You are familiar with the system of measuring plane angles in degrees, but the unit of angular measure in the International System of Units (SI; the metric system) is the radian (rad). A radian is that unique angle which subtends an arc length which is equal to the radius of the circle.
The formula for the circumference of a circle is:
C = 2p r where r = radius of the circle
Since one radian subtends an arc length equal to the radius, and there are 2p radiuses around a circle, there must be 2p radians in a circle. There are also 360 degrees in a circle, so we can relate radians to degrees and vice versa:
Question 1. How many degrees are there in one radian?
Question 2. How many radians are there in one degree?
Note that this angular measure has no dimensions; it is the ratio of a length (the arc) to a length (the radius).
In remote sensing, we frequently express resolving power in terms of the angle (in radians) subtended between the imaging system and two targets spaced at the minimum resolvable distance:
Although we report the angular resolving power in terms of radians, in reality the minimum resolvable distance we measure is NOT the arc length, but rather the chord length (which is the minimum resolvable distance).
Also, as you can see in the diagram, the reported altitude is NOT actually the radius. How serious are these discrepancies?
Let's find out by assuming a 10 degree field-of-view instrument is flown at an altitude of 1000 meters.
a = 10 degrees
10 degrees = 0.1745 radians
alt = 1000 m
r = ------ = -----------
cos a/2 cos 5 degrees
r = radians r = 1003.82 m
0.1745 rad = -------------
arc length = 0.1745 x 1003.82 m
arc length = 175.17 m
The chord length (c) can be calculated as follows:
b (opposite side)
tan -- = -------------------
2 alt (adjacent side)
b = tan 5 degrees x 1000 m
since b = 1/2 c (the chord length)
c = 2(tan 5 degrees x 1000 m)
chord length = 174.98 m
So, you can see that the chord length is actually 0.19 m shorter than the arc length. But, this is only a 0.1% difference, which is not significant for most remote sensing applications. If we reduced the angle (a) to 1 degree -- and most scanners use small angles -- the difference becomes an even more insignificant 0.001%!
Most non-photographic remote sensing systems have fields-of-view which are too narrow to conveniently use radians. Instead, we express angular resolving power in milliradians (mrad, 10-3 radians). A useful relationship to remember is that at an altitude of 1000 units, a 1 mrad system can resolve 1 unit of length. The formula is simple.
minimum resoluble distance
angular resolution (in mrad) = --------------------------------- x 1000
For example, a sensor with an angular resolution of 1 mrad could resolve high-contrast targets which were 10 m apart from an altitude of 10 km, but could also resolve targeets spaced 1 m apart from an altitude of 1000 m.
10 m 1 m
1 mrad = ---------- = ----------
10,000 m 1000 m
Question 3. Using the formula for angular resolution, calculate the resolving power of your eyes in milliradians (mrad). Assume you can resolve 1 mm lines from 5 meters distance.
Question 4. The minimum ground resolved distance for the Thematic Mapper instrument on Landsat 4 and 5 is 30 meters from its nominal altitude of 705 km. What is the angular resolution of this system in millradians?
Question 5. What is the angular resolution of the Thematic Mapper expressed in microradians (mrad, 10-6 radians)?
Question 6. The minimum ground resolved distance for the Panchromatic band of the HRV instument on SPOT 1,2, and 3 is 10 meters from their niminal orbital altitude of 830 km. what is the angular resolution of this system in milliradians?
Question 7. Compare your answer to Question 3 with those you calculated fro Questions 4 and 6. How much better is the angular resolving power of the Thematic Mapper compared to your eyes? _____________ times improvement (TM vs your eyes)
Question 8. How much better is the angular resolving power of the SPOT PAN data compared to your eyes? _____________ times improvement (SPOT PAN vs your eyes)