Uniform pavement sections: a defensible starting point for investigation

Let the boundary earn its place

A neat vertical line on a chart is a hypothesis. Does the evidence actually prefer it?

South African rehabilitation guidance is catalogued as TRH12. Start with the applicable original and the project investigation brief rather than assuming any dashboard grouping is a design section. (Department of Transport, n.d.)

Ask the residuals where the split belongs

Governing relationship: SSE = Σ (yi − meansection)².

yi: interval roughness in m/km; k: candidate split after an interval.

All six intervals are 100 m. The objective measures within-section variation.

The film first plots iri (m/km), then follows the calculation and plots within-section sse ((m/km)²). The minimum is 0.100 at 300 m. Means are 2.30 and 6.00 m/km; engineering evidence must still justify the boundary.

Segmentation experiment

Choosing a section boundary

The lowest SSE is a statistical candidate, not a design boundary.

Download example data (CSV)

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

Understand the model

A candidate boundary divides the same measurements into two groups. The squared residuals quantify how tightly measurements cluster within each group.

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

yᵢ

Roughness observation for interval i (m/km).

S; nS

One candidate section and the number of observations in it.

ȳS; ȳL; ȳR

Mean roughness in a section, or in the left/right groups (m/km).

k; SSE

Candidate split index; sum of squared residuals ((m/km)²).

Values used in this film

• yi: interval roughness in m/km; k: candidate split after an interval.
• All six intervals are 100 m. The objective measures within-section variation.

The mathematical relationships

1. Equation 1

   Source and scope
   $$\bar y_S=\frac{1}{n_S}\sum_{i\in S}y_i$$

   Compute the representative mean within each candidate section. The example uses equal 100 m support lengths.

2. Equation 2

   Source and scope
   $$SSE(k)=\sum_{i\leq k}(y_i-\bar y_L)^2+\sum_{i>k}(y_i-\bar y_R)^2$$

   Subtract the relevant section mean from each observation, square each residual and sum. Lower SSE means less within-group variation for this fixed two-section comparison.

Work through the example

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

1. Left mean

   (2.2 + 2.4 + 2.3) / 3 = 2.3

   m/km

2. Right mean

   (5.8 + 6.2 + 6.0) / 3 = 6.0

   m/km

3. Left squared residuals

   0.01 + 0.01 + 0 = 0.02

   (m/km)²

4. Right squared residuals

   0.04 + 0.04 + 0 = 0.08

   (m/km)²

5. Total objective

   0.02 + 0.08 = 0.10

   (m/km)²

Read the output

The minimum is 0.100 at 300 m. Means are 2.30 and 6.00 m/km; engineering evidence must still justify the boundary.

Check your understanding

Recompute every candidate rather than selecting a visually attractive split. Retain the no-split error as a baseline and examine whether the chosen boundary coincides with located construction, drainage or distress evidence.

Assumptions and limits

The lowest SSE is a statistical candidate, not a design boundary.

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

Timestamped film transcript 32 short entries · select a time to revisit the film

1. 0:00

   Choosing a section boundary. Six equal 100 m intervals: 2.2, 2.4, 2.3, 5.8, 6.2, 6.0 m/km.

2. 0:01

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

3. 0:07

   yi: interval roughness in m/km; k: candidate split after an interval. All six intervals are 100 m. The objective measures within-section variation.

4. 0:14

   INPUT · six interval measurements · read the axis units and the numerical values.

5. 0:23

   Split at 100 m: within-section SSE = 16.072 (m/km)².

6. 0:26

   Split at 200 m: within-section SSE = 10.368 (m/km)².

7. 0:29

   Split at 300 m: within-section SSE = 0.100 (m/km)².

8. 0:32

   Split at 400 m: within-section SSE = 9.227 (m/km)².

9. 0:35

   Split at 500 m: within-section SSE = 16.528 (m/km)².

10. 0:39

    Minimum at 300 m: means 2.30 and 6.00 m/km; SSE = 0.100 (m/km)².

11. 0:43

    OUTPUT · error for each candidate boundary · read the axis units and the numerical values.

12. 0:52

    The minimum is 0.100 at 300 m. Means are 2.30 and 6.00 m/km; engineering evidence must still justify the boundary.

13. 0:59

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

14. 1:05

    Left mean: (2.2 + 2.4 + 2.3) / 3 = 2.3 m/km

15. 1:08

    Right mean: (5.8 + 6.2 + 6.0) / 3 = 6.0 m/km

16. 1:11

    Left squared residuals: 0.01 + 0.01 + 0 = 0.02 (m/km)²

17. 1:14

    Right squared residuals: 0.04 + 0.04 + 0 = 0.08 (m/km)²

18. 1:17

    Total objective: 0.02 + 0.08 = 0.10 (m/km)²

19. 1:20

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

20. 1:29

    Boundary 100 m: 16.072 (m/km)² SSE. Same six observations; change only the candidate split.

