In our discussion of measuring distances using patterns of numbers, we assumed that you and your friend started counting numbers simultaneously. If your watch is one second off, this will translate into 344 meters of error in measuring distance, because sound travels 344 meters in one second. With satellites, the electronic signals travel about 300,000,000 meters per second (the speed of light). So the errors in the satellite clock and the receiver clock contribute profoundly to errors in distance measurements.
Satellite Clock
One billionth of a second (one nanosecond) of inaccuracy in a satellite clock results in about 30 centimeters (one foot) of error in measuring the distance to that satellite. For this reason, the satellites are equipped with very accurate (Cesium) atomic clocks. Even these very accurate clocks accumulate an error of 1 billionth of a second every three hours. To resolve the satellite clock drifts, they are continuously monitored by ground stations and compared with the master control clock systems that are combinations of more than 10 very accurate atomic clocks. The errors and drifts of the satellites' clock are calculated and included in the messages that are transmitted by the satellites. In computing the distance to the satellites, GPS receivers subtract the satellite clock errors from the reported transmit time to come up with the true signal travel time.
Even with the best efforts of the control centers in monitoring the behavior of each satellite clock, their errors cannot be precisely determined. Any remaining satellite clock errors accumulate typically to about a few nanoseconds, which cause a distance error of about one meter.
Receiver Clock
Similar to satellite clock errors, any error in the receiver clock causes inaccuracy in distance measurements. However, it is not practical to equip receivers with very accurate atomic clocks. Atomic clocks weigh more than 20 kilograms, cost about US$50,000, and require extensive care in temperature control.
Assume that at a given time our receiver clock has an error of one millisecond, causing a distance error of about 300,000 meters. If the distances to all satellites are measured exactly at the same time, then they are all off by the same amount of 300,000 meters. We can, therefore, include the receiver clock error as one of the unknowns that we must solve for. Remember from Chapter 1 that we had three unknowns (X, Y, Z) for the position. Now we have four unknowns: three components of position and the new unknown of receiver clock error. We will need four equations in order to solve for the four unknown. Measuring distances to four satellites can provide us such four necessary equations. Instead of three satellites before, now we need four, but in return we can use inexpensive clocks in our GPS receivers.
Note that the concept of receiver clock being one of the unknowns is valid only if we take measurements to all satellites exactly at the same time. If distances to all satellites are not measured at the same time, then for each measurement we have a different clock.
Making simultaneous measurements to four satellites, we not only compute the three dimensions of our position, but we also find the error in our receiver clock with very good accuracy. A typical clock has a drift of about 1000 nanoseconds every second, but we can now adjust the receiver time to the accuracy of the GPS clock. This will make the inexpensive clock of the receiver as good as an atomic clock. Receivers correct their clock every second and provide a corrected tic signal for outside use for those who need accurate time. If we put a receiver in a precisely known location, then we need to track only one satellite to continuously calculate the receiver clock error and adjust it.
Four is the minimum number of satellites that we need to compute position and time. The more satellites we have the more accurate results we can get. This is discussed later in the GDOP section.
Satellite Orbit Error
As we discussed before, the accuracy of our computed position also depends on how accurately we know the location of the satellites (the points of references). The orbits of satellites are monitored continuously from several monitoring stations around the earth and their predicted orbital information is transmitted to the satellites, which they in turn transmit to the receivers. The history of GPS has shown, thus far, that the accuracy of the orbital prediction is in the order of a few meters. This will create about a few meters of error in computing our position. In the next chapter we shall see how to remove this error.
Atmospheric Errors: Ionosphere and Troposphere
Ionosphere
In computing distances to satellites, we first measure the time it takes for the satellite signal to reach the receiver and then we multiply this by the speed of light. The problem is that the speed of light varies due to atmospheric conditions. The upper layer of the atmosphere, called the ionosphere, contains charged particles that slow down the code and speed up the carrier.
The magnitude of the effect of the ionosphere is much more during the day than during the night. The magnitude also has a cyclical period of 11 years that reaches a maximum and a minimum. For the current cycle, the ionosphere will reach its peak magnitude in 1998 and its minimum in 2004. The cycle will then be repeated. The effects of the ionosphere, if not mitigated, can introduce measurement errors greater than 10 meters.
Some receivers use a mathematical model for the effects of the ionosphere. With the approximate knowledge of the density of the charged particles in the ionosphere (broadcast by satellites), the effect of the ionosphere can be reduced by about 50%. The remaining error is still significant.
