Steady, accelerating, stepping: how to read the three shapes of an InSAR time series

22 October 2026 · about 9 min read

By the senior geotechnical engineer · insarwatch

Two points on a deformation map can carry the same velocity and mean opposite things. One has been settling at −11 mm/yr for three straight years, smooth as a drawing. The other averaged −11 mm/yr too — but only because it did nothing for eighteen months and then dropped 80 mm in a single season. The map, which stores one number per point, cannot tell these stories apart. Only the cumulative displacement curve can, and reading that curve is a small skill with an outsized payoff: three shapes cover almost every time series an engineer will meet, and each shape points to a different physical process and a different next action. This article describes the three shapes, the physics behind each, the traps that fake them, and how to fold the reading into a routine.

Shape one: the steady line — and what "steady" still hides

The steady shape is a straight line on the cumulative plot: roughly constant rate, no curvature, no jumps. It is the signature of processes that behave like background rather than events — regional groundwater-drawn subsidence, long-term consolidation of soft clays that has already organized itself, slow thermal-and-load cycles that average out. Most points in most scenes are steady, and that is exactly why the shape deserves more than a glance: it defines the baseline against which the other two shapes are anomalies.

Two things hide inside steady lines. The first is seasonality: swelling and shrinking clays, groundwater pumping cycles and soil-moisture loading all imprint annual waves on the curve. A line fitted through four quarters of data absorbs the wave into the slope and inflates or deflates the rate depending on where the window starts and ends — a fit from January to December and the same data fitted March to February can differ by several mm/yr. The practical rule is to fit over whole years, or to state the window whenever the span is shorter. The second is slow curvature: a genuinely decelerating process viewed over a short window looks flat. Neither trap changes the classification; both change the number, and the number is what gets compared to thresholds.

Shape two: the accelerating curve

Acceleration is curvature: the second half of the record descends faster than the first. The physics behind it is the physics of positive feedback — as a consolidation bowl deepens, more of the aquitard compacts; as a mining panel advances toward a point, the subsidence trough's edge reaches it; as a slope creeps toward failure, displacement rate grows toward collapse. Acceleration is therefore the shape that most often justifies escalation, because it marks a process that is not merely persisting but gathering.

Reading it honestly requires splitting the record. Fit a rate to the first half and the second half separately, and compare: the screening rule this workbench's time-series tool applies by default flags a point when the second-half rate exceeds the first-half rate by 3 mm/yr or more. The threshold is deliberately modest — it is a tripwire for looking, not an alarm in itself — and it is adjustable, because a 3 mm/yr intra-record increase means something different at a baseline of −5 mm/yr than at −60. What acceleration never licenses is extrapolation: a quadratic fitted to two years of accelerating settlement and projected five years out is a fiction with error bars nobody can state. The honest deliverable reports both segment rates, the date of the split, and the physical hypothesis the acceleration supports.

Shape three: the step jump

A step is a single-interval displacement far larger than the record's typical month-to-month change. The default screening threshold in the time-series tool is 15 mm in one interval — an order of magnitude above the residual noise of a healthy series — and when a step passes that bar, the first question is never "how bad is it" but "is it real". Steps have two faces. The engineering face is abrupt ground response: mine collapse, sinkhole initiation, a retaining structure's sudden movement, earthquake displacement. The data face is a processing artifact, above all an unwrapping error, which enters a single epoch and then persists as an offset in every value after it.

Telling the faces apart is a spatial exercise, not a temporal one. A real step is coherent: the points around it step too, in a pattern consistent with a mechanism — a scarp, a bowl edge, a slide's boundary. An unwrap error is spatially incoherent: isolated points or speckled patches step while their neighbours do not, often along decorrelation boundaries such as vegetation edges or water margins. This is where the workbench's structure matters — the time-series tool marks the step and the earlier detection stage of the hazard classifier separates isolated points from gradient-consistent ones, so the two tools checked together answer "real or artifact" faster than either alone.

Shapes meet ground truth

A curve classification is a hypothesis until something on the ground confirms it, and GNSS is the something most projects already have. The comparison is most valuable exactly where the shapes are interesting: a GNSS monument inside an accelerating bowl should show the same curvature — and if it shows a steady line instead, the InSAR acceleration is more likely atmospheric or orbital than geological. The GNSS validation tool pairs the two datasets on distance and compares rates, but the same logic applies by eye at the curve level: overlay the GNSS displacement series on the InSAR cumulative plot and check whether shape, not just slope, agrees. Shape agreement is the stronger test, because it is harder for a correlated artifact to imitate.

One caution from the geometry discussion in the previous article: GNSS measures its own component (typically horizontal plus vertical at the monument), while InSAR measures along the LOS. Comparing them before the geometry correction has been applied injects a systematic difference into every pair, and the validation will "fail" for reasons the data cannot fix.

Making the reading routine

The workflow that turns this from knowledge into practice is short. Import the per-epoch displacement series, not just the fitted velocity — the fitted velocity is the summary, the series is the evidence. Split-fit every point of interest into two halves and let the 3 mm/yr tripwire nominate the accelerators. Scan the differenced series for single-interval moves beyond 15 mm and adjudicate each candidate step spatially. Check seasonality by re-fitting across whole-year windows. Then put the outcome in writing: the report generator assembles per-point classifications — steady, accelerating, stepping, or flagged — into an interpretation note that states the thresholds used, so the next reader knows exactly which rules produced the labels. A classification that arrives with its rulebook attached survives review; one that arrives as a colour-coded opinion does not.

Curves, in the end, are where InSAR earns its keep. The velocity map tells you where to look; the time series tells you what you are looking at — a process at rest, a process gathering, or a process that has already moved. Reading the three shapes fluently is the cheapest early-warning capability a monitoring programme can acquire.

The cost of the discipline is low and the payoff is asymmetric: the same series that reads as noise in a velocity map becomes a documented acceleration when split-fit, and a documented acceleration is what justifies deploying an inclinometer or a GNSS monument — the expensive instruments. Reading the series first is how the monitoring budget gets spent where the evidence points, and the habit pays back on every campaign.