Question

A country has 7 permanent Global Navigation Satellite System stations covering its territory. Their surveying organization generates a network solution after applying double differencing to the observations. These 7 permanent stations can view 5 to 10 common satellites at any given epoch. What is the range (minimum, maximum) of the number of independent double differenced observables possible?

• A. (35, 70)
• B. (30, 56)
• C. (28, 48)
• D. (24, 54)

Hint:

For GNSS double differencing, use the formula \((n-1)(s-1)\) to estimate independent DD observables, where \(n\) is the number of stations and \(s\) is the number of satellites. This reduces clock and atmospheric biases efficiently.

Answer 1

The Correct Option is D

Double differencing is a GNSS technique that uses differences between satellite observations at two receivers and then differences between pairs of satellites, effectively reducing errors such as satellite and receiver clock biases.

To compute the number of independent Double Differenced (DD) observables:

Number of baselines between \( n \) stations is given by: \[ \binom{n}{2} = \frac{n(n-1)}{2} \] For \( n = 7 \): \[ \frac{7 \times 6}{2} = 21 \text{ independent baselines} \] For each baseline, the number of independent DD observables is \( (s - 1) \), where \( s \) is the number of common satellites observed

\[ \text{Min DD observables} = 21 \times (5 - 1) = 84 \quad \text{(but must consider only independent DDs)} \] But we must count independent DD observables. The actual number is: \[ \text{Independent DD observables} = (n - 1)(s - 1) \] For minimum: \( (7 - 1)(5 - 1) = 6 \times 4 = 24 \)
For maximum: \( (7 - 1)(10 - 1) = 6 \times 9 = 54 \)

Thus, the correct range is \( \boxed{(24, 54)} \).