One barometer cannot explain an entire loading field
A geodetic station moves vertically while local air pressure changes. It is tempting to fit one conversion factor between the two curves and call the result an atmospheric-loading correction. That shortcut can miss the central physical issue: the deformation depends on a distributed load, not only on the pressure measured beside the antenna.
This guide uses an original three-region model to show how spatial pressure anomalies combine into a displacement vector. We then change the reference pressure, introduce shared uncertainty, and separate a physical loading signal from the decision to remove it. The numbers are invented teaching coefficients, not a model for any actual station.
Atmospheric loading is not the same as atmospheric radio-path delay or barometric compensation of a water-level logger. Those processes can all involve pressure while acting on different quantities. Keeping the pathways separate helps a reader understand what an observed coordinate change can establish and which additional evidence is still needed.
Geodesy subject guide · Learning that transfers into life · How X Works hub
Choose your reading route
Route 1 · Chapters 1–2: Define the distributed load
Route 2 · Chapters 3–5: Calculate a displacement vector
Route 3 · Chapters 6–8: Audit reference and correction choices
Route 4 · Chapters 9–10: Inspect uncertainty and time sampling
Route 5 · Chapters 11–12: Transfer and report
Test attribution rather than accepting a convincing correlation
Full chapter index · Independent worked case · Questions and practical checks
Chapter index
Define the distributed load
Distinguish deformation from two other pressure effects
Define pressure anomalies and the spatial response
Calculate a displacement vector
Compute one original three-region displacement
Show why the local barometer does not determine the answer
Change spatial aggregation without losing the response weighting
Audit reference and correction choices
Translate a changed reference pressure consistently
Decide whether loading is signal or a contribution to remove
Keep reference-frame motion separate from local deformation
Inspect uncertainty and time sampling
Propagate pressure uncertainty and shared errors
Respect sampling intervals, product components and time labels
CHAPTER 1 OF 12 · Define the distributed load
1. Distinguish deformation from two other pressure effects
Atmospheric-pressure loading concerns deformation associated with changes in atmospheric mass load. A GNSS propagation correction concerns the radio signal’s path through the atmosphere. Barometric compensation of an absolute water-pressure sensor subtracts the air-pressure contribution to its reading. The same pressure record can be relevant to several processes without making their corrections interchangeable.
The International Mass Loading Service explains loading as a spatially distributed problem and describes displacement products and their reference accounts. Its methodological account supplies the physical context for this guide. The small weighted-region model below is our own educational reduction; it is not the service’s computational algorithm or a substitute for its documented products.
A pressure change can therefore be associated with both an atmospheric delay change and a movement of the observing station. Correcting one pathway does not automatically correct the other. A complete interpretation should identify which quantity each model changes, what has already been included and how the resulting observations are compared.
For a practical reading check, draw the chain in words: pressure field, load response, station displacement, observation geometry, fitted coordinates. Contrast it with pressure sensor, hydrostatic delay estimate, signal-path correction. The chains can share inputs but have different transformations. A common cause does not justify reusing one numerical correction in another part of the analysis.
CHAPTER 2 OF 12 · Define the distributed load
2. Define pressure anomalies and the spatial response
Let pj be the pressure anomaly assigned to region j relative to a stated reference field. In an idealised linear model, a station displacement component is d = ΣKjpj. The coefficient Kj represents the integrated sensitivity of that station component to that region. It carries the units needed to turn pressure into length.
For our teaching examples, pj uses hectopascals and Kj uses millimetres per hectopascal. Region area and the chosen response weighting are already included in each stipulated coefficient. Multiplying by area again would count it twice. In a genuine spatial integration, the definition of the kernel and area element must be checked explicitly.
Different displacement components have different spatial sensitivities. A pressure pattern can produce Up, east and north motion under a model, not only a vertical scalar. Signs depend on the response and direction definitions. The fictional coefficients below are selected to make these relationships calculable; they must not be copied as empirical barometric factors for a real site.
A pressure anomaly is also not absolute pressure. Subtracting a reference field changes the zero used to describe the load variation. The time span and spatial definition of that reference matter when displacement products are compared. An anomaly series averaging to zero over its defining interval need not average to zero over another selected interval.
