = 5178256, X2 = 515495, Y2 = 5180687, X3 = 518168, Y3 = 5181112, X4 = 518738, Y4 = 5179680, X5 = 517766, Y5 = 5178184.

Find the ground area of the polygon.

Solution:

Using equation 7.25, the ground area of the polygon can directly be computed as:

½|[(514852)(5180687) + (515495)(5181112) + (518168)(5179680) + (518738)(5178184) + (517766)(5178256)] - [(515495)(5178256) + (518168)(5180687) + (518738)(5181112) + (517766)(5179680) + (514852)(5178184)]| = 851.50 ha

If the polygon is delineated by arcs instead of straight segments, the above formula can still be used to estimate the area of the polygon. In this case, the boundaries of the polygon should be divided into small arcs that could approach a straight segment (figure 7.19). The line segments shown in figure 7.19 are purposely drawn large to better illustrate the concept of this technique. Usually, the curved boundary is divided into small arcs that approach straight segments. The shorter the arcs, the more accurate the area estimation.
 
 

Figure 7.19. Diagram illustrating area estimation by coordinates when the surface boundaries are curves instead of straight lines. 

However, a more accurate method to determine the area of a "non-straight-line-boundary-polygon" is the use of integration, planimeters or computers. All of these techniques are based on the coordinate method described above to determine the area of a polygon.

7.10.2 Area by integration

This technique supposes that the plane surface is bounded by known functions (figure 7.20). In this figure, the area of the shaded surface may be determined by using the area bounded by the X-axis and ordinates at x = a and x = b or by using the area bounded by the Y-axis and ordinates y = c and y = d. The area of this plane surface can then be calculated using the definite integral:

Where dx and dy are the small portions of the curve that approach a straight segment. This technique is rarely used in practical photography area measurement and, therefore, will not be discussed further.
 
 

Figure 7.20. Estimating polygon area by integration.

8.10.3. area by weighing

A surface area of a polygon on plane sheet of uniform thickness can be estimated using the weighing procedure. First, the area and weight of the entire plane sheet are determined; the weight of the map of the photograph paper will be directly proportional to its surface area. Then, the total photo surface and weight are used as the basis to determine the ratio of area to unit weight. For example, if the plane sheet is a 23 cm by 23 cm (9 inches by 9 inches) aerial photograph, its area is 23´23 = 529 cm2. The photo area is then converted to ground surface by multiplying the value of the photo area (529 cm2 in this case) by the square value of the photo scale reciprocal (i.e., PSR2):

Ground Area = Photo Area ´ (PSR)2

For example, if a 23 cm by 23 cm aerial photo weighs 20 grams, then the ratio of area to unit weight is 529 cm2 ¸ 20 g = 26.45 cm2/g. In other words, 1 g of paper would be equivalent to an area of 26.45 cm2 on the photo or 26.45 cm2 ´ PSR2 on the ground. Areas of polygons of any configuration can then be cut out and weighed. Each gram will be equivalent to 26.45 cm2 ´ PSR2 ground area.

Suppose we have interpreted and delineated the units of interest on a set of photographs. The units can then be cut out and sorted by category of information (e.g., all the polygons that represent bare soil are cut and put in the same pile, all the polygons that represent forested areas are cut out and put in a separate pile, etc.). Then, by weighing the piles of paper for each category, the total area of each category can be determined and the percentage area may be compared to the entire project area. The total project area will also serve for checking against the total weights of the individual categories.

Example 7.21

suppose we interpreted an aerial photograph at a nominal scale of 1:20,000 for a land use project. If the aerial photograph is 23 cm by 23 cm and weighs 20 grams, find the ratio of area to the unit weight and the area of a lake that was cut out and found to weigh 4 grams.

Solution:

20 g is equivalent to an area of 23 cm ´ 23 cm = 529 cm2 on the photograph or 529 ´ (20,000)2 cm2 = 2116 ´ 108 cm2 on the ground = 2116 ´ 104 m2 on the ground (because 1 m2 = 10,000 cm2) = 2116 ha on the ground (because 1 ha = 10,000 m2).

If 20 grams = 2116 ha, therefore, 1 gram = 105.8 ha, which represents the ratio of area to the unit weight. If the polygon corresponding to the lake weighs 4 grams, its surface is:

4 ´ 105.8 = 423.2 ha or 20 % ([423.2 ¸ 2116] ´ 100) of the entire area covered by the photo.

