Why Did Nanjing’s GPS Fail? Unveiling the Math Behind Satellite Navigation

Event Overview

On the evening of December 17, 2025, Nanjing experienced a large‑scale satellite navigation signal anomaly. Major map services such as Baidu, Amap, and Tencent Maps stopped working, affecting delivery riders, ride‑hailing drivers, and shared‑bike users, with locations drifting tens of kilometers.

Core Principle: Geometry and Equation Solving

Trilateration Basics

The fundamental idea behind satellite positioning is trilateration. In a 2‑D plane, if the coordinates of three known points (anchor nodes) and the distances from an unknown point to each of them are known, the unknown point’s location can be uniquely determined.

Extending to 3‑D Space

In three dimensions, satellites act as anchor nodes. Each satellite broadcasts its precise 3‑D coordinates; a receiver measures the signal propagation time to compute its distance to each satellite.

In theory, three satellites suffice to locate a point in 3‑D space, but practical considerations require more measurements.

Pseudorange Measurement: Converting Time to Distance

Definition

GNSS determines distance by timestamping the transmitted signal. The receiver subtracts the transmission time from the reception time and multiplies the result by the speed of light, yielding the so‑called pseudorange.

The pseudorange differs from the true geometric distance because of satellite and receiver clock biases, atmospheric delays, and other noise sources.

Observation Equation and Error Sources

The complete pseudorange observation equation incorporates several error terms:

True geometric distance

Receiver clock bias

Satellite clock bias

Ionospheric delay

Tropospheric delay

Other measurement noise

Why Four Satellites?

Solving for the receiver’s three spatial coordinates (x, y, z) and its clock bias requires four independent equations, thus at least four satellites must be observed.

Linearizing the Non‑Linear Equations

Linearization Process

The pseudorange equations are nonlinear due to square‑root terms. In practice they are linearized using a first‑order Taylor series expansion around an approximate receiver position.

Matrix Form and Least‑Squares Solution

After linearization, the equations are expressed in matrix form Δρ = H·Δx + ε, where:

Δρ is the vector of observed‑minus‑computed pseudorange residuals

H is the design matrix composed of partial derivatives

Δx is the vector of unknown corrections (position and clock bias)

ε is the residual error vector

When more than four satellites are available, the system becomes over‑determined and is solved by the least‑squares method, iteratively updating the approximate position until the correction magnitude falls below a preset threshold.

Error Analysis and Accuracy Factors

Geometric Dilution of Precision (GDOP)

Even with a perfect solution, positioning accuracy depends on the geometric distribution of the satellites. The GDOP metric quantifies this effect: a smaller GDOP indicates a favorable satellite geometry and higher accuracy, while a larger GDOP reflects poor geometry and degraded precision.

Signal Interference in the Nanjing Incident

The Nanjing disruption was caused by a deliberate jamming signal transmitted on the same frequency as GNSS. The receiver mistakenly treated the jamming signal as a legitimate satellite, inserting erroneous pseudorange values into the solution and producing locations far from the true position.

Carrier‑Phase Measurement: Pursuing Millimeter‑Level Accuracy

Pseudorange accuracy is limited by the code chip width (e.g., GPS C/A code ≈300 m). To achieve higher precision, carrier‑phase measurements use the carrier wavelength (≈19 cm for GPS L1). Resolving phase differences to 1 % yields theoretical millimeter‑level positioning.

The carrier‑phase observation equation includes the integer ambiguity term (the number of whole carrier cycles), which must be resolved by specialized algorithms.

Conclusion

The Nanjing GNSS outage illustrates that modern satellite navigation relies on precise time‑of‑flight measurements, sophisticated mathematical modeling, and robust error mitigation techniques such as GDOP analysis and carrier‑phase ambiguity resolution. While the technology is mature, its performance remains vulnerable to intentional interference and requires continual advances in signal processing and algorithm design.