A firmly installed station can still move with the Earth
A geodetic antenna is attached securely to a monument. Its coordinates can nevertheless vary because the solid Earth deforms under tidal forcing. This is not the antenna slipping on its mount, nor is it ocean water rising beside the station. The physical reference point itself participates in a changing deformation field.
This guide builds a small directional tide model, separates radial and transverse motion, and follows the resulting station displacement into an observation. Original examples expose sign, axis and timing mistakes. A separate convention exercise explains why removing a time-varying signal is not the same operation as choosing how permanent tidal deformation is represented.
The directions, response factors and displacement scales used below are fictional teaching inputs. They are not predictions for a real station or a substitute for an approved geodetic model. The aim is to make the physical and mathematical interfaces understandable before interpreting a coordinate difference as unexplained ground movement.
Geodesy subject guide · Learning that transfers into life · How X Works hub
Choose your reading route
Route 1 · Chapters 1–2: Understand the moving reference point
Route 2 · Chapters 3–5: Work the directional response
Route 3 · Chapters 6–8: Translate displacement into observations
Route 4 · Chapters 9–10: Audit uncertainty and conventions
Route 5 · Chapters 11–12: Transfer and report
Full chapter index · Independent worked case · Questions and practical checks
Chapter index
Understand the moving reference point
Distinguish body tides from ocean loading
Define a deliberately simplified vector response
Work the directional response
Calculate radial and transverse components explicitly
Use limiting directions to test the equation
Change coordinates without changing the physical deformation
Translate displacement into observations
Project station motion with the receiver-side sign
Keep modelled instantaneous coordinates distinct from a reduced series
Understand why an oscillation can resemble a trend over a short window
Audit uncertainty and conventions
Propagate uncertain response factors under a stated model
Separate periodic reduction from permanent-tide convention
CHAPTER 1 OF 12 · Understand the moving reference point
1. Distinguish body tides from ocean loading
Solid-Earth body tides concern deformation associated with external tidal forcing, principally from the Moon and Sun. Ocean-tide loading concerns deformation caused by the changing distribution of ocean-water load. Both can affect a station, but they are different model contributions. Accounting for one does not establish that the other has already been included.
ESA Navipedia introduces radial and transverse body-tide displacement, while IERS treats reference-point displacement models in a dedicated chapter. The IERS Conventions Centre also distinguishes its registered 2010 edition from later updated material under development. Model provenance matters: an accessible updated page should not silently be described as a newly approved convention.
A station can also move because of atmospheric or land-water loads, tectonic motion, local monument effects and other processes. These are not interchangeable explanations for every periodic coordinate pattern. A useful analysis keeps a contribution ledger so an apparent improvement after one model change is not mistaken for proof that every remaining discrepancy has the same cause.
Start by naming the physical point and the coordinate quantity. A monument point, antenna reference point and electromagnetic phase-centre model are not automatically the same endpoint. A valid body-tide displacement attached to one point does not repair an incompatible antenna-height account. Use the existing antenna-reference guide for that different interface.
CHAPTER 2 OF 12 · Understand the moving reference point
2. Define a deliberately simplified vector response
Let r be a unit vector from Earth’s centre towards the station, and b a unit vector towards a fictional tide-raising body. Define q = r·b. The quantity q is the cosine of the angle between them. Both vectors must share compatible axes and describe the intended common event before their dot product has this geometric meaning.
For one teaching contribution, define d = S{h r[1.5q²−0.5]+3lq[b−qr]}. The scale S has length units, while h and l are dimensionless stipulated response factors. This is a stripped-down degree-two-style expression used to examine geometry. It omits refinements required by complete real models and carries no claimed operational accuracy.
The first bracketed term is parallel to r. The vector b−qr is perpendicular to r because r·(b−qr) = q−q = 0. The second term is therefore transverse to the station’s geocentric radius. Its direction and magnitude depend on the external-body direction as well as the response factor.
Calling the first term radial is precise. Calling it local Up without qualification can be imprecise because ellipsoidal Up and the geocentric radial direction generally differ. The distinction may be small in some uses, but it is a geometric definition, not a matter to settle by rounding the resulting displacement.
