Height Measurement

There are a number of methods for measuring the heights of objects using aerial photography. I will briefly discuss three that I feel are important for photo interpreters to be familiar with. Two deal with the measurement of height from a single photo; while the third deals with the determination of height by measuring parallax differences. The types of parallax being measured here are:
  1. Absolute Parallax; and,
  2. Differential Parallax.
Remember that according to the American Society of Photogrammetry and Remote Sensing's Manual of Photogrammetry, 3rd.ed.:

Parallax:  The apparent displacement of the position of a body, with respect to a reference point or system, caused by a shift in the point of observation.

Absolute Parallax: Considering a pair of aerial photographs of equal principle distance, the absolute parallax is the difference of a point is the algebraic difference of the distances of the two images from their respective photo nadirs measured in a horizontal plane and parallel to the air base.

Differential Parallax:  The difference in the absolute stereoscopic parallaxes of two points imaged on a pair of photographs. This is usually employed in the determination of the differences in the elevation of objects.

Single photo methods of height determination

Shadow Height Method:

Basically, if the shadow cast by an object can be measured and the sun angle causing the shadow is known or can be derived (from latitude, date and time) then the height of the object can be calculated using simple trigonometry, as follows:
h = Ls x tan
Where:
tan = the tangent of the sun angle from the ground surface
Ls = Length of the shadow.

Here we assume that the shadow on which the ground falls is level and that the object is vertical. The object's top must be sharply defined so that it creates a distinct image.  There are many sources of error here, shadows are not always distinct, and the calculation of the sun angle is an involved process.

Displacement Method:

In this method of height determination from a single aerial photo we:
  1. Accept the principle point as the photo nadir (were assuming a true vertical photo);
  2. Must precisely know or be able to determine the altitude from which the photo was acquired.
  3. Both the top and the bottom of the object to be measured should be clearly visible.
  4. The degree of image displacement must be great enough to be accurately measured with available equipment.
If the above conditions are met, the formula for the displacement method of height determination from a single aerial aerial photo can be written as:

Ho =    Ha x D

           R

Where:
Ho = Height of the object;
Ha = Altitude above the surface where the photo is taken;
D = Length of the displaced image;
 

Figure 9-1. Displacement Method

R = Radial distance from the photo nadir to the top of the object.

Basically, its important that you know that these techniques exist. That they can be accomplished in a practical fashion if necessary.  If you ever have to do this you can always go back and cookbook it.

Parallax Height Measurement

This is the most used method of measuring heights on air photos. There are many forms of the parallax equations. Avery and Berlin give one; Paine in his book lists three: 1) for mountainous terrain; 2) for level terrain; and, 3) the short cut equation.

What I will give here is the basic form of the equation:

Ho = Ha x dP

     Pb + dP

Where: Ho = The height of the object of interest;

Ha = Platform height or altitude above datum;
dP = Differential Parallax; and
Pb = Absolute Parallax.

So if the altitude of the aircraft above datum is: 1. known or can be calculated; and, 2. if, from the available stereo pairs, we can calculate the differential and the absolute parallax; then, 3. We can ascertain the heights of objects in the photos.

Important things to remember here include:


Now for a small trick.  If we can assume that:

  1. Photo tilt is less than 3;
  2. Both negatives or positive transparencies of the stereo pair were taken from the same flying height;
  3. Both nadirs and principle points are at essentially the same ground elevation; and,
  4. The base of the objects to be measured are, essentially, at the same elevation as that of the principle point.


Then, we can substitute the average photobase of the stereo pairs being used can be substituted for Pb (absolute parallax).

Let's say, as Avery and Berlin do in their book (5th. ed.) on Pg. 78 and 79; that we are going to measure the height of the Washington Monument from a stereo pair.

The nominal photo scale we have is 1:4,800. We have gone in and corrected this in the area of the monument monument to 1:4,600 at the base of the Monument. The camera focal length was 12 inches. So the flying height was what? 4,600 feet. The average photobase (P) of the stereopair is calculated to be: 4.40 inches.

Absolute stereo parallax at the base; and at the top of the monument is measured parallel to the line of flight with an engineers scale. The difference is: 2.06 in. - 1.46 in. This gives a dP of 0.60 in. So, 0.60 inches is the differential parallax of the displaced images.

Substituting these values into a form of our formula:

Ho = [H] dP

         Pb + dP

Ho = 4,600 Ft. 0.60 in.           = 552 feet

               4.40 in. - 0.60 in.

The actual height of the Washington Monument is 555 feet. This is a very accurate measurement for this type of exercise. For example if we had used the nominal photo scale of 1: 4,800; instead of the corrected 1: 4,600 scale we would have gotten a height of 576 feet. A 21 foot error as opposed to the 3 foot error we did get. It just goes to show that the more time you put in setting up the problem and the more precise the instruments are the better ( up to a point) the measurements that you can achieve. Well that's essentially it for stereo and height measurement.

  Title Page