Incremental displacement shows the contribution of a single segment between the current measurement and the reference. Cumulative displacement is the sum of successive contributions from the adopted fixed base upward, and it describes the profile shape relative to that anchor. The first helps locate the change, the second assesses geometry; in practice, the decision usually requires both plots.
In short
- A segment increment is not the displacement of the whole profile point.
- Any error in the lower segment is carried into all cumulative values above it.
- Axes A and B must be read separately, and the resultant as a magnitude without directional information.
- Calculations are relative to the zero measurement and the adopted anchor.
- The maximum resultant value without profile context is not enough for diagnosis.
Two quantities, two engineering questions
Segment incremental displacement is the local change in the horizontal component of the length of a given segment relative to its reference state.
Cumulative displacement at a node is the sum of the increments of all segments from the adopted base to that node.
These definitions look similar, but they answer different questions:
- incremental: "which segment contributed the change and in which axis?",
- cumulative: "where is the given node relative to the base after summing the changes below?".
GEOKON in its inclinometer interpretation guide describes the incremental plot as the contribution of each interval, and the cumulative plot as the sum from the bottom upward. The Polish Geological Institute shows the same relationship for borehole measurements: local displacements of the intervals are summed successively from the bottom of the hole.
Where the segment increment comes from
For a vertical chain with segment length L, the change in one axis can be described as the difference in horizontal length components:
ΔAᵢ = 1000 × Lᵢ × [sin(αᵢ) − sin(αᵢ,0)]
where:
- ΔAᵢ - incremental displacement of segment i in axis A, in millimeters,
- Lᵢ - segment length in meters,
- αᵢ - current angle in axis A,
- αᵢ,0 - angle of the same segment in the reference measurement,
- 1000 - conversion from meters to millimeters.
ΔBᵢ is calculated analogously for the perpendicular B axis. The exact sign convention, axis orientation and angle unit must come from the device configuration. If the angles are in degrees, the calculation function must receive the correct conversion; if they are in radians, it must not be applied a second time.
An important subtlety: the difference between sines of angles is not always the same as the sine of an angle difference. In a correct model, the horizontal components of the current and reference orientations are compared in line with the sensor method. Do not simplify the formula just because the results seem similar for small angles.
How the cumulative profile is formed
If the base is taken as A₀ = 0 and B₀ = 0, the successive nodes are calculated recursively:
Aᵢ = Aᵢ₋₁ + ΔAᵢ
Bᵢ = Bᵢ₋₁ + ΔBᵢ
The resultant at a node is:
Rᵢ = √(Aᵢ² + Bᵢ²)
The resultant answers the question about the magnitude of the horizontal vector, but it loses directional information. Two points with R = 5 mm may lie on opposite sides of axis A or differ in their contribution from axis B. For that reason, the report should show A, B and R, not just one number.
Summation has a qualitative consequence. If the first segment from the base has an error of +0.6 mm in axis A, every point above it inherits +0.6 mm. That does not mean that all upper segments changed. It means that their cumulative position was calculated on the basis of an erroneous contribution.
Full numerical example: five segments
Illustrative example. Consider a vertical chain of five 1.0 m segments, numbered from the fixed base upward. For clarity, the A and B increments are already the result of converting the angles relative to the zero measurement. We calculate the profile after one session.
| Segment from base | Node level above anchor | ΔA [mm] | ΔB [mm] | A cum. [mm] | B cum. [mm] | R cum. [mm] |
|---|---|---|---|---|---|---|
| 1 | 1.0 m | +0.20 | -0.10 | +0.20 | -0.10 | 0.22 |
| 2 | 2.0 m | +0.40 | +0.10 | +0.60 | 0.00 | 0.60 |
| 3 | 3.0 m | +1.50 | +0.60 | +2.10 | +0.60 | 2.18 |
| 4 | 4.0 m | +0.30 | +0.20 | +2.40 | +0.80 | 2.53 |
| 5 | 5.0 m | -0.10 | +0.10 | +2.30 | +0.90 | 2.47 |
Step-by-step calculation
For the first node:
- A₁ = 0 + 0.20 = 0.20 mm,
- B₁ = 0 - 0.10 = -0.10 mm,
- R₁ = √(0.20² + (-0.10)²) ≈ 0.22 mm.
For the second:
- A₂ = 0.20 + 0.40 = 0.60 mm,
- B₂ = -0.10 + 0.10 = 0.00 mm,
- R₂ = 0.60 mm.
For the third:
- A₃ = 0.60 + 1.50 = 2.10 mm,
- B₃ = 0.00 + 0.60 = 0.60 mm,
- R₃ = √(2.10² + 0.60²) ≈ 2.18 mm.
