Calculating scale – aerial photography

Introduction

There are a number of methods for calculating scale depending on what information is already known:

• focal length and flying height for flat terrain
• focal length and flying height for variable terrain
• photo distance and corresponding map or ground distance.

There are two basic types of aerial camera used in Australia and they have three different focal lengths:

1. wide angle photography (f = 152mm and f = 306mm)
2. super-wide angle photography (f = 88.5mm and f = 152mm).

Possible issues and errors

Issues

Mean scale is an important value, which is often needed before practical use can be made of the aerial photography. It should also be apparent that the calculated mean scale is approximate, and effective only for images of objects at mean terrain height.

Ground objects above or below the mean terrain elevation are imaged at varying scales so it is not possible to obtain completely reliable measurements from single aerial photographs. Also, if the calculated mean terrain height is not representative of the photographic area and the aircraft altimeter is slightly incorrect then the calculated mean scale could be reduced in accuracy.

Errors

Answers obtained in solving problems will inevitably contain errors, some by failure of certain assumptions to be met and others by random errors in measured quantities. Common sources of error and ways of minimising them are listed below.

Error	Way to minimise error
Errors in photographic measurement.	Use precise equipment and caution.
Errors in identification of photographic images.	Take care when identifying photo and map positions.
Shrinkage or expansion of film, photo paper or map.	Generally not much can be done about the source.
Slightly tilted photos when truly vertical photos are assumed.	If the tilt is less than 3 degrees that is an acceptable error. If the tilt is greater than 3 degrees the photography is rejected as vertical photography.
Errors in determination of mean terrain height.	Under average conditions, these errors are generally small.
Errors in aircraft altitude.	Under average conditions, these errors are generally small.

Determining information for calculations

Most of the information that you need to calculate the scale of a photograph can be found around the edge of the photograph.

Modern aerial photographs

With many photographs, the flying height and the focal length are included in the information in the title strip at the bottom of the photograph. It is usual for the flying height to be expressed in feet, as the international system for expressing flying height was in feet. Before you can start your calculations you need to change the feet to metres. 1 foot = 0.3048m, therefore to change feet to metres, multiply by 0.3048. In this case the flying height is 2400 x 0.3048 = 371.5m.

Older aerial photographs

Some older photography has the focal length shown in inches and the exposure height in feet. Before you can complete the calculations, you will need to convert all measurements into the same units (ie all feet, all inches, all metres or all millimetres). For conversion, 1 inch = 25.4mm, 1 foot = 304.8mm and 1 foot = 12 inches.

Often the focal length of the camera is recorded in a data box on the top of the photograph.

Most modern aerial photographs have the scale printed in the title strip.

Single aerial photograph

The aerial photograph is an image that is obtained by projecting the image of the terrain through a lens and onto a film on the negative plane in the camera. The geometry and terminology of the single aerial photo are set out in the following figure.

It can be seen that all light rays from the ground pass through the lens at L, before reaching the negative plane at the back of the camera. The point where the optical axis intersects the ground plane, denoted by O, is called the ground principal point, while the same optical axis intersects the image at o and is simply called the principal point.

The focal length of an aerial camera is very important, because it’s used in all calculations. Focal length is the distance from the lens to the centre of the photograph at o, and is denoted by the abbreviation f.

The figure above shows the square format of the aerial photograph and two different types of fiducial marks. The fiducial marks are generated from within the camera and are exposed on the negative as each photo is taken.

When opposite marks are joined, they:

1. intersect at the principal point, and
2. intersect at right angles.

The figure above shows that accepted geometry of a truly vertical aerial photograph. Since the negative is a reversal in tone and geometry it is convenient for the user of aerial photographs to think in terms of the positive instead of the negative plane.

The positive plane is a plane parallel to the negative plane and below the lens by a distance equal to the focal length.

The distance between the mean terrain height (MTH) and mean sea level is denoted by H'.

The aircraft taking the aerial photography is flown at an altitude above mean sea level so the exposure altitude is H. We are particularly interested in the distance from the lens to the mean terrain height, this is represented by (H-H') and is called the flying height.

Scale over flat terrain

This figure shows a truly vertical aerial photograph over horizontal terrain. As mentioned previously we will work on the positive plane.

The scale of the aerial photo is the ratio of the distance on the photograph, ab, compared to the equivalent distance on the ground, AB, and may be represented by:

This scale may also be expressed in terms of focal length (f) and flying height (H - H'). This gives us the relationship:

So from this, it follows that:

In other words the scale of the aerial photography is calculated by equating the camera focal length and the flying height.

Calculating scale over flat terrain

The calculation of scale for a photograph may be more easily appreciated if we assume a set of values as an example.

Example 1

Given that f = 152mm and (H-H') = 2000m

Then:

Which may be rounded to 1:13160

Example 2

Given that f = 88.5mm and (H-H') = 2000m

Then:

Which may be rounded to 1:22600

These examples show that different types of cameras may be used to obtain different scales of photography from the same flying height above mean ground level. Note that 1:13160 is the larger scale, and 1:22600 is the smaller scale.

Scale over variable terrain

This figure shows variable terrain where the ground point A, has an elevation of H_A and the ground point B has an elevation of H_B. On the positive plane, these points will appear at a and b respectively. Since A lies in a horizontal plane closer to the camera lens than B, it is reasonable to assume that the photographic image of a will be at a larger scale than b.

Calculating scale over variable terrain

The variation in scale on a photograph may be more easily appreciated if we assume a set of values as an example.

Given:

Then:

Scale at A is 1:9840

Scale at B is 1:11160

Thus in this example, a scale variation from 1:9840 to 1:11160 occurs. Note that the maximum scale occurs at the maximum ground elevation above mean sea level.