The network can know its shape without knowing where its origin belongs
A network adjustment reports that its equations are singular. That does not automatically mean every observation is useless. The measurements may determine the relationships between stations while leaving a common shift undetermined. The missing information concerns the reference in which the coordinates are written, rather than the shape that the measurements can resolve.
This guide builds a small original network whose ambiguity is visible by hand. We will select two different origins, show which differences remain unchanged, derive their covariance and then introduce an uncertain external reference. A second case adds redundant observations and a disconnected station, separating a datum freedom from an actual gap in the measured network.
All coordinates, observations and uncertainties are fictional teaching fixtures. The scalar examples deliberately isolate translation; real survey networks can have other datum freedoms depending on their observations. This is not a universal control-selection procedure, an operational tolerance or an instruction to disregard qualified survey review.
Geodesy subject guide · Learning that transfers into life · How X Works hub
Choose your reading route
Route 1 · Chapters 1–2: Find the missing reference
Route 2 · Chapters 3–5: Solve the same shape twice
Route 3 · Chapters 6–8: Follow relative uncertainty
Route 4 · Chapters 9–10: Challenge the network
Route 5 · Chapters 11–12: Practise and report
Full chapter index · Independent worked case · Questions and practical checks
Chapter index
Find the missing reference
Name what the observations actually measure
Expose the translation that the design cannot see
Solve the same shape twice
Select one origin and compute a transparent solution
Select a centred datum without changing the observations
Distinguish datum selection from adding physical information
Follow relative uncertainty
Derive covariance under a fixed-origin convention
Carry the same uncertainty into a centred solution
Add an uncertain reference without losing shared dependence
CHAPTER 1 OF 12 · Find the missing reference
1. Name what the observations actually measure
Imagine three stations A,B and C arranged along a stipulated coordinate axis. The observations are signed differences from A to B and from B to C. They do not include a coordinate for A in an external reference system. Their mathematical job is to connect positions to one another, not to locate the entire group against an unobserved origin.
Let the unknown coordinates be xA,xB,xC, and let observations be d1=xB−xA and d2=xC−xB. If one candidate solution fits, adding the same constant a to all three coordinates leaves both differences unchanged. The observations cannot distinguish the shifted candidate from the original. That ambiguity is present even with perfect arithmetic and noiseless observations.
It helps to separate three questions. Which relative quantities are identifiable? Which reference freedom remains? Which uncertainty model describes the observations? A solver needs enough information about all three. Simply adding more decimal places to the same differences answers none of the missing-reference questions.
This is the simplest example of a datum defect: the observation model has a freedom that must be addressed before unique coordinate values can be reported. It is not synonymous with an inaccurate instrument, a bad benchmark or a corrupted file. Those problems can also exist, but they require separate evidence.
The GNU Gama network-adjustment documentation explains why a free geodetic network can have singular equations and how constraints select a solution. Here we derive our own small cases rather than prescribe a software setting. The useful first question is not how to make the warning disappear, but what transformation of the unknowns leaves every modelled observation unchanged.
CHAPTER 2 OF 12 · Find the missing reference
2. Expose the translation that the design cannot see
Collect the unknowns as x=(xA,xB,xC). The observation matrix A has rows(−1,1,0) and(0,−1,1), so Ax predicts the two differences. Let z=(1,1,1). Multiplication gives Az=(0,0). Therefore A(x+az)=Ax for any scalar a. The vector z describes the undetectable common translation.
In ordinary least squares with positive definite observation weights P, the normal matrix is N=A transpose P A. Because Az=0, Nz=0 as well. N cannot have an ordinary inverse on the full coordinate space. This is a property of the stated model, not a numerical accident caused by insufficient computer precision.
Our A has rank two: its two rows are independent, but its three coordinate columns cannot independently determine all three unknowns. The relative separation xC−xA is identifiable because it equals d1+d2. The absolute coordinate xA is not identifiable from these observations alone. One missing origin does not erase the two measured relationships.