CHAPTER 3 OF 12 · Work the directional response
3. Calculate radial and transverse components explicitly
Choose fictional r = (1,0,0), b = (0.6,0.8,0), S = 0.20 metres, h = 0.6 and l = 0.08. Both direction vectors have unit length and q = 0.6. These round inputs were selected for transparent arithmetic; they are not an ephemeris, station location or published set of response constants.
The radial factor is 1.5×0.6²−0.5 = 0.04. Its displacement is 0.20×0.6×0.04 r = (0.0048,0,0) metres. The apparently small radial term here results from the selected angle being near a zero of that directional factor, not from a universal statement that radial body tides are small.
For the transverse term, b−qr = (0,0.8,0). The scalar multiplier is 0.20×3×0.08×0.6 = 0.0288 metres. Multiplication gives (0,0.02304,0). The complete teaching displacement is therefore d = (0.0048,0.02304,0) metres: 4.8 millimetres radial and 23.04 millimetres along the selected transverse axis.
Check the decomposition by dotting d with r. It returns 0.0048 metres. Subtracting that radial vector from d leaves a vector perpendicular to r. Checking these geometric relationships catches a term assembled in the wrong direction even when its scalar magnitude has been calculated correctly.
CHAPTER 4 OF 12 · Work the directional response
4. Use limiting directions to test the equation
Place the body direction along the station radius, b = r, so q = 1. The transverse vector b−qr is zero, and the radial directional factor is one. With the same teaching S and h, d = 0.12r metres. An implementation returning transverse motion in this aligned case fails the declared formula.
Now choose b perpendicular to r, for example b = (0,1,0). Then q = 0, so the transverse term again vanishes, while the radial factor is minus one half. The displacement becomes −0.06r metres. The sign reversal is a feature of this angular response; it is not obtained by declaring all tidal forcing to point towards the external body.
Reverse the body direction to b′ = −b. Then q′ = −q, and b′−q′r = −(b−qr). Their product q′[b′−q′r] equals q[b−qr], while q′² = q². The degree-two teaching displacement is unchanged. This is a useful parity check, not a statement that every term in a complete tidal expansion has the same symmetry.
Finally choose q² = 1/3. The radial factor vanishes even though a transverse contribution can remain. This limiting case demonstrates why a zero radial result does not mean zero three-dimensional deformation. It also gives a more discriminating synthetic test than comparing only the length of a vector after a general calculation.
CHAPTER 5 OF 12 · Work the directional response
5. Change coordinates without changing the physical deformation
Rotate the entire description by ninety degrees about its z axis. The original r = (1,0,0) becomes r′ = (0,1,0), while b becomes b′ = (−0.8,0.6,0). Their dot product remains 0.6. A correctly transformed response must be d′ = (−0.02304,0.0048,0) metres.
Evaluate the teaching equation again using the rotated directions. The radial contribution now lies along positive y, and the transverse contribution along negative x. Their scalar magnitudes are unchanged. This independent re-evaluation is stronger than merely rotating the final vector and assuming the input interpretation was consistent.
The vector norm is invariant under a common orthogonal rotation. However, a norm check alone cannot expose every axis mistake. Rotating the body direction while accidentally leaving the station direction in the old axes changes q and the physical geometry being represented. The resulting answer can still have a plausible length.
This is the same representation discipline used in satellite orbit-error projection: directions, vectors and uncertainty matrices must travel together between axes. The existing orbit-error guide develops the satellite-side version. Here the displaced endpoint is the station, so the observation sign requires a separate check rather than copying the satellite formula unchanged.
CHAPTER 6 OF 12 · Translate displacement into observations
6. Project station motion with the receiver-side sign
Let u point from receiver to satellite. With the satellite held fixed and the receiver displaced by d, the first-order geometric-range change is δρ ≈ −u·d. The minus sign expresses that movement towards the satellite shortens the separation. A satellite displacement under the same u definition enters with the opposite endpoint sign.
Use the original fictional d = (0.0048,0.02304,0) metres and a unit line of sight u = (0.8,0.6,0). The dot product is 0.00384+0.013824 = 0.017664 metres. The corresponding receiver-displacement range change is minus 17.664 millimetres, not the norm of the station displacement.
Change the line of sight to u₂ = (0.8,−0.6,0). Its dot product with d is 0.00384−0.013824 = −0.009984 metres, so the range change becomes plus 9.984 millimetres. The same station displacement can shorten one satellite path while lengthening another. A single scalar correction applied equally to all observations would miss that geometry.