21. 1:32

    Boundary 200 m: 10.3675 (m/km)² SSE. Same six observations; change only the candidate split.

22. 1:36

    Boundary 300 m: 0.1 (m/km)² SSE. Same six observations; change only the candidate split.

23. 1:39

    Boundary 400 m: 9.2275 (m/km)² SSE. Same six observations; change only the candidate split.

24. 1:43

    Boundary 500 m: 16.528 (m/km)² SSE. Same six observations; change only the candidate split.

25. 1:46

    Check: Recompute every candidate rather than selecting a visually attractive split. Retain the no-split error as a baseline and examine whether the chosen boundary coincides with located construction, drainage or distress evidence.

26. 1:52

    ONE ROW · A neat vertical line on a chart is a hypothesis. Does the evidence actually prefer it?

27. 2:10

    PARAMETER MAP · 6100 evaluated cells. The colour limits stay fixed across every row.

28. 2:20

    Generate 101 equal intervals with rᵢ = 2.3 + Δ·1(i ≥ 50) + 0.2 sin(i), using zero-based i. At each candidate split, recompute both section means and sum squared residuals. As the

29. 2:31

    prescribed step grows, a clearer trough develops near the actual transition. At small Δ the oscillation competes with the boundary signal.

30. 2:39

    VALUE LANDSCAPE · Height and colour encode SSE ((m/km)²). This is a graph over the two input axes, not automatically a physical pavement shape.

31. 2:51

    This optimises one split in one equally spaced synthetic signal. Minimum section lengths, multiple splits and corroborating material, drainage and construction evidence remain separate decisions.

32. 3:00

    The lowest SSE is a statistical candidate, not a design boundary.

Download the timed captions (VTT)

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

Compare evidence on one reference

Our suggested approach aligns observations, structural response, construction history and relevant measurements along chainage. Record why a boundary was proposed and which evidence supports it. Preserve the ungrouped data.

A boundary moves

Imagine readings clustered around one level before chainage 500 m and another afterwards. Consider where a proposed boundary would sit relative to those observations. A convenient split may hide an important local feature; use the summary to prompt investigation.

Keep statistical and design boundaries separate

A statistical change is not proof of a material boundary. Do not claim a segmentation algorithm has identified uniform pavement structure without corroborating evidence.

Record the reason for each proposed boundary

Draft TRH12 describes dividing a road into uniform sections while considering its functional and structural condition. That is a broader task than splitting a spreadsheet into equal lengths. (Department of Transport, 1997)

An original boundary register can hold chainage, the evidence suggesting a change, the proposed interpretation and any conflicting observations. One boundary may follow a known construction change; another may be provisional because only one measurement series changes.

Before summarising a section, inspect local extremes and missing coverage. Two sections with similar averages can contain different spatial patterns. Keep a way to return from the summary to the underlying records.

The calculation laboratory compares five candidate boundaries using recomputed section means and residual errors. Its minimum identifies a statistical candidate for investigation, with engineering evidence still needed to support the grouping.

Calculate the contrast, then question the boundary

Use six equal 100 m intervals with synthetic roughness values 2.2, 2.4, 2.3, 5.8, 6.2 and 6.0 m/km. The first three average 2.3 m/km; the last three average 6.0 m/km. Across the whole 600 m, the descriptive mean is 4.15 m/km. A proposed split at 300 m makes the contrast visible.

Case review

Make the proposed boundary compete with its neighbours

Is a split after the third interval supported by this response profile?

1 · 2.2 m/km

The discriminating check

Compare candidate splits, then check whether materials, construction, drainage or traffic evidence supports the same location.

Evidence supports the comparison

Where independent evidence agrees, record a provisional section boundary and the evidence supporting it.

Evidence is missing or incompatible

Where signals disagree, preserve the competing boundaries and investigate the mismatch.

What you can conclude. The best split in a single signal is a candidate for engineering review, not an automatic design section.

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

That calculation supports a question: does the change align with construction history, other measurements or observed distress? It does not establish that the pavement structure changes exactly at 300 m. Draft TRH12 considers functional and structural condition when identifying uniform sections; a roughness series is one strand of evidence. (Department of Transport, 1997)

Spatial view · synthetic data

Follow the measurement along chainage

0–100 m2.2m/km100–200 m2.4m/km200–300 m2.3m/km300–400 m5.8m/km400–500 m6.2m/km500–600 m6.0m/km

0–100 m · 2.2 m/km

Now move the proposed split to 400 m. The first group's mean becomes (2.2 + 2.4 + 2.3 + 5.8) / 4 = 3.175 m/km; the remaining group's mean is 6.1 m/km. One higher interval has been absorbed into the first section. The new summaries are mathematically correct, but the boundary may be less helpful for explaining the spatial pattern.

