A tide gauge is not a ground-displacement model
The sea level changes, and a coastal geodetic station also changes position. The two quantities are related through a physical system, but they are not interchangeable measurements. Ocean-tide loading concerns the solid Earth’s response to redistributed water load. A water-level value at one gauge does not directly provide the station’s three-dimensional displacement.
This guide develops a small harmonic displacement model using invented coefficients. You will reconstruct phases, translate west/south components into east/north axes, change the time origin consistently, and test how an incorrect phase sign survives an apparently successful check. The examples isolate interpretation rather than recommend coefficients for a real site.
Read the body-tide companion for direct solid-Earth tidal forcing. Here the new question is how a site-dependent ocean-loading response becomes a timed displacement vector and, ultimately, a contribution to a geodetic observation. Every numerical coefficient below is fictional unless explicitly identified as a source description.
Geodesy subject guide · Learning that transfers into life · How X Works hub
Choose your reading route
Route 1 · Chapters 1–2: Define the loading quantity
Route 2 · Chapters 3–5: Reconstruct harmonic displacement
Route 3 · Chapters 6–8: Translate time and component conventions
Route 4 · Chapters 9–10: Inspect dependence and sampling
Combine harmonics with their dependence, not just amplitudes
Route 5 · Chapters 11–12: Transfer and report
Do not transfer a nearby station’s coefficients without evidence
Full chapter index · Independent worked case · Questions and practical checks
Chapter index
Define the loading quantity
Separate water elevation from displacement of the station
Define one harmonic and retain the phase sign
Reconstruct harmonic displacement
Reconstruct a two-harmonic teaching component
Expose a phase-sign error with a discriminating epoch
Convert amplitude/phase into sine and cosine coefficients
Translate time and component conventions
Change the time origin without changing the displacement
Translate radial/west/south components before projecting them
Follow the station vector into a receiver observation
CHAPTER 1 OF 12 · Define the loading quantity
1. Separate water elevation from displacement of the station
Ocean tides redistribute water mass and change the load exerted on the solid Earth. The associated station displacement depends on the spatial load and Earth response, not merely on the water level at the closest shoreline. This is a different mechanism from a receiver being physically raised by a platform floating on the water.
IERS describes site-specific harmonic amplitudes and phases for ocean loading, and ESA Navipedia presents the same broad interpretation. The institutional descriptions provide the physical context. Our original two-harmonic examples intentionally omit the detailed constituent and astronomical machinery of a complete model, so they should not be treated as an operational loading implementation.
A tide gauge commonly observes water level relative to its own reference, which may itself move. Ocean loading is a displacement of the solid reference point in a specified coordinate account. Comparing them requires matching quantities and references. A shared unit such as millimetres does not make a sea-surface height and a crustal-displacement component the same observable.
For a practical reading check, underline the noun attached to each number: water elevation, loading amplitude, instantaneous ground displacement or receiver range contribution. Those are different stages in the chain. Confusion often begins when a table retains the units but drops the quantity and the model that connects it to another stage.
CHAPTER 2 OF 12 · Define the loading quantity
2. Define one harmonic and retain the phase sign
Use a fictional displacement component d(t) = A cos[θ(t)−φ]. The amplitude A is nonnegative and has length units, θ is the defined time-varying argument and φ is a phase lag under this exact convention. Angles must use compatible units. The expression is a complete sign definition, not merely a suggestive label called phase.
At θ = φ, the component reaches its positive maximum A. At θ = φ+180 degrees, it reaches minus A. These two relationships are useful synthetic tests of whether a stored phase was interpreted as a lag or a lead. A zero-time check alone can fail to expose a phase-sign mistake because cosine is an even function.
Real ocean-loading synthesis uses documented astronomical arguments and site-dependent response coefficients. A number called phase in one product cannot be transferred into an unrelated argument convention by retaining only its degree symbol. Preserve constituent identity, argument definition, component axis, phase convention and time account together before comparing or combining coefficients.
The teaching expression also separates amplitude from instantaneous displacement. A component with amplitude ten millimetres can be zero at a particular event. A small instantaneous value does not show that the underlying constituent is negligible, and an amplitude is not a correction to add with the same sign at every epoch.
