VCI mathematics: from defect ratings to a reproducible index

Begin with the mathematical definition

Why does a one-step change in a rating sometimes move the index much further than the previous step?

A Visual Condition Index is a transformation of assessed defect ratings. To explain a result, we need the contribution of each defect, the normalising denominator and the final mapping. Counting completed assessments does not perform that calculation.

This worked study uses the modified method in Technical Addendum A.3 of Draft TRH22 (1994). It is a specific historical method variant, with its own weights, extent exponents and small-degree factors. It must not be silently substituted for another adopted VCI method. The equations and numerical table were checked against the scanned source pages, including superscripts that are missing from some text extractions. (CSRA, 1994)

Define the inputs before aggregating them

For defect n, Dₙ is its degree rating, Eₙ its extent rating, Yₙ the exponent applied to extent, Wₙ its weight and Sₙ the applicable small-degree factor. These ratings are dimensionless categories; E = 3 does not mean three percent. The degree and extent assessment needs to follow the selected assessment procedure. (COTO, 2016b)

In this variant, the contribution is Fₙ = min(Dₙ × Eₙ^Yₙ, Mₙ) × Wₙ × Sₙ. The cap M is 25 for other defects and 12 for functional defects. Functional degree is on 0–4 with extent fixed at 3; other degree and extent ratings are on 0–5. S takes its tabulated small-degree value at degree ≤ 1 for functional items or degree ≤ 2 for other items; above that boundary S = 1. (CSRA, 1994)

The printed formula gives Dₙ × Eₙ^Yₙ and separately states maxima of 25 and 12. This demonstrator enforces those maxima as the explicit min operation above. The four worked inputs stay below the caps, so their reported result does not depend on that boundary interpretation. Keep the cap policy in any implementation record.

The exponent matters. 3^1.3 = 4.17117; it is not 3 × 1.3 = 3.9. The order of operations also matters: cap the degree–extent term before multiplying by W and S. Mixing those operations produces a different index.

TRH22 A.3 · worked mathematics

VCI: ratings to a calculated index

Specified Draft TRH22 (1994) A.3 variant. Confirm the adopted method and assessment mapping for each project.

Download example data (CSV) · Full normalisation ledger (CSV)

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

Understand the model

VCI transforms a complete set of rated defects into an index. The defect contributions, fixed denominator and nonlinear transformation are separate calculation stages.

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

D; E

Degree and extent categories (dimensionless); E is not percent coverage.

Y; W; S

Extent exponent, defect weight and applicable small-degree factor.

M; Fn

Cap on the degree–extent term; weighted defect contribution.

K; P; VCI

Full-list normalisation; preliminary index; transformed final index.

Values used in this film

• D: degree category; E: extent category, not percentage.
• Y: extent exponent; W: defect weight; S: small-degree factor.
• M: cap on the degree–extent term (25 other, 12 functional).
• K: full-list normalisation, 2534.1666667; P: preliminary index.

The mathematical relationships

1. Equation 1

   Source and scope
   $$F_n=\min(D_n E_n^{Y_n},M_n)W_nS_n$$

   Apply the extent exponent before multiplying by degree, cap the result, then apply weight and the applicable small-degree factor. Functional degree is 0–4 with E = 3; other ratings are 0–5. S uses its tabulated value at D ≤ 1 for functional items or D ≤ 2 for other items; above that boundary S = 1.

2. Equation 2

   Source and scope
   $$\begin{gathered}K=\sum_n F_{n,\max}\\P=100\left(1-\frac{\sum_n F_n}{K}\right)\end{gathered}$$

   Keep K fixed for the complete declared item list, including assessed-absent items. Average alternative normalising weights rather than counting alternatives as separate defects.

3. Equation 3

   Source and scope
   $$VCI=(0.04P+0.0006P^2)^2$$

   Apply both powers in the transformation. P = 50 gives VCI = 12.25, so P and VCI must not be treated as interchangeable scales.

Work through the example

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

1. Cracking

   3 × 3^1.3 × 10 = 125.13503

   weighted contribution

2. Rutting

   2 × 3 × 8 × 0.5 = 24

   weighted contribution

3. Potholes and riding quality

   36.93433 + 33 = 69.93433

   weighted contributions

4. Full-list denominator

   25 × 94.1666667 + 12 × 15 = 2534.1666667

   normalisation

5. Preliminary index

   100 × (1 − 219.0693577 / 2534.1666667) = 91.35536898

   P

6. Transformation

   (0.04P + 0.0006P²)² = 75.02499188

   VCI

Read the output

The contribution sum 219.0693577 gives P = 91.35536898 and VCI = 75.02499188. P is an intermediate value; VCI is the transformed output.

