What is Pseudorange?

Definition

A pseudorange is the signal travel time between a satellite and a receiver, multiplied by the speed of light, with the receiver clock error included. The GPS performance standard defines the pseudorange measurement in two steps. It takes the difference between the code time of reception and the code time of transmission. It then multiplies the difference by the speed of light. The receiver clock sets the first time. The satellite clock sets the second.

A pseudorange looks like a distance, and it has the unit of a distance. It is not the true distance. The standard calls the true distance between the satellite and the receiver the geometric range.

How a receiver measures it

Each GPS satellite sends a known pseudorandom code. The C/A code is 1,023 chips long and runs at 1.023 megachips per second. The receiver makes a copy of this code. It adjusts the delay of the copy until the copy matches the received code. The adjustment is the travel time of the signal, as read on the receiver clock (NASA, slides 35 to 37).

The unit of the code is the chip. One chip is 293 m long. The speed of light divided by the chip rate gives this length. The sum is 299,792,458 m/s divided by 1,023,000 chips per second, which is 293.05 m. The receiver measures the code to a small fraction of a chip. This gives a pseudorange at about the meter level (NASA, slide 36).

Time and distance

The speed of light turns a time into a distance. It is the link between the two:

• 1 microsecond of time is 299.79 m of range.
• 1 millisecond of time is 299.79 km of range.
• 1 nanosecond of time is about 0.3 m of range.

The same link shows how a clock error acts. A receiver clock that is 1 microsecond wrong adds 299.79 m to every pseudorange. This is why the receiver clock is an unknown to solve for.

The travel time of a GPS signal is short. Use a spherical Earth with a radius of 6,378.137 km (the WGS 84 semi-major axis). Use a circular GPS orbit with a radius of 26,559.8 km (the semi-major axis). A satellite directly overhead is then about 20,182 km away, and the signal needs about 67 milliseconds. A satellite at the horizon is about 25,783 km away, and the signal needs about 86 milliseconds. These are calculated values and not values from the standard.

Why it is not a true range

The receiver clock is not set to GPS time. The receiver cannot know its clock error before it has a position. So each pseudorange holds the same clock error, and the clock error is the main reason for the name.

The measurement has the following terms (NASA, slide 38):

• True range. The geometric range from the satellite to the receiver. The satellite position is known from the navigation data.
• Clock terms. The satellite clock error is known from the navigation data. The receiver clock error is unknown. The symbol c is the speed of light.
• Atmosphere. The ionosphere and the troposphere slow the signal.
• Other errors. These are the errors in the satellite orbit and clock data, multipath and receiver noise.

The GPS standard also defines a corrected pseudorange. The receiver adjusts the raw measurement to GPS time with the satellite clock correction from the navigation data. The standard calls this the SPS signal ranging measurement.

Pseudoranges and position

Each pseudorange places the receiver on a sphere around the satellite, and the clock error widens every sphere by the same amount. The receiver finds the point and the clock offset that fit all the pseudoranges. A receiver has four unknowns, which are three position coordinates and the clock offset. It needs pseudoranges from at least four satellites. If the altitude is known, three are enough (SPS PS, PVT solution).

Trilateration is the basic method of finding a point from distances to known points. GPS positioning explains the full calculation, the satellite geometry and the accuracy. This page covers the measurement that the calculation uses.

Errors in a pseudorange

NASA gives a typical error budget for a code measurement:

The GPS performance standard commits to a signal-in-space user range error (URE) of 7.0 m at 95 percent. This is a global statistic over all ages of data (Table 3.4-1). The user range error is the error of the satellite orbit and clock data along the line of sight. It does not include the atmosphere, multipath and receiver noise.

The standard adds the receiver errors to the signal errors. The root sum square of a 9.7 m signal error and a 5.5 m receiver error is 11.2 m at 95 percent. This is the user equivalent range error (UERE). The position error is about the UERE times the dilution of precision (Appendix B.2.1). A good satellite geometry keeps the multiplier low. The page on HDOP explains the horizontal part.

Multipath is an error of the receiver site. A signal that reflects from a building or the ground arrives late and distorts the code measurement. NGS notes that multipath causes pseudorange noise that limits the estimate of the integer ambiguities.

Pseudorange and carrier phase

A receiver measures the carrier wave as well as the code. The carrier phase measurement has the same terms as the pseudorange, with three changes (NASA, slide 38):

• It adds an unknown whole number of cycles, called the integer ambiguity.
• The ionosphere delay has the opposite sign.
• The noise is millimeters and not meters.

The two measurements work together. A receiver uses the pseudorange and the carrier phase to estimate the integer ambiguities. This step alone can give positions at the sub-meter level (NGS, p. 14). The carrier phase page covers the integer ambiguity, the cycle slips and the differencing.

What a receiver reports

NASA lists the observables that a tracking receiver gives: the pseudorange, the Doppler shift, the carrier phase and the carrier-to-noise density. Positioning software for surveys uses the first three.

For RTK work, a base station and a rover both log these observables. Differences between the two receivers remove the errors that they share. Differential GPS uses the same idea with pseudorange corrections. The base must stay near the rover so that both see the same errors. The GNSS page covers the other satellite systems that give the same type of measurement.