The impact of the ionosphere on electronic signals depends on the frequency of the signal. The higher the frequency, the less is the impact. So if we transmit the patterns simultaneously via two different frequencies, the ionosphere may delay the code on one frequency, for example, by 5 meters and on the other frequency, say, by 6 meters. We cannot measure the magnitude of these delays, but we can measure their difference by observing the difference on their arrival time, which in this case translates into 1 meter of effective distance between them. By measuring this difference and using known formula for frequency dependency of the ionosphere delay, ionosphere effect can be removed.
It is exactly for this reason that all GPS satellites transmit information in two frequencies, called L1 and L2. Precision receivers track both signals to remove the effect of the ionosphere. All non-precision receivers track only the L1 signal. This is one of the main distinguishing features between different types of receivers. The L1 receivers are also called single frequency receivers, while the receivers that track L1 and L2 are called dual frequency receivers. Dual frequency receivers practically remove the ionosphere effects.
Since the L2 signal is not entirely available to the general public, sophisticated techniques have been implemented in receivers to extract the code and carrier information, even with the partial availability of the L2 signal. These techniques fully satisfy the requirements of the users for non-military applications, while not compromising the Anti Spoof policy and security objectives of the US Department Of Defense (DOD).
There has been some discussion on allowing a different frequency for civilian applications to separate the DOD and civilian requirements. We believe, however, that the existing system fully satisfies the civilian requirement, particularly since advancements in electronics integration have made the technology affordable for broad civilian applications.
Troposphere
The lower level of the atmosphere, which contains water vapors, is called the troposphere. It has the effect of slowing down both code and carrier. The effects of the troposphere cannot be removed using dual frequency systems. The only way to remove the effects of the troposphere is by measuring its water vapor content, temperature and pressure, and applying a mathematical model that can compute the delay of the troposphere.
Multipath
In measuring the distance to each satellite, we assume that the satellite signal travels directly from the satellite to the antenna of the receiver. But in addition to the direct signal, there are reflected signals, from the ground and the objects near the antenna, that also reach the antenna through indirect paths and interfere with the direct signal. The compound signal creates an uncertainty about the true signal arrival time, much the same way as the echo from nearby mountains may cause uncertainty in the exact time you hear your friend's voice. If the indirect path is considerably longer than the direct path (more than 10 meters) such that the two patterns of signals can be separated, then the multipath effect can be substantially reduced by signal processing techniques.
Receiver Errors
Receivers may introduce some errors by themselves in measuring code or carrier. In high quality receivers, however, these errors are negligible (less than one millimeter) for carrier phase and a few centimeters for code phase.
Geometric Dilution Of Precision (GDOP)
We have been talking about the errors in measuring distances to satellites, which are commonly referred to as ranging or range errors. The question is what is the relationship between the range error and the error in computed position. Or, in other words, how many meters of error are introduced in our computed position as a result of one meter of error in measuring distances to the satellites?
The answer is that it depends on the number and the geometry of the satellites used. If four satellites are clustered near each other, then one meter of error in measuring distance may result in tens or hundreds of meters of error in position. But if many satellites are scattered around the sky, then the position error may be less than 1.5 meters for every meter of error in measuring distances. The effect of the geometry of the satellites on the position error is called Geometric Dilution Of Precision (GDOP), which can roughly be interpreted as the ratio of the position error to the range error.
Imagine the tetrahedron that is formed by lines connecting the receiver to each satellite used. The larger the volume of this tetrahedron, the smaller (better) the GDOP. In most cases, the larger the number of satellites the smaller the GDOP.
Selective Availability: The Man-Made Errors
Errors in the satellite clock, the satellite orbit, the ionosphere, the troposphere, the multipath, and the receiver typically amount to less than 10 meters of range error which, under typical GDOPs of about 2, results in a position accuracy of about 20 meters.
The US Department Of Defense has determined that providing this level of precision to the general public is against the US national interest. Therefore, DOD has introduced man-made intentional errors to degrade the position accuracy of GPS to about 100 meters. This intentional degradation is called Selective Availability (SA) and is implemented by tethering the satellite clocks and reporting the orbit of the satellites inaccurately. Military receivers are equipped with special hardware and codes that can mitigate the effect of SA. SA can be turned ON or OFF through ground commands by the GPS system administrators.
Summary
In this chapter, we learned:
We can adjust the receiver time to the accuracy of the GPS clock. This will make the inexpensive clock of the receiver as good as an atomic clock.
Four is the minimum number of satellites that we need to compute position and time. (But the more satellites we have the more accurate results we can get.)
Dual frequency receivers practically remove the ionosphere effects.
If the signal indirect path is considerably longer than the direct path, more than 10 meters, then the multipath effect can be substantially reduced by signal processing techniques.
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