Chainage, direction and wheelpath: reconciling survey records

The join is only as good as its origin

Two clean-looking spreadsheets can put the same pothole a hundred metres apart.

Positioning is treated explicitly in TMH13 Part B. It belongs in the measurement record rather than being left as an unspoken map assumption. (COTO, 2016c)

Turn the origin around, carefully

Governing relationship: sreverse = L − sforward = 1000 − 250 = 750 m.

L = 1,000 m reference length; sf = forward coordinate; sr = reverse coordinate.

The slope −1 means each metre forward removes one metre from reverse chainage.

The film first plots forward chainage (m), then follows the calculation and plots reverse chainage (m). At sf = 250 m, sr = 750 m. The physical point is unchanged; its coordinate depends on the selected origin.

Coordinate transformation

One point, two chainages

Match the reference origin and direction before joining observations.

Download example data (CSV)

Model, calculations and transcriptEquations, worked steps, source notes and a timed reading guide

Understand the model

One physical point can have two coordinates when the origin and travel direction change.

Work through the inputs and equations below, then compare the calculated output with the film. All numerical inputs in this study are synthetic.

Inputs and symbol key

sf; sr

Forward and reverse chainage coordinates (m).

L

Length of the common reference alignment (m).

Values used in this film

• L = 1,000 m reference length; sf = forward coordinate; sr = reverse coordinate.
• The slope −1 means each metre forward removes one metre from reverse chainage.

The mathematical relationships

1. Equation 1

   Source and scope
   $$s_r=L-s_f$$

   Subtract the forward coordinate from the reference length to obtain the reverse coordinate.

2. Equation 2

   Source and scope
   $$\frac{ds_r}{ds_f}=-1$$

   The negative derivative means one metre of increasing forward coordinate produces one metre of decreasing reverse coordinate.

Work through the example

Use the relationships above with consistent units. These substitutions are our worked example, not values quoted from the source.

1. Forward reference

   sf = 250; L = 1000

   metres

2. Reverse transform

   sr = L − sf = 1000 − 250 = 750

   metres

3. Round trip

   1000 − 750 = 250

   metres

4. Misread coordinate

   750 − 250 = 500

   metres apparent displacement

Read the output

At sf = 250 m, sr = 750 m. The physical point is unchanged; its coordinate depends on the selected origin.

Check your understanding

Apply the transform twice and recover the starting coordinate. Verify endpoints 0 and L, transformed interval length, and a located control point away from the midpoint, where an origin reversal can otherwise be missed.

Assumptions and limits

Match the reference origin and direction before joining observations.

Return to the full references and method notes to check the cited edition and its applicability.

Timestamped film transcript 24 short entries · select a time to revisit the film

1. 0:00

   One point, two chainages. One straight 1,000 m reference; opposite origins.

2. 0:01

   MATHEMATICS · define the relationship and the meaning of each symbol before calculating.

3. 0:07

   L = 1,000 m reference length; sf = forward coordinate; sr = reverse coordinate. The slope −1 means each metre forward removes one metre from reverse chainage.

4. 0:14

   INPUT · forward coordinate · read the axis units and the numerical values.

5. 0:23

   One straight 1,000 m reference; opposite origins.

6. 0:43

   OUTPUT · transformed coordinate · read the axis units and the numerical values.

7. 0:52

   At sf = 250 m, sr = 750 m. The physical point is unchanged; its coordinate depends on the selected origin.

8. 0:59

   CALCULATION · substitute the defined inputs; keep units through each intermediate step.

9. 1:05

   Forward reference: sf = 250; L = 1000 metres

10. 1:08

    Reverse transform: sr = L − sf = 1000 − 250 = 750 metres

11. 1:11

    Round trip: 1000 − 750 = 250 metres

12. 1:14

    Misread coordinate: 750 − 250 = 500 metres apparent displacement

13. 1:17

    SENSITIVITY · change one declared input · read the axis units and the numerical values.

14. 1:26

    L = 995 m: 745 m reverse coordinate. Keep forward coordinate 250 m; shorten the reference length.

15. 1:29

    L = 1000 m: 750 m reverse coordinate. Reference coordinate transformation.

16. 1:33

    L = 1005 m: 755 m reverse coordinate. Keep forward coordinate 250 m; length grows by 5 m.

