Least Squares Estimation of GPS Satellite and Receiver Bias

MIT Lincoln Lab
Millstone Hill Radar (image courtesy of MIT)

Space Surveillance

Accurate tracking and identification of objects in space is an important component of our space program. For example, satellite tracking helps NASA verify whether a satellite has been deployed into orbit successfuly. The LINEAR program uses satellite tracking technologies to identify asteroids that come too threatening close to our planet. Furthermore, tracking information tells satellite operators if their satellite has swayed from its orbit or come too close to another space object.

The Space Surveillance Network (SSN) currently consists of a global network of 21 ground-based optical and radar sensors and one space-based sensor. It has been tracking space objects since the launch of Sputnik 1 in 1957. Currently, it tracks more than 8,000 man-made orbiting objects, 7% of which are operational satellites (the rest are space debris such as inactive satellites, used rocket bodies, fragments of stuff, and wrenches lost by astronauts that happen to catch into orbit.) The US Strategic Command has a good page with more detailed info. NASA has a fun website called JTrack that allows you to track your favorite space object.

Earth-based Sensors and Atmospheric Effects

The Millstone Hill Radar at MIT Lincoln Lab's Space Control Division is one of the earth-based satellite trackers in the SSN. Basically, it works like any radar. We just use it to ping satellites as they come over the horizon. By measuring the delay of our radar ping, we can calculate the distance to the satellite. (This is an oversimplication, but it's the basic idea.) The radar dish rotates and shifts as we track the satellites in time.

As in any earth-based observatory, these earth-based sensor require that we cancel the atmospheric effects in order to get more accurate range calculations. Waves travelling through a turbulent atmosphere will be delayed at different speeds, thus causing our ping to slow down at an unknown rate. Usually, the two main sources of error come from the troposphere and the ionosphere. The troposphere, which is the innermost layer of atmosphere, contains different amounts of water vapor and air pressure. The ionosphere contains electrons ionized by the sun's energy. These variables must be determined concurrently while the radar is tracking satellites so that we can subtract their effects from the range calculations.

The ionosphere is especially tricky to predict. The main source of ionization comes from high energy particlese from the sun striking gas molecules in the region(so the ionization peaks during the day and falls at night), but other sources include cosmic rays and atmospheric disturbances. The electron content of the ionosphere varies drastically (and mysteriously sometimes), so our goal is to estimate these values in real-time. (To learn more about the ionosphere, Univ of Alaska has a good reference.

Using GPS to estimate ionosphere electron content

It turns out that we can use the Global Positioning System (GPS) to measure ionosphere electron content. This is not what the GPS is designed for, but it turns out that we can use GPS signal for our purposes (this is similar to the philosophy of passive radars). There are other ways to measure the ionosphere electron content. NOAA has a good website about ionosondes.

The GPS satellies broadcast their signals at two different frequencies (soon, GPS modernization will give us three.). This is the basic insight that led to exploiting GPS signals for ionosphere monitoring. We know from physics that two waves of different frequencies travelling through a dialectric will be diffracted/delayed at different rates. If we know the time two signals from the GPS was transmitted, and record one signal's relative delay to the other, we can calculate the dialectric constant. This is proportional to the number of electrons in the ionosphere, the answer we want. GPS signals are the perfect candidate for this because they not only broadcast at two different frequencies, but also the signals contain the time of transmission. So all we need to do is to record the timestamps when the receiver receives the two signals, do some math, and we'll know the number of electrons in the part of the ionosphere in question.

The catch: GPS biases

However, due to the circuitry in the GPS satellite and receiver hardware, the signals of two frequencies are not exactly synchronous. In other words, even though the two signals are supposed to be transmitted from the satellite at the same time, they might be slightly off, due to circuitry differences. Such a bias is even worse at the receivers, which are cheaper and don't usually have the atomic clocks that satellites have. When our signals are travelling at the speed of light, little clock discrepancies may cause big errors. These kinds of errors are called "L1-L2 satellite and receiver bias" in the literature. We must estimate these errors so that we can calibrate our calibrations for ionosphere modeling. (This was the project I was in while at MIT Lincoln Labs.)

If you're interested in GPS, GPS World is a good magazine. Peter Dana at UT-Austin has a good reference too.

Last updated: Feb 21, 2004