CHAPTER 3 OF 12 · Calculate a displacement vector
3. Compute one original three-region displacement
Choose fictional pressure anomalies p = (10,−4,6) hectopascals in three regions. Assign Up sensitivities KU = (−0.20,−0.10,−0.05) millimetres per hectopascal. The three contributions are −2.0,+0.4 and−0.3 millimetres, so the modelled Up displacement is −1.9 millimetres. Positive Up is declared outward in this local coordinate account.
For east, choose KE = (0.03,−0.02,0.01). Its contributions are +0.30,+0.08,+0.06, summing to +0.44 millimetres. For north, choose KN = (−0.01,0.04,−0.02). Its contributions are −0.10,−0.16,−0.12, summing to −0.38 millimetres. The complete local vector in east/north/up order is (0.44,−0.38,−1.90).
Recompute each result from the original pressures and coefficient row rather than copying the intermediate contributions. That reverse calculation checks region order and signs. A negative pressure anomaly multiplied by a negative Up sensitivity contributes positively here; replacing signed anomalies with their magnitudes would alter the physical pattern represented by the model.
The resulting vector is a teaching output, not a measured displacement or a prediction from a real atmospheric analysis. Its purpose is to make a nonlocal linear account inspectable. A real pressure-loading model needs validated atmospheric fields, Earth-response assumptions, spatial treatment, time interpolation and a compatible reference frame.
CHAPTER 4 OF 12 · Calculate a displacement vector
4. Show why the local barometer does not determine the answer
Suppose region one contains the station and its pressure anomaly remains ten hectopascals. The local barometer then reports the same value in many possible spatial pressure fields. In the first field, the other anomalies were minus four and plus six, giving Up displacement minus 1.9 millimetres under our stipulated sensitivity row.
Now choose a second field p = (10,6,6), changing only region two. The Up result becomes −2.0−0.6−0.3 = −2.9 millimetres. The local pressure is unchanged, but the predicted displacement differs by one millimetre. A single local pressure-to-displacement conversion cannot distinguish these two fields.
Choose a third field p = (10,−4,−14). Its Up displacement is −2.0+0.4+0.7 = −0.9 millimetres. Again the local barometer reads the same anomaly while the modelled deformation differs. The counterexample does not depend on the chosen numerical scale: any nonzero remote-region sensitivity allows remote pressure differences to affect the result.
A local regression can still describe an empirical relationship over a particular dataset. The limitation is its transfer. If the spatial pressure patterns associated with local pressure change, the fitted relationship can change. A useful report distinguishes a local statistical proxy from a distributed physical loading model and tests its performance on genuinely independent spatial and temporal conditions.
CHAPTER 5 OF 12 · Calculate a displacement vector
5. Change spatial aggregation without losing the response weighting
The correct regional contribution is the sum of each pressure anomaly multiplied by its own sensitivity. An unweighted mean pressure need not preserve that contribution. In the original example, the mean anomaly is four hectopascals. Multiplying it by the sum of Up sensitivities, minus 0.35, gives minus 1.4 millimetres rather than the original minus 1.9.
The discrepancy arises because the pressure pattern and sensitivities are not uniform. Replacing several regions with one number discards their relationship. For two regions with combined sensitivity K1+K2 nonzero, an effective pressure preserving that one component is (K1p1+K2p2)/(K1+K2). This is a component-specific weighted summary, not necessarily a physical regional average.
For regions one and two of the original Up model, the combined contribution is −1.6 millimetres and combined sensitivity is −0.30. The preserving effective anomaly is 5.33333 hectopascals. Their ordinary mean is three. The effective summary reproduces the selected Up contribution because it was constructed using that particular response weighting.
The same summary generally will not preserve east and north contributions, whose coefficient rows differ. A single compressed scalar is not guaranteed to retain an entire three-dimensional response. When regridding or aggregating a real field, evaluate the target quantities and approximation error rather than assuming that a visually similar pressure map preserves every loading component.
CHAPTER 6 OF 12 · Audit reference and correction choices
6. Translate a changed reference pressure consistently
Let the original anomaly be p = P−P0. If a new reference field is P0′ = P0+b, the new anomaly is p′ = p−b. Under the same linear response, d′ = Kp′ = d−Kb. The reported displacement zero changes even though the instantaneous physical pressure field P did not change.
Use a fictional uniform reference shift b = (2,2,2) hectopascals. For the original Up row, KUb = −0.70 millimetres, so dU′ = −1.90−(−0.70) = −1.20. For east, KEb = +0.04, giving dE′ = +0.40. For north, KNb = +0.02, giving dN′ = −0.40.
