A straight line needs a time model and an uncertainty model
A station plot slopes upward, and a fitted line reports a rate with a small uncertainty. Before calling that rate steady ground motion, ask how the seasons were sampled and how errors persist through time. A dense series can contain less independent information about a long-term velocity than its point count suggests.
This guide develops the estimation problem behind a geodetic velocity. It distinguishes endpoint rates, fitted linear trends and time-varying motion; explains annual terms and temporal covariance; and shows why a robust estimator is useful without being a universal solution. It is not another rule for joining two sensor records or correcting a hardware step.
All coordinates, rates, noise levels and observation schedules below are invented. No actual station velocity, required record length or professional acceptance threshold is supplied. The aim is to make the reasoning visible enough to check and transfer.
Geodesy subject guide · Learning that transfers into life · How X Works hub
Choose your reading route
Route 1: Name the velocity claim
Route 2: Separate seasons and sampling
Route 3: Work uncertainty through time
Route 4: Compare robust methods and limits
Route 5: Transfer and communicate
Practice with a changed schedule and changed error structure
Full chapter index · Independent worked case · Questions and practical checks
Chapter index
Name the velocity claim
Decide which rate the question requires
Write the deterministic model before choosing the noise
Separate seasons and sampling
Work a seasonal endpoint trap
Inspect seasonal coverage rather than counting observations
See why fitted parameters trade off
Work uncertainty through time
Use slope weights to propagate uncertainty
Do not reduce temporal correlation to one slogan
Distinguish a residual plot from a noise model
Compare robust methods and limits
Understand what interannual robust slopes contribute
Test whether constant velocity remains the useful quantity
CHAPTER 1 OF 12
1. Decide which rate the question requires
Velocity is change of position with time in a specified reference frame. In a constant-velocity model it is the slope of a straight line. Real station histories may also contain periodic motion, discontinuities, transient deformation and changes in long-term behaviour. A fitted line is a useful summary of a chosen interval, not a guarantee that the point followed that line at every instant.
An endpoint difference uses only two positions: v = (x₂ − x₁)/(t₂ − t₁). A least-squares trend uses a collection of positions and a stated fitting model. An instantaneous velocity is a derivative of a time-varying position model. These can differ without any arithmetic error. Before comparing published numbers, identify which quantity each source actually estimated.
For a fictional point with x(t) = 5 + 2t millimetres, with t in years, all three ideas agree at 2 millimetres per year. Add a temporary displacement that later reverses and the endpoint rate can remain 2 while short-window slopes vary. Add acceleration and even a perfectly measured long-interval line becomes an average description rather than the present derivative.
The reference matters as well. Motion relative to a global terrestrial frame is not the same quantity as motion after removing a plate model. The existing coordinate-epoch guide develops that comparison. Here we hold the coordinate reference fixed so that the new difficulty is time-series estimation. Keeping one layer stable lets the learner diagnose whether disagreement comes from the target rate, sampling, model or uncertainty.
CHAPTER 2 OF 12
2. Write the deterministic model before choosing the noise
A useful introductory model is x(t) = a + vt + A cos(2πt) + B sin(2πt) + ε(t), with time measured in years from a stated origin. The intercept a locates the line at that origin, v is its linear rate, and the sine and cosine terms describe an annual component of arbitrary phase. The residual ε contains what this model does not explain.
This equation is a candidate model, not a requirement to fit four parameters to any available series. If observations cover too little time or repeat only a narrow seasonal phase, several parameter combinations can explain them almost equally well. A solver returning numbers does not establish that those numbers are independently identified. Inspect the sampling and parameter covariance.
Known discontinuities may require additional terms, while transient physical motion may require a different functional form. Equipment records can support a candidate step epoch but should not dictate that every event is instrumental. The carrier-continuity and antenna-reference guides help investigate observation changes before they become parameters in a trend fit. Preserve the original series and the evidence for each modelling decision.
NASA JPL’s public time-series description distinguishes positions, velocities, breaks and seasonal parameters. Blewitt and colleagues’ MIDAS paper explains why seasonality, offsets, outliers and heteroscedasticity complicate automatic velocity estimation. Those sources motivate careful modelling. None makes a single model universally correct for a particular station or supplies permission to remove inconvenient observations until a desired slope appears.