Check your understanding

At P = 0, 50 and 100, VCI is 0, 12.25 and 100. Changing rutting degree from 2 to 3 switches S from 0.5 to 1, changes its contribution from 24 to 72 and gives VCI = 70.23177.

Assumptions and limits

Specified Draft TRH22 (1994) A.3 variant. Confirm the adopted method and assessment mapping for each project.

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

Timestamped film transcript 20 short entries · select a time to revisit the film

1. 0:00

   VCI: ratings to a calculated index. Four assessed defects; all other items assessed absent. K = 2,534.1667 for the full default item list.

2. 0:01

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

3. 0:07

   D: degree; E: extent rating; Y: extent exponent; W: defect weight. S: small-degree factor; M: cap (25 other, 12 functional); K = 2,534.1667. Selected method: Draft TRH22 (1994), Addendum A.3; all other defects assessed absent.

4. 0:14

   INPUTS · blue is degree D; mustard is extent E. Both are ratings; neither is a percentage.

5. 0:26

   Extent is a rating, not a percentage. Functional extent is fixed at 3. Rutting at D = 2 uses S = 0.5.

6. 0:33

   CALCULATION · raise the extent rating to Y, apply the cap, then multiply by W and S.

7. 0:46

   INTERMEDIATE OUTPUT · weighted defect contributions · read the axis units and the numerical values.

8. 0:55

   Sum F = 219.0694; K = 2534.1667. K includes all items, not only these four.

9. 1:13

   OUTPUT TRANSFORMATION · preliminary P to final VCI · read the axis units and the numerical values.

10. 1:22

    The mapping is nonlinear: P = 50 gives VCI = 12.25; P = 100 gives 100. P and VCI are not interchangeable.

11. 1:27

    SENSITIVITY · change rutting degree only · read the axis units and the numerical values.

12. 1:36

    D = 2 → 3 removes the small-degree reduction: rutting contribution rises 24 → 72; VCI falls 75.02 → 70.23.

13. 1:51

    ONE ROW · Why does a one-step change in a rating sometimes move the index much further than the previous step?

14. 2:04

    PARAMETER MAP · 36 evaluated cells. The colour limits stay fixed across every row.

15. 2:14

    All 36 cells are valid integer combinations of rutting degree and extent. Other active contributions remain cracking 125.13502533, potholes 36.93433240 and riding quality 33; K

16. 2:23

    remains 2,534.1666667. The final transform is VCI = (0.04P + 0.0006P²)². Crossing degree two changes the small-degree factor as well as D. Follow one extent row to separate those

17. 2:34

    effects.

18. 2:39

    VALUE LANDSCAPE · Height and colour encode Final VCI. This is a graph over the two input axes, not automatically a physical pavement shape.

19. 2:51

    Discrete draft TRH22 A.3 example with the method interpretation described above. No fractional ratings or interpolated response surface is implied. Other defects are explicitly assessed absent, not missing.

20. 2:59

    Specified Draft TRH22 (1994) A.3 variant. Confirm the adopted method and assessment mapping for each project.

Download the timed captions (VTT)

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

Specify a complete example

Four defects are active in this synthetic example. All remaining items in the declared list are assumed to have been assessed and found absent. They are not missing observations. No project data is used.

Active defect	D	E	Y	W	S applied	F
Crocodile cracking	3	3	1.3	10	1	125.13503
Rutting	2	3	1.0	8	0.5	24.00000
Failures/potholes	1	2	1.3	15	1	36.93433
Riding quality	2	3	1.0	5.5	1	33.00000

For rutting, min(2 × 3^1, 25) × 8 × 0.5 = 24. For cracking, min(3 × 3^1.3, 25) × 10 = 125.13503. Riding quality uses S = 1 because its functional degree of 2 is above the small-degree boundary of 1. The contribution sum is 219.06936; contributions are intermediate index arithmetic, not percentages of damaged pavement.

Fix the denominator independently of the observed defects

Let K = ΣFₙ(max) and C = 1/K. The normalisation covers the complete declared item list. Removing absent defects from K would make the index depend on which rows happen to be displayed.

For the default items used here, the sum of the non-functional normalising weights is 94.1666667, and the functional sum is 15. The former includes the average weight where alternative categories are provided: aggregate loss (4 + 2)/2 = 3; block/stabilisation cracks (8 + 6 + 5)/3 = 6.3333333; longitudinal/slip cracks (4.5 + 4.5 + 4)/3 = 4.3333333; patching (8 + 8 + 8)/3 = 8. This follows the source's normalisation convention for alternative weights. (CSRA, 1994)

Thus K = 25 × 94.1666667 + 12 × 15 = 2,534.1666667. Zero-weight texture and voids entries contribute zero. The downloadable full item ledger records all 21 entries, their normalising weights, maximum terms and example contributions. Do not replace those alternative averages with the sum of every alternative row; that would count one distress several times.