Record why a boundary was proposed, not just where it was drawn. A known construction change, a consistent shift in several measurements, and a convenient round chainage are different reasons. They deserve different levels of confidence and different follow-up actions.

Keep local extremes and missing coverage visible within each proposed section. A summary is meant to support an investigation, not to erase inconvenient observations. Where evidence conflicts, a provisional boundary with a clear review question is more useful than a final-looking line whose basis is unknown.

Calculation laboratory

Does the minimum error justify a section boundary?

Calculation step	Substitution or result	Units and interpretation
Left mean	(2.2 + 2.4 + 2.3) / 3 = 2.3	m/km
Right mean	(5.8 + 6.2 + 6.0) / 3 = 6.0	m/km
Left squared residuals	0.01 + 0.01 + 0 = 0.02	(m/km)²
Right squared residuals	0.04 + 0.04 + 0 = 0.08	(m/km)²
Total objective	0.02 + 0.08 = 0.10	(m/km)²

Squaring removes the sign of a residual and penalises larger deviations more strongly. The section mean minimises the sum of squared residuals for a fixed group, which makes it the natural representative value for this particular objective. It does not make the mean the correct statistic for every engineering decision: local maxima and spatial patterns may remain important.

The plotted sensitivity is a boundary search, not a time history. Each point recomputes both group means from the same six observations. Moving a boundary to 200 m forces the low third observation into the high-response group; moving it to 400 m forces a high observation into the low group. The larger errors quantify this loss of internal consistency.

Unrestricted segmentation can drive residual error to zero by giving every observation its own section. A useful investigation therefore needs constraints such as a declared minimum section length and corroborating evidence, and may need a complexity penalty. This article deliberately compares exactly one split at five candidate positions. Equal 100 m intervals justify the unweighted calculation; unequal support lengths require a stated weighting convention. A statistical change in roughness suggests an investigation boundary but does not prove a change in pavement construction or material. (Department of Transport, 1997)

Controlled sensitivity experiment

Controlled case	Calculated output	What changes
Boundary 100 m	16.072000 (m/km)² SSE	Same six observations; change only the candidate split.
Boundary 200 m	10.367500 (m/km)² SSE	Same six observations; change only the candidate split.
Boundary 300 m	0.100000 (m/km)² SSE	Same six observations; change only the candidate split.
Boundary 400 m	9.227500 (m/km)² SSE	Same six observations; change only the candidate split.
Boundary 500 m	16.528000 (m/km)² SSE	Same six observations; change only the candidate split.

CALCULATION LABORATORY

Does the minimum error justify a section boundary?

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

Sensitivity case1. Boundary 100 m2. Boundary 200 m3. Boundary 300 m4. Boundary 400 m5. Boundary 500 m

Boundary 100 m: 16.072 (m/km)² SSE. Same six observations; change only the candidate split.

Reproduce and challenge the result

Recompute every candidate rather than selecting a visually attractive split. Retain the no-split error as a baseline and examine whether the chosen boundary coincides with located construction, drainage or distress evidence.

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.

Watch the boundary evidence sharpen

Generate 101 equal intervals with rᵢ = 2.3 + Δ·1(i ≥ 50) + 0.2 sin(i), using zero-based i. At each candidate split, recompute both section means and sum squared residuals. As the prescribed step grows, a clearer trough develops near the actual transition. At small Δ the oscillation competes with the boundary signal.

Reproduce the experiment. SSE(k, Δ) = Σleft(ri−meanleft)² + Σright(ri−meanright)². The downloadable table contains 6,100 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 · 6,100 cells

Let the boundary earn its place

A neat vertical line on a chart is a hypothesis. Does the evidence actually prefer it?

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

This optimises one split in one equally spaced synthetic signal. Minimum section lengths, multiple splits and corroborating material, drainage and construction evidence remain separate decisions.

References and method notes

1. Department of Transport (n.d.). Roads: Engineering Standard Guidelines and Manuals. Official catalogue; accessed 5 September 2026. Original source (Accessed: 5 September 2026).TMH22 V8b Draft; TRH12; TRH15; TRH24 catalogue entries
2. Horak (2008). Benchmarking the structural condition of flexible pavements with deflection bowl parameters. Journal of the South African Institution of Civil Engineering, 50(2), pp. 2–9. Original source (Accessed: 5 September 2026).Relative structural benchmarking with deflection bowls
3. Department of Transport (1997). TRH12: Flexible Pavement Rehabilitation Investigation and Design. Draft TRH12, Pretoria: Department of Transport for COLTO. Original source (Accessed: 5 September 2026).Chapter 1: rehabilitation design process; investigation, evaluation, analysis and economic appraisal

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