CHAPTER 3 OF 12
3. Work a seasonal endpoint trap
Use the fictional position x(t) = 2t + 4 cos(2πt) millimetres. The long-term linear term is +2 millimetres per year and the annual amplitude is 4 millimetres. At t = 0, x = 4. At t = 0.5, x = 1 − 4 = −3. The endpoint difference is −7 millimetres over half a year, giving −14 millimetres per year.
That negative endpoint rate does not mean the stipulated linear component is negative. The seasonal contribution changed by −8 millimetres while the line increased by 1. Both terms are part of the position history. Calling the endpoint rate an erroneous calculation would also be wrong: it correctly describes those two endpoints, but it is a poor substitute for the specified secular component.
Now observe at t = 1. The position is 2 + 4 = 6 millimetres. The change from t = 0 is 2 millimetres over one year. The annual component cancels because these endpoints have exactly matching phases in this constructed model. That cancellation depends on the assumed stable annual cycle and the actual sampling phases, not merely on writing two calendar years in the report.
Change the annual amplitude in the second year from 4 to 5 millimetres at the matching phase. The later position becomes 7 and the one-year endpoint rate becomes 3. The same-phase shortcut no longer isolates the original linear term. This is why a useful worked example must include its failure condition. Periodic structure can help design a comparison while still requiring evidence that the periodic model transfers.
CHAPTER 4 OF 12
4. Inspect seasonal coverage rather than counting observations
Imagine 500 observations concentrated into the first month of each of two years. They can describe those brief periods closely, but they do not directly observe the rest of either annual cycle. Another series with fewer points spread through the year may distinguish phase and amplitude more effectively. Observation count, elapsed span and phase coverage answer different questions about information.
Plot observation times within the year as well as against absolute time. Mark long gaps, changing cadence and intervals with different precision. If an early period samples mainly one season and a later period another, a trend can absorb part of the difference. A histogram of observation dates can expose a limitation that a smooth fitted curve hides.
Repeated annual sampling at exactly one phase can cancel a perfectly stable annual term in interannual differences. It cannot estimate the full annual cycle from those observations alone. If the scientific question concerns seasonal amplitude or its evolution, that schedule is inadequate even when its linear rate is useful under additional assumptions. Evaluate the schedule against the target parameter, not against an abstract ideal of more data.
The remote-sensing temporal-frame guide addresses joining records with different seasonal sampling and reference epochs. This article goes further into the covariance of a single station’s velocity estimator. Use the two explanations together when a velocity is obtained from reconciled sensors: uncertainty in the reconciliation must enter the final time-series model rather than vanish after the curves are placed on one plot.
CHAPTER 5 OF 12
5. See why fitted parameters trade off
Over a short interval, part of a sine wave can resemble a straight line. A fitting procedure may compensate for a larger seasonal amplitude by changing the velocity. The parameters are then statistically coupled. Their separate printed uncertainties should be interpreted with their covariance and with the observation schedule that created the trade-off.
For an extreme example, two observations cannot uniquely estimate an intercept, a velocity and two annual coefficients without additional information. Many models can pass through both points. Adding more observations at essentially the same two epochs improves precision about those endpoint positions but does not create the missing temporal shapes. Replication and identifiability are different sources of information.
A longer interval containing several well-observed cycles usually helps distinguish a linear trend from annual terms, but no universal duration follows from that statement. Gaps, transient motion, changing seasonal behaviour and the required precision matter. An arbitrary rule such as “one year always resolves a cycle” confuses a calendar duration with evidence for the chosen parameterisation.
Inspect sensitivity by fitting defensible alternative windows and supported model terms, then explain why estimates differ. Do not select only the model giving the smallest uncertainty. A very restrictive model can be precise about the wrong quantity. A very flexible model can absorb the signal of interest. The useful choice balances physical meaning, identifiable parameters and residual evidence, with assumptions stated rather than hidden in a software default.
CHAPTER 6 OF 12
6. Use slope weights to propagate uncertainty
For ordinary least squares with an intercept, the slope can be written v̂ = Σwᵢxᵢ, where wᵢ = (tᵢ − t̄)/Σ(tⱼ − t̄)². The weights sum to zero, so a constant coordinate offset does not change the slope. Their time-weighted sum is one, so an added linear drift passes directly into it. This small formula reveals which error shapes a rate estimate is sensitive to.
