Road-surface noise: the statistical pass-by method

A road surface makes no noise of its own, yet the same traffic can be up to 15 dB louder on one surface than on another (the Introduction of ISO 11819-1 gives that range). Texture and porosity both change how much noise the tyres make, and porosity also changes how it propagates close to the ground. ISO 11819-1 is the method that measures this, and ranks surfaces by it. It does so without choosing any vehicles: it measures the traffic that happens to use the road, one isolated vehicle at a time, and lets statistics take the vehicles out.

One site, three categories of vehicle, three straight lines in the logarithm of speed, and the three points on them that the index is made of.

Every vehicle measured is put in one of three categories (3.4): cars (1), dual-axle heavy vehicles (2a), trucks, buses and coaches with two axles and more than four wheels, and multi-axle heavy vehicles (2b), those with more than two axles. Vans, cars with trailers and motorcycles are not used, because they add nothing about the surface (clause 4), and a vehicle that cannot be classified without doubt is discarded (7.1).

The road is put in one of three speed categories (3.3), by the average speed of its traffic: low, 45 km/h to 64 km/h, usually urban; medium, 65 km/h to 99 km/h, suburban roads and rural highways; high, cars at 100 km/h or more, usually motorways. Each speed category fixes a reference speed per vehicle category, the speed every level is brought to, and the share each category is given in the index (Table 1).

The table is environment.SPB_REFERENCE_SPEEDS_KMH and environment.SPB_WEIGHTING_FACTORS, keyed by road speed category and then by vehicle category.

One microphone beside one lane of real traffic: every vehicle that passes on its own gives one point, its category, its speed and its maximum level.

For each vehicle category the pass-bys are fitted by least squares with a straight line in the logarithm of speed (9.1),

and the line is read at the reference speed of Table 1. That ordinate is the vehicle sound level $L_{\mathrm{veh}}$ of the category (9.2): what a typical vehicle of it emits at that speed on this surface. The logarithm of speed is the axis the clause fits in, which is why the figure draws speed on a logarithmic scale, where each cloud is a band and its fit a straight line.

import numpy as np
from phonometry import environment

rng = np.random.default_rng(11819)
# A medium-speed site: per category, the number of pass-bys, the mean speed
# (km/h), the spread of lg v, and the line and scatter (dB) they are drawn about.
site = {
    "1": (120, 88.0, 0.055, 16.6, 32.6, 1.4),
    "2a": (40, 76.0, 0.050, 46.5, 18.8, 2.0),
    "2b": (50, 74.0, 0.045, 34.5, 26.7, 2.0),
}
categories, speeds, levels = [], [], []
for category, (n, mean_kmh, spread, a, b, scatter) in site.items():
    lg_v = np.log10(mean_kmh) + spread * rng.standard_normal(n)
    categories += [category] * n
    speeds += (10.0**lg_v).tolist()
    levels += (a + b * lg_v + scatter * rng.standard_normal(n)).tolist()

result = environment.statistical_pass_by(
    categories, speeds, levels, road_speed_category="medium"
)
print(dict(result.reported_vehicle_sound_levels_db))
# {'1': 78.9, '2a': 81.0, '2b': 83.7}   dB, at 80, 70 and 70 km/h
cars = result.regressions["1"]
print(round(cars.slope_db_per_decade, 2), round(cars.correlation, 2))
# 28.22 0.75
print(round(cars.residual_standard_deviation_db, 2), round(cars.mean_speed_kmh, 1))
# 1.4 86.9

Each regression carries what clause 13 asks to be reported about it (item 29): the slope and the intercept, the standard deviation of the speeds and of the residuals about the line, and the mean speed. The mean speed is $10^{\overline{\lg v}}$, the mean of the variable the line is fitted in turned back into km/h, which is also what Annex E prints under “Value converted from the logarithm of speed”. The standard deviation of the speeds is given in the same variable, lg_speed_standard_deviation, because that is the one the line is fitted in and the one 9.3 is judged in.

