A moving antenna does not automatically mean a moving foundation
A station rises during warm weather. Did the ground rise, did its supporting monument expand, or did the measurement acquire a temperature-related error? Those possibilities can produce similar-looking plots while requiring different evidence. Begin with the physical point that moved, not with a correlation that happens to look convincing.
This guide builds a small thermal-expansion model from length, temperature change and an explicitly chosen material coefficient. We then heat different parts by different amounts, delay the temperature response, compare two monuments and follow uncertainty into a corrected coordinate. Each change tests a different assumption behind the original calculation.
All dimensions, temperatures, coefficients and observation histories below are fictional teaching inputs. They are not specifications for concrete, steel, an actual GNSS station or a regional correction product. The aim is to distinguish a transparent mechanical example from a complete explanation of real station motion.
Geodesy subject guide · Learning that transfers into life · How X Works hub
Choose your reading route
Route 1 · Chapters 1–2: Identify the moving point
Separate the station coordinate from the foundation coordinate
Route 2 · Chapters 3–5: Build the thermal model
Route 3 · Chapters 6–8: Test sampling and attribution
Route 4 · Chapters 9–10: Carry uncertainty and references
Propagate temperature uncertainty without hiding common errors
Route 5 · Chapters 11–12: Transfer and communicate
Full chapter index · Independent worked case · Questions and practical checks
Chapter index
Identify the moving point
Separate the station coordinate from the foundation coordinate
Distinguish a mechanism from a seasonal resemblance
Build the thermal model
Calculate one freely expanding column in consistent units
Heat two sections differently instead of averaging by habit
Let geometry and restraint challenge the scalar model
Test sampling and attribution
Measure the temperature that the model actually needs
See what an air-only regression can and cannot recover
Keep thermal deformation separate from other corrections
Carry uncertainty and references
Propagate temperature uncertainty without hiding common errors
Compare two monuments without assuming their expansions cancel
CHAPTER 1 OF 12 · Identify the moving point
1. Separate the station coordinate from the foundation coordinate
Imagine a foundation mark, a four-metre support and an antenna fixed to its top. A positioning solution concerns a specified antenna-related reference, connected to the foundation through that support. If the support length changes while its base remains fixed, the top moves. Calling the entire result “ground motion” would silently merge two physical points.
Write the simplified vertical connection as z_top(t) = z_base(t) + L(t). Between two epochs, the change is Δz_top = Δz_base + ΔL. This is a bookkeeping identity for a vertical support with the stated endpoints. It does not yet explain either term. An independent observation of the base would answer a different question from a second calculation using the same antenna record.
An antenna’s electrical response adds another part to the measurement chain. A mechanical reference point, a signal-dependent antenna model and a foundation benchmark should not acquire the same name merely because they are nearby. First state which reported coordinate is under discussion and which connection has already been applied. The existing antenna-reference-point guide develops that separate geometry.
For the first worked model, hold the base fixed, keep the column vertical and allow it to expand freely. These conditions deliberately remove foundation movement, restraint and bending. They make the calculation checkable; they are not evidence that a real installation satisfies them. A useful first-principles example states what it leaves out before its neat answer becomes a tempting universal correction.
CHAPTER 2 OF 12 · Identify the moving point
2. Distinguish a mechanism from a seasonal resemblance
Thermal expansion is a physical change in a body’s dimensions. Temperature-driven deformation below the surface is a further pathway, while pressure or water loading concerns the mechanical response to redistributed mass. They can share seasons without being the same process. One observed annual wave cannot, by its shape alone, allocate the displacement among these contributions.
The 2026 TED research paper explicitly distinguishes above-ground monument expansion from subsurface thermoelastic displacement and models the latter with layered properties and temperature forcing. That is the specialist context for this guide, not a ready-made correction for our fictional column. Its dataset and model assumptions should be read before using any associated product.
Suppose an antenna rises whenever its surroundings warm. This observation is consistent with expansion but also with other changes that occur at similar times. A prediction becomes more discriminating when it uses independently documented dimensions, relevant temperatures, timing and material behaviour. If those quantities predict the wrong sign or phase, a good-looking seasonal match should not rescue the proposed explanation.
Keep three statements separate: the coordinate varies; a particular model predicts part of that variation; the remaining series is suitable for a specific interpretation. Establishing the first does not establish the second, and subtracting a model does not establish the third. This distinction transfers to any instrument whose mounting, environment and measured target can all change together.
