Selecting independent baselines for GNSS network adjustment

Baseline selection can affect the quality of a GPS/GNSS control geodetic network. Choosing independent baselines – the minimal set of baseline vectors – is crucial for a robust network adjustment.

Below we describe what independent baselines are, why they matter, and how to select them.

What are independent GNSS baselines?

In a GNSS survey session with multiple receivers, you can form baseline vectors between every pair of simultaneously occupied receivers. However, many of those baselines are redundant (linearly dependent).

A set of baselines is considered independent if no baseline can be derived by a linear combination of the others. In practical terms, independent baselines are the smallest subset of baselines that still connect all points in the network. Think of it like a spanning tree connecting all receivers.

For example, in a session with N receivers, there are N(N–1)/2 possible baselines, but only (N–1) of them are independent. Those N–1 baselines are the non-redundant (also called non-trivial) baselines needed to compute the network solution.

In a 4-receiver GNSS session, only 3 of the 6 possible baselines are independent (solid lines). The other 3 baselines (dashed, not shown) are dependent or “trivial” because they don’t add new information. Only three baselines (the solid lines connecting the 4 stations) are used as independent baselines, while the other three possible connections would form a closed loop using the same session’s data – thus they are dependent baselines that add no new information.

This illustrates the general rule: avoid selecting baselines that form a single-session closed loop. All baselines derived from the same simultaneous observations share the same errors, so any loop composed solely of those baselines will close with zero error, contributing nothing to network strength or redundancy. In other words, including them would only recycle the same information.

Why only independent baselines matter

When performing a network adjustment (least-squares adjustment of the baselines to solve for station coordinates), using independent baselines is essential. According to adjustment principles, the input observations should be independent – using dependent (redundant) baselines violates this, leading to misleading results. Dependent baselines are essentially combinations of others, so they introduce false redundancy and can make the adjustment appear unrealistically consistent (over-optimistic precision). Once the independent baselines are processed, “processing the remaining dependent baselines merely adds false redundancy and causes the adjustment statistics to be overly optimistic.” For this reason, dependent baselines must not be used in the final network adjustment. The key of network adjustment is to use the minimum number of high-quality baselines needed for a unique solution. Think of it this way: if you include all baselines from one session (including those that form a closed loop), your adjustment might report zero loop misclosure for that session – not because your network is perfect, but because you’ve added redundant data from the same session that mask the errors. Such dependent lines cannot improve accuracy; they only give an illusion of redundancy. Real redundancy in GNSS networks comes from having overlapping baselines measured in different sessions (or with independent setups), not from adding all baselines from one session.

Moreover, network design guidance recommends that each station be connected via at least two independent baselines (to two different stations) across the network, ensuring the network forms interconnected loops spanning multiple sessions rather than a single “chain” (avoiding a hinged network). This provides true redundancy and robustness: if one baseline is problematic, the network is still held together by alternate paths.

How to select independent baselines (best practices)

1. Determine the number of independent baselines: For any single simultaneous observing session with N receivers, plan to use N–1 baselines in the adjustment (e.g. 4 receivers equals to 3 independent baselines).

2. Choose baselines that avoid closed loops: Ensure the selected baselines do not form a closed loop within the same session. In practice, this usually means selecting baselines in a tree structure. A common approach is to pick one receiver as a reference (or a hub) and use baselines from it to all other receivers, or otherwise connect receivers in a way that each new receiver adds a new baseline without completing a loop. The trivial baselines that would close the loop (the dependent ones) should be left out of the adjustment (they can still be measured for quality check, but not used in the final computation).

3. Prefer most accurate baselines: When multiple choices of independent baseline sets are possible, it’s wise to choose baselines that are likely to give the most accurate results. Often, the shorter, direct baselines (with good satellite geometry and low errors) are preferred as the independent set, and longer or noisy baselines are treated as dependent/redundant. In our 4-receiver example above, the planners chose the three shortest vectors as the independent baselines, eliminating the longer ones as trivial. However, this is a rule of thumb – if a short baseline’s data turned out poor (e.g. due to multipath or cycle slips), one might swap in a different baseline.

4. Use redundant measurements across sessions for QC: Although we exclude dependent baselines from the final adjustment, it’s good practice to measure extra baselines (creating loops across sessions) for quality control. Loop closures that involve baselines from different sessions provide a meaningful check on survey consistency. You can measure all possible baselines in the field to detect blunders, but plan to only include an independent set in the adjustment. Any loop closure error should come from multi-session loops; a zero closure error in a single-session loop is a red flag that you’re using trivial (same-session) data in the loop.

5. Consider an efficient workflow: An insightful strategy from research is to handle baseline selection in two phases. Wei et al. (2011) suggest first including all candidate baselines in a preliminary unconstrained network adjustment to identify any bad observations, and then selecting the independent set from the vetted baselines for the final (possibly constrained) adjustment. This two-step approach saves time: rather than painstakingly picking baselines beforehand, you let the initial adjustment statistics reveal which baselines are outliers or causing inconsistency. After removing or fixing those, you then choose an independent subset from the remaining good baselines and run the final adjustment. This method was found to meet precision requirements with higher efficiency than manually selecting baselines from the start. In fact, the research concluded that “selecting all the passed baselines as a whole for network adjustment and then selecting the independent baselines for the constrained adjustment… is not only more efficient, but also achieves higher accuracy”.

In summary, independent GNSS baselines are the backbone of a reliable network solution. Select one fewer baseline than the number of receivers per session, avoiding internal loops. Exclude dependent (trivial) baselines from your final computations to prevent misleadingly optimistic results. Instead, build redundancy by re-occupying stations in multiple sessions rather than by adding all baselines in one session. By carefully choosing a strong independent baseline set – and using a smart workflow to validate data – you’ll ensure your GNSS control network adjustments are both accurate and efficient.

References

• Drawing the baselines