Two instruments at one site do not automatically measure the same point
A radio telescope, a laser-ranging telescope and a satellite-navigation antenna stand at the same observatory. Their measurements can help describe the same Earth, but their reference points occupy different positions. Sharing an address does not make those positions identical. A local tie is the surveyed connection that allows the different reference points to be related.
This guide follows that connection from accessible survey targets to instrument reference points, from local coordinates to a common basis, and from a measured vector to its uncertainty. You will calculate original fictional ties, deliberately reverse an endpoint convention and test a closed survey loop that still conceals an error.
The examples are educational, not an observatory survey specification or an engineering acceptance test. They use invented coordinates and uncertainties. Actual instrument reference-point definitions, survey methods and reporting requirements come from the relevant technical documentation and qualified practitioners. No fictional offset or precision below is a coefficient for a real instrument.
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Choose your reading route
Route 1 · Chapters 1–2: Define the physical connection
Route 2 · Chapters 3–5: Build the tie vector
Route 3 · Chapters 6–8: Audit the geometry and uncertainty
Route 4 · Chapters 9–10: Check epochs and combination
Route 5 · Chapters 11–12: Transfer and report
Transfer the checks to orientation, common shifts and changed endpoints
Full chapter index · Independent worked case · Questions and practical checks
Chapter index
Define the physical connection
Separate a shared site from a shared reference point
Define vector direction before using a distance
Build the tie vector
Connect accessible targets to the intended endpoints
Rotate a measured vector into compatible axes
Infer an inaccessible point without treating the model as an observation
Audit the geometry and uncertainty
Test network geometry rather than counting observations
Propagate target and offset uncertainty through the whole path
Use closure while recognising the errors it cannot detect
Check epochs and combination
Keep the tie epoch distinct from a global frame label
Treat a local tie as information with weight, not a command
CHAPTER 1 OF 12 · Define the physical connection
1. Separate a shared site from a shared reference point
Co-location means that different geodetic techniques operate near one another. It does not mean that their coordinate solutions refer to one common material point. The location assigned to a technique follows its reference-point definition and modelling convention. Before subtracting coordinates, identify the actual endpoints each coordinate represents.
The International Laser Ranging Service explains why this matters for combining techniques into a terrestrial reference frame. Its site-survey guidance describes vectors connecting instrument reference points and notes that an inaccessible reference point may need to be inferred from measurements to accessible targets. The NOAA site-survey programme likewise treats local ties as measured information supporting co-located techniques.
Imagine two desks in the same classroom. Saying that both are in the classroom tells a visitor where to start looking, but does not supply the vector from one desk corner to the other. An observatory name plays a similar organisational role. The measured tie supplies the geometrical relationship that the name cannot.
Start a working record with endpoint A, endpoint B, vector direction, epoch, coordinate basis and units. Include the instrument configuration that makes each reference point meaningful. If a report gives a distance between two visible structures without establishing that those structures represent the required reference points, it has not yet supplied the intended local tie.
CHAPTER 2 OF 12 · Define the physical connection
2. Define vector direction before using a distance
Let X_A and X_B be positions in one common Cartesian basis at one epoch. Define the tie from A to B as d_AB = X_B − X_A. Its three components retain orientation. Its length |d_AB| is only one summary. Many vectors with different directions share the same length.
Choose fictional local east-north-up coordinates A = (10,20,2) metres and B = (13,24,2.5) metres. The tie is (3,4,0.5) metres. Its horizontal length is five metres and its three-dimensional length is √25.25, approximately 5.024938 metres. Reporting “five metres apart” loses both the horizontal direction and the vertical separation.
Reverse the endpoint order and the vector becomes d_BA = (−3,−4,−0.5) metres. The length does not change. This is why a distance-only check cannot expose a reversed tie. A combination that expects B minus A cannot safely accept A minus B simply because the magnitude looks plausible.
Check subtraction by reconstruction: X_A + d_AB must equal X_B, component by component. Check the reverse relation d_AB + d_BA = 0. These are inexpensive tests to include in teaching code and practical data handling. They validate the declared algebraic convention, not the correctness of the survey or the endpoint definitions.
CHAPTER 3 OF 12 · Build the tie vector
3. Connect accessible targets to the intended endpoints
Often the point that can be observed conveniently is not the point required by the geodetic model. Let P_A and P_B be accessible survey targets. Define their reference-point offsets as e_A = X_A − P_A and e_B = X_B − P_B, expressed in the same basis. Then d_AB = (P_B − P_A) + e_B − e_A.