These changes do not represent an instantaneous physical movement caused by rewriting the reference file. They describe another anomaly baseline. If a coordinate baseline is adjusted consistently with the displacement definition, predictions can remain equivalent. Mixing one displacement zero with another coordinate baseline can instead introduce a constant mismatch.
This three-region construction is not a universal prescription for real global reference-pressure changes. It isolates the algebra of an anomaly definition under a fixed response operator. A real product may have additional mass-conservation, ocean-response and frame conventions. Translate its documented complete account rather than applying this toy reference shift blindly.
CHAPTER 7 OF 12 · Audit reference and correction choices
7. Decide whether loading is signal or a contribution to remove
A coordinate series containing loading-induced motion is not necessarily wrong. The physical question determines the target. A study of environmental deformation may want the loading signal retained for comparison. A study seeking another motion component may model and remove it, provided the correction and uncertainty are appropriate and its inclusion history is known.
For a one-component teaching ledger, write y = s+d, where y is the observed coordinate change, s the target non-loading contribution and d the modelled loading displacement. Choose fictional s = +3.0 and d = −1.9 millimetres. Then y = +1.1. Subtracting the negative d gives s = 1.1−(−1.9) = +3.0.
Subtracting the same model twice gives +4.9 rather than +3.0. Omitting it gives +1.1. The arithmetic demonstrates how a duplicate or missing contribution changes the target estimate. It does not show that a real model perfectly predicts loading or that every unexplained residual belongs to the target process.
Retain the before-and-after series and a ledger identifying each included model. A phrase such as pressure corrected is incomplete unless it identifies which pathway and transformation were applied. In particular, a hydrostatic radio-delay correction does not establish that coordinate deformation was removed, and a loading subtraction does not certify the propagation model.
CHAPTER 8 OF 12 · Audit reference and correction choices
8. Keep reference-frame motion separate from local deformation
A displacement product needs a stated reference frame. Redistributing fluid mass can be described relative to different reference origins, and a translation between origins can alter reported station displacements. The underlying physical configuration need not have changed merely because the origin account changed. This is separate from changing a pressure-anomaly baseline.
For a fictional Cartesian example, suppose a displacement in frame A is dA = (1,−2,3) millimetres. Define a translation relation for this exercise as dB = dA−t, with t = (0.5,0.2,−0.1). Then dB = (0.5,−2.2,3.1). The signs follow the declared relation, not a universal rule attached to the letter B.
The translation vector is Cartesian in this example. It cannot simply be subtracted from a list labelled east/north/up without translating the axes at the site. A local basis depends on location. Mixing a global-origin displacement with untransformed local components creates a representation error even when every number is expressed in millimetres.
Use the product’s documented frame and any supplied origin-motion terms when comparing outputs. The existing coordinate-epoch guide addresses another reference interface, while the ocean-loading companion develops component and harmonic conventions. Together they encourage an explicit account of representation instead of treating all reported displacement vectors as immediately comparable.
CHAPTER 9 OF 12 · Inspect uncertainty and time sampling
9. Propagate pressure uncertainty and shared errors
Return to the original Up sensitivities and choose independent pressure-anomaly standard uncertainties (1,2,1) hectopascals. Under a fixed linear response, Var(dU) = ΣKUj² Var(pj). The contributions are 0.04,0.04 and0.0025 square millimetres, giving variance 0.0825 and standard uncertainty approximately 0.287228 millimetres.
Now introduce an additional shared pressure error a with standard uncertainty two hectopascals, added equally to all regions and independent of the previously stipulated errors. Its Up contribution is (ΣKUj)a = −0.35a. Its variance is 0.35²×2² = 0.49 square millimetres. The combined variance is 0.5725, giving standard uncertainty approximately 0.756637 millimetres.
Averaging more grid cells does not automatically remove this common contribution. It is shared rather than independent. Conversely, particular spatial contrasts can cancel some shared errors under their coefficient pattern. The correct result follows the full covariance and sensitivity account, not a general claim that spatially separated pressure values must be independent.
This propagation treats the response coefficients as fixed and the chosen errors as an adequate stochastic description. It omits uncertainty in the Earth model, spatial resolution, ocean treatment and other interfaces. A small formal number from pressure noise alone must not be presented as an independent accuracy certificate for the complete loading product.