More generally, a linear combination l transpose x is invariant to this translation when l transpose z=0. Its coefficients sum to zero. For example, the contrast(−1,0,1) gives xC−xA and satisfies that condition. The contrast(1,0,0) gives xA and does not. This is a direct way to test whether a requested result depends on the unresolved origin.
Keep this criterion attached to the actual observation matrix. A different network can leave rotations, scale or disconnected-component shifts unresolved. There is no universal rule that every geodetic network has exactly one defect, or that fixing one coordinate will always solve it. Inspect which freedoms the observations truly constrain before selecting the corresponding datum definition.
CHAPTER 3 OF 12 · Solve the same shape twice
3. Select one origin and compute a transparent solution
Give the fictional differences values d1=10 metres and d2=20 metres. Impose xA=0 as an exact coordinate convention for this example. Then xB=10 and xC=30 metres. The fitted differences recover the observations exactly. We have selected one member of an entire family of equally fitting coordinate triples.
Now choose xA=100 instead. The solution becomes(100,110,130), while the fitted differences remain10 and20. The stations did not move between these two mathematical descriptions. Their origin label changed. Reporting the second result as a100-metre physical displacement would confuse a datum choice with new observation evidence.
Our two observations and two relative degrees of freedom leave no residual redundancy. Zero residuals therefore do not independently establish that the observations are correct. If the first difference has an unrecognised bias, these coordinates can reproduce that bias exactly. The reference convention cannot reveal an error that the measurement design has no redundancy to test.
The condition xA=0 is often called fixing a coordinate, but its interpretation must be explicit. Here zero is an arbitrary gauge choice. It is not a claim that station A has a perfectly known authoritative external coordinate. A true surveyed control value would carry a different evidence account and usually its own uncertainty.
This distinction matters when software displays a zero uncertainty for a fixed coordinate. Under a hard mathematical constraint, that coordinate does not vary within the constrained solution. The displayed zero describes what was held constant in that calculation. It does not prove that the real monument’s location is known without error in the national or global reference system.
CHAPTER 4 OF 12 · Solve the same shape twice
4. Select a centred datum without changing the observations
A second convention imposes xA+xB+xC=0. Begin with the solution(0,10,30), whose mean is40/3 metres. Subtract that mean from every coordinate. The centred solution is(−40/3,−10/3,50/3), approximately(−13.333333333,−3.333333333,16.666666667) metres. Its coordinate sum is zero.
Check the observations again. xB−xA=(−10+40)/3=10 and xC−xB=(50+10)/3=20. The A-to-C separation is30. Every measured difference and its least-squares objective is unchanged. We selected the origin by centring the coordinates rather than attaching it to station A.
This constraint removes only the common-translation freedom of the stated network. In that limited sense it is a minimum datum condition: it supplies just enough information to select a coordinate representation. It does not introduce an additional measured separation. A constraint that instead changes the model’s identifiable relative shape has done something more than choose the origin.
The centred values are awkward fractions, but that makes them useful teaching numbers. They discourage the intuition that a coordinate close to zero must be physically stable or accurately located. Station B being near the centre is a consequence of the chosen convention and geometry, not an independent measurement of its absolute position.
GNU Gama also documents a regularisation criterion based on selected coordinate corrections. That is related to choosing the reference of a free solution, but it should not be reduced to a claim that every software option implements our simple coordinate-sum constraint. Read the specific programme’s definition, approximate-coordinate convention and supported observation model before equating a checkbox with a mathematical condition.
CHAPTER 5 OF 12 · Solve the same shape twice
5. Distinguish datum selection from adding physical information
Suppose an independent source supplies xA=100 metres. Adding that observation can connect the network to an external coordinate reference. It is a different operation from declaring xA=0 by convention. The source may also have uncertainty, an epoch, a physical endpoint and a frame realisation that need to match the intended comparison.
