Course · ← 0016 The Third Dimension · 0018 Imagery Pipelines → · depth track
Lesson 0017 · ~18 minutes
Lesson 0013 said the parcel fabric computes coordinates by least squares. This is what that sentence contains: weights, redundancy, residuals, error ellipses, and the statistical test that finds the one measurement somebody typed wrong.
Measure a network of survey points and you will always take more measurements than you strictly need. Those extra measurements contradict each other — not because anyone erred, but because every observation carries random error. There is no set of coordinates that satisfies all of them exactly.
Least squares picks the coordinates that come closest, in a specific and defensible sense. Observation equations relate measured values to their residual errors and the unknown parameters; the equations are weighted according to the precision of the observations, and solved by minimising the sum of the weighted squared residuals.1
observation = f(unknowns) + residual
minimise Σ w·v² w = weight, v = residual
weight w = 1 / σ² precise observations pull harder
Weighting is the whole game. A GNSS baseline good to 5 mm and a tape measurement good to 50 mm are not equal evidence, and an unweighted average pretends they are. Weight by 1/σ² and each observation influences the answer in proportion to how much it deserves to. Get the weights wrong and the mathematics is still perfect — it is just answering a question about a network that does not exist.
Redundancy is the difference between the number of observations n and the minimum number needed to determine the model uniquely.1 With zero redundancy there is exactly one solution, no residuals, and no way to know whether any measurement is wrong. Every additional observation buys two things at once: a slightly better estimate, and — more importantly — the ability to detect a blunder.
This is why measuring a traverse "twice for safety" is weaker than measuring cross-ties. Duplicate observations of the same quantity add redundancy in one place. A braced network spreads it everywhere, which is what makes a bad observation stand out from its neighbours.
Two adjustment configurations, matching the free/constrained pair from lesson 0013:
| Configuration | Holds fixed | Tells you |
|---|---|---|
| Free / minimally constrained | The bare minimum to fix position and orientation | Whether the measurements agree with each other — internal quality, uncontaminated by control |
| Constrained / weighted | Control points, at their own stated precision | How well the network fits the existing framework — absolute accuracy, and whether the control itself is suspect |
Run free first. If the free adjustment is clean and the constrained one is ugly, your measurements are fine and the control is wrong — a conclusion you cannot reach if you only ever run constrained.
An adjustment report is where most people's confidence collapses. Four numbers carry nearly all the meaning.
Residuals (v). How far each observation moved to fit the solution. Large ones are suspects, but size alone is not proof — a weak part of the network lets a bad observation hide by pushing the residual onto its neighbours.
The variance factor (reference variance). Roughly: did the residuals come out the size your stated precisions predicted? Near 1 means your error estimates were honest. Much greater than 1 means either a blunder is present or you claimed better precision than your instrument delivers. Much less than 1 means you were pessimistic — a real result, and one that quietly inflates every error ellipse downstream.
Redundancy numbers. Per observation, how much of its own error the network can absorb and reveal. Near zero means the observation is effectively unchecked: a blunder in it will pass through into the coordinates undetected. These numbers are a design tool — they tell you where the network is blind before you go to the field, not after.
Error ellipses. The uncertainty of each adjusted point, as a shape rather than a number. Direction matters: a point can be tightly held across a line of sight and loose along it. Reporting one figure for "accuracy" throws away the part a surveyor needs.
Baarda's data snooping, from a series of papers in the mid-to-late 1960s, remains the standard method for identifying blunders in geodetic data.2 The refinement that makes it work: rather than depending on the estimated error of the observation, it computes the uncertainty in the residual of that observation.3
w = v / σ_v residual, divided by the uncertainty OF THAT RESIDUAL
not v / σ_observation ← the naive version, which flags well-checked
observations and misses unchecked ones
Normalise, test the largest, reject one observation, re-adjust, repeat. One at a time — a single blunder distorts every other residual, so rejecting a batch on one pass usually throws away good data along with the bad. Baarda's tests use the known a-priori variance; the tau criterion is the variant for when you must use the a-posteriori estimate instead.3
Adjustment output is evidence, not a build artefact. When a boundary is later disputed, the defensible answer is "this coordinate came from these observations, with this reliability, adjusted on this date against this control" — which means the observations, weights, control set and adjustment report must be stored and versioned like everything else in lesson 0004. A system that keeps only the adjusted coordinates has kept the conclusion and thrown away the argument.
NOAA Technical Memorandum NOS NGS 74 — On Least-Squares Adjustments. A national geodetic survey's own treatment: rigorous, free, and written for people who must defend the numbers:
ngs.noaa.gov — NOAA TM NOS NGS 74 (PDF)
Gentler entry points: xyHt's "What Is a Least Squares Adjustment Anyway?" and its data snooping and tau criterion instalment, which is the clearest short account of why the normalisation matters.