The line is only read at a speed the measured traffic actually covers: the shaded window is where 9.3 allows the reference speed to lie.

import matplotlib.pyplot as plt

result.plot()                    # the three categories and the index
result.regressions["1"].plot()   # one category, its window and L_veh
plt.show()

Two conditions, judged and not enforced. A regression is only a fair way to move levels to a reference speed if the traffic was near that speed. Clause 9.3 therefore asks the reference speed to lie within one standard deviation of the measured mean speed for heavy vehicles and within one and a half for cars. The library judges it in the variable of the fit, $\overline{\lg v}\pm k{s}_{\lg v}$, reports the window in km/h as speed_window_kmh and the verdict as reference_speed_in_window. Clause 7.3 asks for 100 cars, 30 vehicles of each heavy category and 80 heavy vehicles in all when the purpose is to classify the surface. Both are kept on the result and, when either fails, emitted as StatisticalPassByWarning rather than raised: a before-and-after study of one site (6.6) is still a pass-by measurement, just not a classification.

The three vehicle sound levels are added on an energy basis into one number, the Statistical Pass-By Index (9.5):

The weights are the assumed proportions of the three categories, and the speed ratios are there because the index stands for the energy of a flow: a lorry at 70 km/h is heard for longer than a car at 80 km/h, in proportion to the ratio of the two speeds. The index is therefore close to the mean of the three levels but is not an equivalent level of any real traffic (9.5 NOTE); it compares surfaces under a standard mix and does not predict what a street will measure.

print(round(result.index_db, 3), result.reported_index_db)
# 80.14 80.1   dB

The index adds the vehicle sound levels as 9.2 reports them, to one decimal: 9.5 defines $L_1$, $L_{2a}$ and $L_{2b}$ as the vehicle sound levels “according to 9.2”, and 9.2 has every level “calculated to two decimal places and rounded to one decimal place”. The regressions are carried at full precision and each level is rounded once, half up (reported_vehicle_sound_levels_db, 78,9, 81,0 and 83,7 dB here); index_db is the index of those three levels, rounded once more when it is reported. The index of the unrounded levels stays on the result as full_precision_index_db, 80,153 dB for this site.

Table 1 is the standard mix. 9.5 recognises that national mixes differ and allows other factors, provided they are stated; weighting_factors takes them and refuses factors that are not proportions (negative, or not adding up to 1). Because every $L_{\mathrm{veh}}$ has to be reported, anyone can recompute the index for a mix of their own from a report, with statistical_pass_by_index.

In many cases the index is used as a difference from a reference surface (9.5, clause 10), and clause 10 offers four: a real dense asphalt concrete of 11 mm to 16 mm chippings, at least a year old, with a macrotexture depth of 0,50 mm to 1,00 mm measured by ISO 10844 or ISO 13473-1 and its air voids or absorption within the requirements of ISO 10844 (10.1, the general case); a normalized reference case (10.2), a fictitious surface whose levels are set by convention, usually the average of many such surfaces; the same general surface at the same age as the one tested (10.3); and any surface the tester chooses (10.4), which only compares the surfaces measured together.

Annex D shows how a normalized case is built, from seven dense bituminous surfaces for the medium speed range (four asphalt concrete and three stone mastic asphalt), and prints their average. normalized_reference_levels averages any set of surfaces the same way, and the Annex D rows and their average are published as SPB_ANNEX_D_SURFACES_DB and SPB_NORMALIZED_REFERENCE_DB. Hand the reference to statistical_pass_by as an index, used as given, or as its three levels, whose index is computed from them to one decimal with the same weighting factors:

result = environment.statistical_pass_by(
    categories,
    speeds,
    levels,
    road_speed_category="medium",
    reference_db=environment.SPB_NORMALIZED_REFERENCE_DB,
)
print(round(result.reference_index_db, 2), round(result.difference_db, 2))
# 78.92 1.22   dB: this surface is 1.2 dB louder than the Annex D reference

The Annex D surfaces are an example of how to build a reference, not a reference the standard sets: the annex asks that the surfaces behind every version of a reference be kept on record, so that a difference quoted against it can be traced.

