Three dates can belong to one position without meaning the same thing
Two coordinate tables describe the same monument, but their numbers disagree. One table was measured recently and reported at an earlier epoch. Another uses a different frame whose transformation parameters have their own reference date. Choosing whichever date looks newest will not make the positions comparable. First identify what each date actually describes.
Geodesy connects positions to reference systems, their realisations and time-dependent models. This guide develops the next step beyond simply adding velocity times elapsed time: it separates the observation clock, coordinate epoch and parameter epoch, then checks whether a frame change and a time change have been applied consistently.
Every coordinate, velocity, reference frame and transformation below is fictional. The deliberately simple one-dimensional models make the relationships independently checkable. They are not actual ITRF, EPSG or national transformation parameters, and they do not establish a survey procedure or a tolerance for an engineering decision.
Geodesy subject guide · Learning that transfers into life · How X Works hub
Choose your reading route
Route 1: Identify the dates · Chapters 1–3
Route 2: Follow a moving-frame calculation · Chapters 4–6
Route 3: Check limits and uncertainty · Chapters 7–8
Route 4: Validate the comparison · Chapters 9–10
Route 5: Transfer and deliver · Chapters 11–12
Full chapter index · Independent worked case · Questions and practical checks
Chapter index
Identify the dates · Chapters 1–3
Name the event behind every date
Separate a reference system, its realisation and a coordinate epoch
Define the motion model before carrying a coordinate through time
Follow a moving-frame calculation · Chapters 4–6
Give a time-dependent frame offset its own reference epoch
Transform velocity as well as position in a moving-frame model
Compare operation paths without confusing consistency with truth
Check limits and uncertainty · Chapters 7–8
Stop a linear model at unsupported discontinuities
Propagate uncertainty with the coordinate model
Validate the comparison · Chapters 9–10
Avoid averaging incompatible epochs or frames
Use an adversarial epoch fixture
Transfer and deliver · Chapters 11–12
Transfer the reference clocks into an API or exported table
Deliver a comparison that states its time basis
CHAPTER 1 OF 12
1. Name the event behind every date
A measurement can be observed in2024, processed in2025 and reported as a coordinate at epoch2020. A transformation used during processing can have parameters stated at yet another reference time. These dates are not automatically inconsistent. They describe different events or modelling conventions. The problem begins when a reader substitutes one for another without following the calculation that connects them.
Make a temporal ledger before comparing coordinates. Include observation period, reported coordinate epoch, reference-frame identity, parameter reference epoch where relevant and processing version. Leave unknown entries unresolved rather than filling them with the file creation date. A recent file timestamp is evidence about a file event, not proof that its coordinates represent the monument at that instant.
Think of a photograph taken today but used to estimate the position of a moving object yesterday under a stated motion model. The photograph date and the estimated-position date have different jobs. Geodetic processing is more demanding than this analogy, but the distinction helps explain why the time of observation need not be the epoch attached to a reported coordinate.
Ask the supplier what the coordinate represents, not simply when the survey was done. A record that identifies only the campaign year may still lack the information needed for a precise comparison. Conversely, an older coordinate epoch does not necessarily mean the underlying observations are old. Trace the reporting model before drawing conclusions about freshness.
CHAPTER 2 OF 12
2. Separate a reference system, its realisation and a coordinate epoch
The IERS describes the International Terrestrial Reference System as conventions and modelling for origin, scale, orientation and time evolution. Its realisation through a terrestrial reference frame uses estimated station coordinates and velocities. This distinction matters because a label describing the reference is not itself a measurement of every point at every later time.
A coordinate epoch identifies the time associated with a particular coordinate value. A frame or transformation parameter reference epoch identifies the time at which specified model parameters apply. Those roles can use different dates. A frame’s name or publication year should not be silently treated as the coordinate epoch of every point expressed in that frame.
For a classroom model, define frame F and state that point P has coordinate1000.000 metres at2020.0. Separately define a transformation whose offset parameters are expressed at2010.0. The coordinate date belongs to P’s reported position; the parameter date anchors the transformation’s own time dependence. Neither can replace the other merely because both are called an epoch.
Real reference products have more structure than our one-dimensional fixture, including axes, scale, rotations, relevant time conventions and validity limits. Use the authoritative product definitions for actual work. The transferable habit is to keep the object being dated explicit: a coordinate, an observation, a parameter set and a published document are not the same object.