Use an accessible-target difference of (3.010,3.980,0.520) metres. Let e_A = (0.020,−0.010,0.030) metres and e_B = (0.010,0.010,0.010) metres. Adding e_B and subtracting e_A yields the intended reference-point tie (3.000,4.000,0.500) metres. The offsets correct the endpoints, not the definition of the coordinate frame.
A common mistake is to add both offsets because both are called corrections. That would give (3.040,3.980,0.560) metres here. The words do not determine the signs. The definitions e_A = X_A − P_A and e_B = X_B − P_B do. Draw or describe the directed path from A to P_A, then P_B, then B if the equation feels uncertain.
Preserve the original target observations and offset definitions alongside the derived tie. This allows another analyst to check whether an instrument configuration changed, an offset was applied twice or an opposite direction was supplied. A final vector without its endpoint chain can be numerically precise yet difficult to interpret or update responsibly.
CHAPTER 4 OF 12 · Build the tie vector
4. Rotate a measured vector into compatible axes
A survey may determine coordinates in a local working basis, while another product uses geocentric Cartesian axes. A difference vector must be expressed in compatible axes before it can be compared or combined. Translation of the coordinate origin cancels from a vector difference, but rotation of the basis does not disappear.
In a planar teaching rotation through ninety degrees counterclockwise, use R = [[0,−1],[1,0]]. It transforms a local horizontal tie (3,4) into (−4,3). Both have length five. If another analyst retains (3,4) while claiming to have changed to the rotated basis, the unchanged length will not reveal the directional error.
Use the same rotation for every term in the endpoint equation. Rotating the target difference but leaving e_A and e_B in their original basis adds components along different directions. All arrays may be labelled metres, yet their component sums no longer represent one physical vector. Matching units is necessary but not sufficient.
The local ENU companion develops the connection between local and Cartesian axes. Here the important invariant is |Rd| = |d| for an orthogonal rotation, with the inverse transformation recovering d. State the basis and rotation convention explicitly. A local orientation uncertainty is separate from the exact coordinate rotation used to describe an already known vector.
CHAPTER 5 OF 12 · Build the tie vector
5. Infer an inaccessible point without treating the model as an observation
The ILRS guidance notes that the reference point of a laser-ranging mount can be inaccessible and that survey procedures and modelling are used to connect it to observable points. This is a different task from simply measuring a straight line between two visible marks. A geometrical model is part of the inference chain.
Consider a fictional horizontal rotating assembly with an accessible target at a known lever arm l = (0.2,0) metres relative to a fixed centre O. At orientation θ, its target position is P(θ) = O + R(θ)l. A measurement of P is therefore not automatically a measurement of O. The reference-point estimate follows O = P − R(θ)l under this declared rigid model.
At zero degrees, the offset from O is (0.2,0). At ninety degrees, it is (0,0.2). Subtracting the unrotated lever arm from both measurements makes the inferred centre move even though O stayed fixed in the teaching model. The error arose from orientation handling, not from a physical translation of the whole instrument.
This example does not prescribe how to locate the invariant point of a real telescope. Actual instruments can have additional geometry, axis offsets, deformation and configuration effects. Its practical lesson is narrower: identify which parts are directly observed, which depend on mechanical information and which are inferred through a model. Carry uncertainty in that model rather than declaring the inferred point exact.
CHAPTER 6 OF 12 · Audit the geometry and uncertainty
6. Test network geometry rather than counting observations
Suppose an unknown point is inferred from ranges to known surrounding points. Each range is sensitive primarily along its line of sight. Repeating a range from one direction can improve information about that direction, but it does not automatically determine a perpendicular component. Observation count and geometrical independence answer different questions.
In a simple two-dimensional linearised example, observations with rows [1,0] constrain only an east coordinate correction. Ten copies of that row still leave the north correction undetermined. Adding a row [0,1] provides the missing direction. Nearly parallel rows instead give a formally solvable problem that may be weak across the narrow angle between them.
An inferred instrument reference point can inherit similar weaknesses from the orientations and targets used in its geometrical model. A small residual at the observed configurations does not demonstrate that an unobserved orientation is well predicted. A held-out configuration provides a stronger challenge than another repetition of the same geometry.
