Why Four GPS Satellites Are Needed for Accurate Positioning and Altitude

In the world of satellite navigation, GPS (Global Positioning System) is a crucial technology for determining precise positions and altitudes. However, a common question arises: why are four GPS satellites required to accurately locate a position? This article explores the underlying principles of GPS and explains why this quantity is necessary.

Understanding GPS

GPS is designed to provide accurate location and time information by using a network of satellites that orbit the Earth. Each satellite transmits a signal, which includes the precise time the signal was sent and the satellite’s precise position in space. Ground-based receivers utilize these signals to compute their own position.

The Role of Trilateration

Instead of triangulation (determining a point by measuring angles from known points), GPS uses trilateration. This method involves calculating positions based on the time differences of the signals received from multiple satellites.

The Four-Satellite Requirement

To accurately position and time-track an object, at least four satellites are required. This is due to the underlying mathematics and the dimensions involved in the GPS system:

Trilateration and Dimensions

Trilateration requires solving a system of equations to find a point in three-dimensional space. Mathematically, this requires four data points. Here’s a detailed breakdown:

First Satellite: Only one spherical shell centred on the satellite is defined (distance to the satellite). Second Satellite: The intersection of two spheres defined by the signals from the first two satellites forms a volume. Third Satellite: The intersection of three spheres defined by the signals from the first three satellites forms a single point. Fourth Satellite (and Time): The fourth satellite data point, combined with the time information, allows for the precise calculation of the position and time.

This explains why four satellites are required to solve the four-dimensional (3 spatial dimensions time) problem accurately. The fourth satellite is essential to determine the exact time, which in turn helps in calculating the actual distance and position.

Practical Implications

Understanding the need for four satellites is crucial in various applications:

Aircraft and Rough Terrain

When flying an airplane or navigating through rough terrain, knowing the altitude is critical. GPS can provide this information if four or more satellites are available. The altitude information helps in navigating safely through mountains, valleys, and other challenging terrains.

Reliability and Redundancy

In real-world scenarios, having signals from multiple satellites ensures greater accuracy and redundancy. This is particularly important in environments where a signal might be temporarily lost or attenuated. By relying on multiple satellites, the system can continue to provide accurate positioning even if a single satellite fails.

Complexity and Simplification

The need for four satellites is a result of the inherent complexity of the problem. GPS operates in a four-dimensional space (3D position time), and solving this problem requires at least four data points for a unique solution. If you have a perfect clock, you could theoretically solve with three satellites, but in practice, the time uncertainty requires a fourth satellite.

Conclusion

In summary, four GPS satellites are essential for precise and reliable positioning and altitude measurement. The mathematical principles of trilateration and the four-dimensional nature of GPS determine this requirement. Understanding the role of trilateration and the need for accurate time information helps explain why four satellites are necessary.