A clock coefficient has meaning only beside its reference epoch
A satellite clock record supplies three coefficients and a reference epoch. Copying the coefficients correctly is not enough. The polynomial describes an offset relative to that epoch, in stated units and under a particular clock convention. Changing the epoch without transforming the coefficients changes the model, even when the printed numbers still look reasonable.
This guide develops that interface with original fictional values. It explains offset, slope and curvature; evaluates a clock polynomial; shifts its reference epoch without changing the underlying clock history; and tests how a partial reference change creates an error. It then adds uncertainty, record changes and limits on extrapolation.
The examples are not navigation-message parameters or receiver settings. They isolate mathematical relationships that a validated implementation must respect. Use the existing emission-time guide for the separate question of connecting satellite and receiver clock labels to physical signal events.
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Choose your reading route
Route 1: Define the clock quantity
Route 2: Evaluate and differentiate
Route 3: Change references consistently
Route 4: Audit corrections and records
Keep the polynomial contribution separate from additional corrections
Route 5: Transfer and report
Full chapter index · Independent worked case · Questions and practical checks
Chapter index
Define the clock quantity
Name the offset rather than merely the clock
Give each coefficient the units it needs
Evaluate and differentiate
Evaluate the polynomial one contribution at a time
Separate instantaneous drift from average drift
Distinguish a change of offset from a range-like effect
Change references consistently
Re-centre the reference without changing the history
Expose the error caused by changing only the epoch
Transform uncertainty with the coefficients
CHAPTER 1 OF 12
1. Name the offset rather than merely the clock
Define δ(t) as a satellite clock label minus the chosen system-time label at the same event. Under this definition, a positive offset means the satellite clock reads ahead. Other interfaces may supply a correction to add rather than an offset to subtract. Translate the definition before translating its numerical sign; the word correction alone does not fix the convention.
ESA Navipedia describes broadcast satellite-clock modelling using coefficients relative to a reference epoch. In a simplified quadratic form, δp(t) = a₀ + a₁(t−t₀) + a₂(t−t₀)². The subscript p reminds us that this is the polynomial contribution. The complete applicable satellite-clock account may include other documented terms.
The reference epoch t₀ is not the signal reception event simply because it appears in the same record. It is the origin about which the coefficients parameterise the clock model. The epoch at which the model is evaluated must follow the applicable observation and navigation-message definition. A mathematically correct value evaluated at the wrong event remains unsuitable.
A complete data record therefore contains the clock convention, time scale, coefficient units, reference epoch and validity information together. Retaining only three numbers is not enough to reconstruct the model. This is the same principle as keeping a map’s coordinate reference with its coordinates: the numerical values do not carry their own interpretation.
CHAPTER 2 OF 12
2. Give each coefficient the units it needs
Let δp be expressed in nanoseconds and the local time difference Δ = t−t₀ in seconds. Then a₀ has units of nanoseconds, a₁ of nanoseconds per second and a₂ of nanoseconds per squared second. Multiplying each coefficient by its corresponding time power produces an offset in nanoseconds. The three contributions can then be added meaningfully.
For our fictional model, choose a₀ = 4 nanoseconds, a₁ = 0.2 nanoseconds per second and a₂ = 0.01 nanoseconds per squared second. These values were invented to make the arithmetic visible. They are not representative performance specifications and should not be entered into a real receiver or used to certify a positioning result.
The quadratic coefficient is not numerically identical to the second derivative under this polynomial definition. Differentiating twice gives δp″ = 2a₂. Some descriptions use rate terminology informally, so the explicit equation is the reliable guide. A coefficient placed beside Δ² contributes half the second derivative only when that relationship is interpreted consistently.
Unit checks are especially valuable when one file uses seconds and another display uses nanoseconds. One nanosecond is 10⁻⁹ second. A coefficient conversion must preserve its time denominator as well as its offset unit. Converting a₀ but leaving a₁ and a₂ in incompatible units can produce a plausible result at the origin and rapidly incorrect values away from it.
CHAPTER 3 OF 12
3. Evaluate the polynomial one contribution at a time
Set the fictional reference epoch to t₀ = 100 seconds and evaluate at t = 130 seconds. The local difference is Δ = 30 seconds. The constant contribution is 4 nanoseconds, the linear contribution is 0.2 × 30 = 6 nanoseconds, and the quadratic contribution is 0.01 × 30² = 9 nanoseconds. Their sum is 19 nanoseconds.