A hard constraint can be appropriate as an explicitly chosen mathematical datum. It is less honest to use one merely because a reference coordinate was supplied without its uncertainty. Turning an uncertain value into an exact condition can make the resulting absolute coordinate covariance smaller than the evidence supports.
There is another distinction: a datum-only translation should leave every fitted difference unchanged, while extra independent control can test the network. If a second external point coordinate disagrees with the measured separation, the combined solution must reconcile that disagreement under stated weights, or flag an inconsistency. It is no longer simply moving the whole network intact.
For example, our measured A-to-C separation is30 metres. External values xA=100 and xC=131 imply31 metres. No common translation can remove that one-metre discrepancy. The cause could involve observation error, endpoint identity, epoch mismatch or control information. Calling it a datum defect would avoid the actual incompatibility rather than explain it.
Start the review by listing what is a convention and what is an observation. Put uncertainty beside each observation. Then ask which quantities become identifiable when it is added. This small evidence ledger prevents a numerical constraint from silently acquiring the authority of an independent survey, while still allowing a useful free-network solution to be reported within its proper limits.
CHAPTER 6 OF 12 · Follow relative uncertainty
6. Derive covariance under a fixed-origin convention
Give d1 and d2 independent standard uncertainties1 and2 millimetres, respectively. Their variances are1 and4 square millimetres. These uncertainties apply to errors around the fictional metre-valued differences; using millimetres for the covariance makes the arithmetic readable without changing the model.
Under xA=0, xB=d1 and xC=d1+d2. The coordinate covariance has rows(0,0,0),(0,1,1),(0,1,5), in square millimetres. In particular, B and C are correlated because the uncertain first difference contributes to both. Their marginal standard uncertainties are1 andsqrt(5) millimetres under this fixed-origin description.
Now recover the variance of C−B. It is5+1−2×1=4 square millimetres, matching the variance of d2. If you discard the off-diagonal covariance, you report6 instead. A coordinate table containing only marginal standard deviations is therefore insufficient for every later difference calculation.
The variance of C−A is5 because the two independent differences add. The variance of B−A is1. These are identifiable relative quantities and have a direct relationship to the observation covariance. Their values are more useful than a superficial comparison of which coordinate column appears to carry the smallest uncertainty.
The zero row and column for A follow from the exact constraint. Do not attach that zero uncertainty to a physical claim outside the constrained model. If an uncertain reference coordinate is introduced later, its contribution belongs in the absolute-coordinate account. The relative variances can remain unchanged even while every absolute coordinate acquires additional uncertainty shared across the network.
CHAPTER 7 OF 12 · Follow relative uncertainty
7. Carry the same uncertainty into a centred solution
Express the centred coordinates directly in terms of the observations: xA=−(2d1+d2)/3, xB=(d1−d2)/3 and xC=(d1+2d2)/3. These linear relationships let us compute covariance without attempting to invert the singular normal matrix as though it were nonsingular.
With observation variances1 and4, the centred covariance has diagonal entries8/9,5/9 and17/9 square millimetres. Its off-diagonal entries are cov(A,B)=2/9, cov(A,C)=−10/9 and cov(B,C)=−7/9. The row sums are zero because the coordinate sum is constrained to remain exactly zero.
Check C−B using the full matrix:17/9+5/9−2×(−7/9)=36/9=4. Check B−A:5/9+8/9−2×2/9=1. Check C−A:17/9+8/9−2×(−10/9)=5. They match the relative variances under the fixed-A convention exactly.
The individual coordinate variances changed, yet the identifiable differences retained their uncertainties. This is not a contradiction or a free improvement in survey precision. We changed the coordinate reference convention. An absolute-coordinate variance must be interpreted with that convention before comparing it with a value from a differently constrained solution.