Take three invented observations at t = 0, 1 and 2 years. The slope weights are −0.5, 0 and +0.5 per year. Thus v̂ = (x₂ − x₀)/2. The middle point helps determine the intercept and residual pattern but has zero direct weight in this particular equally spaced three-point slope. That fact is a property of the design, not a reason to discard middle observations in general.
If the observation-error covariance matrix is C, the variance of this linear slope estimator is wᵀCw. For independent errors with standard uncertainty 2 millimetres at each epoch, C has diagonal entries 4 square millimetres and zero off-diagonal entries. The slope variance is 0.25×4 + 0.25×4 = 2 square millimetres per square year, giving standard uncertainty √2 millimetres per year.
This is conditional uncertainty under the stated error model and deterministic fit. It does not include an omitted seasonal component, a wrong frame or a physical transient unless those uncertainties are represented. The formula is valuable because it exposes missing assumptions. A small result from an incomplete C is not evidence that the omitted errors are absent.
CHAPTER 7 OF 12
7. Do not reduce temporal correlation to one slogan
Time-correlated errors have relationships across epochs. Their covariance affects a slope through both the size and sign of its weights. Positive correlation often means fewer effectively independent pieces of information, but it does not follow that every positively correlated model increases every estimated quantity’s variance compared with independent errors of the same marginal size. The target and covariance shape matter.
Keep the three-epoch design and marginal standard uncertainty of 2 millimetres. Stipulate Cᵢⱼ = 4×0.8 raised to the separation in observation indices. The endpoint covariance is 4×0.8² = 2.56 square millimetres. The slope variance becomes (4 + 4 − 2×2.56)/4 = 0.72 square millimetres per square year. Its standard uncertainty is about 0.849 millimetres per year, smaller than √2 in this particular comparison.
Why? The slope subtracts endpoints, so their positive shared variation partly cancels. The mean would respond differently. This constructed covariance is not a measured GNSS noise spectrum and is not offered as a realistic universal alternative to independent errors. It is a counterexample to the claim that the word correlated alone determines the direction of every uncertainty change.
Now add an uncertain linear drift q t with standard uncertainty 3 millimetres per year, independent of the original independent errors. Since Σwᵢtᵢ = 1, the drift contributes variance 9 to the slope. The total becomes 2 + 9 = 11, with standard uncertainty about 3.317 millimetres per year. A noise component aligned with the target trend is especially consequential. Model its structure, not just its name.
CHAPTER 8 OF 12
8. Distinguish a residual plot from a noise model
White noise is an idealisation with no temporal covariance between distinct observations under its model. Coloured noise has frequency-dependent structure; flicker-like and random-walk-like contributions are examples discussed in geodetic time-series analysis. A residual plot containing long runs or slow variation can prompt investigation, but visual resemblance does not uniquely identify a stochastic process or its parameters.
First remove only the deterministic terms justified by the question and evidence. An unmodelled annual signal can appear as correlation in residuals. An overly flexible deterministic fit can remove low-frequency variation and make the remaining noise look less persistent. Noise estimation and signal modelling therefore interact; they should not be treated as two unrelated boxes completed once in arbitrary order.
Blewitt and colleagues explicitly discuss autocorrelation and effective independent information when developing MIDAS velocity uncertainty. Their method includes dependence from reusing coordinate observations to form slopes and an empirically tested uncertainty treatment. That does not license applying a universal multiplier from the paper to any unrelated estimator or sampling design. Use the method’s actual definition and validation domain.
For a practical review, compare uncertainty results under defensible temporal models, retain the model specifications and explain important sensitivity. When resampling is used, it must respect relevant dependence; shuffling individual daily values as if all dates were interchangeable can destroy the very structure under examination. No resampling scheme automatically fixes a misspecified physical model or supplies observations in a missing season.
CHAPTER 9 OF 12
9. Understand what interannual robust slopes contribute
A median-of-slopes estimator can reduce sensitivity to a minority of extreme slope values. Selecting observation pairs roughly one year apart can also reduce a stable annual contribution. MIDAS, developed by Blewitt and colleagues, combines particular pairing and robust-adjustment choices with an uncertainty procedure. Its published method is more than taking an arbitrary median of whatever differences happen to be convenient.