A reverse check begins at the origin. At t = 100, Δ is zero and the model returns exactly a₀ = 4 nanoseconds. At t = 110 it gives 4 + 2 + 1 = 7 nanoseconds. These smaller cases help identify an implementation that accidentally uses t rather than t−t₀, or that squares the coefficient instead of the time difference.
Horner’s arrangement writes the same value as a₀ + Δ(a₁ + Δa₂). At Δ = 30, the inner value is 0.2 + 0.3 = 0.5 nanoseconds per second. Multiplying by 30 gives 15 nanoseconds and adding four returns 19. Agreement between the expanded and nested forms checks the arithmetic pathway.
Numerical rearrangement does not change the physical assumptions. Both forms will evaluate the wrong model if their input epoch uses a different time scale or their coefficients belong to another satellite. Test the units and record identity separately rather than treating a successful calculation as proof that the correct record was selected.
CHAPTER 4 OF 12
4. Separate instantaneous drift from average drift
Differentiate the fictional model with respect to its seconds-valued argument: δp′(t) = a₁ + 2a₂Δ. At the reference epoch the slope is 0.2 nanoseconds per second. At Δ = 30 it is 0.2 + 0.6 = 0.8. The offset is 19 nanoseconds at that event, but its instantaneous rate of change is a different quantity.
The average slope between Δ = 0 and Δ = 30 is (19−4)/30 = 0.5 nanoseconds per second. It differs from both endpoint slopes because the clock polynomial has curvature. For a quadratic, the average slope over an interval equals the instantaneous slope at the interval midpoint; here Δ = 15 gives 0.2 + 0.3 = 0.5.
Check with a centred difference around Δ = 30. At Δ = 31, the offset is 4 + 6.2 + 9.61 = 19.81. At Δ = 29 it is 4 + 5.8 + 8.41 = 18.21. Their difference divided by two seconds is 0.8 nanoseconds per second, matching the derivative exactly for this quadratic.
A finite difference need not be an exact derivative for an arbitrary real clock history, especially across a discontinuity or record change. The exact agreement here is a property of the stipulated polynomial. State what was assumed before using a numerical derivative as a check on a measured or estimated clock series.
CHAPTER 5 OF 12
5. Distinguish a change of offset from a range-like effect
A clock-offset term expressed in seconds can be multiplied by the physical speed of light to form a length-equivalent contribution. The sign with which that contribution enters an observation equation depends on the declared clock and observation conventions. Do not infer an add-or-subtract instruction solely from a positive number in the clock model.
For scale only, one nanosecond multiplied by 299,792,458 metres per second is 0.299792458 metres. Our fictional 19-nanosecond polynomial value therefore has a length-equivalent magnitude of approximately 5.696057 metres. This is not a prediction that the receiver’s coordinate is wrong by that amount. The full observation model determines how a clock contribution enters the solution.
A satellite-specific clock error and a receiver-clock term also play different roles. A common receiver offset can be estimated alongside coordinates under an appropriate geometry, whereas individual satellite contributions affect observations according to their identities and model. A single time-to-length conversion does not show which estimated parameter will absorb an omitted term.
The useful check is to reconstruct the observation equation with the clock definition written beside it. If a report gives an applied adjustment rather than the underlying offset, its sign can be opposite without any disagreement about the clock. Preserve that distinction before comparing two implementations or deciding that one corrected the data in the wrong direction.
CHAPTER 6 OF 12
6. Re-centre the reference without changing the history
Suppose the same polynomial is to be expressed about a new reference epoch t₁ = t₀ + s. Define u = t−t₁, so Δ = u+s. Substitute into a₀ + a₁Δ + a₂Δ² and collect powers of u. The new coefficients are b₀ = a₀ + a₁s + a₂s², b₁ = a₁ + 2a₂s and b₂ = a₂.
Choose s = 10 seconds. For the fictional coefficients, b₀ = 4 + 2 + 1 = 7 nanoseconds, b₁ = 0.2 + 0.2 = 0.4 nanoseconds per second and b₂ = 0.01 nanoseconds per squared second. The new reference epoch is 110 seconds. The coefficients changed because their reference changed, not because the clock suddenly changed.