The example also shows why a negative covariance is not automatically an invalid uncertainty matrix. Centre-constrained coordinates can compensate one another. Validity concerns the whole matrix being symmetric and positive semidefinite, not every entry being positive. Here it is produced by a linear transformation of a valid observation covariance, and its exact zero-sum direction reflects the imposed constraint.
CHAPTER 8 OF 12 · Follow relative uncertainty
8. Add an uncertain reference without losing shared dependence
Let an independent external reference a estimate station A with variance9 square millimetres. Keep the same relative observations and assume a is independent of their errors. Then xA=a, xB=a+d1 and xC=a+d1+d2. The absolute-coordinate covariance has rows(9,9,9),(9,10,10),(9,10,14).
All three coordinates share the reference uncertainty. Marginal variances are9,10 and14, larger than under the exact xA=0 convention. Nevertheless C−B has variance14+10−2×10=4, and C−A has variance14+9−2×9=5. The common uncertain reference cancels from the differences under the stipulated model.
This cancellation is conditional on the dependence model. If the external coordinate and the relative observations share processing inputs, the assumption of independence may be inappropriate. Cross-covariance between a and the observed differences would then enter the propagation. The word external does not automatically mean statistically independent.
Now imagine station A suffers a physical shift between epochs while the remaining stations keep their relative positions. A datum convention that fixes A independently at both epochs can transfer that shift into every reported coordinate. Identifying a convention-induced common pattern does not establish which monument actually moved. Monitoring needs a stable reference strategy and independent evidence appropriate to the physical interpretation.
The practical lesson is not to prefer free or constrained adjustment universally. It is to distinguish the reference uncertainty, relative measurement uncertainty and physical reference stability. A clear report can show both the internal shape precision and the external tie, allowing the reader to decide which uncertainty matters for the intended comparison rather than assuming one scalar represents both.
CHAPTER 9 OF 12 · Challenge the network
9. Add redundancy and inspect what it can test
Introduce a third independent observation of C−A, with value31 metres for a deliberately exaggerated inconsistency. Give all three observed differences equal weights for this case. Write p=B−A and q=C−B. The objective is(p−10)²+(q−20)²+(p+q−31)².
Differentiating with respect to p and q gives2p+q=41 andp+2q=51. The solution is p=31/3 and q=61/3 metres, approximately10.333333333 and20.333333333. Their sum is92/3, approximately30.666666667. Residuals, defined as predicted minus observed, are(+1/3,+1/3,−1/3) metre.
A common translation still leaves those predictions and residuals unchanged. Redundancy supplied information about the compatibility of the relative observations; it did not supply an absolute origin. The network retains its translation defect even though its relative shape is now overdetermined and its residuals are nonzero.
Do not use this example as a recommended way to distribute a real one-metre discrepancy. We deliberately made the arithmetic large enough to inspect. Actual analysis requires observation units, uncertainty magnitudes, possible blunders and the relevant professional quality procedures. Least squares can always distribute some inconsistencies; that does not establish that averaging is the right disposition.
This guide’s new issue is datum identifiability, distinct from the existing levelling-loop article’s variance-weighted corrections. That companion is useful when the question is how redundant differences should be reconciled under a specified uncertainty model. Here the key additional test is whether shifting every coordinate changes the observation predictions. If not, an origin still needs an explicit definition.
CHAPTER 10 OF 12 · Challenge the network
10. Distinguish a disconnected station from the shared origin freedom
Add station D without any observation connecting it to A,B or C and without an external coordinate observation. Its coordinate does not appear in the design equations. The original three stations still share one common translation freedom, while D has another independent freedom. One datum condition cannot generally remove both.
Setting xA=0 leaves xD unconstrained. Setting the sum of all four coordinates to zero leaves one degree of freedom because translating A,B,C together can be balanced by a compensating change in D. The solver’s singularity is now partly a connectivity issue, not merely a shared origin convention for one connected network.
Adding a genuine observation from C to D joins the components and can reduce the defect. Adding an arbitrary exact coordinate for D can also produce a unique numerical result, but it does not create the missing measured relationship. The two actions have different scientific meanings even when both make a matrix factorisation succeed.