CHAPTER 3 OF 12 · Reconstruct harmonic displacement
3. Reconstruct a two-harmonic teaching component
Define an original fictional Up component U(θ) = 10 cos(θ−30°)+4 cos(2θ+60°), in millimetres. The second term can equally be written with phase lag minus sixty degrees. The argument θ is an invented common phase variable for this exercise; the terms are not presented as measured coefficients for named ocean tides.
At θ = 0, the first contribution is 10 cos30° = 5√3, approximately 8.66025 millimetres. The second is 4 cos60° = 2. Their sum is approximately 10.66025. Add the signed instantaneous contributions, not their amplitudes ten and four.
At θ = 90 degrees, the first term becomes 10 cos60° = 5. The second becomes 4 cos240° = −2. Their sum is three millimetres. Both amplitudes are unchanged, yet the total has changed because the time arguments moved through different phases.
At θ = 180 degrees, the first contribution is 10 cos150° = −5√3, while the second returns to two. The total is approximately minus 6.66025 millimetres. Reconstructing several distinct arguments helps test the full synthesis. A table containing only amplitudes would not reveal any of these signed event-specific totals.
CHAPTER 4 OF 12 · Reconstruct harmonic displacement
4. Expose a phase-sign error with a discriminating epoch
Suppose an implementation mistakenly uses 10 cos(θ+30°) for the first term while retaining the second term correctly. At θ = 0, it still returns 10.66025 millimetres because cos(+30°) = cos(−30°). Passing that zero-argument check is therefore insufficient evidence that the phase lag was handled correctly.
At θ = 90 degrees, the mistaken first term is 10 cos120° = −5 instead of +5. The second remains minus two, so the incorrect total is minus seven rather than plus three. The difference is minus ten millimetres. A deliberately chosen nonzero argument exposes the error immediately.
This example explains why a useful reference test includes epochs where competing conventions make different predictions. Repeating a symmetric test with more decimal places cannot make it discriminate a sign ambiguity. Select test inputs for the failure mode being investigated rather than assuming that any available example exercises every convention.
Do not repair the result by globally reversing every displacement sign. That would change both harmonics and their intended component interpretation. The error lies in one phase convention. The appropriate repair translates that convention consistently, then checks several epochs and components. Localising the interface prevents a compensating mistake from hiding the original one.
CHAPTER 5 OF 12 · Reconstruct harmonic displacement
5. Convert amplitude/phase into sine and cosine coefficients
Using cos(θ−φ) = cosθ cosφ+sinθ sinφ, write a harmonic as C cosθ+S sinθ, where C = A cosφ and S = A sinφ. These coefficient pairs can be convenient for linear combinations and uncertainty propagation. They still need their argument and component definitions; two unnamed coefficients are not self-describing.
For A = 10 millimetres and φ = 30 degrees, C = 5√3, approximately 8.66025, and S = 5. Evaluating at θ = 0 returns C, while θ = 90 returns S. Those two events independently identify the pair in this idealised noiseless construction.
Reverse the transformation with A = √(C²+S²) and a quadrant-aware phase φ = atan2(S,C). A simple inverse tangent of S/C can lose the quadrant. For example, C = −3 and S = 4 give amplitude five and a phase in the second quadrant, not a negative first-quadrant angle chosen solely from their ratio.
When A is zero, phase is not identifiable: every φ gives the same zero harmonic. Near zero amplitude, phase can be very unstable even if C and S are estimated reasonably. Do not report a highly precise-looking phase for a nearly absent component without an appropriate uncertainty interpretation and a clear reason that phase remains useful.
CHAPTER 6 OF 12 · Translate time and component conventions
6. Change the time origin without changing the displacement
Suppose θ(t) = ωt and define a new time coordinate τ = t−t0. Then θ = ωτ+ωt0. To keep A cos(θ−φ) unchanged, the phase relative to τ is φ′ = φ−ωt0. Changing the time origin requires changing the phase representation; it does not represent a new physical tidal response.
Use a fictional angular rate ω = 30 degrees per hour and t0 = one hour. With original A = 10 and φ = 30 degrees, the transformed phase is zero. At old t = two hours, θ = sixty degrees and displacement is 10 cos30°. At new τ = one hour, the transformed expression gives the same value.