Describe the actual sensitivity before reporting a precise-looking three-component tie. State what constrains each component and how model parameters are separated. The free-network companion explains datum ambiguities in a broader setting; the lesson here is that a local survey also needs an identifiable design. Constraints can choose coordinates, but they should not be presented as measurements of an otherwise invisible direction.
CHAPTER 7 OF 12 · Audit the geometry and uncertainty
7. Propagate target and offset uncertainty through the whole path
The tie equation combines several uncertain quantities. With independent target-difference and offset errors, its covariance is the sum of their covariance matrices because changing the sign of an independent term does not change its variance. When terms are correlated, cross-covariance terms are also needed. Independence must be stated, not inferred from separate column headings.
For one fictional component, let the target-difference standard deviation be two millimetres, and each endpoint-offset standard deviation be one millimetre. If the three errors are independent, tie variance is 2² + 1² + 1² = 6 square millimetres. Its standard deviation is √6, approximately 2.449490 millimetres.
Now suppose the two endpoint offsets have covariance +0.5 square millimetres, while remaining independent of the target difference. Since the tie contains e_B − e_A, its variance becomes 4 + 1 + 1 − 2×0.5 = 5 square millimetres. The standard deviation is √5, approximately 2.236068 millimetres. The signs in the measurement equation determine how shared uncertainty enters.
These are conditional model uncertainties, not realised survey errors. A misidentified reference point or an omitted orientation effect is not automatically included because a small formal covariance was computed. Keep the complete covariance and the evidence supporting the model. A standard deviation stripped from its definitions cannot explain which parts of the measurement chain it covers.
CHAPTER 8 OF 12 · Audit the geometry and uncertainty
8. Use closure while recognising the errors it cannot detect
For three reference points, consistent ties satisfy d_AB + d_BC + d_CA = 0. This closure follows directly from coordinate subtraction. It provides a valuable compatibility check when independently measured connections form a loop, but it is not a proof that every reference point and offset is correct.
Use fictional vectors d_AB = (3,4,0.5), d_BC = (−1,2,−0.2) and d_CA = (−2,−6,−0.3) metres. Their sum is zero. Now suppose the inferred position of B has an unrecognised offset q. The A-to-B vector changes by +q and the B-to-C vector by −q. Their sum still closes, even though both affected connections are wrong for the intended reference point.
This is a useful distinction between an internally consistent coordinate construction and an independent survey check. If all reported ties were calculated from the same fitted coordinates, their exact algebraic closure may be guaranteed by construction. It does not provide the same evidence as additional independently acquired observations that were not forced into that coordinate solution.
A nonzero closure also needs uncertainty and dependence information before it is judged. Summing three covariance matrices assumes independent vector errors; shared endpoints often violate that assumption. Use closure to ask a precise question about compatibility, then inspect independent reference-point and configuration evidence. Do not assign a geological or instrumental cause to a loop discrepancy from its magnitude alone.
CHAPTER 9 OF 12 · Check epochs and combination
9. Keep the tie epoch distinct from a global frame label
A local tie describes a connection at an epoch or over a stated interval. A terrestrial frame may have its own reference epoch, while station coordinates and instrument configurations can change over time. These labels do not automatically refer to the same physical state. The coordinate-epoch companion explains the broader distinction.
Write d_AB(t) = X_B(t) − X_A(t). If both reference points share exactly the same translation velocity, that contribution cancels and the tie stays constant. If their velocities differ, the tie changes at rate v_B − v_A. A stable site name does not decide which of these physical situations applies.
For a fictional tie with vertical component 0.500 metres at t_0, suppose B moves upward relative to A at two millimetres per year. After three years, the vertical tie becomes 0.506 metres in a constant relative-velocity model. Reusing the older tie without the epoch distinction would introduce a six-millimetre discrepancy under these invented assumptions.
Real surveys should not adopt that linear model merely because it makes extrapolation easy. Equipment changes, monument motion and configuration-dependent offsets may require different treatment. Keep survey dates, endpoint histories and any propagation model together. If the evidence does not establish a valid extrapolation, report the limitation instead of turning an old measurement into a supposedly current exact constraint.
CHAPTER 10 OF 12 · Check epochs and combination
10. Treat a local tie as information with weight, not a command
A tie can connect coordinate solutions from different techniques. Its role is to contribute measured information about their relative endpoints. It does not logically require every technique estimate to agree exactly, and it does not remove every systematic effect in those techniques.