Consider five fictional one-year slope samples: 2, 3, 2, 1 and 22 millimetres per year. Their mean is 6 while their median is 2. One large value strongly changes the mean but not the median. If that value arose from an unhandled step crossing one pair, the contrast helps explain why robust approaches can be useful. It does not identify the cause of the large slope or prove that 2 is the true rate.
The slopes may share observations. Differences x₁ − x₀ and x₂ − x₁ both use x₁, so they are not independent even if the underlying positions were independent. Counting each slope as a wholly new measurement would overstate information. The same principle appears when one calibration or template is reused across many corrected values.
Use a robust estimator as a defined scientific method and as a comparison with other justified estimates, not as a way to avoid looking at station history. The MIDAS paper discusses limits for sparse records, non-annual periodic signals and nonconstant velocity. Robustness is resistance to specified disturbances under specified conditions. It is not immunity to arbitrary missing data, changes of target quantity or unsupported causal interpretation.
CHAPTER 10 OF 12
10. Test whether constant velocity remains the useful quantity
Suppose a fictional station rises steadily for three years and then follows a decaying transient after an independently documented event. A single line over the entire period compresses those histories into one rate. That rate can be a legitimate summary if labelled as such, but it is not necessarily the pre-event velocity, the post-event instantaneous velocity or a forecast of future motion.
Changing the fitting window will change the balance of pre-event and transient observations. The difference between rates can therefore reflect a change in estimand rather than an unstable algorithm. Define the scientific purpose before deciding which interval to use. If the task concerns a current deformation process, a historical average may not answer it even when its formal uncertainty is small.
Do not infer a physical cause from a preferred functional fit alone. Equipment, monument stability, environmental loading, reference effects and genuine deformation can produce temporal structure. Independent observations and event histories help distinguish them. The existing relative-sea-level guide explains why a land-motion rate also needs spatial and temporal relevance to a tide gauge before it is combined with coastal measurements.
A useful evidence check asks what would change the interpretation: another season of coverage, a reliable equipment history, an independent nearby station, or a different physical observable. Specify that need concretely. “Collect more data” is less useful when the main uncertainty is an unidentified step, an incompatible reference or an unsupported assumption that motion remained linear throughout the period.
CHAPTER 11 OF 12
11. Practice with a changed schedule and changed error structure
Return to x(t) = 2t + 4 cos(2πt). A learner now receives only times t = 0.25 and 1.25. The cosine terms are both zero, so the endpoint rate is 2 millimetres per year. Explain that result without claiming the annual amplitude was estimated from those two values. The cancellation works for this model and schedule; it leaves the unobserved seasonal shape untested.
Next keep the equally spaced three-epoch slope but add a common uncertain coordinate offset with standard uncertainty 10 millimetres. Its contribution to the slope is zero because the slope weights sum to zero. Its contribution to absolute position is not zero. This is a direct reminder that uncertainty must remain attached to the target quantity rather than being copied unchanged from one calculation to another.
Now add the uncertain linear drift from chapter seven instead. It does not cancel, because its time shape matches the slope. An explanation that says “all common errors cancel in differences” fails this transfer. Constant common offsets and common drifts are different models, even though both are shared across observations.
For a wider analogy, a student’s progress score can mix a long-term learning change with seasonal test difficulty. A neat trend needs comparable assessments and a suitable error account. The analogy illustrates sampling and reference discipline, not an equivalence between educational assessment and GNSS physics. Transfer means recognising the structure and then checking the new domain’s assumptions, not importing a formula without rebuilding its meaning.
CHAPTER 12 OF 12
12. Report the model, interval and remaining uncertainty together
A defensible station-rate statement identifies the monument, coordinate component, frame, time origin, observation interval, sampling and relevant processing version. It states whether the estimate is an endpoint rate, a linear fitted component or a parameter of a transient model. Units such as millimetres per year belong beside both the rate and its uncertainty.
Describe seasonal terms, supported discontinuities, exclusions and the stochastic model. Explain whether the printed uncertainty is a conditional formal standard uncertainty, an empirically validated error estimate, a confidence interval or another quantity. These labels cannot be swapped merely because all use a plus-or-minus sign. Include important covariance or sensitivity information needed for the downstream calculation.
For the fictional three-point independent-error case, the slope standard uncertainty is √2 millimetres per year under the stipulated model. Under the alternate covariance it is about 0.849. With an added uncertain drift it becomes about 3.317. Reporting these values side by side makes the assumption dependence visible. None is a general accuracy claim for a GNSS station.