In a nonlinear distance-only network, additional freedoms can involve rotations and, depending on observation types, scale. Degenerate station geometry can also leave particular motions weakly determined or unobservable. The correct count is a property of the complete model and geometry. Our scalar chain isolates the idea rather than supplying a universal count for three-dimensional networks.
Use a connectivity drawing and a null-space test together. A drawing reveals an isolated component; the algebra reveals freedoms hidden in otherwise connected geometry. When an algorithm reports a rank problem, do not automatically tighten every constraint until it stops complaining. First identify whether the missing information is a coordinate convention, a physical connection, an observation type or an unsuitable station arrangement.
CHAPTER 11 OF 12 · Practise and report
11. Design independent checks for the identifiable quantities
A useful first test shifts all coordinates by the same arbitrary number, then recomputes every predicted difference. For the connected scalar model, each must remain unchanged. Next solve under two admissible datum conventions and compare fitted observations, residuals and the covariance of identifiable differences. These tests target the datum claim directly.
A third test asks for a quantity that is not invariant, such as xA. The result should change when the datum convention changes. If a report treats that change as evidence of ground motion, the interpretation test has failed even if both numerical solutions are correct. Reference labels belong with the coordinates, not only in a software log.
For a fourth test, add an uncertain external coordinate with known fictional variance and verify that its shared contribution enters every absolute coordinate. Then form station differences and confirm the appropriate cancellation. This checks the distinction between relative precision and absolute-reference uncertainty without relying on an actual benchmark being error-free.
For a fifth test, remove the C-to-D observation in a four-station extension and check whether the rank analysis detects the disconnected freedom. This adversarial test is valuable because a solver that silently assigns an arbitrary value to D may output plausible-looking coordinates while concealing an unconstrained quantity.
Finally compare against independent control only when it is compatible in endpoint, frame and epoch. An internal residual test evaluates the observation model’s consistency. An external comparison evaluates another relationship with its own uncertainties. Neither should be advertised as a universal proof of positional accuracy, and a clean residual pattern cannot certify the physical stability of all reference monuments.
CHAPTER 12 OF 12 · Practise and report
12. Report the reference separately from the measurement result
A reusable result identifies the observation model, station connectivity, datum freedoms, imposed constraints, weighting and covariance treatment. It says which constraints merely select a reference and which incorporate external information. It also separates fixed mathematical values from measured control coordinates whose uncertainties may matter outside the adjustment.
For our first fixture, the most defensible compact result is that B−A=10 metres and C−B=20 metres under the stated observations, with relative variances1 and4 square millimetres in the independent-noise case. Coordinates(0,10,30) use the exact xA=0 convention; centred coordinates use the zero-sum convention. Neither set alone supplies an authoritative global origin.
For the reference-tied extension, state the external estimate a, its variance9 and its assumed independence from the relative observations. Give shared covariance when later differences may be formed. A coordinate’s small marginal uncertainty and a baseline’s small uncertainty answer different questions; the report should not force the reader to guess which one was meant.
For repeated epochs, preserve the datum strategy and control history before comparing coordinates. A changed reference selection can produce a common shift without changing the measured network shape. Conversely, aligning away a real common displacement can conceal a scientifically important signal if the target question requires an external frame. The purpose of the comparison determines which freedoms are nuisance and which carry information.
The transferable insight is simple: a unique number can come from a convention as well as a measurement. Good geodesy keeps those sources of uniqueness distinguishable. You can then use a free solution honestly, tie it to appropriate external evidence when needed, and explain why resolving an algebraic singularity is only one step toward a trustworthy statement about positions.
Independent worked case and evidence checks
Try a new chain with observed B−A=6 metres and C−B=9 metres. Under xA=0, coordinates are(0,6,15). Their mean is7, so a zero-sum convention gives(−7,−1,8). Check both differences directly and explain why the changed coordinates do not imply that every station moved seven metres.
