Why Does GNSS Require at Least 4 Satellites for Positioning?

Have you ever wondered this: when using satellite navigation, we notice that the receiver can pick up signals from dozens of satellites. Yet, geometrically, only three satellites should be enough to determine the three-dimensional coordinates (X, Y, Z). So why does GNSS positioning require at least four satellites?

The answer lies in that fourth crucial variable: time. Today, we’ll uncover the mathematical and physical principles behind this—starting from how GNSS pseudorange positioning actually works.

The core of pseudorange positioning lies in using the autocorrelation property of ranging codes to measure time. Specifically, after receiving the carrier signal transmitted by a satellite, the receiver first demodulates the modulated ranging code embedded within. At the same time, driven by its own internal clock, the receiver generates a local replica code identical in structure to the satellite's ranging code. By progressively delaying this local replica code and performing a correlation operation with the received ranging code, the receiver searches for the state at which the correlation between the two reaches its peak. When the replica code is delayed by a specific time interval (Δt) and achieves optimal alignment with the received ranging code—where the autocorrelation coefficient is maximized—this delay Δt is taken as the signal propagation time from the satellite to the receiver.

According to the principle of "distance = speed × time", the propagation time Δt and the speed of light are used to calculate an initial distance value. However, this distance value is not the true geometric distance, which is why it is referred to as the "pseudorange." The fundamental reason lies in the fact that the entire measurement process relies on the receiver's own imperfect local clock. As a result, the measured propagation time Δt inevitably incorporates the error of the receiver’s clock. Additionally, propagation delays caused by the signal passing through the ionosphere and troposphere also introduce further errors.

Let’s now use the calculation formula to understand the underlying principle.

The pseudorange ρ can be obtained by the following formula.

ρ = c (t_u−t_s) = cΔt

Here, c represents the speed of light, which is a known quantity; Δt denotes the propagation time of the ranging code from satellite transmission to receiver reception, also a known quantity. t_u is the time at which the receiver receives the ranging code, and t_s is the time at which the satellite transmits the ranging code.

We assume the receiver clock error is t₁, the satellite clock error is t₂, the error caused by signal propagation through the ionosphere is t₃, and the error caused by propagation through the troposphere is t₄. Then the true distance r, taking these factors into account, can be calculated using the following formula:

In the equation, (x₁, y₁, z₁) represents the satellite coordinates derived from the navigation message ephemeris, which are known values; (x, y, z) denotes the receiver position, which is an unknown variable; c is the speed of light, a known constant; the pseudorange ρ is a calculable quantity; t₂ is provided in the navigation message, making it a known value; while both t₃ and t₄ can be computed using corresponding models, rendering them calculable quantities.

From the above formula, it can be seen that there are four unknown quantities x, y, z, and t₁, so four equations are required for solving. Precisely because the pseudorange contains this key unknown—the clock error—the geometric intersection of only three satellites is insufficient to determine the user's position. To calculate the receiver’s three-dimensional coordinates (X, Y, Z) while simultaneously eliminating the influence of its own clock error, at least four independent pseudorange observation equations must be established. This is why, in actual satellite positioning, it is necessary to observe at least four satellites simultaneously. By solving this system of equations, the system can not only accurately calculate the user's position but also calibrate the receiver’s local clock. This is exactly where the ingenuity of global satellite navigation systems lies in achieving high-precision positioning and time synchronization.

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