CHAPTER 3 OF 12 · Build the thermal model
3. Calculate one freely expanding column in consistent units
For a uniform, freely expanding column in a small-change model, take ΔL = αLΔT. Here L is its reference length, α is the chosen linear expansion coefficient, and ΔT is the body’s temperature change relative to the same reference condition. The temperature difference is expressed in kelvins; a change of ten degrees Celsius has the same numerical size as ten kelvins.
Choose L = 4 m, α = 0.000010 per kelvin and ΔT = 10 K. Multiplication gives ΔL = 0.000010 × 4 × 10 m = 0.0004 m = 0.4 mm. The positive sign means a longer column. With a fixed base and vertical alignment, its top rises by 0.4 mm in this model.
Check the units before trusting the decimal: per kelvin times metres times kelvins leaves metres. Check scale independently: each kelvin produces 0.04 mm for this fictional support, so ten kelvins produces 0.4 mm. Entering four thousand millimetres into a calculation whose output is still labelled metres would create a thousandfold error, not a subtle geophysical discovery.
Reverse the change as another test. Cooling the same body by 10 K gives −0.4 mm relative to its reference length. Doubling the temperature difference doubles the answer only while α, free-expansion behaviour and geometry remain appropriate. The coefficient here was selected to make the arithmetic visible. It must not be copied into a real station record as an undocumented material property.
CHAPTER 4 OF 12 · Build the thermal model
4. Heat two sections differently instead of averaging by habit
Now divide the four-metre support into two two-metre sections made from the same fictional material. Let the upper section warm by 10 K and the lower section by 2 K. Their length changes are 0.2 mm and 0.04 mm, respectively. Adding them gives 0.24 mm, not the 0.4 mm obtained by assigning the upper temperature to the entire column.
For sections with different lengths or coefficients, use ΔL = Σ α_i L_i ΔT_i. Each term represents a particular part of the structure. A simple arithmetic average of two thermometer readings works only under special weighting conditions. Equal section lengths and equal coefficients happen to supply those conditions in this example; unequal sections do not.
A continuous version writes ΔL = ∫ α(z)ΔT(z) dz over the support’s reference length, still under the free axial-expansion assumptions. This is a compact way of adding many small sections. It does not account for sideways temperature gradients, restraints or elastic interactions merely because it uses an integral. A more sophisticated symbol cannot repair an incomplete physical model.
Try a changed geometry: three metres warm by 2 K and one metre warms by 10 K. The answer is 0.06 + 0.10 = 0.16 mm. The unweighted average temperature change is still 6 K, but using it for all four metres would predict 0.24 mm. The disagreement exposes the missing length weights. Always identify what the average is meant to preserve.
CHAPTER 5 OF 12 · Build the thermal model
5. Let geometry and restraint challenge the scalar model
The equation for ΔL describes axial extension, not an unrestricted three-dimensional displacement. If a straight support has fixed unit direction e, its top-to-base vector is Le. With direction unchanged, the vector change is ΔL times e. Only a vertical e makes the entire result an up displacement. An inclined support has horizontal and vertical components.
If direction also changes, the first-order connection becomes Δr ≈ eΔL + LΔe. The first term is extension; the second is a small rotation of the support. They have different causes and can reinforce or oppose each other in one coordinate. A temperature difference across a structure may motivate a bending model, but the axial equation alone cannot calculate that bending.
Restraint creates another limit. A body prevented from expanding freely can develop stress and interact with its attachments. Substituting a free-expansion coefficient into a fully restrained installation does not automatically give the endpoint displacement. To move beyond the teaching column, one would need the actual structural configuration and an appropriate mechanical model, not just a more accurate air thermometer.
This chapter supplies a useful diagnostic question: is the proposed correction a change in length, a change in orientation, movement of the base, or some combination? Label each contribution and its endpoint. A single “temperature correction” column can hide several models with incompatible meanings. The same distinction matters when translating local survey ties into a station’s global coordinate history.
CHAPTER 6 OF 12 · Test sampling and attribution
6. Measure the temperature that the model actually needs
The model uses the body’s relevant temperature field. A nearby air measurement is a possible input to another model, not automatically that field itself. Sunshine, exposure, depth and thermal response can make an environmental proxy differ from the temperatures of the supporting structure. For an actual installation, the relationship must be investigated rather than assumed.
