Different station velocities do not automatically mean the ground is stretching
Four stations move by different amounts. Is the region translating, rotating, stretching or doing several of these at once? Looking at arrows on a map is a useful start, but it cannot settle the question. A rigid rotation can produce different arrows without changing the distances between material points.
This guide builds a local planar velocity model from a small fictional network. You will recover its gradient, separate symmetric strain rate from rigid rotation, test directional length changes and investigate what station geometry leaves unmeasured. The focus is inference from spatially separated observations, not simply recognising a matrix formula.
All coordinates, velocities and uncertainties are invented teaching inputs. The calculations are not a deformation assessment of a real site, a structural safety check or an earthquake forecast. Real geodetic strain estimation also depends on reference frames, spatial scale, interpolation choices and the uncertainty of the station velocities.
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Choose your reading route
Route 1 · Chapters 1–2: Define a local motion model
Route 2 · Chapters 3–5: Recover and interpret the gradient
Recover a fictional gradient from three noncollinear stations
Route 3 · Chapters 6–8: Challenge the network geometry
Route 4 · Chapters 9–10: Audit uncertainty and references
Route 5 · Chapters 11–12: Respect limits and transfer
Do not use infinitesimal formulas as finite-rotation identities
Full chapter index · Independent worked case · Questions and practical checks
Chapter index
Define a local motion model
Name position, velocity and the quantity being inferred
Use baseline differences to remove a common translation
Recover and interpret the gradient
Recover a fictional gradient from three noncollinear stations
Separate symmetric strain rate from antisymmetric rotation
Derive directional extension rather than guessing from arrows
Challenge the network geometry
Read principal directions, area rate and shear conventions
Test rank before trusting a fitted strain map
Use an extra station to test the affine assumption
Audit uncertainty and references
Carry the shared reference station into covariance
Separate reference translation, rotation and scale effects
CHAPTER 1 OF 12 · Define a local motion model
1. Name position, velocity and the quantity being inferred
Let x and y be local east and north coordinates in metres on a small planar teaching region. Let v_E and v_N be velocity components in metres per year. Position tells us where a station is; velocity tells us how its position changes with time in a stated reference. A strain rate describes a spatial variation of that velocity field, not the velocity of one station.
We approximate the local field by v(r) = t + Hr, where r = (x,y)ᵀ, t is a translation velocity and H is a two-by-two velocity-gradient matrix. Its entries are H_11 = ∂v_E/∂x, H_12 = ∂v_E/∂y, H_21 = ∂v_N/∂x and H_22 = ∂v_N/∂y. With consistent length units, each entry has units of inverse time.
The approximation says the velocity field is affine over the region and interval considered. It does not claim that the Earth is globally flat or that a broad plate can be described by one tangent-plane matrix. Choose a spatial scale and coordinate convention that make the approximation meaningful, and test whether observations support it.
A station moving east at five millimetres per year tells us nothing by itself about local stretching. Every nearby station could share that translation. Strain inference needs differences across known spatial separations, with uncertainty and reference effects carried through. Begin with the physical target: a local rate of relative length change under an explicitly stated model.
CHAPTER 2 OF 12 · Define a local motion model
2. Use baseline differences to remove a common translation
For two stations separated by Δr, the affine model gives Δv = HΔr. The translation t cancels. This is why relative station motion can reveal local gradients even when all absolute velocity arrows share a large common component. The cancellation is an algebraic property of the model, not a claim that every reference problem has disappeared.
If B lies L metres due east of A, then Δr_AB = (L,0) and Δv_AB = L(H_11,H_21). This baseline measures the first column of H. A northward baseline instead measures its second column. A single eastward pair cannot determine how velocity varies northwards, regardless of how accurately its endpoints are measured.
Keep millimetres and metres compatible. A one-millimetre-per-year velocity difference across one kilometre means 0.001/1000 = 10⁻⁶ per year. We may call this one microstrain per year when it is a longitudinal strain-rate component, but not every velocity-gradient entry alone is a longitudinal strain rate. Some entries also participate in local rotation.
The helpful first calculation is therefore not a colourful regional contour map. It is a pairwise check: which component difference, across what baseline vector, in what common reference, with what covariance? This identifies the actual sensitivity before an interpolated field makes the result look spatially complete.