The same procedure can be applied for the rest of the categories interpreted on the photograph or a set of photographs or a map.

This simple technique was developed by the U.S. Soil Conservation Service and has been found to be both efficient and accurate. The accuracy is of course dependent, among others, upon the homogeneity of the thickness of the paper, the precision of cutting the polygons, and the scale used to weigh the paper.

7.10.4. Area by transects

The transect method is based on the assumption that linear measurements vary directly as do areal measurements, provided the sampling technique is unbiased and the number of samples is sufficient. Like the weighing method, the transect method is basically a technique for proportioning the total acreage of a known area among the constituent categories of unknown areas. The transect method uses regularly spaced lines that are drawn directly on the area of interest or on a transparent overlay that can be laid on top of the area (figure 7.21). Using a graduated scale, the distance of all the segments passing through each unit are measured to the nearest desired unit (e.g., 0.1 mm) and added up for each category constituting the area of interest. The area of each category is then determined by relating the total measure of a given area to the total linear distance covering the area:

where a is the area of the category to be determined, A is the total area of all the categories (i.e., the project area), l is the sum of all the segments (transects) passing through the category, and L is the sum of all the transects inside the project area considered. For example, in figure 7.21, L = S Li = L1 + L2 + ... +L11, l = S li = l1 + l2 + l3 + l4, A = x ´ y, and a = (A ´ l) ¸ L.
 
 

Figure 7.21. Determining polygon area from total area using the transect technique.
For example, if 15 equally spaced and parallel lines were overlaid parallel to the side of a 23 cm by 23 cm photograph, the total transect length would be 23 cm x 15 = 345 cm. If wetlands areas were crossed for a total of 146 cm, this category would occupy an area of 146 ¸ 345 or 42.32 percent of the total area of the 23 cm by 23 cm photograph. On a 1:15840 scale photograph, this percentage would be equivalent to: 0.4232 ´ (0.23 ´ 0.23) m2 ´ 158402 = 561.7 ´ 104 m2 or 561.7 ha.

The number of and spacing between transects used depend upon the size and arrangement of the polygons to be measured; the higher the number, the more accurate the measurements. Generally, at least 10 transects should be used and a higher number for accuracy, depending on the size of the area to be measured. The transect lines may randomly be laid across the area to be measured. However, for regularly shaped units (e.g., rectangular or linear), the transects lines should be laid across the short dimensions to obtain the best sample (figure 7.22). The transect lines must not be laid parallel to the boundaries of long polygons.
 
 

Figure 7.22. Diagram illustrating the proper orientation of the transects in respect to the orientation of the polygons; the transects must not parallel any long straight polygon boundary.

One of the procedures of using the transect method may be as follows: Starting about half a centimeter from an appropriate boundary (so that the transect lines are not parallel to the sides of long polygons as described in the previous paragraph) of the area to be measured, transect lines are drawn 1 cm apart until the opposite boundary of the area (figure 7.23). The same process may be performed by overlaying a transparency on which the transects lines are already drawn. Next, the line segments intercepting the boundaries of the polygons are measured and the transect line distances are summarized for each polygon and each category (Table 7.1). The sum of proportions of all categories can then be verified to make sure they total one hundred percent (or 1.00) and the sum of the areas of individual categories totals the total area of the entire project area considered.
 
 
 
 
Table 7.1. Control table for area determination of individual categories using the transect technique.
Polygon type
(1)
 Total distance in a category 
(2)
 Proportion to total distance in the entire area
(3)
 Area of a category
(4)
 Correction factor

(5)

 Adjusted area
(6)


A
B
C
.
.
Z


S liA
S liB
S liC
.
.
S liZ


S liA / S Li
S LiB / SLi
S liC / S Li
.
.
S liZ / SLi


Equation
7.27
=
=
=
=


Equation
7.28
=
=
=
=


(4) + (5)
=
=
=
=
=


 


EL


100%


PAT


 


TA


 
 

Figure 7.26. Illustration of (a) Mechanical planimeter, (b) Electronic planimeter, and (c) schematic representation of a mechanical planimeter.
Verification of the paper plane

Measurements from paper may be subject to errors introduced from shrinkage or dilatation. When making measurements from a grided map, errors may also be introduced from defective griding. In either case, these errors must be known prior to area measurement. The dimensions of the grid on a map are usually known and errors due to paper deformation or defective griding may be determined by measuring the dimensions of the grid on the map and comparing them to the actual dimensions. For example, if a 100 mm by 100 mm grid were measured to be 100.2 mm and 100.3 mm, its area is 10050.06 mm2. Therefore, a correction factor of -50 mm2 per 10000 mm2 (or -5/1000) should be applied to the measured surfaces.

