1. How GPS Turns Radio Wave Travel Time Into Distance
GPS satellites fly in medium Earth orbit at an altitude of about 20,200 km (12,551 miles), circling the Earth twice a day. [3] The smartphone measures the time it takes for a radio signal to travel from the satellite to the receiver. Once that time is known, it can be converted into distance. [2]
The calculation is simple multiplication. Radio waves travel at the speed of light: 299,792,458 meters per second, which is roughly 300,000 km per second. [1] This speed allows the wave to circle the Earth 7.5 times in one second. [1]
For a satellite directly overhead, the signal travels 20,200 km. Dividing this distance by the speed of light gives a travel time of about 0.067 seconds, or 67 milliseconds. [1] This is the shortest time for a satellite directly overhead. Satellites at an angle are farther away, so their signals take longer to arrive. [1]
2. How One Distance Limits Your Location to a Sphere
Imagine a flat map first. If you know you are 5 km (3.1 miles) from Station A, your location is somewhere on a circle with a 5 km radius around that station. [1] If you also know you are 3 km (1.9 miles) from Station B, your location narrows to where two circles intersect. [1] A third distance usually pinpoints one specific spot. [1]
In real space, height matters too. So instead of circles, we use the surface of spheres. Each satellite’s distance defines a sphere centered on that satellite. Your phone is located where these spheres intersect. [1]
If the receiver’s clock were perfectly accurate, three satellites would be enough to determine your position. [1] Position has three values: horizontal, vertical, and height. Three distances provide three equations, which seems sufficient. [1]
However, the receiver’s clock has slight errors. [1] How much does this "slight" error matter?
3. How a One-Microsecond Clock Error Becomes a 300-Meter Mistake
Because radio waves travel at the speed of light, any error in time measurement becomes an error in distance. [1] If the clock is off by one millionth of a second (1 microsecond), multiplying that by the speed of light results in an error of about 300 meters (984 feet). [1]
For perspective, a 1-millisecond error causes a 300 km (186 mile) mistake. A 1-nanosecond error (one-billionth of a second) causes an error of about 30 cm (12 inches). [1] This is the distance light travels in one nanosecond. While this time is invisible to humans, it creates significant position errors in GPS. [1]
Satellites carry highly precise atomic clocks. [3] In contrast, the receiver in your phone uses a much cheaper, less accurate clock. [5] Since putting an atomic clock in a phone is not practical, GPS systems treat the receiver’s clock as "off" and calculate that error as part of the solution. [5]
4. How Four Satellites Solve for Both Position and Clock Error
The phone needs to find four unknown values: three for position (horizontal, vertical, and height) and one for its own clock error. [5] Each satellite provides one distance equation. To solve for four unknowns, you need four equations. [5] This is why a fourth satellite is necessary. [5]
The clock error affects all measurements equally. Whether the clock is fast or slow, that same amount of error is added to every satellite’s distance measurement. [5] Therefore, the measured distance is not the "true distance," but the "true distance plus the clock error." [5] These measurements are called pseudoranges. [5]
The system uses the equations from four satellites to find the position and the clock error together. [5] This process not only finds your location but also calculates a time close to the satellite’s time. [5] The accuracy of this time depends on the receiver and conditions. [5]
5. How the Satellite Network Keeps Four Satellites in View Almost Everywhere
According to GPS.gov, the United States promises to maintain at least 24 operational satellites 95% of the time. [3] The Space Force has operated 31 satellites for over 10 years. [3] The orbits are arranged so that at least four satellites are visible from almost any point on Earth. [3] This design matches the mathematical need for four satellites. [3]
In Japan, the Michibiki system (Quasi-Zenith Satellite System) began service with four satellites in November 2018. [6] It works together with GPS to ensure enough visible satellites. [6] When the system reaches seven satellites, at least four will always be above Japan. [6] This allows for continuous positioning using Michibiki alone. [6]
6. How Accurate Is the Blue Dot on Your Phone's Map?
GPS.gov states that GPS-enabled smartphones typically have an accuracy within a 4.9 m (16 feet) radius in open areas. [4] Accuracy decreases near buildings, bridges, or trees. [4] This 4.9 m figure is a typical value, not a guarantee. [4]
The Geospatial Information Authority of Japan states that standalone positioning with one receiver has an error of about 10 m (33 feet). [2] The difference in numbers comes from different conditions and receivers, not because one is wrong. [2]
To reduce error, relative positioning uses two or more receivers to measure the same satellite signals simultaneously and subtract common errors. [2] Using carrier phase, the relative position between two receivers can be found with an accuracy of one-millionth, or 1 cm over 10 km. [2] This is the accuracy of the relationship between two devices, not the absolute position of one phone. [2]
Users with dual-frequency receivers or augmentation systems can achieve real-time positioning of a few centimeters, and millimeter-level accuracy with long-term observation. [4]
7. Activity: Comparing GPS Signal Near a Window and in the Middle of a Room
With a family member, open a map app. You may see a faint circle around the blue dot. The appearance varies by app. [1]
Compare the size of the circle when you are near a window versus deep inside a room or near the center of a building. [1] If the circle changes size, it may reflect differences in how easily the phone receives satellite signals. [1] However, the reason for the change is not always just signal reception. [1] You can observe whether the size of the circle changes depending on your location. [1] When checking, do not walk; stand still in a safe place. [1]
Also, hold a 30 cm (12 inch) ruler to visualize the distance radio waves travel in one nanosecond. [1] This helps you grasp how clock errors become distance errors. [1]