CHAPTER 3 OF 12 · Recover and interpret the gradient
3. Recover a fictional gradient from three noncollinear stations
Choose A = (0,0), B = (1000,0) and C = (0,1000), in metres. Give their velocities in millimetres per year as v_A = (5,−2), v_B = (7,−3) and v_C = (8,2). At the chosen origin t = v_A. The eastward and northward separations provide independent spatial directions.
The A-to-B velocity difference is (2,−1) millimetres per year. Dividing its metre-per-year form by 1000 metres gives the first gradient column (2,−1)×10⁻⁶ per year. The A-to-C difference is (3,4) millimetres per year, giving the second column (3,4)×10⁻⁶ per year.
Thus H = [[2,3],[−1,4]]×10⁻⁶ per year. Substitution verifies each original station velocity. For instance, at C the gradient contribution is (3,4) millimetres per year; adding t = (5,−2) gives (8,2). Reconstructing the inputs catches unit and column-order errors before interpreting the tensor.
Now predict a fourth station D = (1000,1000). Its gradient contribution is (5,3) millimetres per year, so the predicted velocity is (10,1). Keep D out of the fit if it is intended as an independent model check. Once its observation is used to fit the same model, its residual no longer has the same held-out interpretation.
CHAPTER 4 OF 12 · Recover and interpret the gradient
4. Separate symmetric strain rate from antisymmetric rotation
Split H into E = (H + Hᵀ)/2 and W = (H − Hᵀ)/2. The symmetric part E is the infinitesimal strain-rate tensor for this local velocity description. The antisymmetric part W describes the local rigid rotational contribution. Together they reconstruct the full gradient: H = E + W.
For our network, E = [[2,1],[1,4]]×10⁻⁶ per year and W = [[0,2],[−2,0]]×10⁻⁶ per year. The scalar angular rotation rate under the east-right, north-up convention is ω = (H_21 − H_12)/2 = −2×10⁻⁶ radians per year. Its negative sign indicates clockwise local rotation in this convention.
Notice that H_12 is three while H_21 is minus one, in micro-per-year units. Neither cross derivative alone is the symmetric shear term. Their average is one; their antisymmetric difference supplies rotation. Reading the full northward change of east velocity as “shear strain rate” would mix these mechanisms.
Reconstruct the gradient as a check: the upper-right entries add as 1 + 2 = 3, and the lower-left as 1 − 2 = −1. This decomposition is mathematical, not a diagnosis of the geological cause. A symmetric component may be inferred under a model without establishing which fault, material response or environmental process produced the observed field.
CHAPTER 5 OF 12 · Recover and interpret the gradient
5. Derive directional extension rather than guessing from arrows
Take a small material baseline l with unit direction n = l/|l|. Its velocity difference is Hl. Differentiating its squared length gives d(l·l)/dt = 2lᵀHl. Since lᵀWl = 0 for an antisymmetric matrix, the instantaneous fractional length rate is (1/|l|)d|l|/dt = nᵀEn. Rotation does not contribute to this first-order length rate.
For an eastward line, n = (1,0), the result is 2×10⁻⁶ per year. For a northward line it is 4×10⁻⁶ per year. For a northeast line n = (1,1)/√2, the result is (2 + 2×1 + 4)/2 = 4×10⁻⁶ per year. Direction matters even though the tensor is unchanged.
A one-kilometre eastward baseline therefore has an instantaneous length rate of two millimetres per year in the model. The northeast baseline from A to D has length 1000√2 metres, so its predicted length rate is 4×10⁻⁶×1000√2 metres per year, approximately 5.656854 millimetres per year. Its velocity-difference magnitude is a different quantity.
This is a useful transfer test: a relative velocity component perpendicular to a baseline initially changes its direction, not its length to first order. Always project onto the baseline before calling a speed a length-change rate. The tensor formula performs that directional calculation consistently for any n.
CHAPTER 6 OF 12 · Challenge the network geometry
6. Read principal directions, area rate and shear conventions
The symmetric matrix E has orthogonal principal directions. Its eigenvalues give the largest and smallest instantaneous directional extension rates. For [[2,1],[1,4]] in micro-per-year units, they are 3 + √2 and 3 − √2, approximately 4.414214 and 1.585786. Both are positive in this fictional example, so every planar direction has a positive instantaneous extension rate.
The maximum direction is 67.5 degrees counterclockwise from east; its perpendicular direction has the smaller rate. These directions are not necessarily the directions of station velocity arrows. They describe the orientation dependence of local relative extension after the translation and rotational contribution have been separated.