Calculate the preliminary and final outputs

The preliminary index is P = 100(1 − ΣF/K). Substitution gives P = 100(1 − 219.0693577 / 2,534.1666667) = 91.35536898.

The A.3 transformation is VCI = (0.04P + 0.0006P²)². Substituting P gives VCI = 75.02499188, reported as 75.02. Keep full precision until the reporting step. Rounding every contribution first introduces avoidable disagreement between implementations.

The two outputs have different meanings. P is the intermediate normalised result; 75.02 is the transformed VCI in this example. At P = 50, the transformation gives (2 + 1.5)² = 12.25, not 50. At P = 0 and P = 100 it gives 0 and 100 respectively. These endpoint checks catch missing powers and misplaced constants, but do not by themselves validate an implementation.

Change an input and follow the consequence

Hold every other input fixed and increase rutting degree from 2 to 3. The small-degree factor changes from 0.5 to 1. The rutting contribution therefore rises from 24 to 72, rather than from 24 to 36. The total contribution becomes 267.06936, P becomes 89.46126 and VCI becomes 70.23177.

Controlled input experiment · TRH22 A.3

Change the rating and inspect the calculation

Other defect ratings and K = 2,534.1666667 remain fixed. The line connects discrete rating cases; it does not define fractional ratings.

Rutting degree DRutting extent rating E

D = 2, E = 3, S = 0.5. Rutting F = 24.00; ΣF = 219.06936; P = 91.35537; VCI = 75.02499.

Rutting degree	S	Rutting contribution	Final VCI
1	0.5	12	76.25977
2	0.5	24	75.02499
3	1	72	70.23177
4	1	96	67.92110
5	1	120	65.66652

The film plots this sensitivity explicitly. The large change at degree 3 comes from a rule boundary, not from arbitrary animation timing. Ratings are discrete; intermediate frames must not be interpreted as valid degree 2.4 observations.

For a smooth change in the preliminary index, the transformation has derivative dVCI/dP = 2(0.04P + 0.0006P²)(0.04 + 0.0012P). A change in contribution changes P by ΔP = −100ΔF/K. These relationships explain why equal contribution increments do not necessarily produce equal changes in final VCI. The discrete S switch must still be applied before using that local sensitivity argument.

Interpret the result without losing its evidence

The result describes the selected index calculation for the assumed ratings. It does not specify a rehabilitation treatment, establish structural capacity or recover the spatial arrangement of distress. The full calculation record should retain the assessment, method variant, chosen category for each alternative item, missing-data state, denominator, intermediate contributions and rounding rule.

Do not replace an unassessed item with degree zero. This worked example can use zero for the remaining contributions because absence was explicitly assumed. A partially observed section needs the project's approved missing-data policy before a reportable index is calculated. That distinction keeps a numerical result connected to the evidence it claims to summarise.

Read all thirty-six rating combinations

All 36 cells are valid integer combinations of rutting degree and extent. Other active contributions remain cracking 125.13502533, potholes 36.93433240 and riding quality 33; K remains 2,534.1666667. The final transform is VCI = (0.04P + 0.0006P²)². Crossing degree two changes the small-degree factor as well as D. Follow one extent row to separate those effects.

Reproduce the experiment. F = min(DE,25)·8·S; S=.5 if D≤2, else 1; P=100(1−ΣF/K). The downloadable table contains 36 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 · 36 cells

The jump belongs to the rule

Why does a one-step change in a rating sometimes move the index much further than the previous step?

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

Discrete draft TRH22 A.3 example with the method interpretation described above. No fractional ratings or interpolated response surface is implied. Other defects are explicitly assessed absent, not missing.

Case review

One rating step crosses a rule boundary

Why does rutting degree 2 → 3 cause a larger index change?

Degree 1 · 76.26 Final VCI

The discriminating check

Confirm the method variant, integer ratings, assessed-absent versus missing states, and fixed normalisation.

Evidence supports the comparison

With a complete compatible record, show the contribution change, preliminary index and final transform together.

Evidence is missing or incompatible

With an unassessed item, apply the approved missing-data policy instead of silently substituting degree zero.

What you can conclude. The index change follows this rule set. It neither proves a physical step-change in the pavement nor prescribes rehabilitation.

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

References and method notes

1. CSRA (1994). Draft TRH22: Pavement Management Systems. Pretoria: Department of Transport, 1994. Original source (Accessed: 5 September 2026).Technical Addendum A.3, equations A.3.1–A.3.3 and Table A.3.1, printed pp. 125–126; Section 6.2.2, printed pp. 55–56, for alternative-weight normalisation
2. COTO (2016b). TMH9: Manual for Visual Assessment of Road Pavements. Part A: General. Committee Draft Final, May 2016. Original source (Accessed: 5 September 2026).Section A.2: degree and extent

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