For the fourth:
- A₄ = 2.10 + 0.30 = 2.40 mm,
- B₄ = 0.60 + 0.20 = 0.80 mm,
- R₄ ≈ 2.53 mm.
For the fifth:
- A₅ = 2.40 - 0.10 = 2.30 mm,
- B₅ = 0.80 + 0.10 = 0.90 mm,
- R₅ ≈ 2.47 mm.
What can be seen here? The largest local contribution in this session comes from segment 3: +1.50 mm in axis A and +0.60 mm in axis B. The cumulative profile increases clearly from that point, but the largest R value in the table appears at node 4, not in segment 3 itself. In segment 5, the negative A increment slightly reduces the cumulative A component, while B still increases.
This single session should not be used to declare a shear surface automatically. Further dates, a repeatable trend, geometry information and comparison with geotechnical conditions are needed. All numbers are illustrative.
What the incremental plot shows, and what the cumulative plot shows
| Question | Incremental | Cumulative |
|---|---|---|
| Which segment changed orientation? | directly | indirectly as a change in shape |
| What is the profile relative to the base? | no | yes |
| Where does a local peak appear? | clearly | may be spread by summation |
| What does the head movement relative to the anchor look like? | requires summation | directly |
| How does the error in the lower segment propagate? | remains local | affects all points above |
| Is the direction visible? | in A and B | in A and B |
| Is the resultant alone enough? | no | no |
The incremental plot is usually more sensitive to a single sensor problem. A one-time peak in one segment may result from interference, an incorrect status or a real local change. The cumulative profile will show the displacement of the whole part above that level, which can visually look more serious than a single faulty reading suggests.
On the other hand, increments do not answer the question about the total movement of the head. Five small positive values may not look alarming on their own, but their sum creates a significant profile displacement.
Axes A, B and resultant: how not to lose direction
Axis A should have its orientation documented relative to the object, for example perpendicular to the wall or in the direction of the expected movement. Axis B is perpendicular to it. If the orientation was not recorded or changed during installation, the sign and direction become difficult to interpret.
GEOKON points out that groove and axis orientation are fundamental to interpretation. In oblique movement, axis A alone may underestimate the scale. The resultant R shows magnitude, but it does not say whether the movement occurred toward the excavation, along the wall or in the opposite direction.
A practical report layout includes:
- incremental A profile,
- incremental B profile,
- cumulative A profile,
- cumulative B profile,
- resultant profile,
- time trend for selected nodes,
- axis orientation description on the plan.
Not every audience needs to see all plots on the main screen. They must, however, remain available to the person diagnosing the alarm.
The anchor is not a cosmetic assumption
The cumulative profile is relative to the adopted base. If the base moves, the entire summation loses the expected reference point. GEOKON indicates that movement of the bottom causes underestimation of cumulative displacement when a fixed anchor is assumed.
In practice, document:
- where the lower point is located,
- why it was considered stable,
- whether the installation geometry reaches beyond the expected movement zone,
- whether there is independent control of the head or base position,
- how to proceed if the data suggest anchor movement.
The Crossrail case shows the importance of this assumption. In the Shape Accel Arrays publication, the authors describe corrections based on independent geodetic measurement because assuming an immobile base did not match the actual wall behavior. This is not a universal recipe for correction, but it is an important lesson: the reference system must be confirmed with independent data.
How to distinguish a local change from a segment error
Do not make a decision based on a single peak. Use the sequence:
- Assess the quality status, axis completeness and time.
- Compare the same segment in several successive samples.
- Check whether the change is consistent in A, B and R.
- Compare neighboring segments and other instruments.
- Look at temperature and construction events.
- Assess whether the cumulative profile retains a physically possible shape.
- If the risk is high, trigger field verification or an independent measurement.
A repeatable increase in the increment across successive dates has different weight than an isolated jump that returns in the next sample. That does not mean the single jump can be ignored; it means its first interpretation is "an event requiring verification", not a ready diagnosis.
What it looks like in Inclify
Inclify supports automatic chain inclinometers. The vertical profile is calculated from the configured fixed base upward, taking into account the length of each segment and its reference angles. The results include A, B and resultant components, and the dashboard shows the profile and time trends.
The configuration defines the node order, lengths, orientation and reference point. Changing the active setup creates a new version, so the earlier profile keeps its own context. The platform applies to automatic data; it is not intended for importing campaigns from a manual probe.