The gold standard is a velocity that another person can interpret, reproduce and use within its limits. A fitted rate is most useful when the reader knows what could change it. Preserve the actual time history so that a future analyst can ask a different question without having to treat one compressed slope as the entire story of the station.
Independent worked case and evidence checks
An independent fictional review receives observations at t = 0, 1 and 2 years with positions 1, 4 and 5 millimetres. The ordinary least-squares slope is (5 − 1)/2 = 2 millimetres per year. The fitted intercept is 4/3 millimetres, giving fitted positions 4/3, 10/3 and 16/3. Residuals are −1/3, +2/3 and −1/3 millimetres. Check that the residuals sum to zero and their time-weighted sum is zero.
These small residuals do not determine the full uncertainty model. If the three positions share an unknown linear drift, that drift can be absorbed by the fitted slope without leaving a large residual. The data alone cannot distinguish that component from the target rate. Independent information about the processing or reference is needed to bound it.
Changed context: the middle observation is repeated a hundred times at exactly t = 1 with the same value. Symmetry keeps its direct slope weight at zero in this ordinary least-squares construction. The count rises dramatically without extending the time lever arm. This example concerns a particular design; real unequal precisions and different times require their own calculation. Explain the design before using point count as an argument for a precise long-term velocity.
Worked comparison
| Quantity or model | Fictional result | Interpretation |
|---|---|---|
| Half-year endpoint rate with annual term | −14 mm/year | Includes seasonal change |
| Same-phase one-year endpoint rate | +2 mm/year | Stable annual term cancels in this construction |
| Three-epoch independent-error slope uncertainty | √2 mm/year | Conditional on diagonal covariance |
| Same marginal errors with stated correlation | √0.72 mm/year | Endpoint covariance partly cancels |
| Independent errors plus uncertain linear drift | √11 mm/year | Trend-shaped error passes into the slope |
Questions and practical checks
Does a long record guarantee a reliable velocity?
No. Longer span helps separate some temporal shapes, but gaps, changing motion, reference problems and unsupported corrections can remain. Inspect the actual interval and sampling rather than only the first and last dates.
Does positive correlation always increase slope uncertainty?
No. It depends on the covariance structure and estimator weights. The three-point example shows cancellation of positively correlated endpoints; an uncertain drift instead increases slope uncertainty. Use the actual model.
Can annual differencing remove every periodic effect?
No. Stable exactly annual contributions can cancel at matching phases. Changing amplitudes, mismatched phases and other periods need separate treatment. An algebraic cancellation is conditional, not universal.
Is a median of slopes the complete MIDAS method?
No. MIDAS specifies pairing, robust adjustment and an uncertainty procedure, with published tests and limitations. The five-slope example explains a principle, not a substitute implementation.
Can I use a GNSS velocity directly in a sea-level calculation?
Only with compatible signs, frames, intervals, uncertainties and a justified physical tie to the gauge’s land reference. A nearby monument may not share the same local movement. Follow the dedicated coastal guide.
Continue with the connected subjects
Establish compatible references
Coordinate epochs, frame epochs and velocity transformations
Tracking continuity before interpreting a displacement step
Antenna setup changes and the physical reference point
Connect temporal models with their applications
Temporal frame ties and seasonal sampling across sensors
Relative sea-level trends and GNSS land-motion uncertainty
Sources and scope
Blewitt and colleagues (2016): MIDAS, robust GPS velocities and uncertainty limits
Nevada Geodetic Laboratory: time-series, velocity and station-history product documentation
NASA JPL: GNSS time-series and associated velocity, break and seasonal products
NASA JPL: processing description and separate fitted parameter classes
IERS: station coordinates, velocities and terrestrial reference realisation
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 observation checks
These companions examine different transformations of the evidence. Follow the unresolved term: repeated propagation, long-term estimation, frequency dependence or shared references. Passing one check does not settle the others.
Multipath: repeated geometry and held-out correction checks. Test repeated signal-path structure without treating repeatability as independent accuracy evidence.
Ionosphere-free combinations: cancellation and surviving noise. Check what a frequency combination removes, preserves and amplifies.
Double differences: reference satellites and shared covariance. Trace shared reference observations through clock cancellation and covariance.