CHAPTER 10 OF 12 · Inspect uncertainty and time sampling
10. Respect sampling intervals, product components and time labels
A gridded or station displacement series samples a model at specified epochs. It does not necessarily resolve every shorter event. For a fictional test, suppose loading displacement is zero at hours zero and six but reaches minus six millimetres at hour three. A straight interpolation between only the two endpoint samples returns zero and misses the excursion.
The example does not assert that every real pressure-loading product has that sampling or waveform. It demonstrates why the sampling interval and interpolation method belong to a claim about short-time behaviour. Denser output times generated from the same coarse information do not automatically add independent physical resolution.
Some loading products separate selected harmonic components from a residual time series. The service documentation describes such decomposition. Before adding components, determine whether the supplied series already includes them and whether the frame, model and time conventions are compatible. A ledger prevents both omission and duplicate addition of a named periodic contribution.
For a comparison with GNSS coordinates, align effective epochs and the observation interval. A coordinate estimate averaged over an interval and an instantaneous loading sample do not automatically describe the same quantity. Use a comparison that respects how the observations were weighted, and state the uncertainty introduced by any temporal approximation.
CHAPTER 11 OF 12 · Transfer and report
11. Test attribution rather than accepting a convincing correlation
A correlation between local pressure and a coordinate component can motivate investigation, but it does not uniquely identify the physical pathway. Atmospheric propagation, spatial loading, instrument environment and other correlated conditions can all participate. The three-region counterexample already shows that local pressure alone does not determine the modelled deformation.
A useful test compares predictions across independent intervals with different spatial pressure configurations. Examine whether a distributed model explains direction, timing and amplitude better than a local proxy under comparable uncertainty assumptions. Retain held-out evidence rather than choosing a coefficient only because it removes a desired feature from the same series used to fit it.
Also inspect the other displacement components and relevant independent observations. A fit to Up alone does not establish that the full predicted vector is correct. If a model predicts east and north changes that are absent under sufficiently informative independent evidence, that discrepancy deserves investigation rather than being hidden by reporting only the favourable component.
The existing groundwater-barometric guide explains how pressure can change well observations through another mechanism. The repeat-microgravity guide explains how changing mass, reference heights and drift can overlap in a different observation. These comparisons help refine the causal question; they do not allow a coefficient calibrated for one observable to be imported into another without a physical model.
CHAPTER 12 OF 12 · Transfer and report
12. Report the field, response and correction ledger together
A reproducible loading statement identifies the pressure-field source, anomaly reference, spatial treatment, response model, ocean assumptions, component axes, reference frame, effective epochs and any separately supplied harmonic terms. It also states whether the output is being retained as a physical signal or removed to estimate another target quantity.
For the original fictional three-region example, the model vector is east +0.44, north −0.38 and Up −1.90 millimetres. The local pressure value by itself does not determine that vector. Under a separate stipulated pressure-noise account, the Up standard uncertainty changes markedly when a shared error is restored. These are conditional teaching conclusions, not field measurements.
If an observed residual remains after applying a model, report what was compared and which uncertainties remain. Do not relabel the remainder as tectonic movement, structural settlement or water-storage change merely because an atmospheric-loading term was subtracted. Removing one candidate contribution does not independently establish every alternative cause.
The transferable lesson is to keep spatial structure and reference definitions attached to the correction. A loading result is a modelled response to a field, not a magic barometer multiplier. Understanding that chain lets a reader distinguish real environmental motion, a changed representation and a bookkeeping error before making a stronger claim about the ground.
Independent worked case and evidence checks
Build an independent two-region model with Up sensitivities (−0.15,−0.05) millimetres per hectopascal and east sensitivities (+0.02,−0.03). Let pressure anomalies be (8,−2) hectopascals. The Up displacement is −1.2+0.1 = −1.1 millimetres. The east displacement is +0.16+0.06 = +0.22. Reconstruct each from its coefficient row.
Hold local region-one pressure at eight and change only region two to plus four hectopascals. Up becomes −1.4 millimetres and east becomes +0.04. The local barometer has not changed, but the distributed model predicts changes in both magnitude and directional balance. A local conversion fitted to the first field is not a complete description of the second.
Now shift both reference pressures upwards by one hectopascal without changing the physical pressure field. The anomalies become (7,−3). Up becomes −1.05+0.15 = −0.90, while east becomes +0.14+0.09 = +0.23. These differences follow the anomaly-baseline transformation, not a sudden physical station movement.
