TRH12 rehabilitation investigation: build the evidence before the design

A perfect fit can still leave a question

If two different input pairs draw the same bowl, which pavement have you identified?

Draft TRH12 (1997) presents rehabilitation as a process of investigation, evaluation, analysis, design and economic appraisal. An observation is a starting point for that investigation, rather than an automatic treatment instruction. Confirm the edition and project requirements before applying the detailed methods. (Department of Transport, 1997)

Fit the bowl, then question the fit

Governing relationship: RMSE = √[Σ (wmodel − wreference)² / n].

Reference vector: five modelled sensor readings, generated at E = 200 MPa.

Candidate E varies from 100 to 400 MPa; radius, load and ν remain fixed.

The film first plots reference displacement (μm), then follows the calculation and plots rmse (μm). RMSE is zero at the generating modulus of 200 MPa. This checks numerical consistency; real measurements can have model mismatch.

Inverse-model experiment

Testing competing stiffness hypotheses

Recovering a synthetic modulus does not identify the layers of a real pavement.

Download example data (CSV)

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

Understand the model

An inverse experiment compares predicted readings with reference readings. A small residual measures agreement with that reference, not completeness of the pavement explanation.

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

E

Candidate Young’s modulus (MPa in the candidate list; Pa in the solver).

wi(E); wi,ref

Predicted and reference displacement at sensor i (μm).

n; RMSE

Number of sensors; root-mean-square residual (μm).

Values used in this film

• Reference vector: five modelled sensor readings, generated at E = 200 MPa.
• Candidate E varies from 100 to 400 MPa; radius, load and ν remain fixed.

The mathematical relationships

1. Equation 1

   Source and scope
   $$RMSE(E)=\sqrt{\frac{1}{n}\sum_i[w_i(E)-w_i^{ref}]^2}$$

   Subtract each reference sensor reading from its prediction, square, average across sensors and take the square root. RMSE retains the reading unit.

2. Equation 2

   Source and scope
   $$w(E)\propto E^{-1}\quad\text{for this fixed-load half-space}$$

   With load, radius and Poisson ratio fixed, the homogeneous-model displacement scales as 1/E. This proportionality does not establish uniqueness in a multilayer inverse problem.

Work through the example

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

1. Reference generation

   wref = w(E = 200 MPa)

   five synthetic μm readings

2. Candidate scaling

   w(100 MPa) = 2 wref

   same load and geometry

3. Residual vector

   ri = wi(E) − wi,ref

   μm

4. Objective

   RMSE = √(Σri² / 5)

   μm; zero at 200 MPa

Read the output

RMSE is zero at the generating modulus of 200 MPa. This checks numerical consistency; real measurements can have model mismatch.

Check your understanding

Plot signed residuals by sensor offset as well as RMSE. Perturb the assumed load and test whether the fitted modulus follows it. Keep synthetic recovery tests separate from validation against independent field measurements.

Assumptions and limits

Recovering a synthetic modulus does not identify the layers of a real pavement.

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

Timestamped film transcript 31 short entries · select a time to revisit the film

1. 0:00

   Testing competing stiffness hypotheses. Reference half-space E = 200 MPa; candidates 100–400 MPa.

2. 0:01

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

3. 0:07

   Reference vector: five modelled sensor readings, generated at E = 200 MPa. Candidate E varies from 100 to 400 MPa; radius, load and ν remain fixed.

4. 0:14

   INPUT · synthetic reference readings · read the axis units and the numerical values.

5. 0:23

   E = 100 MPa; RMSE = 348.36 μm relative to the synthetic reference.

6. 0:26

   E = 150 MPa; RMSE = 116.12 μm relative to the synthetic reference.

7. 0:29

   E = 200 MPa; RMSE = 0.00 μm relative to the synthetic reference.

8. 0:32

   E = 300 MPa; RMSE = 116.12 μm relative to the synthetic reference.

9. 0:36

   E = 400 MPa; RMSE = 174.18 μm relative to the synthetic reference.

10. 0:39

    E = 200 MPa; RMSE = 0.00 μm relative to the synthetic reference.

11. 0:42

    OUTPUT · residual versus candidate modulus · read the axis units and the numerical values.

12. 0:51

    RMSE is zero at the generating modulus of 200 MPa. This checks numerical consistency; real measurements can have model mismatch.

13. 0:58

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

14. 1:04

    Reference generation: wref = w(E = 200 MPa) five synthetic μm readings

15. 1:07

    Candidate scaling: w(100 MPa) = 2 wref same load and geometry

16. 1:10

    Residual vector: ri = wi(E) − wi,ref μm

17. 1:13

    Objective: RMSE = √(Σri² / 5) μm; zero at 200 MPa

18. 1:16

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

19. 1:26

    E = 100 MPa: 348.361 μm RMSE. Reference generated at 200 MPa; vary one candidate modulus.