At the original query t = 130, the new local time is u = 20. The new expression gives 7 + 0.4 × 20 + 0.01 × 20² = 7 + 8 + 4 = 19 nanoseconds. The value agrees with the old representation. Its derivative also agrees: 0.4 + 2 × 0.01 × 20 = 0.8 nanoseconds per second.
This is an exact algebraic identity for the quadratic model. It is not an instruction to rewrite a broadcast navigation record or claim a new validity interval. Recentring preserves the same polynomial history; it does not improve its agreement with the physical clock or extend the evidence supporting its use.
CHAPTER 7 OF 12
7. Expose the error caused by changing only the epoch
Now make a deliberate mistake: replace the reference epoch 100 with 110 seconds but keep the old coefficients 4, 0.2 and 0.01. At t = 130, this incomplete transformation gives 4 + 4 + 4 = 12 nanoseconds. The correct representation gives 19. The seven-nanosecond discrepancy was introduced by the interface change itself.
At the new reference event t = 110, the incorrect model returns four nanoseconds, whereas the original history gives seven. A check performed only at the old reference epoch would reveal another discrepancy but might not identify its source. Testing the new origin directly makes the missing coefficient transformation especially clear.
The general difference between evaluating the old coefficients at Δ−s and the original polynomial at Δ is −a₁s + a₂(s²−2Δs). It depends on Δ when curvature is present. The error is therefore not always one constant that can be absorbed elsewhere. For our Δ = 30 and s = 10, it is −2 + (1−6) = −7 nanoseconds.
A changed epoch can look like a harmless metadata edit. This example shows why reference metadata can be part of the mathematics. Preserve the relation between coefficients and their origin rather than editing one field to make files appear aligned. If a different product uses another epoch, either evaluate each correctly or perform a complete, documented transformation.
CHAPTER 8 OF 12
8. Transform uncertainty with the coefficients
Recentring also changes how coefficient uncertainties are related. Write the coefficient vector as a = (a₀,a₁,a₂). The transformation b = Ma has rows (1,s,s²), (0,1,2s) and (0,0,1), with units understood. If the original covariance is C, the transformed covariance is MCMᵀ. Keeping only the old diagonal uncertainty labels generally loses information.
Use a simplified fictional linear model first, with a₂ fixed at zero. Suppose a₀ and a₁ are independent, with standard uncertainties 1 nanosecond and 0.1 nanoseconds per second. Re-centre by s = 10 seconds. Then b₀ = a₀ + 10a₁ has variance 1 + 100 × 0.01 = 2 square nanoseconds. The slope b₁ remains a₁.
The covariance between b₀ and b₁ is 10Var(a₁) = 0.1 in the corresponding units. At a query twenty seconds after the new origin, the variance of b₀ + 20b₁ is 2 + 400 × 0.01 + 40 × 0.1 = 10 square nanoseconds. At thirty seconds after the old origin, the original expression also gives 1 + 900 × 0.01 = 10.
If the new coefficients were incorrectly treated as independent, the result would be six rather than ten square nanoseconds. The clock history was preserved algebraically, but its uncertainty would not have been. A reference transformation is complete only when the uncertainty account is transformed consistently with the values it describes.
CHAPTER 9 OF 12
9. Keep the polynomial contribution separate from additional corrections
A satellite-clock model can contain contributions beyond the polynomial. ESA distinguishes the polynomial clock term from an eccentric-orbit relativistic contribution in the applicable account and notes that inclusion conventions differ between navigation messages. The companion eccentric-orbit guide develops that term. The lesson here is to keep an explicit ledger rather than assume every input omits the same correction.
Construct a fictional polynomial contribution of +19 nanoseconds and an additional documented contribution of −3 nanoseconds. If the supplied product excludes the additional term, their total is +16. If a supplied value already equals that total, adding −3 again incorrectly produces +13. A correct formula applied at the wrong stage can double-count a contribution.
A useful ledger records each term’s definition, sign, units, source and whether it is already included. It also identifies the signal or observable convention when relevant. Do not merge unknown contributions into a column labelled final correction and then expect another routine to infer what has already happened.
This is not a recommendation to apply one identical ledger to every constellation or product. Current product and signal documentation controls the actual definitions. The fictional example teaches how to detect duplicate accounting. It does not supply a universal operational correction sequence or replace the applicable navigation-message specification.