CHAPTER 3 OF 12
3. Define the motion model before carrying a coordinate through time
For a stipulated constant-velocity coordinate, write x(t)=x0+v×(t−t0). Here x0 is the coordinate at t0, v is velocity in that same coordinate interpretation, and the elapsed time must use compatible units. The equation is not an instruction to apply any convenient regional speed to a point whose movement history has not been established.
Our fictional point P has xF=1000.000 metres at2020.0 and vF=0.010 metres per year. At2026.0, the model gives1000.000+6×0.010=1000.060 metres. The60-millimetre change follows from the stated assumptions. It does not independently demonstrate that a real monument moved by that amount or that a real local velocity is constant.
Keep the velocity tied to its frame, component and model. An east component in one representation is not automatically an interchangeable input to a different set of axes. Likewise, a velocity describing regional crustal motion may not account for local monument instability. A numerically compatible unit does not establish that the physical quantity belongs in the intended equation.
Decimal-year arithmetic is part of this teaching fixture. Real epoch handling needs the product’s defined time convention and appropriate conversion of actual observation dates. Do not replace that work with an unexplained assumption that every calendar year has the same number of days. The precision of the date representation should match the precision and model requirements of the task.
CHAPTER 4 OF 12
4. Give a time-dependent frame offset its own reference epoch
Define a second fictional frame G through xG(t)=xF(t)+a(t), where a(t)=2.000+0.002×(t−2010.0) metres. The two-metre offset is stated at parameter epoch2010.0. The additional0.002 metres per year describes a changing relation between these invented frames. This is a teaching transformation, not an official geodetic parameter set.
At2026.0, a(t)=2.000+16×0.002=2.032 metres. Combining it with P’s propagated F coordinate gives xG=1000.060+2.032=1002.092 metres. Notice that two elapsed times entered the calculation: six years for P’s coordinate propagation and sixteen years for the transformation offset. They start from different reference events.
If a client treats the transformation offset as a timeless2.000 metres, it returns1002.060 metres instead. The32-millimetre discrepancy is not caused by rounding P’s velocity. It comes from omitting the modelled evolution of the frame relationship. Naming that cause makes the repair precise rather than encouraging another arbitrary adjustment to the final coordinate.
A real frame transformation may include more than a translating origin. Its parameters and their rates require the stated conventions and suitable implementation. Do not generalise our scalar addition into a universal transformation recipe. Its purpose is to expose the separate clocks and dependencies in a small case where every step can be inspected by hand.
CHAPTER 5 OF 12
5. Transform velocity as well as position in a moving-frame model
Differentiate the fictional relation xG=xF+a with respect to time. The corresponding velocity is vG=vF+the rate of a. With our values, vG=0.010+0.002=0.012 metres per year. Keeping the original velocity after changing to this moving frame would make later propagation inconsistent with the transformation that was just applied.
At2020.0, the frame offset is2.020 metres, so P’s G coordinate is1002.020 metres. Propagating that value six years with vG=0.012 gives1002.092 metres, exactly matching the result obtained by first propagating in F and then transforming at2026.0. The agreement is a useful internal consistency check for this complete stipulated model.
Using vF=0.010 after converting the2020 coordinate into G instead gives1002.080 metres. That result misses12 millimetres, equal to six years of the omitted frame-offset rate. The starting transformed coordinate was correct; the later velocity interpretation was not. A check performed only at the original epoch would fail to reveal the problem.
For real multidimensional transformations, velocity handling can involve additional terms and conventions. Establish the actual operation and implementation rather than borrowing our simple derivative. The general lesson remains: when the coordinate relationship itself changes with time, transporting a position and transporting its rate are connected parts of the same model.
CHAPTER 6 OF 12
6. Compare operation paths without confusing consistency with truth
Our two calculation paths agree because their definitions are compatible. Path1 propagates P in F and transforms at the target epoch. Path2 transforms P and its velocity at the starting epoch, then propagates in G. This path comparison tests whether the model was implemented consistently. It does not prove the model represents the actual physical history of a station.
A forward-and-reverse calculation has a similar limit. If the same wrong offset is added and then subtracted, the original value is recovered perfectly. The numerical round trip is reassuring about inversion within that chosen operation, but it cannot establish that the chosen operation was appropriate. An incorrect reference assumption can survive a flawless reversible calculation.