20. 1:29

    E = 150 MPa: 116.12 μm RMSE. Reference generated at 200 MPa; vary one candidate modulus.

21. 1:33

    E = 200 MPa: 0 μm RMSE. Reference generated at 200 MPa; vary one candidate modulus.

22. 1:36

    E = 300 MPa: 116.12 μm RMSE. Reference generated at 200 MPa; vary one candidate modulus.

23. 1:39

    E = 400 MPa: 174.18 μm RMSE. Reference generated at 200 MPa; vary one candidate modulus.

24. 1:42

    Check: Plot signed residuals by sensor offset as well as RMSE. Perturb the assumed load and test whether the fitted modulus follows it. Keep synthetic recovery tests separate from validation against independent field measurements.

25. 1:48

    ONE ROW · If two different input pairs draw the same bowl, which pavement have you identified?

26. 2:06

    PARAMETER MAP · 6161 evaluated cells. The colour limits stay fixed across every row.

27. 2:16

    The reference bowl uses 200 MPa and 40 kN at offsets 0, 0.3, 0.6, 0.9 and 1.2 m. Linear elasticity scales every displacement with P/E. Evaluate 6,161 input pairs and a low-error

28. 2:28

    valley follows P/E = 0.2 kN/MPa. A measured load removes one source of ambiguity; letting both load and stiffness float creates it.

29. 2:36

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

30. 2:48

    One homogeneous elastic parameter, noise-free synthetic reference and fixed radius/Poisson ratio. This is an identifiability demonstration, not multilayer backcalculation or proof that actual field loads are unknown.

31. 2:57

    Recovering a synthetic modulus does not identify the layers of a real pavement.

Download the timed captions (VTT)

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

Keep observations and hypotheses distinct

Our recommended investigation log distinguishes what was observed, what mechanism is suspected, what evidence is missing and what test could resolve it. That structure prevents a preferred treatment from becoming an untested explanation.

Build a hypothesis table

For a synthetic distressed section, list two possible explanations and the evidence each would require. Record which observations support or conflict with each. A later design decision should reference the resolved investigation, not just the original map flag.

Design an investigation that separates explanations

The article is a workflow guide, not a substitute for the complete TRH12 method. No universal rut-depth or IRI treatment threshold is asserted here.

Use a hypothesis table, not a treatment shortlist

The following original template helps organise the investigation described above.

Record	What belongs here
Observation	A located, dated record of what was seen or measured
Hypothesis	A possible explanation that still needs testing
Supporting evidence	Records consistent with that explanation
Conflicting evidence	Records that challenge it
Next question	The uncertainty a further investigation should resolve

Avoid filling the hypothesis column with a preferred rehabilitation product. A treatment is a response to an established problem, while a hypothesis proposes what the problem may be.

When evidence changes the interpretation, preserve the earlier hypothesis and the reason it was revised. The resulting design report can then explain how the investigation progressed, rather than presenting a decision with no traceable basis.

Design the investigation around what would change your mind

Consider a synthetic section with 12 visual observations, six deflection observations, four DCP records and three drainage observations. The inventory tells us what evidence exists; it does not produce a confidence score. Different observations answer different questions, may share the same location, and may not be independent.

Case review

A second parameter can rescue a bad hypothesis

Does matching the bowl identify a unique stiffness?

40 kN / 200 MPa · 0.2 P/E (kN/MPa)

The discriminating check

Use the measured load to constrain the model, then ask which independent observations distinguish remaining explanations.

Evidence supports the comparison

With load and geometry fixed, assess fit, sensitivity and agreement with the investigation evidence.

Evidence is missing or incompatible

If several unsupported inputs can move together, report the ambiguity and collect evidence that constrains them.

What you can conclude. This thought experiment motivates investigation design; it is not a multilayer backcalculation result.

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

A count of 25 records cannot substitute for an investigation rationale. If all deflection observations miss the location where the concern is concentrated, the number six says little about that local uncertainty. Equally, a small number of well-located observations may clarify a specific question without establishing the condition of an entire network.

Draft TRH12 sets out rehabilitation as investigation, evaluation, analysis, design and economic appraisal. Use that process to structure the decision instead of beginning with a preferred treatment and searching for supporting evidence. (Department of Transport, 1997)

Investigation question	Evidence to examine	What would remain unresolved?
Is the apparent change spatially consistent?	Located measurements and repeatable references	Cause of the change
Does a construction change align with it?	Construction records and field evidence	Current material behaviour
Is a moisture-related explanation supported?	Relevant site investigation and water observations	Appropriate drainage design

A strong hypothesis is useful because it can be challenged. State what observation would conflict with it. If the evidence contradicts the preferred explanation, preserve that result and revise the investigation rather than burying it in an appendix.

The eventual design report should make the route from evidence to intervention legible. It should explain the problem being addressed, the alternatives considered and the assumptions carried into appraisal. No generic rut-depth or IRI threshold is needed to demonstrate that reasoning process; the applicable project method supplies the detailed criteria.