The trace of E is six micro-per-year, equal to the planar divergence of the affine velocity field. It gives the instantaneous fractional area rate in the planar model. It is not a universal volumetric strain rate for the three-dimensional Earth; no vertical derivative was measured by this two-dimensional construction.
State the shear convention. The tensor shear-rate entry E_12 is one micro-per-year here, while the engineering shear-rate quantity 2E_12 is two micro-per-year. A factor-of-two discrepancy may therefore be a convention mismatch rather than a numerical error. Label the reported quantity rather than placing an unqualified “shear” beside a number.
CHAPTER 7 OF 12 · Challenge the network geometry
7. Test rank before trusting a fitted strain map
For each velocity component, fitting an affine function uses columns [1,x,y]. To recover its three coefficients uniquely, the station positions must span those columns. Three noncollinear stations provide that rank in the ideal planar problem. The two velocity components then supply six scalar observations for the six entries of t and H.
Put every station on y = 0 instead. The y column is zero, so H_12 and H_22 remain invisible to those observations. Adding more stations on that same line does not restore the missing sensitivity. The network can determine variation along the line while leaving northward derivatives undetermined.
Near-collinearity creates a related but softer problem. If the northward spread is tiny compared with observational uncertainty, the northward derivatives may be formally estimable but very uncertain. Rank and conditioning should both be checked. A matrix that can technically be inverted can still provide weak practical information about a direction.
With exactly three noncollinear stations the affine model fits their velocities exactly, even when those velocities contain error. There are no spare observations to validate the assumed spatial form. More stations can test the model, but their geometry, covariance and use in fitting must be stated. A visually smooth interpolated field should not conceal where the underlying observations are sparse or one-sided.
CHAPTER 8 OF 12 · Challenge the network geometry
8. Use an extra station to test the affine assumption
Keep the three-station fit from Chapter 3 and observe D independently. Suppose its fictional measured velocity is (10,2), rather than the predicted (10,1), in millimetres per year. The observed-minus-predicted residual is (0,1). This challenges either the model, the inputs or their stated uncertainty. It does not identify a particular physical cause by itself.
One possibility is a field with spatial curvature that one constant H cannot capture. Another is an error or unmodelled time-series feature in one station velocity. A reference or coordinate inconsistency can also affect the comparison. Before expanding the model, check whether all velocities describe the same interval and reference and whether their uncertainties support the observed discrepancy.
Changing the neighbourhood changes the estimation question. A triangle spanning a small patch may support a local gradient while a much wider network averages over variations. A very small triangle can instead magnify velocity noise when divided by short baselines. The relevant trade-off is between spatial resolution, uncertainty and the adequacy of the assumed model.
The USGS-hosted research on geodetic strain-rate uncertainty discusses inference from scattered GNSS velocities and the role of interpolation choices. Our affine example exposes the same general challenge without adopting any regional model. Compare defensible spatial models and held-out observations rather than treating one interpolation’s smooth contours as direct measurements at every point.
CHAPTER 9 OF 12 · Audit uncertainty and references
9. Carry the shared reference station into covariance
Suppose the east velocities at A, B and C have independent uncertainties, each with standard deviation 0.1 millimetres per year. The estimated H_11 uses v_EB − v_EA, and H_12 uses v_EC − v_EA. Those two differences share the same uncertain v_EA, so they are correlated even though the original station errors were independent.
Each difference has variance 0.1² + 0.1² = 0.02 square millimetres per square year. Dividing by a 1000-metre baseline and converting consistently gives each gradient standard deviation as approximately 0.141421 micro-per-year. Their covariance in squared micro-per-year units is +0.01 because the common A contribution enters both differences with a minus sign.
Ignoring that covariance changes the uncertainty of combinations such as principal rates or a predicted velocity difference. The appropriate propagation follows the complete linear transformation from station velocities to gradient entries, C_H = J C_v Jᵀ. Real station velocities may already have cross-station correlations, so independence is an assumption to justify, not a default guarantee.
There is another uncertainty layer: the affine spatial approximation itself. A covariance propagated through a fixed H model describes uncertainty conditional on that model. It does not automatically quantify missing spatial curvature, station instability or an incorrect reference transformation. Report statistical propagation and model-sensitivity checks separately instead of asking one small formal uncertainty to certify everything.
CHAPTER 10 OF 12 · Audit uncertainty and references
10. Separate reference translation, rotation and scale effects
Add the same velocity t_0 to every station. All pairwise velocity differences remain unchanged, so the inferred H and E are unchanged in this planar construction. The translation estimate changes. This is a clean invariant and a useful implementation test: a common velocity shift must not manufacture local strain.