Use independent controls that do not merely reuse the same unsupported model. For real work, suitable reference products, station histories, qualified observations and documented transformations can provide the necessary context. The appropriate discrepancy threshold depends on the task and evidence. This article does not assign one universal tolerance to every coordinate comparison.
Keep three verdicts distinct: the arithmetic matches the stipulated equations; the equations are justified for the source and target references; and the resulting position is suitable for the intended use. A passing first verdict does not automatically supply the second or third. A clear report states which of these has actually been established.
CHAPTER 7 OF 12
7. Stop a linear model at unsupported discontinuities
A constant velocity describes one kind of motion history. A step, seasonal loading or post-event relaxation describes another. A long interval does not become safely linear because the input table contains only one velocity column. Read the model’s intended period and the station’s history before carrying a coordinate beyond the evidence that supports the assumption.
Introduce a fictional upward coordinate step of0.020 metres in2023, while preserving the original constant velocity on either side. The simple unbroken F calculation at2026 still gives1000.060. The stipulated step-inclusive history gives1000.080. More decimal places in the original velocity cannot recover the missing step; the model needs another supported component.
A visible jump is not automatically physical displacement. An equipment replacement, changed processing or altered reference description can also produce a discontinuity in reported positions. Preserve those possibilities and investigate them with appropriate evidence. Do not delete a step merely because a smooth line is easier to explain, and do not call every step tectonic movement.
The changed-context test is to ask what new fact would justify changing the model. A documented physical event with coherent independent observations supports a different interpretation from an undocumented software change. The coordinate series is evidence to analyse, not a complete causal explanation. The root Geodesy guide introduces that distinction; here it becomes an explicit limit on epoch propagation.
CHAPTER 8 OF 12
8. Propagate uncertainty with the coordinate model
A predicted coordinate depends on uncertain starting position and velocity. For the simple model, its variance includes position variance, elapsed-time squared times velocity variance, and twice elapsed time times the position–velocity covariance. Ignoring the covariance is an assumption, not an algebraic rule that applies to every estimated station solution.
For an independent fictional case, let the starting standard uncertainty be0.003 metres and velocity standard uncertainty0.001 metres per year, with zero covariance. Six-year propagation gives variance0.003²+6²×0.001²=0.000045 square metres, and standard uncertainty about0.00671 metres. The time term has increased the uncertainty even though the predicted coordinate is printed with more decimals.
Now stipulate a positive position–velocity covariance of0.0000015 square metres per year. The additional term2×6×0.0000015 is0.000018 square metres. The total becomes0.000063, giving about0.00794 metres. These invented covariance values illustrate the dependence; they are not default uncertainty parameters for any real frame or station.
This propagation does not automatically include uncertainty in a separate frame transformation, a missed discontinuity, a wrong station identity or an unsuitable velocity model. State which terms the reported uncertainty covers. A small standard uncertainty calculated within an incomplete model is not proof that an omitted physical or reference error is small.
CHAPTER 9 OF 12
9. Avoid averaging incompatible epochs or frames
Suppose one table supplies P in F at2020 and another supplies P in G at2026. Their mean is not automatically a better position. The two numbers describe different reference-and-time states. Before any combination, decide which common quantity is intended and establish supported operations that express each estimate in that compatible state.
Even after transformation, the estimates may share observations or model errors. Treating them as independent repetitions can understate uncertainty. A reprocessed solution and its predecessor are not necessarily two separate pieces of physical evidence. Preserve provenance and dependence rather than counting every coordinate row as an independent measurement.
For a paper exercise, give two learners our F and G tables without their labels. Ask them to average the values, then restore the labels and repeat the reasoning. The important improvement is not a different arithmetic average; it is recognising that the original operation had no justified target quantity. The labels changed what could responsibly be inferred.
A comparison can still be useful before full harmonisation if its limits are explicit. For example, an observed discrepancy may flag missing reference information rather than establish motion. Record that diagnostic result without over-interpreting its magnitude. Honest incomparability is more informative than a precise average of quantities whose meanings do not match.