For an uncertainty check, assign independent standard uncertainties one hectopascal to both anomalies. Up variance is 0.15²+0.05² = 0.025 square millimetres, giving standard uncertainty approximately 0.158114. East variance is 0.02²+0.03² = 0.0013, giving approximately 0.036056 millimetres.
Their covariance is (−0.15)(0.02)+(−0.05)(−0.03) = −0.0015 square millimetres. The two displacement components are correlated despite independent pressure-input errors because they are derived from the same two inputs. Separate component columns do not establish independent uncertainty.
For a derived quantity z = dU+dE, propagate that covariance. Its variance is 0.025+0.0013+2×(−0.0015) = 0.0233 square millimetres, giving standard uncertainty approximately 0.152643. Ignoring covariance would give a different answer even though each marginal component variance was correct.
Introduce a new shared pressure uncertainty of two hectopascals. Its Up sensitivity is minus 0.20 millimetres per hectopascal, so its Up standard uncertainty is 0.40 millimetres. Its east sensitivity is minus 0.01, so its east standard uncertainty is 0.02. The shared contribution also adds covariance; it should not be independently attached to each output as unrelated noise.
Finally, use a correction ledger. Suppose the target non-loading Up change is +2.0 millimetres and the original two-region loading is −1.1. The observed change in the ideal exercise is +0.9. Removing the model once returns +2.0; removing it twice returns +3.1. Identify which series a supplied file represents before applying either operation.
Ask a second reader to distinguish five changes: another remote pressure pattern, another anomaly zero, a new input covariance, a new coordinate-frame origin and a duplicate model subtraction. All can change the reported output. Only some represent a changed physical pressure-driven deformation. Classifying them prevents a single numerical difference from being overinterpreted.
A practical transfer check should therefore preserve the complete pressure field and model metadata, not just the local barometer and final corrected coordinate. With that information, another analyst can reconstruct the intended target, test an alternative spatial assumption and locate where a discrepancy entered the chain. Without it, a smooth corrected series may be difficult to interpret independently.
For the derived quantity z, the shared pressure sensitivity is the sum of its Up and east sensitivities: minus 0.20 minus 0.01, or minus 0.21 millimetres per hectopascal. The additional variance is therefore 0.21²×2² = 0.1764 square millimetres. Add that once to the earlier 0.0233, giving 0.1997. Propagating the shared input through the derived quantity directly supplies an independent check on the full component-covariance calculation.
Worked comparison
| Change or observation | What it establishes | What it does not establish alone |
|---|---|---|
| Local pressure changes | An input at one location | The complete distributed loading displacement |
| Remote pressure pattern changes | Another spatial forcing under the model | A universal local conversion factor |
| Reference anomaly zero changes | Another representation baseline | An instantaneous physical motion |
| Loading term is subtracted | A defined transformed coordinate target | Correct attribution of every remaining residual |
| Formal pressure-noise covariance | Conditional uncertainty of a derived vector | Accuracy of every model and reference assumption |
Questions and practical checks
Is pressure loading just the GNSS tropospheric correction?
No. One concerns station deformation, the other signal propagation. They can share atmospheric inputs without being the same correction.
Can one barometer determine the loading vector?
Not generally. The original counterexamples retain the same local pressure while changing remote loading contributions.
Does more spatial averaging always reduce uncertainty?
No. Shared errors and lost response weighting can survive averaging or alter the target result.
Should loading always be removed?
The scientific target determines that choice. It may be a signal of interest, or a modelled contribution to remove with a clear ledger.
Does a remaining coordinate residual prove structural movement?
No. Other model, observation and reference uncertainties still require investigation and independent evidence.
Continue with the connected subjects
Separate pressure-related observation pathways
Distinguish radio-path delay from coordinate deformation
Separate sensor barometric compensation from real aquifer response
Preserve the geodetic and mass-change account
Keep coordinate frame and epoch comparisons explicit
Compare transient and seasonal coordinate structure with long-term velocity
Investigate mass, height and reference contributions in repeat microgravity
Sources and scope
International Mass Loading Service: spatial response, pressure anomalies and product decomposition
IERS Conventions Centre: model status and reference-point displacement context
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
Work through solid-Earth tidal deformation and coordinate conventions
Reconstruct ocean-tide loading from harmonic phases and component axes