CHAPTER 10 OF 12
10. Do not smooth over a record change without evidence
Two successive clock records can describe related but not identical estimated histories. They may have different reference epochs, coefficients, validity information or model updates. Comparing a₀ values alone is not a meaningful continuity test when their reference epochs differ. Evaluate both models at a common compatible epoch within an appropriate comparison interval.
For a fictional comparison, let record A be our original quadratic. Let record B use the correctly recentered coefficients plus a constant two-nanosecond offset. At t = 130, A gives 19 and B gives 21. Their difference is two at every event in this constructed example. The discrepancy is not caused by recentering, because the exact recentering itself preserves the history.
Now change B’s slope as well. Its discrepancy becomes time-dependent. A one-epoch match cannot establish that two records describe the same interval. Compare value and, when relevant, derivative at more than one supported event, and inspect the documented record selection rather than stitching values together because the resulting plot looks smooth.
A discontinuity in a record sequence is not automatically a physical clock jump, just as a smooth join is not proof of physical continuity. The update may reflect estimation, time-tag or interpretation changes. Preserve the original records and the selection rule so the distinction can be investigated rather than erased by an undocumented smoothing step.
CHAPTER 11 OF 12
11. Test a changed time unit and a changed argument
Re-express the fictional local time in minutes: v = Δ/60, so Δ = 60v. The same history becomes δp = a₀ + (60a₁)v + (3600a₂)v². The new coefficients are 4 nanoseconds, 12 nanoseconds per minute and 36 nanoseconds per squared minute. At Δ = 30 seconds, v = 0.5 minute.
Evaluate: 4 + 12 × 0.5 + 36 × 0.25 = 4 + 6 + 9 = 19 nanoseconds. The derivative with respect to minutes at that event is 12 + 72 × 0.5 = 48 nanoseconds per minute. Dividing by sixty returns 0.8 nanoseconds per second. Both value and derivative preserve the same physical history under a complete unit conversion.
If only the local-time input were changed to minutes, while the old per-second coefficients remained, the value would be 4 + 0.1 + 0.0025 = 4.1025. The arithmetic would execute normally but describe an incompatible model. A successful programme run is therefore a weaker check than invariance under a correctly specified reference and unit transformation.
For timing data stored modulo a cycle or week, the applicable epoch interpretation also matters. Do not invent a nearest-wrap rule from this polynomial example. Establish the proper time scale and record conventions from the relevant product specification. Reference arithmetic is useful only after the intended event label has been recovered correctly.
CHAPTER 12 OF 12
12. Report the evaluated model rather than an unsupported clock truth
A careful conclusion says which polynomial was evaluated, at which compatible event, with which reference epoch and units, and which additional contributions were included. It distinguishes that evaluated model from a direct independent observation of the physical satellite clock. The number of digits returned by the calculation does not establish the model’s accuracy.
For our fictional inputs, the value at 130 seconds is 19 nanoseconds whether represented about 100 or 110 seconds, provided coefficients and argument are transformed consistently. Its derivative is 0.8 nanoseconds per second. Those are exact properties of the declared quadratic within rounding, not claims about any actual navigation satellite.
The model’s validity interval remains a separate question. A polynomial can be evaluated far beyond the interval supporting it, but mathematical evaluability is not evidence of physical predictability. The quadratic term can dominate at large time differences even when the true clock follows another history. Recentring does not create new observations or extend that interval.
The transferable habit is to preserve the reference beneath the coefficients. This applies to calibration curves, economic baselines and local coordinate models as well as clocks. When a reference changes, transform the whole relationship, including its units and uncertainty, then verify that the same event still produces the same intended quantity.
Independent worked case and evidence checks
Use a new fictional polynomial with a₀ = −2 nanoseconds, a₁ = 0.3 nanoseconds per second and a₂ = −0.005 nanoseconds per squared second, referenced to t₀ = 200 seconds. At t = 220, Δ = 20 and the offset is −2 + 6 − 2 = +2 nanoseconds. Its derivative there is 0.3 − 0.01 × 20 = 0.1 nanoseconds per second.