CHAPTER 10 OF 12
10. Use an adversarial epoch fixture
A practical fixture should include a nonzero coordinate velocity, a nonzero transformation rate and different reference epochs for their parameters. A test in which every rate is zero cannot reveal whether the software handles time correctly. Choose asymmetric values and a target date far enough from both reference dates to make omitted terms visible.
Test both operation paths against independently calculated expected values. Then deliberately freeze the transformation offset and deliberately reuse the wrong velocity after transformation. In our fixture those mistakes create different32-millimetre and12-millimetre discrepancies. The test is stronger because it identifies distinct failure signatures instead of merely reporting that the final numbers differ.
Add a model-limit case with a documented fictional step. A client that implements the linear equation perfectly should still be described as incomplete for that changed history unless it applies the additional supported component. This prevents implementation correctness from being confused with adequacy of the selected model.
Keep the test inputs, expected outputs, source and target definitions, epoch convention and uncertainty assumptions together. A screenshot of two coincident map points cannot reveal which parameter date was used. The evidence should let another person reproduce the calculation without guessing the reference event behind any of its timestamps.
CHAPTER 11 OF 12
11. Transfer the reference clocks into an API or exported table
A service request can negotiate output coordinates while the underlying source has its own reference and epoch. A correct output CRS declaration does not by itself establish that the relevant time-dependent operation was appropriate. The connected feature-API guide explains input and output reference roles; add the epoch and motion information required by the actual task rather than assuming the output label contains everything.
A plain table can preserve the necessary information if its columns and accompanying definitions remain explicit. Keep coordinate epoch separate from observation date and processing date. Identify the frame realisation, applicable motion model and any transformations. A column simply labelled year can make a valid coordinate record impossible to interpret later.
For transfer practice, ask a colleague to compare a new point Q using our two-frame model. Supply the definitions but not the answer. They should identify the coordinate’s epoch, evaluate the frame offset at the target epoch and use the appropriate transformed velocity if following the alternative path. Their explanation matters as much as the final number.
Do not promise that every receiving format can preserve every element of a precise geodetic model. Inspect the actual export and recipient contract. A successful byte transfer can still lose the reference meaning needed for the next analysis. If information must travel in an accompanying record, make that dependency visible in the handoff.
CHAPTER 12 OF 12
12. Deliver a comparison that states its time basis
A useful final statement names the physical point, source coordinate and frame, coordinate epoch, target frame and epoch, transformation and motion-model versions, validity limits and covered uncertainty terms. It also says whether the result is a modelled coordinate at the target epoch or an independent observation made at that time. Those claims are not interchangeable.
Ask the recipient which date would change if the same observations were reprocessed, which date would change if the output epoch were changed, and which date anchors the transformation parameters. A reader who can answer those questions has separated the clocks. A reader who calls every date the survey date still needs the reference model clarified.
Retain unresolved limits beside the result. A plausible time-dependent correction does not establish that the monument was stable, that an antenna setup was measured correctly or that phase observations retained continuity. The connected instrument guides address those dependencies. A position comparison is strongest when its separate controls are connected without being collapsed into one blanket accuracy claim.
The endpoint is a coordinate whose time meaning survives a changed question. You can explain why its numbers changed, which model transported them and what independent evidence supports the comparison. That is more useful than choosing the newest-looking date or treating a frame label as a substitute for the temporal relationships beneath it.
Independent worked case and evidence checks
Try an independent fictional point Q. In F, Q is200.000 metres at2022.0 with velocity−0.004 metres per year. Define G by the same teaching offset a(t)=2.000+0.002×(t−2010.0). Find Q in G at2027.0. Begin by naming the separate coordinate and parameter reference epochs.
Propagate in F for five years:200.000−5×0.004=199.980 metres. Evaluate the frame offset after seventeen years from its own reference epoch:2.000+17×0.002=2.034 metres. The G coordinate is202.014 metres. A negative point velocity and positive frame-offset rate coexist without contradiction because they describe different relationships.
Check the alternative path. At2022.0 the offset is2.024, so Q is202.024 metres in G. Its G velocity is−0.004+0.002=−0.002 metres per year. Five-year propagation gives202.014 metres again. Keeping the F velocity after changing frames would instead produce202.004, missing10 millimetres of frame-relative evolution.