17. 1:36

    Check: Apply the transform twice and recover the starting coordinate. Verify endpoints 0 and L, transformed interval length, and a located control point away from the midpoint, where an origin reversal can otherwise be missed.

18. 1:42

    ONE ROW · Two clean-looking spreadsheets can put the same pothole a hundred metres apart.

19. 2:00

    PARAMETER MAP · 12261 evaluated cells. The colour limits stay fixed across every row.

20. 2:10

    The central row uses the same 60 km alignment with opposite origins. Rows above and below introduce a declared origin offset δ. Every point moves by the same amount: a constant

21. 2:21

    registration error survives even when all reported chainages have three decimal places. The sloping plane is a coordinate transformation, not terrain.

22. 2:29

    VALUE LANDSCAPE · Height and colour encode Reverse chainage (km). This is a graph over the two input axes, not automatically a physical pavement shape.

23. 2:41

    This affine example assumes the same route geometry and distance basis. Curved alignment revisions, wheelpaths, carriageways and survey dates need explicit identifiers; a subtraction does not reconcile them.

24. 2:49

    Match the reference origin and direction before joining observations.

Download the timed captions (VTT)

The transcript records the on-screen wording. Formula provenance and interpretation are explained in the source note above.

Make the reference reproducible

Our recommended record includes route identifier, reference-line version, start origin, direction, coordinate system, units and wheelpath. Keep the original coordinate and the derived chainage together. A later geometry correction should be traceable.

Metres and kilometres

A sample at 1.25 km is at 1,250 m, not 1.25 m. Use one internal unit and convert only at a named boundary. For a reversed survey, compare against the same reference origin rather than merely sorting timestamps.

Preserve the reference system

Nearest-point projection can be ambiguous near junctions, loops and parallel carriageways. The diagram is an explanation, not a road-matching algorithm.

A reversal example

Suppose a synthetic reference line is 1,000 m long and chainage increases from its western origin. A measurement at reference chainage 250 m retains that location even if the survey vehicle approached from the eastern end. Under this simple straight-reference example, it is 750 m from the other end. The timestamp order does not redefine the reference origin.

Store survey travel direction separately from reference direction. Also state how “left” and “right” wheelpaths are defined. Those words become ambiguous when a pass is reversed without an explicit convention.

When joining two datasets, inspect a few known landmarks and section endpoints before trusting a full-table match. A successful join on a road name is insufficient if that name refers to different reference geometries.

Join the location conventions before joining the tables

Take a synthetic reference line 1,000 m long, with chainage increasing from west to east. A sample at chainage 250 m remains there whether the survey vehicle travelled eastward or westward. Its distance from the eastern end is 1,000 − 250 = 750 m. Confusing travel distance with reference chainage would relocate the observation by 500 m in this example.

Case review

The same point can be 250 m and 750 m

Are the two teams disagreeing about location or merely about the origin?

Forward origin · 250 Chainage (m)

The discriminating check

Check route identity, alignment version, origin, increasing direction and lane/wheelpath identifiers.

Evidence supports the comparison

On the same reversed reference, transform s to 1,000 − s and retain the original coordinate alongside it.

Evidence is missing or incompatible

If the alignment or distance basis differs, establish a mapping between references; subtraction alone is insufficient.

What you can conclude. A location join needs a defensible reference transformation before it needs more decimal places.

Original worked case. Read the source editions and method notes.

The example is deliberately simple: it assumes the same reference geometry and endpoints. On a real route, do not assume that reversing a distance array reconciles different carriageways, alignments or reference versions. Positioning belongs to the measurement method, as reflected in TMH13 Part B. (COTO, 2016c)

Record field	Synthetic value	Purpose
Reference length	1,000 m	Defines the example's extent
Reference chainage	250 m from west	Establishes location
Distance from east end	750 m	A different distance convention
Travel direction	Westbound	Describes the pass
Wheelpath convention	Relative to travel direction	Removes left/right ambiguity

A join can appear successful while remaining physically wrong. Matching a road name and a numeric chainage does not prove that the two rows refer to the same point. Check origin, direction, carriageway, units and geometry version. Inspect known features and endpoints before trusting an entire joined dataset.

