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Two sensors can track the same place yet start from different zero dates, sample different seasons and straddle different events. Joining their time series requires a temporal model, not a convenient vertical shift.
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Chapter contents
Choose the route closest to the question, or read from Chapter 1 for the complete argument. The chapters remain open on the page; these grouped routes are an index.
Route 1 · Name each temporal frame
Route 2 · Work offsets, steps and seasons
Route 3 · Reconcile sensors and uncertainty
Route 4 · Test evidence and limits
Route 5 · Check and continue
CHAPTER 1 OF 12 · NAME EACH TEMPORAL FRAME
1. Every relative time series has a temporal frame
An InSAR displacement series is normally relative to a reference date and a spatial reference. Another sensor may use a different first epoch, processing window, coordinate frame or zero convention. Subtracting each series’ first value makes both start at zero, but it does not place them in one common temporal frame.
Write the observation from sensor s as
ys(t) = cs + bt + A cos(2πt/T) + B sin(2πt/T) + Σj hj I(t ≥ τj) + εs(t).
Here cs is a sensor-specific offset, b is a trend, the sine and cosine terms describe a seasonal component, hj is an event step at time τj, and εs contains measurement error and unmodelled behaviour. Not every dataset supports every term. The equation is a checklist that prevents a datum offset from silently absorbing a trend, a season or an event.
CHAPTER 2 OF 12 · NAME EACH TEMPORAL FRAME
2. Temporal alignment is not the same as spatial alignment
Coregistration makes locations correspond. Line-of-sight projection makes observation directions comparable. Temporal alignment asks a different question: what physical state does zero represent, and how are epochs before and after events connected?
Two series can be spatially collocated yet temporally incompatible. One may report displacement relative to 1 January, another relative to 1 July. One may average monthly values, another observe on irregular satellite passes. A GNSS daily solution can resolve a step that a sparse radar series only brackets.
A defensible merge records acquisition times, reference dates, averaging windows and known equipment or event epochs. Convert time standards and units before fitting. “Same year” is not a temporal tie when seasonal amplitude is large or an abrupt event occurs between observations.
CHAPTER 3 OF 12 · WORK OFFSETS, STEPS AND SEASONS
3. Work a clean overlap offset
Suppose sensor A observes months 0, 3, 6 and 9 with relative displacements 0, 3, 6 and 9 millimetres. Sensor B uses month 6 as its own zero and observes months 6, 9 and 12 as 0, 3 and 6 millimetres. The shared physical motion is a steady 1 millimetre per month.
At the common month 6, A reports 6 and B reports 0. Adding 6 millimetres to every B value places B in A’s temporal datum, giving 6, 9 and 12. The overlap at month 9 confirms the same correction.
| Month | Sensor A | Sensor B raw | Sensor B tied to A |
|---|---|---|---|
| 0 | 0 | — | — |
| 3 | 3 | — | — |
| 6 | 6 | 0 | 6 |
| 9 | 9 | 3 | 9 |
| 12 | — | 6 | 12 |
CHAPTER 4 OF 12 · WORK OFFSETS, STEPS AND SEASONS
4. Event steps can be mistaken for datum shifts or trends
Change the example. Sensor B still starts at month 6, but an event at month 8 adds a permanent 4-millimetre step. Its raw values at months 6, 9 and 12 are now 0, 7 and 10. Adding the clean pre-event offset of 6 gives 6, 13 and 16.
If the event is ignored, a line through months 0 and 12 suggests 16/12 ≈ 1.33 millimetres per month instead of the true background trend of 1. Using B alone from month 6 to 12 suggests 10/6 ≈ 1.67. The same four-millimetre step contaminates the fitted rate differently because the windows differ.
The remedy is not automatically a more flexible curve. Use an event indicator only when timing evidence supports it, and require enough observations on both sides to distinguish a step from noise or a short pulse. If the event falls in a data gap, report an interval for its epoch rather than invented precision.
CHAPTER 5 OF 12 · WORK OFFSETS, STEPS AND SEASONS
5. Changed context: seasonal sampling creates a false offset
Assume there is no sensor offset and no long-term trend, only a seasonal displacement s(t) = 3 cos(2πt/12) millimetres. Observations in January-like phases are near +3, while observations six months later are near −3. If sensor A samples mostly the first phase and sensor B mostly the second, their averages can differ by about 6 millimetres even though both use the same datum.
Aligning the two means with a constant shift would erase the seasonal sampling difference and create a false frame correction. A linear trend can also absorb part of a seasonal cycle when the record covers only a fraction of a year or repeatedly samples the same phase.
This transfer check asks whether the overlap spans enough phases. Plot observation day within the seasonal cycle for each sensor. Fit or bound seasonal terms before declaring a constant offset, and test how the inferred trend changes when those terms are included.
CHAPTER 6 OF 12 · RECONCILE SENSORS AND UNCERTAINTY
6. Cross-sensor ties need common physical support
Different sensors may observe different footprints, wavelengths, incidence angles and scattering populations. A pixel centre match does not guarantee that both measurements represent the same moving object. One series may be dominated by a building, another by surrounding soil or vegetation.
Choose stable tie locations using coherence, land-cover and structural evidence. Aggregate only after establishing comparable support. For InSAR, confirm sign convention and line-of-sight direction. For optical or thermal products, confirm that the quantity itself is commensurate before discussing temporal offsets.