Now label the observation campaign2025 without altering the stipulated reported coordinate epoch2022.0. The input position remains a2022 coordinate under the fixture. The campaign date cannot replace t0 in the calculation unless the coordinate value is also changed through a justified model. This tests whether a client understands the record or merely selects the most recent year.
For a changed history, add a supported fictional displacement of−0.015 metres in2024. The target F coordinate becomes199.965 and G becomes201.999 metres. The old linear calculation remains internally reproducible but is no longer complete for that history. Explain which new model component caused the changed answer.
Finally reverse the original step-free calculation from2027 to2022 using the same consistent definitions. It recovers the starting coordinate. That is a useful consistency check, but it does not independently establish that the chosen F-to-G relationship is physically justified. Add appropriate source evidence before describing the toy’s round-trip success as real geodetic validation.
For an additional uncertainty check, return to P with position variance0.000009 square metres, velocity variance0.000001 square metres per year squared and covariance0.0000015 square metres per year at2020. Re-expressing the same model at2022 changes the position variance to0.000019 and the position–velocity covariance to0.0000035. These are changed parameter relationships for the same stipulated process, not new observations.
Propagate from2022 to2026 using four years. The variance is0.000019+16×0.000001+8×0.0000035=0.000063 square metres, matching direct propagation from2020. If the revised covariance is discarded, the result becomes0.000035 instead. Changing a reference epoch without carrying the associated uncertainty relationships can therefore change the reported precision even though the underlying model is meant to remain the same.
This last check is useful because the coordinate itself can still agree while the uncertainty does not. Compare both quantities during a handoff. A model reparameterisation should not silently acquire extra confidence simply because a new table has a more convenient reference date. Real products require their complete documented covariance treatment; the fixture only makes the dependency visible with small numbers.
Ask which parts of this final exercise were new observations. None were: only the parameter reference date changed. That distinction prevents a reformatted model from being counted as independent evidence or advertised as an additional measurement of the station.
Worked comparison
| Date or quantity | What it anchors | What it cannot replace |
|---|---|---|
| Observation period | When measurements were acquired | Reported coordinate epoch |
| Coordinate epoch | Time represented by the coordinate | Frame-parameter reference epoch |
| Parameter reference epoch | Time of stated transformation parameters | Target date of every output |
| Velocity in a frame | Rate under that coordinate model | A rate in another moving frame without conversion |
| Processing date | When a solution was produced | Time the monument occupied the reported position |
| Round-trip agreement | Consistency of tested operations | Independent physical correctness |
Questions and practical checks
Is the observation date always the coordinate epoch?
No. A solution can report coordinates at another epoch through an appropriate documented model.
Can I use the frame name as the epoch?
Do not infer that. Identify the coordinate epoch and any parameter reference epoch explicitly.
Does a frame change leave velocity unchanged?
Not generally. Our moving-offset fixture shows why the corresponding velocity must change consistently.
Does a perfect reverse transformation prove the model is right?
No. The same unsupported assumption can be applied and reversed consistently. Independent evidence remains necessary.
Can a constant velocity cross an earthquake unchanged?
Only if the complete applicable model supports that treatment. A discontinuity or other nonlinear history may require additional components.
Continue with the connected subjects
Preserve reference and time meaning
Geodesy overview: systems, realisations and a moving planet
Feature APIs: input bounds, output coordinates and recipient reference contracts
Observation time versus metadata maintenance: a separate temporal contract
Check the observation chain
RINEX records: preserve observation epochs, time systems and antenna metadata
GNSS–acoustic comparisons: reference frames, campaign history and displacement evidence
Sources and scope
IERS: reference-system conventions and realisation through station coordinates and velocities
NGS antenna calibration: consistency of reference-frame products in processing
JCGM: authoritative guidance on measurement models and uncertainty
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.
Connect the geodetic checks
These guides examine different links in one position claim. A correct time model does not prove the antenna setup, and a correct instrument model does not establish the reference or uncertainty. Follow the unresolved link rather than treating one passing check as complete validation.
Antenna reference points: connect setup geometry with the reported point. Identify the actual physical endpoint and which reductions have already been applied.
Carrier-phase continuity: separate ambiguity changes from displacement. Preserve observation arcs and independent evidence before interpreting a phase jump as ground motion.
Height differences: geoid variation, shared errors and covariance. State the target height difference and supported dependence assumptions before claiming differential precision.