These line-of-sight directions are fictional teaching arrangements. Real satellite visibility and precise event/frame relationships require their own model. The useful point is structural: station deformation is a vector contribution whose range effect depends on the observed direction, not a universal range offset carrying the same sign everywhere.
CHAPTER 7 OF 12 · Translate displacement into observations
7. Keep modelled instantaneous coordinates distinct from a reduced series
Write a simple coordinate account xinst(t) = xbase(t)+dtide(t), with every term in compatible axes and conventions. The baseline might include a frame-compatible position and an adopted secular-motion model. The tide displacement connects that baseline with an instantaneous modelled position; the sign follows this declared relationship.
If an observation-derived coordinate includes the tide effect and the target is the stated baseline, the corresponding reduction subtracts dtide. If the target is an instantaneous predicted antenna position from a baseline coordinate, the construction adds dtide. Both operations can be correct because their starting quantities and target quantities differ.
Use a one-dimensional fictional example: a baseline coordinate component is 1,000.000 metres and the model displacement is +0.020. The instantaneous component is 1,000.020. Subtracting the model returns 1,000.000. Adding it again to an already instantaneous value gives 1,000.040, a duplicate application rather than a more complete correction.
A processing ledger should therefore state whether the input coordinates are instantaneous, already reduced or otherwise regularised under the adopted model. A field named corrected does not settle the question. Preserve the original quantity, direction of the transformation and model version before comparing files that use different reduction histories.
CHAPTER 8 OF 12 · Translate displacement into observations
8. Understand why an oscillation can resemble a trend over a short window
For a separate timing exercise, let one fictional displacement component be d(t) = A cos(ωt), with a stipulated twelve-hour period and amplitude twenty millimetres. This is a simple test waveform, not a complete Earth-tide prediction or an assignment of a named constituent. Its average over exactly a full cycle is zero.
Over only the first three hours, the component decreases from twenty to zero millimetres. A two-endpoint slope is minus 20/3, approximately minus 6.6667 millimetres per hour. That short-window slope does not represent a permanent station velocity. It describes the selected section of an oscillation.
Sampling at the same phase every cycle can make the component appear constant instead. Sampling at changing phases can reveal variation. The observation schedule interacts with the signal. Neither a visually flat sampled series nor a short sloping segment proves the absence or presence of secular deformation without a suitable time model.
The existing station-velocity guide addresses seasonal terms and time-correlated uncertainty in longer coordinate series. Use it for the trend-estimation question. The present example supplies a concrete reason to retain the applied tide model and sampling history when deciding which part of a coordinate record is relevant to a long-term velocity estimate.
CHAPTER 9 OF 12 · Audit uncertainty and conventions
9. Propagate uncertain response factors under a stated model
Treat the directions and S as exact only for this uncertainty exercise, and let h and l be uncertain. In the original geometry, the radial component is 0.008h metres and the transverse component is 0.288l metres. These coefficients follow directly from the selected q and b−qr, so their units and parameter sensitivities can be checked independently.
Choose fictional standard uncertainties σh = 0.01 and σl = 0.002, with the two parameters independent. The radial component standard uncertainty is 0.008×0.01 = 0.00008 metres, or 0.08 millimetres. The transverse component standard uncertainty is 0.288×0.002 = 0.000576 metres, or 0.576 millimetres.
For u = (0.8,0.6,0), the range projection has parameter coefficients −0.0064 for h and −0.1728 for l, in metres per dimensionless parameter. Its variance is (0.0064×0.01)²+(0.1728×0.002)². The standard uncertainty is approximately 0.00035148 metres, or 0.35148 millimetres.
This conditional propagation does not include uncertainty in ephemerides, Earth response structure, timing, omitted terms or the station reference. A complete physical error budget needs its own evidence. The exercise teaches how parameter uncertainty maps into a directional observation; it does not certify that the stripped-down model predicts a real station to that accuracy.
CHAPTER 10 OF 12 · Audit uncertainty and conventions
10. Separate periodic reduction from permanent-tide convention
A model can contain a time-independent contribution as well as time-varying terms. Choosing how the permanent contribution is represented in coordinates is a convention question distinct from removing an oscillation in a time series. The IERS reference-point chapter discusses conventional tide-free and mean-tide coordinates and warns against interpreting the conventional label as a literal Earth with every tidal influence absent.