If the phase were left at thirty degrees after shifting the time origin, the new expression at τ = one hour would give ten rather than approximately 8.66025 millimetres. Both calculations look plausible and use the same amplitude. Their disagreement is a representation mismatch caused by changing only part of the definition.
An operational model’s astronomical argument is more involved than this constant-rate teaching example. Use the documented implementation and time scales rather than replacing it with a casual local-clock conversion. The transferable point is exact: a phase is defined relative to an argument, so modifying the argument requires a consistent translation of the accompanying phase.
CHAPTER 7 OF 12 · Translate time and component conventions
7. Translate radial/west/south components before projecting them
IERS expresses an ocean-loading displacement scheme using radial, west and south components. A downstream local east/north/up account can therefore require sign and direction translation. West is opposite east and south is opposite north. Radial and ellipsoidal Up also need their geometric distinction retained where it matters; they are not universally identical by definition.
For a fictional site where the radial direction is stipulated to coincide with the chosen Up axis, let a model return radial = 10, west = 2 and south = −3 millimetres at one event. Its east/north/up vector is then E = −2, N = +3, U = 10. A negative south component means movement towards north under the original component convention.
Check the squared vector length in both representations. The original gives 10²+2²+(−3)² = 113 square millimetres; the transformed vector gives (−2)²+3²+10² = 113. This invariant is useful, but it does not by itself validate every axis sign. A reversed east component preserves length while changing a directional observation.
A second independent check projects a synthetic displacement known to lie west. It should have negative east in the new description. Combining a magnitude invariant with known-direction tests is more discriminating than relying on either alone. Preserve the transformation and site geometry as part of the loading record, not as undocumented software assumptions.
CHAPTER 8 OF 12 · Translate time and component conventions
8. Follow the station vector into a receiver observation
Let u be the unit vector from receiver towards satellite, expressed in the same local axes as the station displacement d. The first-order range contribution from receiver motion is −u·d. This is an observation-specific projection. It is not generally equal to the station’s Up component or to its three-dimensional displacement magnitude.
Using d = (−2,3,10) millimetres and a fictional u = (0.6,0,0.8), the dot product is −1.2+8 = 6.8 millimetres. The receiver-motion range change is therefore minus 6.8. Using west = +2 as though it were east would instead give minus 9.2, exposing a component-convention error.
A satellite due in the opposite horizontal direction but with the same vertical coefficient has u₂ = (−0.6,0,0.8). Its range contribution is minus [1.2+8] = minus 9.2 millimetres under the correct vector. Equal elevation does not guarantee equal loading range effects when the station displacement contains horizontal components.
The satellite geometry and precise state/event relationships still need their full processing account. These vectors are artificial arrangements chosen to test component translation. The lesson is that a displacement model enters a measurement through the endpoint geometry, so a scalar label such as loading correction should not conceal whether it is a vector or a projected range term.
CHAPTER 9 OF 12 · Inspect dependence and sampling
9. Combine harmonics with their dependence, not just amplitudes
Two contributions at the same frequency can reinforce or cancel. If each has amplitude ten millimetres but their phases differ by one hundred eighty degrees under the same argument, their sum is zero at every epoch. Adding their amplitudes to report twenty would describe neither the combined waveform nor its instantaneous value.
In sine/cosine representation, add C values and S values first, then reconstruct the resultant amplitude and phase. For coefficients (3,4) and (−3,−4), the sum is (0,0). Each individual amplitude is five, yet the combined amplitude is zero. The result follows from the signed vectors, not from a general rule that loading contributions are small.
For an uncertainty exercise, let C and S have variances one and four square millimetres with covariance 0.5. The displacement variance at argument θ is cos²θ Var(C)+sin²θ Var(S)+2cosθ sinθ Cov(C,S). At forty-five degrees this is 0.5+2+0.5 = three, giving standard uncertainty √3 millimetres.
Ignoring the covariance would give variance 2.5 instead. A consistently propagated formal uncertainty still does not cover wrong phases, missing constituents or an unsuitable spatial model unless those uncertainties have been included. Keep coefficient uncertainty, structural model uncertainty and independent validation evidence distinct rather than asking one standard deviation to represent every possible discrepancy.
CHAPTER 10 OF 12 · Inspect dependence and sampling
10. Inspect the observation window before claiming an average cancels tides
A single ideal harmonic has zero continuous mean over an exact integer number of its complete cycles. That statement has conditions. A shorter interval, irregular weights or a sum of different frequencies can retain a nonzero mean. An observation labelled daily does not automatically establish cancellation of every contribution in an ocean-loading model.