There is also a distinction between an original coordinate and a derived location. A new reference geometry may change the projected chainage of an old coordinate. Preserve both and record the transformation. That lets a reviewer reproduce the match and understand why a later report differs without assuming that the observation itself moved.

For the engineer, this is more than data housekeeping. A misplaced measurement can create a false local pattern, obscure a real one, or associate a treatment decision with the wrong section. The first engineering check is therefore often the identity of the place being discussed.

Calculation laboratory

How does a coordinate error propagate through a route comparison?

Calculation step	Substitution or result	Units and interpretation
Forward reference	sf = 250; L = 1000	metres
Reverse transform	sr = L − sf = 1000 − 250 = 750	metres
Round trip	1000 − 750 = 250	metres
Misread coordinate	750 − 250 = 500	metres apparent displacement

The negative derivative means direction reversal changes the sign of coordinate increments. A forward interval from 200 m to 300 m maps to reverse coordinates 800 m and 700 m; reorder the endpoints for an ascending interval record while retaining the travel-direction attribute. Simply relabelling the axis leaves a reversed interval or a misplaced observation.

Uncertainty in the reference length contributes directly to the reverse coordinate. The exact error relation is δsr = δL − δsf. A five-metre change in L shifts every transformed coordinate by five metres when sf is unchanged. If both errors are correlated, treating their uncertainty contributions as independent would lose the covariance term.

This one-dimensional example assumes both coordinates describe the same physical alignment and endpoints. Real route revisions, chainage equations, ramps and carriageway separation need a segment mapping rather than a global subtraction. Direction also does not determine wheelpath identity by itself: retain carriageway, lane, travel direction and the convention used to name each wheelpath. The positioning record should allow another engineer to reconstruct the transformation, not merely view a marker at a plausible location. (COTO, 2016c)

Controlled sensitivity experiment

Controlled case	Calculated output	What changes
L = 995 m	745.000000 m reverse coordinate	Keep forward coordinate 250 m; shorten the reference length.
L = 1000 m	750.000000 m reverse coordinate	Reference coordinate transformation.
L = 1005 m	755.000000 m reverse coordinate	Keep forward coordinate 250 m; length grows by 5 m.

CALCULATION LABORATORY

How does a coordinate error propagate through a route comparison?

Compare the cases on a fixed scale. Select a case to inspect its input change and calculated output.

Sensitivity case1. L = 995 m2. L = 1000 m3. L = 1005 m

L = 995 m: 745 m reverse coordinate. Keep forward coordinate 250 m; shorten the reference length.

Reproduce and challenge the result

Apply the transform twice and recover the starting coordinate. Verify endpoints 0 and L, transformed interval length, and a located control point away from the midpoint, where an origin reversal can otherwise be missed.

These are original synthetic calculations. The sensitivity cases vary the stated input while retaining the other assumptions; they are not field observations or treatment thresholds. The film’s calculation and sensitivity chapters use the same numerical cases as this table.

A constant offset across sixty kilometres

The central row uses the same 60 km alignment with opposite origins. Rows above and below introduce a declared origin offset δ. Every point moves by the same amount: a constant registration error survives even when all reported chainages have three decimal places. The sloping plane is a coordinate transformation, not terrain.

Reproduce the experiment. s_reverse = 60 − s_forward + δ. The downloadable table contains 12,261 evaluated cells, with the two input coordinates and output value retained on every row. These are parameter combinations, not that many independent field observations. The equation extends the worked example above; it is original teaching arithmetic, not a new standard or calibrated prediction.

Computed parameter atlas · 12,261 cells

The join is only as good as its origin

Two clean-looking spreadsheets can put the same pothole a hundred metres apart.

Download every calculated cell (CSV) · Open full-size explorer · Equation source and tests (ZIP)

This affine example assumes the same route geometry and distance basis. Curved alignment revisions, wheelpaths, carriageways and survey dates need explicit identifiers; a subtraction does not reconcile them.

References and method notes

1. COTO (2016c). TMH13: Automated Road Condition Assessments. Part B: Positioning. Committee Draft Final, May 2016. Original source (Accessed: 5 September 2026).Positioning and distance measurement guidance

Use the source edition and project requirements that apply to your work. Educational examples are not site-specific design advice.