Use a scalar teaching combination. Two technique products imply a relative separation of 10 millimetres with variance four square millimetres. A local survey implies 14 millimetres with variance one square millimetre. If their errors are independent, zero-mean and correctly described, an inverse-variance weighted estimate is (10/4 + 14/1)/(1/4 + 1) = 13.2 millimetres. Its formal variance is 1/(1/4 + 1) = 0.8 square millimetres.
The combined estimate does not equal both inputs. Their difference of four millimetres remains an important piece of evidence. Its standard deviation under the independent model is √5 millimetres, so the standardised difference is approximately 1.788854. Interpreting that number requires the assumed error distribution, the validation context and any multiple comparisons; it is not a universal acceptance rule.
Setting the tie variance to zero would replace a measured relationship with an exact constraint. That may force one form of agreement while concealing the survey’s uncertainty. Conversely, reducing a technique’s weight until disagreement vanishes is not a diagnosis. Preserve the distinction between estimating a common result and investigating why the separate inputs disagree.
CHAPTER 11 OF 12 · Transfer and report
11. Transfer the checks to orientation, common shifts and changed endpoints
Change all site coordinates by the same translation vector. The tie d_AB must remain unchanged. Change the basis by a common orthogonal rotation and both the tie and covariance must transform consistently, while its physical length stays unchanged. These invariants are useful tests of an implementation that claims to perform only a coordinate relabelling.
Change one endpoint definition instead. The tie generally must change, even if no physical instrument moved. For example, replacing an accessible target by its inferred reference point adds the declared eccentricity. This is a change of represented quantity, not simply a frame transformation. State that distinction when comparing old and new survey records.
Orientation uncertainty grows into transverse position uncertainty through a lever arm. For a fictional 100-metre horizontal vector and a small uncertain orientation of one arcsecond, convert the angle to π/(180×3600) radians. The first-order transverse uncertainty contribution is approximately 0.000484814 metres, or 0.484814 millimetres. This is a sensitivity calculation, not an accuracy specification for an instrument.
Keep this separate from random range uncertainty and mechanical endpoint uncertainty. A direction error may create correlated component errors rather than independent additions along each axis. The covariance rotation and horizontal-ellipse companions explain how to describe the resulting geometry. The wider transfer is to follow every uncertainty through the physical equation that connects it to the final quantity.
CHAPTER 12 OF 12 · Transfer and report
12. Write a report that distinguishes observation, inference and interpretation
An auditable local-tie report identifies reference-point definitions, accessible marks, eccentricities, instrument configurations, survey epochs, coordinate basis, vector direction, units and uncertainty. It preserves the original observations and documents which values came from direct measurements, mechanical information, fitted geometry or extrapolation.
It also identifies the independent checks: reverse-vector reconstruction, held-out configurations, repeated surveys and any genuinely redundant loop observations. A check guaranteed by the fitted coordinate construction should be labelled as internal consistency, not independent validation. This lets another reader understand what the evidence actually challenges.
When two techniques disagree after applying a tie, report the residual and its uncertainty model before interpreting a cause. The discrepancy might involve the survey, endpoint handling, time evolution, a technique-specific model or their dependence. A local tie makes that comparison better defined; it does not automatically assign responsibility to one instrument.
The gold standard is a complete connection from physical points to a reproducible vector, with uncertainty and limits visible. The result is useful because someone else can reconstruct it, test it and update it when an endpoint changes. That discipline is what allows measurements made by different instruments to contribute to one coherent account of a changing Earth.
Independent worked case and evidence checks
Independent exercise: use new fictional accessible-target coordinates P_A = (0,0,0) metres and P_B = (5,−2,1) metres. Define reference-point offsets e_A = (0.010,0.020,−0.010) and e_B = (−0.020,0.010,0.030) metres, with each offset pointing from its accessible target to its intended reference point.
The tie is P_B − P_A + e_B − e_A = (4.970,−2.010,1.040) metres. Reconstruct the endpoints independently: X_A = (0.010,0.020,−0.010) and X_B = (4.980,−1.990,1.030). Subtracting these reconstructed positions reproduces the tie. This second path is an arithmetic check, not an independently acquired survey.
Reverse the tie and verify that adding it to X_B recovers X_A. Then change the sign convention of both offset fields in the input file, so that each now points from reference point to accessible target. The equation must change accordingly: subtract the B offset and add the A offset under the new definitions. Keeping the old equation after changing the field definition would create an error even though the physical geometry stayed identical.