For a fictional component A cos(ωt−φ), the mean between t1 and t2 is A[sin(ωt2−φ)−sin(ωt1−φ)]/[ω(t2−t1)], using consistent radian units. This equation makes the interval dependence visible. The numerator vanishes over complete cycles but need not vanish over an arbitrary selected window.
As a discrete teaching example, sample 10 cosθ at θ = 0,90,180,270 degrees. The equal-weight mean is zero. Retain only θ = 0 and90 and the mean becomes five millimetres. The signal did not change; the sampling and denominator changed. A missing interval can therefore change a summary without indicating a new loading mechanism.
For a real coordinate product, preserve the effective observation weighting and the loading treatment in the estimation process. A post-hoc arithmetic average of model values may not reproduce the effect of that model inside a weighted adjustment. The appropriate comparison follows how the observations entered the solution, not merely how the final output interval was named.
CHAPTER 11 OF 12 · Transfer and report
11. Do not transfer a nearby station’s coefficients without evidence
Ocean-loading coefficients describe a response at a chosen site under a specified model and reference account. A neighbouring station is not automatically equivalent. The spatial distribution of ocean load and Earth response matters, and a local gauge cannot replace that information by supplying a single water-level curve.
Imagine two fictional sites whose same-frequency Up coefficients are (C,S) = (3,4) and (4,3) millimetres. Both have amplitude five. At θ = 0 their displacements differ by one millimetre; at ninety degrees the difference has the opposite sign. Equal amplitude does not establish interchangeable time series.
A reference-frame convention can also affect reported displacement components. IERS discusses the relationship between the total Earth system’s centre of mass and motion associated with redistributed fluid mass. A loading file therefore needs its reference definition, not just three component columns. Use the product’s documented frame rather than inferring it from the station name.
A meaningful transfer test compares reconstructed vectors over an informative time span, translates compatible conventions, and examines independent evidence where available. If coefficients are unavailable for the intended site, report that limitation. Reusing a nearby file without a defensible spatial model makes the resulting precision claim stronger than the evidence permits.
CHAPTER 12 OF 12 · Transfer and report
12. Report coefficients, reconstructed motion and residual evidence separately
A reproducible result identifies the ocean model, station location and endpoint, constituent/argument definitions, amplitude and phase conventions, component axes, time account and reference frame. It also states whether the loading contribution was already included in the supplied coordinate series or is being applied in the present comparison.
For the two-harmonic teaching example, amplitudes ten and four millimetres produce 10.66025 millimetres at zero argument and three at ninety degrees. Those event-specific values are not tide-gauge heights, full vector norms or universal coastal loading magnitudes. The quantities remain meaningful because their definitions and fictional scope are attached.
The existing station-velocity guide explains how periodic terms can interact with longer coordinate trends. The repeat-microgravity guide explains another observation affected by changing reference points and environmental contributions. Use those connected routes when the question changes, rather than repurposing a displacement coefficient as though it directly answered every geodetic or hydrological inference.
The transferable lesson is that a phase is a relationship and a displacement is a vector at an event. Keep those relationships intact as values move through files, axes and time origins. Then a loading correction becomes an inspectable physical contribution instead of an unexplained number whose apparent precision hides its reference assumptions.
Independent worked case and evidence checks
Construct an independent fictional harmonic with C = 6 and S = 8 millimetres. Its amplitude is ten, and its lag under A cos(θ−φ) lies in the first quadrant, with φ = atan2(8,6), approximately 53.1301 degrees. At θ = 0 the displacement is six; at ninety degrees it is eight. Those values independently reconstruct the original coefficient pair.
At θ = 180 degrees the displacement is minus six. At θ = 270 it is minus eight. A phase-sign reversal preserves the zero and one-hundred-eighty-degree values but reverses the sine contribution, so the ninety and two-hundred-seventy-degree checks are essential. This reproduces the earlier failure mechanism with new numbers rather than a copied answer.
Add a second same-frequency term with coefficients (−2,1). The combined pair is (4,9), giving amplitude √97, approximately 9.84886 millimetres. Adding the individual amplitudes ten and √5 would not produce the resultant amplitude because the contributions have different phases. Reconstruct at zero and ninety degrees to check the vector addition.