The system can publish the maximum resultant value of the profile, but within the current scope we do not promise a separate automatic "maximum depth" field. Interpretation of the change location requires a view of the profile, nodes, A/B axes and installation context. Before using alarms, agree on anchor stability, axis orientation, segment lengths and the response to a single faulty reading. Raw data and the configuration version remain the basis for later reconstruction of each profile point and for change audit. If you are selecting a method for a project, first compare chain inclinometer and manual probe.
Profile interpretation checklist
- [ ] The base point is identified and its stability is justified.
- [ ] The segments are arranged in the correct order from the base.
- [ ] The length of each segment matches the installation.
- [ ] The zero measurement comes from an approved state.
- [ ] Axes A and B have an orientation on the plan.
- [ ] The angle unit and sign are unambiguous.
- [ ] A and B increments are shown, not only the resultant.
- [ ] The cumulative A and B profile is shown.
- [ ] The resultant is interpreted without assigning direction.
- [ ] Peaks are compared across successive dates.
- [ ] Statuses and data gaps are checked.
- [ ] The effect of temperature and works is assessed.
- [ ] The result is compared with neighboring segments and devices.
- [ ] Movement of the base or head is considered.
- [ ] The configuration change has the correct version.
- [ ] The alarm leads to a procedure, not an automatic diagnosis.
Method and presentation limitations
The profile is relative to the baseline and the anchor. It is not an absolute measurement of position in a global coordinate system unless the project provides a separate transformation and independent reference. The resultant does not preserve sign or direction. The increment localizes the contribution of the segment, but its spatial resolution is limited by the length and spacing of the segments.
ISO 18674-3 covers the measurement of displacements along a line using inclinometers, among other uses in excavation walls, slopes, dams and tunnels. Compliance of the device or procedure with the project requirements must be confirmed separately; the plot alone does not establish standard compliance.
The GEOKON Model 6140 manual describes a specific automatic string and the method for building the profile. Its accuracy parameters, procedures or limitations should not be transferred to another product. PIG describes manual measurements in a column; the summation principle is useful, but the data collection workflow is different.
The choice of reference was discussed in the guide zero measurement for structural monitoring, and the effects of changing segment length or number were covered in the article chain inclinometer configuration versioning.
FAQ
Is incremental displacement the change from the previous measurement?
Not always. In this article, "incremental" means the local contribution of a segment calculated relative to the approved zero measurement. Some reports use the word "increment" for the difference between two consecutive dates. The specification must therefore explicitly state the time reference, segment length, unit and exact calculation method.
Why does an error in the lower segment affect the whole profile?
Because cumulative values are created by summing from the base. An error in the first segment is part of the sum for every node above it. On the incremental plot it remains local, but on the cumulative plot it shifts the whole upper part of the profile. That is why both views should always be analyzed together.
Is it enough to monitor resultant R?
No. R shows the magnitude of the horizontal vector, but it removes direction and sign information. An alarm on R may be useful, but diagnosis requires the A and B components, axis orientation, incremental and cumulative profiles, and comparison over time with other independent measurements.
Does the largest increment indicate the slip surface?
It may indicate a zone of concentrated change, but a single result is not decisive. You need a repeatable trend, a quality assessment, geology, segment length and independent data. Sensor error, a bad reference or anchor movement can create a similar pattern and require additional independent field verification of the project.
What happens when the chain base is not fixed?
The whole profile will be described relative to a moving point, so cumulative values may underestimate or change the picture of the actual movement. The project should justify anchor stability and, when needed, use independent geodetic control or another reference system and an explicit, documented calculation correction.
Can segment length be changed without affecting history?
Length appears directly in the conversion from angle to displacement. Overwriting it would change the meaning of the results. A physical change requires a new configuration version from the correct moment, and any historical recalculation should be a separate, deliberate operation with full validation, a clearly defined scope and result audit.
Sources and further reading
- ISO 18674-3:2017 - Measurement of displacements across a line: Inclinometers, ISO.
- Model 6140 Vertical MEMS In-Place Inclinometer String - Instruction Manual, GEOKON.
- Manual Inclinometer Probe - Data Interpretation Guide, GEOKON.
- Inclinometer measurements - deep monitoring of landslides, Polish Geological Institute - PIB.
- The Use of Shape Accel Arrays for Measuring Retaining Wall Deflection, Crossrail Learning Legacy.
Next step: test the profile on one known event
Prepare the segment configuration, approved baseline and a sample from a controlled construction stage. We will show the increments, the A/B profile and the resultant separately, and then record the acceptance criteria without pretending to make an automatic diagnosis. Schedule a chain inclinometer pilot.