For a purely fictional bookkeeping example, define coordinate systems A and B by xB = xA+p, where p is a stated constant vector. A periodic displacement d(t) can be represented relative to either baseline if the constant offset is handled consistently. Changing A to B does not imply a physical station movement at the conversion instant.
Let one component have p = 0.030 metres and periodic d = −0.010. A baseline A value of 100.000 becomes baseline B = 100.030, while corresponding instantaneous representations retain their documented relation. Combining baseline A with a displacement definition intended for B can introduce a constant discrepancy even when both files use metre units.
The invented p is not an official tide-system conversion. Real transformations require the relevant convention and geodetic quantity; coordinate, gravity-potential and height conventions must not be conflated. The safe lesson is to retain the tide-system label and transformation provenance, not to derive a universal conversion from the round bookkeeping numbers used here.
CHAPTER 11 OF 12 · Transfer and report
11. Check model provenance and duplicate contributions
The Conventions Centre provides registered and updated material with an explicit status distinction. A report should state the actual model edition, software implementation and update history used. A recent download date does not prove that an implementation includes every required refinement, and a familiar routine name does not establish which version produced the result.
A useful validation sequence starts with the implementation’s documented reference cases, then adds independent synthetic direction, sign and limiting tests such as those above. Matching one published test is necessary evidence for that case, but does not inspect every interface in a larger processing chain. Units and endpoint conventions can still be wrong outside the routine.
Maintain separate ledger entries for body tide, ocean loading and other relevant displacement terms. If a supplied coordinate product already includes a particular contribution, adding the same term downstream can double it. If two products omit different contributions, subtracting them can reveal a model-definition difference rather than unexplained geophysical motion.
For an independent evidence check, compare like-defined quantities and preserve both before-and-after series. Reduced scatter may support a useful modelling change, but it is not proof that every remaining feature is tectonic or structural. Attribution requires the relevant physical observations and an uncertainty account, especially where a result will inform engineering decisions.
CHAPTER 12 OF 12 · Transfer and report
12. Report physical motion and representational choices separately
A clear result names the station endpoint, coordinate frame, effective epochs, modelled displacement components, permanent-tide convention and reduction direction. It distinguishes physical deformation from an axis transformation, a baseline-convention shift and an instrument-reference change. All can change reported coordinates, but they do not describe the same event.
For the original simplified example, the model produces 4.8 millimetres radial and 23.04 millimetres transverse under explicitly fictional inputs. One receiver-to-satellite direction yields a minus 17.664-millimetre first-order range change. Those are teaching results under a limited equation, not a prediction for a named station or a validated field correction.
When another context changes the body direction, sample timing or target coordinate convention, identify which previous steps remain valid. The vector decomposition may need recalculation; the endpoint sign may remain; the constant reference offset may require translation. Transfer is the ability to revise the right part of the chain without casually discarding every earlier definition.
The central lesson is that stable installation and constant coordinates are different claims. A station can faithfully follow real Earth deformation while its published baseline follows an agreed convention. Understanding the relationship makes coordinate changes more informative: they become quantities to explain with a model and evidence, not automatic signs of either instrument failure or dangerous movement.
Independent worked case and evidence checks
Work a fresh geometry with r = (0,0,1) and b = (0.8,0,0.6), retaining fictional S = 0.20 metres, h = 0.6 and l = 0.08. Here q is again 0.6. The radial displacement is (0,0,0.0048), while the transverse term is (0.02304,0,0). The complete result is (0.02304,0,0.0048) metres.
Use line of sight u = (0.6,0,0.8). Its dot product with d is again 0.017664, and the receiver-side range contribution is minus that value. This equality is expected because the new geometry is a common rearrangement of the previous physical configuration. Check the vectors, not merely the reused numerical answer.
Now change only the body direction to b = (0.6,0,0.8). The dot product becomes 0.8. The radial directional factor is 1.5×0.64−0.5 = 0.46, giving radial displacement 0.0552 metres. The transverse factor produces 0.20×3×0.08×0.8×0.6 = 0.02304 metres along x.
The transverse magnitude happens to remain the same because q√(1−q²) is equal for q = 0.6 and q = 0.8. The radial component changes substantially because it depends on q² through another expression. Equal transverse values do not mean the full geometries or displacement vectors are equivalent.
