There is definitely a lot of satisfaction involved in predicting where and when a satellite will appear and then to actually observe it. Very soon you ask yourself whether you can do more with these satellites than just looking at them. The answer is yes and it is twofold.
Firstly, you can make positional measurements, i.e. measure the position of the satellite with respect to the stars at a certain time. Using these measurements it is possible to determine deviations from the predicted track. These deviations can be due to atmospheric drag (i.e. friction with the air molecules that are still present even at orbital heights), the pressure of the solar radiation, orbital manoeuvres or gravitational resonances.
Using positional observations from all over the world, scientists are able to determine :
In the early days of the space age, not many tracking stations for accurate positional measurements were available. Amateurs were helping professionals to obtain accurate positional measurements. Later, more sophisticated radar equipment became available and was positioned all over the world to get better coverage of the orbits. Due to this evolution, measurements by amateurs became less desired. Over the last twenty years, support for amateur networks by professional research institutes (by providing predictions, etc.) has gradually almost completely disappeared. Nevertheless, research has not completely dwindled and amateur positional measurements remain useful. Observations of the interesting objects aren't necessarily available in abundance, which is one reason visual observations are valued. Another reason is that visual observations are of help in dealing with biases that might be present in more accurate data. And visual observations are quite adequate and appropriate for high drag objects.
During the last few years a unique application of position measurements has become more popular : the determination of the orbital elements of U.S. spy satellites. Orbital elements are required to predict the passage of satellites. NASA distributes orbital elements of virtually all satellites, except for secret U.S. military satellites like the KH (KeyHole) and NOSS Double and Triple Satellite Formations. A rather informal worldwide network of persistent amateurs has specialized in tracking these objects and calculating the orbital elements themselves. This celestial detective work certainly gives an additional stimulus to positional measurements.
Fig. 1: The meteorological satellite DMSP 2-05 of the US Army photographed by Tristan Cools on May 1, 1994. Two long flashes are visible.
Apart from positional measurements, you can also make flash period measurements and details of this and the approach required can be found here, but the requirements for making positional observations can be found here.