Chapter 2. Navigation Satellites

In the previous chapter we discussed that if our distances to several satellites of known locations could be measured, then we can compute our position. We have put forward three major prerequisites:

1. Satellites — We need satellites as our points of reference to which we can measure our distances. At any given time, we also need to know the exact location of each satellite.

A satellite by itself is nothing more than a vehicle. You may call it a space vehicle. The function of each satellite depends on what equipment it carries. If the equipment on board is something like a TV station, then it becomes a TV satellite. If the equipment on board is weather observation equipment, then it becomes a weather satellite whose function is to prepare large-scale images of clouds, storms, hurricanes, etc. Obviously, we need special satellites for our purpose.

2. Coordinate System — How do we express the location of each satellite and how do we express our position? As we recall from algebra, We can express them with sets of numbers related to some coordinate system. We need to define the coordinate system such that they are recognizable by everyone. Thus, we need a universal coordinate system.

3. Distance Measurement — What type of electronic signals must the satellites emit to enable us measure our distances? How do we measure these distances? How accurately can we measure these distances? And how do these measurement errors relate to the accuracy of our calculated position? We will attempt to answer these questions in the remainder of this chapter.

Have you noticed that during a thunderstorm, you hear the sound sometime after you see the light? The reason is that sound waves travel much slower than light waves. We can estimate our distance to the storm by measuring the delay between the time that we see the thunder and the time that we hear it. Multiplying this time delay by the speed of sound gives us our distance to the storm (assuming that the light reaches us almost instantaneously compared to sound). Sound travels about 344 meters (1,130 feet) per second in air. So if it takes 2 seconds between the time that we see the lightning and the time that we hear it, our distance to the storm is 2 x 344 = 688 meters. We are calculating the distance to an object by measuring the time that it takes for its signal to reach us.

In the above example, the time that we see the lightning is the time that the sound waves are generated in the storm. Then we start to measure the delay until the time that we hear the sound. In this example, the light is our start signal. What about the cases for which we don't have a start signal? Consider the next example.

Assume that your friend at the end of a large field repeatedly shouts numbers from 1 to 10 at the rate of one count per second (10 seconds for a full cycle of 1 to 10 count). And assume that you are doing the exact same thing, synchronized with him, at the other end of the field. Synchronization between you and him could have been achieved by both starting at an exact second and observing your watches to count 1 number per second. We assume that you both have very accurate watches. Because of the sound travel time, you will hear the number patterns of your friend with a delay relative to your patterns. If you hear your friend's count with a delay of one count relative to yours then your friend must be 344 meters away from you (1 sec x 344 meters/sec = 344 m). This is because the counts are one second apart.

Now assume that you and your friend count twice as fast, two counts in one second. Then at the same distance between you and your friend you will hear a two-count delay. This is because now each count takes 0.5 seconds and each count delay measures 172 meters. If you could count 100 times faster then each count would take 0.01 seconds and each count delay between you and your friend would measure the distance of 3.44 meter. Counting faster is like having a ruler with finer graduation. Of course in real world, you need appropriate devices and instruments to generate and receive very fast counts.

Next assume that you and your friend are far apart and counting very fast, say each count in 0.01 second (each delay count is 3.44 meters), and, as before, both are repeatedly counting from 1 to 10. Assume when you say 7 you hear your friend's voice say 5. You hear a delay count of 2 but you know your distance is more than 6.88 meters. This is because the delay is not just only 2 counts, but rather 2 counts plus some multiples of 10 counts (i.e. some multiples of the pattern cycle). This is as if your measuring tape is not long enough and there are some multiples of the full length of measuring tape plus some fraction. We refer to this unknown number of full pattern delays as unknown integer. If you and your friend were to count repeatedly from 1 to 1000 (instead of 1 to 10) then you could hear 212 count delays between the numbers that you hear and your numbers, which would produce the distance of 212 count delays x 3.44 meters = 729.28 meters. This is 21 full cycles of the 1-to-10 pattern, plus 2 counts. The number of full cycles, 21, that we were not able to observe with our short pattern is our unknown integer.

What we demonstrated above are the concepts of pattern granularity (fineness of tape marks) and pattern length (tape length).

The concept of measuring distances to satellites is much like what we discussed above, but satellites transmit electronic patterns rather than voice counts. Likewise, our receiver generates similar electronic patterns for comparison with the received patterns from satellites in order to measure the distances to them.