Construct a deliberately simple periodic example. Let the recorded air anomaly be 10 cos(ωt) K, while the effective body anomaly is 8 cos(ωt) + 6 sin(ωt) K. The body signal has amplitude √(8² + 6²) = 10 K. Its phase differs by arctan(6/8), about 0.644 radians. Equal amplitudes therefore do not imply simultaneous warming.
Multiplying the body signal by 0.04 mm/K gives a top displacement of 0.32 cos(ωt) + 0.24 sin(ωt) mm. At a time when the air anomaly is zero, the displacement can be 0.24 mm. An instantaneous air-only rule would predict zero at precisely that time. This is a timing failure, not evidence that the material coefficient changed sign.
The example is a chosen harmonic response, not a universal heat-conduction law or a specified lag for any monument. Its purpose is to show what must be checked when replacing a physical state with an accessible proxy. Record the observation times, averaging windows, reference temperatures and sensor locations. Without that information, apparently paired rows may not describe the same thermal condition.
CHAPTER 7 OF 12 · Test sampling and attribution
7. See what an air-only regression can and cannot recover
Fit displacement against the air anomaly in the harmonic example, using complete cycles with uniform sampling and an intercept. Over those cycles, sine and cosine are orthogonal. The fitted slope is 0.32/10 = 0.032 mm/K, while the unmodelled part is 0.24 sin(ωt) mm. The physical column sensitivity remains 0.04 mm/K.
Why are the slopes different? The fitted coefficient describes how much of this particular displacement record projects onto this particular air record under this sampling scheme. It is not a direct laboratory measurement of αL. Thermal lag moved some signal into a component that the air-only predictor cannot represent. Calling the regression coefficient a material constant would erase that distinction.
Change the observation schedule and the tidy orthogonality need not survive. Sampling mainly during one part of a cycle can make the sine component correlate with the sampled cosine values or with a fitted trend. A coefficient estimated from that schedule can change without any physical alteration of the support. More repeated observations at the same phases do not supply the missing phases.
A useful transfer check is to fit one interval and predict another with different thermal timing. Inspect both residual size and phase. If an independently measured body temperature improves the prediction in the expected way, that supports the mechanism more directly than an air-only correlation does. Even then, check competing effects and sensor stability before claiming that all remaining motion belongs to the ground.
CHAPTER 8 OF 12 · Test sampling and attribution
8. Keep thermal deformation separate from other corrections
Return to Δz_top = Δz_base + ΔL. If the intended output is base motion and a suitable expansion model supplies ΔL, subtract that contribution once from the top motion. If the intended output is the physical antenna trajectory, expansion may be part of the signal to retain. A correction is meaningful only relative to the question being answered.
Suppose the fictional observed top rise is 0.7 mm and the independently modelled support expansion is 0.4 mm. The inferred base rise is 0.3 mm, conditional on the model and the rest of the measurement chain. Subtracting another 0.4 mm because a second file also calls itself “thermal” would change the answer to −0.1 mm without adding any physical evidence.
A subsurface thermal-displacement model, an above-ground expansion model and an atmospheric-loading model may describe distinct contributions. Alternatively, two products may partly represent the same process. Their labels alone cannot decide. Read their definitions, endpoints, forcing, reference conditions and components. Establish whether an existing station product already contains the term before adding a new adjustment.
Keep an explicit account: original quantity, contribution represented, sign, reference condition, version and resulting quantity. This is not busywork. It allows another reader to reconstruct the 0.3 mm result and to reverse the operation if the intended target changes. A smooth corrected series is not a substitute for that account; incorrect subtractions can make a plot look attractively flat.
CHAPTER 9 OF 12 · Carry uncertainty and references
9. Propagate temperature uncertainty without hiding common errors
For the uniform model f = αLΔT, small perturbations give δf ≈ LΔT δα + αΔT δL + αL δ(ΔT). These sensitivities identify which input errors matter at the chosen state. With independent inputs, square each sensitivity, multiply by the corresponding variance, and add. When inputs are correlated, include the cross-covariance terms instead of silently assuming independence.