Apply a ninety-degree horizontal basis rotation. The tie components become (2.010,4.970,1.040) metres when the vertical axis is retained. The horizontal length and three-dimensional length must remain unchanged. Rotate the endpoint offsets and target difference separately and confirm that their recombination produces this same result. This test exposes partial transformations that accidentally mix bases.
For a covariance check, let a single target-difference component have variance nine square millimetres and the two offset components each have variance four square millimetres. With independence the tie variance is 17 square millimetres and its standard deviation is √17, approximately 4.123106 millimetres. If the two offset errors have covariance three square millimetres, while other cross-covariances are zero, the tie variance becomes 9 + 4 + 4 − 6 = 11 square millimetres, giving √11, approximately 3.316625 millimetres.
Now test an epoch change. Give B a relative vertical velocity of minus one millimetre per year with respect to A. After four years the tie’s up component becomes 1.036 metres under the constant-velocity teaching model. Its horizontal components stay unchanged. Neither the original survey date nor a global frame’s reference epoch should be silently replaced by the later comparison date.
Finally add a third reference point C and calculate its ties entirely from the same three fitted positions. The resulting loop will close exactly in arithmetic, even if the input definition of B was wrong. Design one additional check that could challenge that error: for example, an independently modelled reference-point determination or observations at a configuration not used in fitting. Explain what it measures and why it is not simply a restatement of the existing coordinates.
Worked comparison
| Check | Fictional result or invariant | What it does not prove |
|---|---|---|
| A=(10,20,2), B=(13,24,2.5) m | A-to-B tie=(3,4,0.5) m | That these are the required instrument reference points |
| Reverse endpoint order | Negate every vector component | Length alone cannot identify the direction |
| Target difference plus e_B minus e_A | Reference-point tie reconstructed | Offsets are not automatically exact |
| Rotate (3,4) by 90° | (−4,3), same length | Only some terms may be rotated |
| Independent component variances 4,1,1 mm² | Tie variance 6 mm² | Unmodelled endpoint errors are included |
| Closed loop from fitted positions | Zero algebraic closure | Independent survey accuracy |
| Relative vertical rate +2 mm/year for 3 years | Tie up component increases 6 mm | Real motion necessarily stays linear |
Questions and practical checks
Is a local tie just a distance between two instruments?
No. It is a directed relationship between declared reference points, normally represented by a vector with uncertainty and supporting metadata. Distance alone loses direction and cannot expose a reversed endpoint convention.
Why does an inaccessible reference point matter?
The observable target may move with instrument orientation or sit at an offset from the required point. A geometrical inference connects the two. Its assumptions and uncertainty belong in the final tie rather than being treated as exact by default.
Does a closed survey loop prove every endpoint is correct?
No. A shared endpoint error can cancel between adjacent ties, and loops constructed from fitted coordinates close by definition. Distinguish independent redundant observations from internal algebraic consistency.
Can an old tie be reused at a new epoch?
Only with evidence that the represented relationship remains valid or with a justified propagation model. Common translation cancels, but differential motion, configuration changes and endpoint redefinitions may alter the tie.
Does combining techniques remove their systematic errors?
Not automatically. A local tie defines how their endpoints are connected. Combination still depends on the observation models, covariance, reference conventions and independent checks. A smaller formal uncertainty does not by itself explain a disagreement.
Continue with the connected subjects
Establish endpoints and measurement quantities
Separate a GNSS antenna reference point from its phase-centre model
Follow the different endpoints and corrections in laser-ranging records
Understand what a radio arrival-delay measurement constrains
Keep frames, epochs and uncertainty explicit
Transform local vectors and covariance consistently
Distinguish survey and coordinate epochs from frame reference epochs
Separate survey information from exact datum constraints
Interpret covariance geometry without overstating coverage
Sources and scope
International Laser Ranging Service: site surveys and instrument reference-point connections
NOAA National Geodetic Survey: co-location local-tie survey programme
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 1 November 2026.
Continue through observations, reference points and combined information
Use the measurement model to identify what a record represents, the local survey to connect physical endpoints, and the adjustment model to retain the information and uncertainty. These are complementary steps, not interchangeable corrections.
Compare the observation quantities
Radio-source structure: follow complex visibility into a group-delay slope
DORIS phase counts: preserve integration windows and frequency endpoints
Connect the reference and the estimate
Normal-equation combination: retain nuisance uncertainty and avoid duplicate information