The safest tie separates three transformations: spatial support, observation geometry and temporal datum. Document each one. A successful numerical fit cannot reveal which transformation compensated for an error in another.
CHAPTER 7 OF 12 · RECONCILE SENSORS AND UNCERTAINTY
7. Multiple overlaps constrain more than one offset
With several sensors, treat each series as a node and each credible overlap as an edge. A connected overlap graph can propagate a common datum. Closed loops provide a diagnostic: tying A to B, B to C and C back to A should return near zero within uncertainty. A large closure residual signals an incompatible offset, trend, step or support area.
Avoid building the entire frame on one fragile overlap. Estimate offsets jointly when possible, with one explicit reference constraint such as cA = 0. If a sensor has no overlap and no independent control, its offset is not identifiable from the data. A visually smooth concatenation is not evidence of a valid tie.
| Overlap pattern | What can be estimated? | Main risk |
|---|---|---|
| one simultaneous epoch | constant offset only | single-epoch noise |
| several epochs, short span | offset and local consistency | trend–offset trade-off |
| overlap across seasons | offset plus seasonal contrast | changing physical support |
| overlap across an event | offset, trend and step if both sides sampled | step absorbed into frame |
CHAPTER 8 OF 12 · RECONCILE SENSORS AND UNCERTAINTY
8. Offset uncertainty becomes shared covariance
If one estimated offset is added to every value in a series, its uncertainty is shared by all corrected epochs. Treating that uncertainty as independent at each date overstates the information in the joined record. The covariance matrix gains a common component from the temporal tie.
Event time and seasonal parameters can also be correlated with the offset and trend. Report parameter covariance or use resampling that repeats the entire tie procedure. Propagating only per-epoch sensor noise produces intervals that are too narrow.
Validation should be withheld from fitting where possible. Compare the joined result with independent GNSS, levelling, corner reflectors, field logs or a second overlap region. Agreement at the tie points is expected; predictive agreement elsewhere is more informative.
CHAPTER 9 OF 12 · TEST EVIDENCE AND LIMITS
9. What a joined time series does not prove
A continuous curve across sensors does not prove continuous ground motion. Interpolation bridges missing epochs by assumption. Seasonal atmospheric delay, changing scatterers, unwrapping error and reference-area motion can all produce apparent temporal structure.
A detected step does not identify its cause. Earthquakes, groundwater operations, construction, instrument changes and processing discontinuities may create similar shapes. Causal interpretation needs external timing and spatial-pattern evidence.
Long records are not automatically safe. If acquisitions cluster in one season or one local time, systematic delay can bias the trend. The effective temporal coverage is the distribution of observation phases and gaps, not merely the difference between first and last dates.
CHAPTER 10 OF 12 · TEST EVIDENCE AND LIMITS
10. A practical temporal-reconciliation checklist
Preserve acquisition timestamps, time standard, reference date, sensor and track, sign convention, line-of-sight vector, spatial reference and processing version. Mark known earthquakes, maintenance, antenna changes and algorithm transitions. Plot gaps and seasonal phase before fitting.
Begin with the smallest identifiable model. Add a constant offset only with overlap or independent control. Add a trend only with sufficient span. Add seasonal terms only when sampling can resolve them. Add event steps with evidence on timing and observations on both sides. Compare residuals by sensor, season and location.
For an evidence check, perform leave-one-overlap-out validation and loop-closure tests. Refit after shifting plausible event dates within a gap. Report how the trend and offsets change. Sensitivity is part of the result, not an inconvenience to hide.
CHAPTER 11 OF 12 · CHECK AND CONTINUE
11. Questions readers should ask
Can I join two series by subtracting their first observations?
Only if those first observations represent the same physical epoch and support, or a model justifies the difference. Separate zeroing does not create a common datum.
Why can a step bias a velocity?
A line spreads an abrupt displacement across the fitted interval. Different sensor windows therefore convert the same step into different apparent slopes.
Does one year of data resolve an annual cycle?
Usually not robustly when sampling is irregular, gaps are long or trend and seasonal phase are correlated. More cycles and better phase coverage help.
What if two sensors never overlap?
Their relative offset needs an independent tie or a physical model. Without one, the temporal datum is not identifiable from the two series alone.
CHAPTER 12 OF 12 · CHECK AND CONTINUE
12. Continue through the verified remote-sensing ecosystem
Begin with the Remote Sensing root, the Gold Standard of Education and the How X Works hub. Establish the spatial and geodetic side with GNSS Ties, Reference-Frame Alignment and Cross-Track Datum Reconciliation, InSAR Reference Areas, Common-Mode Motion and Relative Displacement Bias and Coregistration, False Change and Boundary Effects.
For sampling and systematic delay, continue to Revisits, Cloud Gaps and Missed Events and Atmospheric Phase Screens, Spatial Correlation and False Deformation Ramps. Authoritative technical reading: the USGS study Characterizing and Estimating Noise in InSAR and InSAR Time Series with MODIS documents seasonal sampling bias and temporal gaps; the EarthScope/UNAVCO InSAR Product Format records required time-series reference dates; and Spatio-Temporal Linking of Multiple SAR Satellite Data addresses tie points and temporal reference alignment across datasets.
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