For a simple isolated calculation, hold L and α exact and assign a standard uncertainty of 1 K to ΔT. The displacement’s standard uncertainty is then 0.04 mm. This is only the stated temperature contribution. It excludes uncertainty in the material model, geometry, temperature representativeness and measured coordinate. Reporting it as the total uncertainty would misdescribe the calculation.
Temperature change itself is a difference: ΔT = T_later − T_reference. A constant sensor offset present equally in both readings cancels. Independent reading noise does not cancel in the same way. A drifting offset can survive the subtraction. Thus a sensor’s absolute calibration uncertainty and the uncertainty of its measured temperature change need not be identical.
For the inferred base motion b = observed top motion − modelled expansion, Var(b) = Var(observed) + Var(model) − 2 Cov(observed, model). If the expansion coefficient was estimated from the same coordinate record, the covariance deserves particular attention. Treating a fitted correction as independent of its training data can produce an uncertainty statement that looks precise while omitting the dependence that created it.
CHAPTER 10 OF 12 · Carry uncertainty and references
10. Compare two monuments without assuming their expansions cancel
Let the observed vertical separation between tops A and B be d = z_B − z_A. Its thermal contribution is ΔL_B − ΔL_A in the simple vertical model. Identical expansion cancels from that relative quantity even though both absolute top coordinates move. Different supports or different temperature histories leave a differential signal.
Use α = 0.000010/K for both supports. A is four metres long and warms by 10 K, giving 0.4 mm. B is two metres long and warms by 10 K, giving 0.2 mm. With both bases fixed, B minus A changes by −0.2 mm. Nearby weather and the same material did not produce cancellation because the mechanical lengths differed.
Now give B a temperature change of 20 K. Its expansion becomes 0.4 mm and the relative thermal change vanishes. This zero does not show that neither support moved. It shows that the chosen difference is insensitive to this particular shared displacement. Change the target to either base coordinate and the missing information matters again.
The uncertainty of a difference likewise depends on shared inputs. One common thermometer-scale error can affect both predictions, while different exposures create separate representativeness errors. Document which uncertainties are shared and how the sensitivities differ. A local tie measured at one thermal state should not be carried to another state as though its connection vector were necessarily unchanged.
CHAPTER 11 OF 12 · Transfer and communicate
11. Design evidence checks that distinguish the explanation
Begin with tests that could falsify the model. Does the predicted sign agree with free expansion under the declared geometry? Are the dimensions tied to the actual moving endpoints? Does the relevant body temperature lead or lag the environmental record as assumed? Does an independent observation of the base behave differently from the top? Each question targets a specific link in the explanation.
Then change the context deliberately. Compare shaded and exposed periods, intervals with different thermal timing, or supports with different documented lengths. Do not interpret these as controlled experiments unless competing conditions really were controlled. They are opportunities to ask whether the model predicts more than the record from which it was fitted.
Inspect residuals against time and the proposed forcing, not just their overall scatter. A reduced average residual can coexist with a remaining phase error, a changed trend or a discontinuity after equipment work. A model may improve one statistic while damaging the quantity needed for the decision. Decide beforehand whether the purpose is an endpoint estimate, a velocity or a local connection.
An evidence summary should separate established facts from remaining hypotheses. “The fictional expansion model accounts for 0.4 mm of a 0.7 mm top displacement” is a conditional calculation. “The ground rose 0.3 mm” is a stronger physical attribution requiring the rest of the chain to be supported. Preserve that distinction when explaining the result to someone who will not see the fitting process.
CHAPTER 12 OF 12 · Transfer and communicate
12. Deliver a thermal-motion result another person can reuse
A useful handover names the foundation and upper reference points, the coordinate directions and epochs, the temperature reference, and the body or proxy actually observed. It states the mechanical model and explains why its assumptions are reasonable for the intended application. Attach dimensions and coefficient provenance rather than leaving a future reader to infer them from the final displacement.
Give the original observation and the modelled contribution separately. Report the corrected result with units, sign and uncertainty scope. If a fitted empirical relation was used, state its training interval and sampling, and avoid presenting its coefficient as an independently measured material property. If a product already supplied the term, identify that fact so it cannot be subtracted again.
Keep model limitations close to the number: fixed or moving base, uniform or distributed temperature, free or restrained expansion, fixed or changing orientation. These choices govern what the result means. A compact report that exposes them is more reusable than a long table of decimal places whose endpoints remain ambiguous.