Add a small rigid rotational velocity field Ωr, with Ω antisymmetric in the same local axes. H changes to H + Ω, but the symmetric part remains E. The reported local rotation changes while instantaneous length rates do not. This statement concerns a consistent local rigid field, not arbitrary subtraction of unrelated map arrows.
A scale-rate field κr behaves differently. It adds κI to H and therefore to E. Every directional extension rate increases by κ. Thus an apparent uniform expansion can be affected by how scale evolution is realised in the reference description. External reference information and consistent transformations are needed before calling such a signal physical ground expansion.
The companion guides on Euler-pole velocities and time-dependent Helmert transformations treat larger reference and motion questions. Use them to establish what was transformed before fitting local gradients. On a curved Earth, vectors at different stations require compatible handling; copying their local east-north numbers into one planar calculation without checking basis and extent can introduce an artificial spatial pattern.
CHAPTER 11 OF 12 · Respect limits and transfer
11. Do not use infinitesimal formulas as finite-rotation identities
Our E is a symmetric velocity gradient giving an instantaneous rate. A finite displacement gradient is a different object. In a small-displacement approximation its symmetric part is often used as infinitesimal strain, but that approximation must not be promoted into an exact finite-deformation measure.
For a finite planar rigid rotation through angle θ, the exact map is F = R(θ). The displacement gradient is F − I. Its symmetric part is (cosθ − 1)I, which is nonzero for a nonzero finite angle. At θ = 0.1 radians it is approximately −0.004995835I. Reading this as exact physical compression would be wrong: the rigid rotation preserves every length.
An exact finite-length check is FᵀF = I. The Green strain measure (FᵀF − I)/2 is therefore zero for that rotation. This does not contradict the instantaneous velocity result: a rigid rotational velocity gradient is antisymmetric and its symmetric rate is zero. The objects and approximation levels differ.
Use the finite-rotation test as a guardrail when comparing coordinate epochs. Multiplying a local strain rate by elapsed time can be a useful first-order approximation over an appropriately small change, but it is not automatically a complete finite map. Accumulated deformation depends on the evolving field, direction and reference. The connected vector-calculus guide develops those mathematical distinctions in depth.
CHAPTER 12 OF 12 · Respect limits and transfer
12. Write a conclusion that preserves the inference chain
A defensible statement identifies the region, interval, coordinate basis, reference, station selection, velocity estimates and affine model. It reports E, the rotation convention, units and uncertainty, then states which extra observations tested the spatial approximation. This makes “local strain” a reproducible inference rather than a label attached to a map.
Keep the evidence hierarchy visible. Measured or estimated station velocities constrain a spatial model. The symmetric gradient describes directional extension rates under that model. Interpreting the causes of those rates requires additional physical evidence. Inferring a future damaging event requires still more; the algebra here supplies no such forecast.
For a changed-context check, remove t from every station and verify that E stays unchanged. Add a purely antisymmetric gradient and verify that only rotation changes. Then add κI and explain why directional extension changes. Finally put all stations on a line and identify exactly which gradient entries are no longer recoverable.
These checks make the learning useful beyond one numerical example. They apply whenever differences across space are used to infer a field: identify the quantity, expose the model, test geometry, carry covariance and separate observed motion from a physical explanation. The gold standard is not a complicated tensor display. It is knowing what the tensor establishes, what the network leaves uncertain and which claim still needs independent evidence.
Independent worked case and evidence checks
Independent exercise: use a new gradient H = [[0,−3],[3,0]]×10⁻⁶ per year and translation t = (2,1) millimetres per year. At A = (0,0), B = (1000,0), C = (0,1000) and D = (1000,1000) metres, the model predicts velocities (2,1), (2,4), (−1,1) and (−1,4) millimetres per year. The four arrows differ substantially.
Recover the columns from A-to-B and A-to-C differences. The first is (0,3) and the second (−3,0), in micro-per-year units after conversion. H is antisymmetric, so E = 0. The local angular rotation rate is +3×10⁻⁶ radians per year. Every directional instantaneous length rate nᵀEn is zero, despite the different velocity arrows.
For the A-to-D diagonal, Δr = (1000,1000) metres and Δv = (−3,3) millimetres per year. Their dot product is zero after using compatible units. The velocity difference is perpendicular to the baseline, so its initial length rate is zero. Its magnitude, 3√2 millimetres per year, must not be reported as extension.