Determining areas using a planimeter

Before making measurement with a planimeter, the following issues must be taken into account.
 

  1. The paper should be taped down on a horizontal table to ovoid displacement of the paper or the sliding of the planimeter.
  2. The drawing paper should be smooth and free from cuts and wrinkles as these anomalies disturb the planimeter's maneuver.
  3. The pivot point, P, should be placed so that the polar arm, PB, is at a right angle with the tracing arm, BT, when T is placed in the middle of the polygon to be measured (figure 7.27).
  4. The base circle must be outside the polygon being measured and all the planimeter's components must be on the drawing paper. If the polygon is too big, it may be arbitrarily divided into smaller surfaces, which are measured separately and added up. Although it is possible to measure large areas with the base circle inside the polygon being measured, this method is more complicated than dividing the polygon into smaller areas so that the base circle is outside the area measured.
  5. In a given tracing direction, the angle PBT must not exceed 180o during the rotation of the wheel. This situation is shown in figure 7.28, which exhibits an incompatible positions of PBT and PB'T'.
  6. The roller edge (i.e., measuring wheel) should not be touched at any time as this may introduce errors in measurements and/or cause it to rust or corrode, which will adversely affect the measuring accuracy.
Figure 7.27. Proper positioning of the planimeter for area measurement. The polar arm, PB, should be perpendicular to the tracing arm, BT, when the tracing point is approximately at the center of the polygon to be measured.
Figure 7.28. Diagram showing an improper positioning of the polar and the tracer arms because the position, from B to B', of these two components exceeds 180o.
Starting at point A (chosen by the operator), an initial reading is taken on the recorder, R, then the tracing point T is moved progressively and carefully around the perimeter until it returns to the starting point A and a second reading is taken on the recorder. As the tracing point T is moved around the perimeter of the polygon, the wheel, W, combine sliding, positive and negative rotations to give the algebraic sum of the rotation, n, which is registered on the recorder. The difference between the initial and the final reading is the actual planimeter reading, n. For convenience, it is usually preferable to set the starting point to zero so that the final reading on the recorder is the planimeter reading, n. For accurate measurements, it is recommended that the planimetric reading be reiterated at least twice. The area, PA, of the polygon is obtained by multiplying the difference between the final and the initial readings by the scale coefficient, k:

PA = n ´ k (7.40)

The value of n is a function of the radius of the wheel and the polar arm of the planimeter and the scale factor, k, is a function of the scale of the plane containing the polygon being measured. The scale factor also varies with the polar arm of the planimeter and is usually provided with the planimeter. A table for unit area may be presented as indicated in table 7.3.

Table 7.3. Table unit for area in metric scale for two tracer arm lengths at different scales.
 
 

Tracer Arm Length 149.5 116.2
Scale Unit Area Unit Area
1:100 0.1 m2 0.08 m2
1:200 0.4 m2 0.32 m2
1:300 0.9 m2 0.72 m2
1:400 1.6 m2 1.28 m2
1:500 2.5 m2 2 m2
1:600 3.6 m2 2.88 m2
1:1000 10 m2 8 m2
1:1500 22.5 m2 18 m2
1:2000 40 m2 32 m2
1:3000 90 m2 72 m2
1:5000 250 m2 200 m2
1:6000 360 m2 288 m2
1:10000 1000 m2 800 m2
1:20000 4000 m2 3200 m2
1:24000 5760 m2 4608 m2
1:25000 6250 m2 5000 m2
1:30000 9000 m2 7200 m2
1:50000 25000 m2 20000 m2
Example 7.27

Assume in figure 7.27, the scale is 1:24,000. After tracing the perimeter of the polygon the difference between the final and the initial readings of the planimeter was found to be 1265. What is the ground area of the polygon?

Solution:

From the table 7.3, if the scale coefficient is 5760 m2 (tracer arm length = 149.5) the area is: 1265 ´ 5760 m2 = 7286400 m2 or 728.64 ha if the scale coefficient is 4608 m2 (tracer arm length = 149.5) the area is:

1265 ´ 4608 m2 = 5829120 m2 or 582.91 ha

7.9 Factors Affected by Scale  |  7.10 Determining Areas from Aerial Photographs  |  References