For the same u, the new range projection is minus [0.6×0.02304+0.8×0.0552] = minus 0.057984 metres. A reader copying the earlier range scalar because one component was unchanged would miss the radial contribution. This case tests whether the observation is recomputed from the entire changed vector.
Next double S while keeping directions and response factors fixed. Both displacement components and their first-order range projection double. Their directions do not change. This linearity test is distinct from the body-direction test: changing the strength scales a fixed pattern, while changing geometry can reshape the component balance.
For a separate temporal check, sample the fictional twenty-millimetre cosine at zero, three, six and nine hours within a twelve-hour period. The values are twenty, zero, minus twenty and zero. Their four-point average is zero. Now retain only the first two samples; their average is ten, although the full-cycle average remains zero.
This simple selection change does not alter the physical waveform. It alters which part was observed. A reported mean over an incomplete interval should therefore carry its interval and sampling definition. Comparing it with a full-cycle mean as though both represented the same quantity would manufacture an apparent offset.
Finally create a duplicate-correction test. Let a fictional measured component be baseline plus a known +8-millimetre model displacement. Subtracting the model once recovers the baseline. Subtracting it twice creates a minus 8-millimetre error relative to that target. Preserve an explicit included/not-included ledger rather than relying on a filename containing the word tide.
Ask another reader to describe each change in plain language: rotated axes, changed forcing direction, stronger response, changed sampling and duplicate reduction. Only some describe a different physical deformation. This classification is an evidence check in its own right because it prevents every numerical coordinate change from being reported as a new ground-motion event.
One further sensitivity test changes the original direction cosine q from 0.60 to 0.61 while retaining the stipulated S and h. The radial response changes from 0.0048 to 0.006978 metres. Its difference is 2.178 millimetres. Differentiating the radial expression with respect to q gives 3Shq, equal to 0.216 metres per unit q at the original point. The first-order prediction for a 0.01 change is therefore 2.16 millimetres, close to the exact finite difference.
This test concerns only the radial component. A physically consistent changed body direction also changes its transverse vector, which must be recalculated. Varying q while retaining an incompatible original body vector would violate q = r·b. Sensitivity checks are most useful when the perturbed inputs remain consistent with the definitions that created the model.
Worked comparison
| Operation | What changes | What should remain clear |
|---|---|---|
| Common coordinate rotation | Vector components | Physical direction and scalar range projection |
| Changed external-body direction | The deformation geometry | Input event and response-model definition |
| Periodic displacement reduction | The target coordinate series | Whether the contribution was already included |
| Permanent-tide convention change | Coordinate representation | This is not an instantaneous physical movement |
| Short-window slope | A summary of the selected interval | It need not represent secular station velocity |
Questions and practical checks
Can a well-fixed antenna move tidally?
Yes. Its supporting Earth can deform; secure installation does not imply an unchanging instantaneous geocentric coordinate.
Are solid-Earth tides the same as ocean loading?
No. The models represent different forcing pathways and require separate contribution checks.
Is radial always local Up?
Not exactly. Geocentric radial and ellipsoidal Up have different geometric definitions.
Should every tide model be added to a supplied coordinate?
No. First identify the coordinate definition and whether that contribution has already been included or removed.
Does conventional tide-free mean an Earth untouched by tides?
No. It is a defined representation, not a literal description of an entirely unforced physical Earth.
Continue with the connected subjects
Establish the coordinate and physical endpoint
Keep monument, antenna-reference and phase-centre quantities distinct
Distinguish coordinate epochs and frame-compatible motion models
Follow a displacement into an observation or time series
Compare vector projection and covariance on the satellite side
Separate periodic coordinate structure from long-term station velocity
See why reference height and tidal contributions also matter in repeat gravity
Sources and scope
ESA Navipedia: body-tide displacement and radial/transverse geometry
IERS Conventions Centre: reference-point models, software and edition status
IERS reference-point displacement chapter, including permanent deformation and loading distinctions
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.
Compare the connected observation pathways
Keep signal-path effects separate from physical station displacement. These companion guides develop the related models with their own worked examples, units and reference checks.
Separate azimuth-dependent tropospheric gradients from horizontal position
Reconstruct ocean-tide loading from harmonic phases and component axes
Trace atmospheric-pressure loading through spatial anomalies and reference ledgers