Satellites generate two types of patterns: One has a granularity of about 1-millimeter and a length of about 20 centimeters. The other has a granularity of about 1 meter and effectively an unlimited length. In satellite terminology, the first pattern is called "carrier" and the second is called "code". The distance measured by carrier is called "carrier phase" and the distance measured by code is called "code phase". Because code pattern is long, the code phase measurements are complete and do not have any unknown integer. We can measure our distance to a satellite as 19,234,763 meters, for example. In contrast, the carrier pattern is short and carrier phase has a large unknown integer. You may think that it is useless to say, for example, that our distance to satellite is 13.2 centimeters plus an unknown number of carrier cycles. The unknown integer is in the order of several tens of millions. You may ask what good will it do to measure the fractional part so accurate when millions of full cycles are missing? We will explain more.

In the previous counting example with a short pattern, assume that you and your friend are standing next to each other and synchronized together counting fast from 1 to 10. You hear no delay because you are standing next to each other. Then your friend starts to move away. The count delays start to grow from 0 (no delay) to 9. After it reaches 9 it will drop back to 0. This is actually 10 and not zero. You know that this is the case (that the zero count delay actually represents one full cycle count) because you have been following the count delays continuously. You will keep in mind, as your friend moves away, to count the whole number of cycles that are being added to your distance. In this case, there is no unknown integer as long as you keep track of him continuously.

If, instead of starting next to each other, you start at some unknown distance, then you are starting from an unknown integer of cycles. However, if after starting your friend moves away from or towards you, you can account for the number of full cycles that must be added to or subtracted from the initial unknown integer. All the distances that you measure every second contains the same initial unknown integer. This is true as long as you keep track of him continuously. If you don't hear him for some period of time, then you don't know how many full cycles he moved and you will have to start with another unknown number of cycles. The point is that as long as you keep track of him you have only one initial unknown integer.

The concepts of code and carrier are very important. Let us use another analogy for better understanding. You may consider that code phase is like a watch that only has an "hours" hand (call it code watch). At any time you can look at this watch and know the time of the day approximately. You may consider carrier phase like a watch that only has a "seconds" hand (call it carrier watch). You can keep track of the elapsed time with this watch with the accuracy of one second as long as you monitor the watch continuously to keep track of the elapsed full minutes. If you somehow can determine the number of full minutes initially (the initial unknown integer when you started looking at this watch) then you can keep track of time very accurately. If you get distracted and lose track of the number of minutes, then you have a new "initial unknown integer" that you somehow must determine again. With code phase watch you always get the time of the day instantly but with the accuracy of not better than 10 minutes by estimating the location of the hour hand. The code watch can narrow the estimate of unknown minutes (integers) of the carrier watch to plus or minus few minutes. You see that there is a gap between the seconds hand and the hours hand. We are missing the minutes hand. GPS manufacturers have developed techniques to narrow the gap such that code phase and carrier phase can make unambiguous and accurate distance measurements as fast as possible. We will explain the reason for the gap later.

The good news is that the integer ambiguity of carrier phase can be determined by tracking satellites for some period of time. This is the fundamental concept in precision applications like geodesy.

With carrier phase, tracking the correct number of full cycles that the distance to satellite is changing is very critical. You will miscalculate this number if you miss a cycle or add an extra cycle. In GPS terminology, this is called a "cycle slip". In our previous example, cycle slips can happen if you don't hear your friend's voice correctly due to noise or other effects, or if he suddenly jumps a very long distance. Cycle slips is like missing the meter marks while you are concentrating on reading the millimeter ticks. It can create large errors. Most GPS systems are able to detect and repair cycle slips.

Note that not all receivers can measure carrier phase. Carrier phases are typically used in high precision receivers.

We can measure the distances to the satellites with the accuracy of 1 meter with code phase and 1 millimeter with carrier phase. This does not mean that we can determine our position with a GPS receiver with the accuracy of one meter or one millimeter. There are several sources that introduce inaccuracies into the GPS measurement. We will discuss them in the next Chapter.

In this chapter, we learned:

• We can calculate the distance to an object by measuring the time that it takes for its signal to reach us.

• Satellites generate two types of patterns, carrier and code. Carrier has a granularity of about 1-millimeter and a length of about 20 centimeters. Code has a granularity of about 1 meter and effectively an unlimited length.

• The integer ambiguity of carrier phase can be determined by tracking satellites for some period of time.