An inverse calculation with a known answer

The film generates a reference deflection vector from the same circular-load half-space model used in the FWD article, with E = 200 MPa, ν = 0.35, load 40 kN and radius 0.15 m. Candidate moduli from 100 to 400 MPa produce alternative vectors at offsets 0, 0.3, 0.6, 0.9 and 1.2 m. The root-mean-square residual is RMSE = √[Σ(wmodel − wreference)² / 5] in μm.

The residual reaches zero at 200 MPa because the reference was generated by that same model without measurement noise. That recovery is a numerical consistency demonstration. Real data can contain noise, model mismatch and multiple parameter combinations with similar residuals; a small fitting error alone does not validate a layer interpretation. Preserve those distinctions when connecting backcalculation to rehabilitation investigation. (Rocscience, n.d.)

Calculation laboratory

Can a perfect fit still be an incomplete explanation?

Calculation step	Substitution or result	Units and interpretation
Reference generation	wref = w(E = 200 MPa)	five synthetic μm readings
Candidate scaling	w(100 MPa) = 2 wref	same load and geometry
Residual vector	ri = wi(E) − wi,ref	μm
Objective	RMSE = √(Σri² / 5)	μm; zero at 200 MPa

This is a closed numerical experiment: the reference and candidate predictions use the same homogeneous half-space equation. Recovering the generating modulus demonstrates internal consistency. It does not establish that a field pavement behaves as a homogeneous half-space, nor that an inverse solution is unique when several layer properties are allowed to vary.

Residuals should remain visible after the scalar objective is calculated. A small overall RMSE can conceal a systematic shape mismatch, particularly if large central deflections dominate the scale. Conversely, normalising every residual by its reading changes the objective and gives relatively more influence to small outer readings. A weighted fit can be appropriate when justified measurement uncertainties are known, but the weights must not be chosen merely to make the fit look better.

In this fixed-geometry model, displacement depends on P/E. A 10% load error can therefore be absorbed by a 10% modulus adjustment while preserving the predicted bowl. The data alone cannot separate those two errors without independent load information. Investigation should address such ambiguities through test context, layer evidence and alternative hypotheses. TRH12 frames rehabilitation as investigation, evaluation and design; an optimisation minimum is one piece of that argument. (Department of Transport, 1997)

Controlled sensitivity experiment

Controlled case	Calculated output	What changes
E = 100 MPa	348.360629 μm RMSE	Reference generated at 200 MPa; vary one candidate modulus.
E = 150 MPa	116.120210 μm RMSE	Reference generated at 200 MPa; vary one candidate modulus.
E = 200 MPa	0.000000 μm RMSE	Reference generated at 200 MPa; vary one candidate modulus.
E = 300 MPa	116.120210 μm RMSE	Reference generated at 200 MPa; vary one candidate modulus.
E = 400 MPa	174.180314 μm RMSE	Reference generated at 200 MPa; vary one candidate modulus.

CALCULATION LABORATORY

Can a perfect fit still be an incomplete explanation?

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

Sensitivity case1. E = 100 MPa2. E = 150 MPa3. E = 200 MPa4. E = 300 MPa5. E = 400 MPa

E = 100 MPa: 348.360629 μm RMSE. Reference generated at 200 MPa; vary one candidate modulus.

Reproduce and challenge the result

Plot signed residuals by sensor offset as well as RMSE. Perturb the assumed load and test whether the fitted modulus follows it. Keep synthetic recovery tests separate from validation against independent field measurements.

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.

Walk along the valley of equally good fits

The reference bowl uses 200 MPa and 40 kN at offsets 0, 0.3, 0.6, 0.9 and 1.2 m. Linear elasticity scales every displacement with P/E. Evaluate 6,161 input pairs and a low-error valley follows P/E = 0.2 kN/MPa. A measured load removes one source of ambiguity; letting both load and stiffness float creates it.

Reproduce the experiment. RMSE(E,P) = RMS(w_ref)·| (P/40)(200/E) − 1 |. The downloadable table contains 6,161 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 · 6,161 cells

A perfect fit can still leave a question

If two different input pairs draw the same bowl, which pavement have you identified?

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

One homogeneous elastic parameter, noise-free synthetic reference and fixed radius/Poisson ratio. This is an identifiability demonstration, not multilayer backcalculation or proof that actual field loads are unknown.

References and method notes

1. 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
2. Rocscience (n.d.). Settle3: Settlement and Consolidation Theory Manual. Theory manual. Original source (Accessed: 5 September 2026).Section 2.3.1: Boussinesq stress kernel and integration over a loaded area
3. University of Cambridge (2005). Engineering Tripos Part IIA: 3D5 Geotechnical Engineering. Examination paper and formula information, 2005. Original source (Accessed: 5 September 2026).Data sheet Sections 3.1 and 3.3, printed pp. 6 and 8: circular-load stress and surface settlement

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