The broader lesson is practical and simple. Before assigning environmental variation to the thing you care about, examine the support, the sensor, the reference and the timing that connect you to it. The same reasoning helps with laboratory mounts, building-monitoring sensors and any measurement whose frame can respond to its surroundings. Carry the mechanism and its limits into the new setting, not just the original conversion factor.
Independent worked case and evidence checks
Try the calculation without reusing the four-metre fixture. A fictional vertical support is 3 m long, expands freely, has α = 0.000008/K, and warms uniformly by 5 K while its base stays fixed. Predict the top displacement before reading on. The sensitivity is 0.024 mm/K, so the rise is 0.12 mm.
Change only the warming to 12 K. The rise becomes 0.288 mm. Now change the target: the observed top rise is 0.5 mm and this model is accepted only for the support contribution. The inferred base rise is 0.212 mm. State that it is conditional; the arithmetic does not validate the model or establish real ground movement.
Finally divide the support into a one-metre section warming by 12 K and a two-metre section warming by 3 K. Expansion is 0.096 + 0.048 = 0.144 mm. An unweighted average of the two temperatures would give a different answer because it ignores section lengths. A satisfactory solution explains why the weights changed, not merely which button produced the result.
Evidence check: what would you request before applying this to a real monument? At minimum, identify the actual endpoints, temperature observations, material and structural assumptions, existing corrections and independent checks. No amount of agreement among the fictional answers supplies that missing site evidence.
Worked comparison
| Fictional condition | Predicted thermal change | What the comparison tests |
|---|---|---|
| 4 m; uniform warming 10 K | +0.40 mm at top | Units and free-expansion sensitivity |
| Two 2 m sections; warming 10 K and 2 K | +0.24 mm at top | Length-weighted temperature |
| 3 m at 2 K; 1 m at 10 K | +0.16 mm at top | Why an unweighted average fails |
| Top B: 2 m; top A: 4 m; both warm 10 K | −0.20 mm in B minus A | Differential motion does not generally cancel |
Questions and practical checks
Can I correct a station using air temperature alone?
Only if an appropriate, independently assessed relationship connects that record to the relevant structural response. Air temperature is not automatically the body’s temperature. Timing, exposure, depth, averaging and the intended reference point matter. The harmonic example shows why equal seasonal amplitudes can still give the wrong instantaneous correction.
Does a seasonal coordinate change prove thermal expansion?
No. Several mechanisms can share a season. Look for discriminating predictions using dimensions, material behaviour, relevant temperatures and independent observations. A correlation is a useful clue, not a complete attribution. Keep thermal expansion, subsurface response, loading and observation errors conceptually separate.
Should thermal displacement always be removed?
No. It depends on whether you need the physical trajectory of the upper point or an estimate of another point, such as the foundation. Name the target first. Check what the existing processing already includes and preserve an account of each subtraction to avoid counting the same contribution twice.
Why can two nearby stations have a changing local tie?
Their supports can differ in length, geometry, exposure and temperature history. Even the same coefficient and weather do not guarantee equal endpoint displacement. A differential observation may cancel a truly common component while retaining unequal thermal responses and uncertainties.
What is the most useful quick check?
Recompute a simple case in consistent units, reverse the temperature change, and then change the spatial or temporal temperature pattern. If your method gives the same answer when the physical length weights or timing change, examine which assumption has been hidden. For real work, follow this with site-specific evidence rather than treating the teaching check as approval.
Continue with the connected subjects
Locate the point and its connection
Separate foundation marks, antenna reference points and signal models
Connect physical instruments through surveyed local ties
Resolve an extension or tilt into stated local directions
Distinguish environmental response from a fitted trend
Test seasonal coverage and time-correlated uncertainty in station velocities
Keep distributed atmospheric loading separate from thermal expansion
Sources and scope
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.
Compare the physical quantities without merging them
A station’s support can change position, a falling body can reveal acceleration, clocks can constrain a potential difference, and a plumb line can establish direction. Read these as complementary measurements with different references, not interchangeable ways to report one height.
Follow the support and the acceleration
Absolute gravimetry: recover acceleration from a calibrated free-fall trajectory
Follow potential and direction
Optical clock comparisons: infer potential differences before converting to height
Deflection of the vertical: distinguish local plumb up from an ellipsoid normal