Re-centre at t₁ = 210. The transformed coefficients are b₀ = −2 + 3 − 0.5 = 0.5, b₁ = 0.3 − 0.1 = 0.2 and b₂ = −0.005, with the corresponding units. The query is now ten seconds after the new origin. Its value is 0.5 + 2 − 0.5 = 2 nanoseconds and its derivative is 0.2 − 0.1 = 0.1.
Change only the reference epoch while retaining the old coefficients. The incorrect result is −2 + 3 − 0.5 = 0.5 nanoseconds. Explain why that happens to equal the correct value at the new reference event rather than the later query. This coincidence makes the error look plausible unless the event labels are tracked explicitly.
Next convert the local argument to minutes. The coefficients become −2,18 and −18 in nanosecond-based minute units. At one third of a minute, the value is −2 + 6 − 2 = 2. The derivative is 18 − 36/3 = 6 nanoseconds per minute, equivalent to 0.1 per second. Reconstruct both the seconds and minutes calculations without changing the event.
Finally add an independently documented fictional contribution of +0.7 nanoseconds. If the polynomial excludes it, the total is 2.7. If the supplied product already contains it, adding again would give 3.4. Identify which documentation or saved ledger would distinguish those cases; arithmetic alone cannot determine what a product already includes.
A reverse transformation provides another check on the recentering. Starting with b₀ = 0.5, b₁ = 0.2 and b₂ = −0.005 at epoch 210, shift the origin back by −10 seconds. The recovered constant is 0.5 − 2 − 0.5 = −2, and the recovered slope is 0.2 + 0.1 = 0.3. The curvature coefficient remains −0.005. Recovering the original coefficients checks the transformation independently of one successful query value.
Now shift the entire common time-label origin by adding 1,000 seconds to every epoch. The reference becomes 1,200, the query becomes 1,220 and their difference remains twenty. No coefficient recentering is required because the local argument did not change. This differs from moving the polynomial reference event while keeping the query event fixed. Both operations can be called changing the epoch informally, but their mathematics is different.
The learner’s final explanation should identify which physical events changed, which labels changed and which model representation changed. If none of the physical events changed, a complete reference transformation should preserve the evaluated offset at the same query. That invariant is a powerful interface test without pretending to verify the physical clock independently.
Worked comparison
| Quantity or operation | Meaning in this guide | Useful check |
|---|---|---|
| a₀ | Offset at the stated origin | Evaluate at Δ = 0 |
| a₁ | Slope at the stated origin | Differentiate the polynomial |
| a₂ | Coefficient of the squared argument | Second derivative is 2a₂ |
| Recentring | Another representation of the same polynomial | Compare value and derivative at a common event |
| Unit conversion | Another scale for the same argument | Transform every time-power coefficient |
| Additional correction | A distinct documented contribution | Check inclusion before adding |
Questions and practical checks
Is the reference epoch the signal reception time?
Not necessarily. It is the polynomial’s parameterisation origin. The event at which the clock model is evaluated follows the applicable observation and product definition.
Is a₂ always the second derivative?
Not for the polynomial written here. With a₂ multiplying Δ² directly, the second derivative is 2a₂. Read the equation rather than relying on an informal coefficient label.
Can I change t₀ without changing the coefficients?
Not while claiming to preserve the same history. The recentering equations show which coefficients must change.
Does recentering make the model valid for longer?
No. It changes the representation, not the evidence supporting the polynomial or its agreement with the actual clock.
Why preserve covariance after changing the reference?
The transformed coefficients can be correlated even if the original ones were independent. Ignoring that dependence can change the uncertainty of an otherwise unchanged evaluated clock value.
Continue with the connected subjects
Connect clock values with the observed signal
Separate clock labels from GNSS signal emission events
Place satellite and receiver clock terms in the wider GPS model
Preserve related reference and combination conventions
Keep RINEX epochs, records and signal identities interpretable
Check surviving clock and bias terms in ionosphere-free combinations
Sources and scope
ESA Navipedia: broadcast clock coefficients, reference epochs and contribution definitions
ESA Navipedia: the distinct eccentric-orbit contribution and inclusion conventions
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.
Continue with the state and clock model
Choose the companion that addresses the next unresolved interface. These are distinct model checks, not substitutes for an independent positioning comparison.
Orbit interpolation: evaluate missing states without creating new observations
Eccentric-orbit clocks: check radial motion, scalar invariance and included terms
Eclipse geometry: separate a shadow classification from model-validity decisions