Now change the argument to θ′ = θ−40 degrees. Keeping the physical waveform requires translating its phase or, equivalently, rotating its coefficient pair. For C = 6,S = 8, the new cosine coefficient is 6 cos40°+8 sin40°, approximately 9.73857, and the new sine coefficient is −6 sin40°+8 cos40°, approximately 2.27163.
The squared amplitude remains one hundred within rounding. At old θ = 40 degrees, new θ′ = 0 and the new cosine coefficient must equal the old waveform value at forty degrees. That is a discriminating origin-change check. Comparing only amplitudes would miss a phase representation left attached to the wrong argument.
For a separate component test, stipulate a site whose radial direction equals the chosen Up axis. Let radial/west/south displacement be (−4,−1,2) millimetres. In east/north/up order it becomes (1,−2,−4). With u = (0,0.6,0.8), the receiver range contribution is minus [−1.2−3.2] = plus 4.4 millimetres.
Reverse only the interpretation of south and the computed range changes, even though the vector length is unchanged. A positive south test vector should point negative north. This is why an axis-direction test belongs beside a norm check in a validation set: each catches a different family of mistakes.
Finally compare two sampling summaries of 6 cosθ+8 sinθ. At θ = 0,90,180,270 degrees, the equal-weight mean is zero. With weights 2,1,1,1, the weighted sum is twelve plus eight minus six minus eight = six, divided by total weight five, giving a mean of 1.2 millimetres. The nonzero result is explained by weighting, not a new constant physical offset.
Write a short interpretation that distinguishes the generating harmonic, its amplitude, the sampled values and the weighted mean. Then state what additional information would be needed to compare that mean with a coordinate product. The answer should retain the observation weighting, compatible frame, model inclusion history and effective time span.
This exercise makes a practical evidence check possible without operating equipment or claiming field accuracy. Another reader can reconstruct every number, alter one convention and identify precisely which invariance fails. That is a stronger learning outcome than memorising that ocean loading is a correction which should somehow be added to every coastal coordinate.
For one last phase-sensitivity check, a small phase error has different first-order effects at different arguments. Near a harmonic maximum, the slope with respect to phase is zero; near its zero crossing, the slope magnitude is the amplitude. A phase test performed only near maxima can therefore understate how strongly that same phase error affects other epochs.
Worked comparison
| Reported quantity | What it describes | What it does not supply alone |
|---|---|---|
| Tide-gauge level | Water level relative to a stated reference | Three-dimensional crustal loading displacement |
| Harmonic amplitude | Size of a component under an argument definition | Its signed value at every epoch |
| Phase lag | Relationship to the defined argument | A meaningful number without its sign/time convention |
| Radial/west/south vector | Displacement in specified component directions | An east/north/up vector without translation |
| Weighted interval mean | A summary of selected events and weights | Guaranteed cancellation of all tidal constituents |
Questions and practical checks
Can I use the nearest tide gauge as a loading correction?
Not directly. Loading depends on spatially distributed ocean mass and a response model, not one water-level value.
Why can a phase-sign error pass a zero-time check?
Cosine is even. A nonzero argument where the sine contribution matters exposes the difference.
Do harmonic amplitudes simply add?
Not in general. Add signed sine/cosine coefficients or reconstruct the contributions with their phases.
Are west and south the same signs as east and north?
No. Translate the opposite axes, and retain the separate radial-versus-Up definition.
Does a daily average eliminate ocean loading?
Not automatically. Frequencies, interval boundaries, gaps and observation weights determine the remaining mean.
Continue with the connected subjects
Preserve the coordinate and endpoint account
Keep coordinate epochs and reference-frame comparisons compatible
Distinguish the station endpoint from antenna-height and phase-centre models
Follow periodic motion into other interpretations
Separate periodic displacement from an estimated long-term station velocity
Compare loading, height and reference issues in repeat microgravity
Sources and scope
ESA Navipedia: site-dependent loading harmonics and component interpretation
IERS Conventions Centre: loading software, reference cases and edition status
IERS reference-point chapter: ocean-loading harmonics and reference-frame 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
Trace atmospheric-pressure loading through spatial anomalies and reference ledgers