Let the two independent difference variances be4 and1 square millimetres. Under fixed A, the B,C covariance block has rows(4,4),(4,5). The variance of C−B is5+4−8=1. The variance of C−A is5. These checks should be recovered from the observation relationships before using a general covariance routine.
Under the centred datum, apply xA=−(2d1+d2)/3, xB=(d1−d2)/3,xC=(d1+2d2)/3. The diagonal variances become17/9,5/9 and8/9. Off-diagonal covariances are−7/9,−10/9 and2/9 forAB,AC andBC. Their row sums are zero; the variance of C−B is8/9+5/9−4/9=1.
Now supply an independent reference coordinate a with variance16 square millimetres. Absolute covariance becomes rows(16,16,16),(16,20,20),(16,20,21). The C−A variance remains5 and C−B variance remains1. If only the marginal variances were exported, a later user could not infer this cancellation reliably.
For a changed-context check, add a direct C−A observation of16 metres with equal weight to the two original differences. The fitted relative values solve2p+q=22 andp+2q=25, givingp=19/3 andq=28/3. Their sum is47/3, and residuals are again(+1/3,+1/3,−1/3). Translation remains unobservable despite the added redundancy.
Before looking at a solution, predict what happens if station labels are swapped, a signed difference is reversed or an isolated station is added. Reversing a difference requires reversing both its model row and observed value. Changing only one creates a different dataset, not another datum convention. That distinction is a useful final check on whether the reference idea has transferred.
Worked comparison
| Question | Datum-only change | New observation or assumption |
|---|---|---|
| Shift all three scalar coordinates | Predicted differences unchanged | No external origin measured |
| Fix A versus centre the network | Individual coordinate variances change | Relative covariance should remain compatible |
| Add uncertain external A | Adds a common reference contribution | Preserve its covariance and independence assumptions |
| Add C−A observation | Does not remove translation freedom | Adds relative redundancy |
| Add isolated D | Creates another unresolved coordinate | One original constraint no longer suffices |
Questions and practical checks
Does singular mean unusable?
No. Some relative quantities can remain identifiable while an overall origin or other datum freedom is unresolved. Identify the null space and requested quantity before judging usability.
Does fixing one station make its true coordinate exact?
No. It makes that coordinate constant within the constrained calculation. Its physical location in an external reference can still be uncertain or change through time.
Why can centred coordinates look more precise?
Their individual variances refer to a different reference convention. Compare identifiable relative quantities with full covariance, not isolated marginal uncertainties from different datums.
Does adding more differences remove every datum defect?
Not necessarily. Redundant relative observations can test consistency while remaining insensitive to a common translation. Observation type and geometry determine what becomes identifiable.
Is a constraint always harmless?
No. A minimum datum constraint selects a representation without changing identifiable shape in the stated model. Additional or incompatible constraints can change the fit and require an explicit evidence account.
Continue with the connected subjects
Keep reference and physical meaning explicit
Geodetic engineering: frames, control and engineering decisions
Coordinate epochs and compatible motion models
Follow adjustment and shared uncertainty
Levelling-loop misclosure and variance-weighted corrections
Height differences and shared-model covariance
Frame maps, inverse checks and common parameter uncertainty
Sources and scope
GNU Gama official manual source: network adjustment, datum constraints and covariance
The numerical cases in this guide are original fictional teaching examples, not measurements from a real site. This is an educational explanation, not a field safety instruction, engineering approval or an official certification. For a real project, use qualified practitioners, current product documentation and site-specific evidence. Sources were checked on 31 October 2026.
Continue through local references, network constraints, physical heights and rates
Local ENU Rotations, Tangent-Plane Origins and Covariance
Geopotential Numbers, Normal Heights and Path-Dependent Levelling
Time-Dependent Helmert Rates, Velocity Transformations and Epoch Recentring