Change context by adding a scale-rate field κr with κ = 2×10⁻⁶ per year. The new gradient is [[2,−3],[3,2]]×10⁻⁶ per year and E = 2I×10⁻⁶ per year. All directions now extend at the same instantaneous fractional rate. The rotation rate remains +3 micro-radians per year. This separates isotropic extension from a rotational contribution in a way arrow inspection alone can miss.
Now remove C and D, and add ten further stations on the east-west line y = 0. Their velocities may determine the first gradient column very well, but the second column remains unobserved. Both the pure rotation and other fields with the same first column can agree along this line. State the missing northward sensitivity instead of assigning zero to an unmeasured derivative.
For an uncertainty check, shorten the eastward baseline from 1000 metres to 100 metres while retaining independent 0.1-millimetre-per-year endpoint uncertainties. The standard deviation of its inferred derivative grows from about 0.141421 to 1.414214 micro-per-year. Shorter spacing improves nominal locality but can amplify the derivative’s observational uncertainty. Whether that trade-off is useful requires a spatial model and independent evidence.
Finally, compare a true finite rigid rotation with one forward displacement step r_new = (I + HΔt)r using the antisymmetric H. The latter is a first-order numerical approximation, not an exact rotation matrix. Its length behaviour need not be exactly rigid for a finite step. This check protects the distinction between a valid instantaneous strain-rate statement and an approximate accumulated displacement calculation.
Worked comparison
| Question | Result for H=[[2,3],[−1,4]] micro/year | Check or limit |
|---|---|---|
| Symmetric strain-rate tensor | E=[[2,1],[1,4]] micro/year | Do not use one cross derivative alone |
| Local angular rate | −2 micro-radians/year | East-right, north-up sign convention |
| Eastward / northward extension | 2 / 4 micro/year | Directional projections |
| Northeast extension | 4 micro/year | nᵀEn with a unit direction |
| Principal rates | 3±√2 micro/year | Not station arrow directions |
| Planar fractional area rate | 6 micro/year | No vertical derivative inferred |
| All stations on y=0 | Northward gradient column undetermined | Repetition does not restore rank |
Questions and practical checks
Can a rotating region have different station velocities but zero strain rate?
Yes. A rigid rotational velocity field varies with position but has an antisymmetric gradient. Its symmetric strain-rate tensor is zero, and relative velocity is perpendicular to each local baseline. Different arrows alone do not prove stretching.
Why are three stations not always enough?
They must be noncollinear for the planar affine design to have full rank. Even then, exactly three stations provide no spare observations to test that model. Their uncertainty, spatial distribution and independent checks still matter.
Is divergence the entire strain tensor?
No. In the planar model it is the trace and gives the instantaneous fractional area rate. Directional extension and shear require the rest of the symmetric matrix. Zero divergence can coexist with extension in one direction and contraction in another.
Does subtracting a reference velocity remove every reference effect?
A common translation cancels from baseline differences. A consistent rigid local rotation changes only the antisymmetric part. Scale-rate and incompatible coordinate transformations can affect inferred extension. Establish the full reference treatment rather than assuming every subtraction is harmless.
Does a high strain-rate estimate predict an earthquake?
No. This guide explains a kinematic inference from a stated velocity model. Hazard interpretation needs specialist physical models, observations and uncertainty assessment. Neither a colourful map nor the toy arithmetic provides a forecast or a safety determination.
Continue with the connected subjects
Establish the velocities and reference first
Estimate station velocities with seasonal terms and time-correlated noise
Distinguish rigid plate velocity predictions from deformation residuals
Keep local vector bases and covariance compatible
Check reference-frame rates before interpreting local expansion
Test the mathematical and network limits
Explore exact material-length changes, finite rotation and the limits of divergence
Separate network information from imposed datum constraints
Sources and scope
ESA Navipedia: formal uncertainty versus measured positioning accuracy
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 1 November 2026.
Continue from measurements to deformation and uncertainty
The velocities entering a deformation model are derived geodetic products. Compare two underlying measurement concepts, then check how covariance geometry becomes a probability statement for a horizontal result.
Compare measurement methods
Radio arrival delays: distinguish baseline projections from clock terms
Laser-ranging records: keep travel times, normal points and endpoints separate
Interpret derived positions
Horizontal uncertainty: attach a justified coverage model to covariance axes