Tyre-road noise depends on temperature, and 9.4 asks for the vehicle sound levels to be corrected to 20 °C air, but says in the same clause that “a suitable method is at present under consideration” and makes only the uncorrected levels mandatory. No correction is invented here. Levels corrected by a method of your choosing go in as corrected_vehicle_sound_levels_db, and the index is computed for them too, from the levels rounded to one decimal (reported_corrected_vehicle_sound_levels_db, corrected_index_db). Clause 13 lists the corrected levels and index as optional report items (26 and 28) beside the mandatory uncorrected ones (25 and 27). Where the air temperature was taken at every pass-by, 9.4 prefers to correct each measured level instead; correct max_levels_db and call the method again.

Table 2 gives the random error pre-normative research found: individual vehicles scatter about $L_{\mathrm{veh}}$ by 1,5 dB for cars and 2,0 dB for heavy vehicles, which leaves a 95 % confidence interval of 0,3 dB and 0,7 dB on the vehicle sound level for 100 cars and 40 heavy vehicles of each type (SPB_VEHICLE_STANDARD_DEVIATIONS_DB, SPB_CONFIDENCE_INTERVALS_DB). They describe the medium and high speed categories; the low one is sensitive to acceleration and should be treated with caution (9.6).

A measurement also carries its own interval. confidence_interval_db is the 95 % interval of the fitted line at the reference speed, from the Student distribution with $n-2$ degrees of freedom, which widens as the reference speed moves away from the mean speed. The standard says the random error of the index is “a combination of these errors according to the chosen weighting factors” and gives no formula; index_confidence_interval_db combines the three intervals through the sensitivity of the index to each level, $\partial \mathrm{SPBI}/\partial L_x$, taking the three categories as independent.

print([round(r.confidence_interval_db, 2) for r in result.regressions.values()])
# [0.3, 0.71, 0.69]   dB, cars and the two heavy categories
print(round(result.index_confidence_interval_db, 2))
# 0.27   dB

Beyond the random error, 9.6 lists the systematic ones: about ±1 dB for a precision instrument system, and 0,3 dB to 0,8 dB for the variation of the vehicle fleet within a country. Annex C adds that results should not be expected to stay valid for more than about five years, as vehicles and tyres change.

Annex E is a complete test report for a Swedish single surface dressing on a medium-speed road. It prints each category’s regression line, the vehicle sound levels 78,5, 81,1 and 83,8 dB, the index 79,9 dB, the index of the temperature-corrected levels (78,8, 81,1 and 83,8 dB) 80,1 dB, and a temperature-corrected difference of 2,8 dB, 80,1 dB less the 77,3 dB of its reference surface. The conformance suite reproduces every one of these numbers, the uncorrected index from pass-bys placed on the printed lines.

Which levels the index adds is settled by the clauses, not left to the example. 9.2 ends “All levels shall be calculated to two decimal places and rounded to one decimal place”, and 9.5 defines the levels of its formula as “the Vehicle Sound Levels for vehicle categories 1, 2a and 2b according to 9.2”: the index adds the levels as 9.2 reports them, to one decimal. That is also what lets anyone recompute it from a report, which 7.4 (“The sound levels Lveh and the SPBI calculated from them”) and 9.5 (the mandatory reporting of every $L_{\mathrm{veh}}$ “allows such SPBI calculations to be made”) rely on. The library follows that chain, and so does the example: its printed levels give 79,946 dB, the 79,9 it prints.

import math

printed = {"1": 78.5, "2a": 81.1, "2b": 83.8}
print(round(environment.statistical_pass_by_index(printed, road_speed_category="medium"), 3))
# 79.946   the levels of 9.2, to one decimal: Annex E prints 79.9
lines = {"1": (16.6, 32.55, 80.0), "2a": (46.5, 18.76, 70.0), "2b": (34.5, 26.74, 70.0)}
unrounded = {k: a + b * math.log10(v) for k, (a, b, v) in lines.items()}
print(round(environment.statistical_pass_by_index(unrounded, road_speed_category="medium"), 3))
# 79.985   the same levels left unrounded: it would print 80.0

Carried unrounded from the printed lines, the levels are 78,546, 81,114 and 83,838 dB and their index 79,985 dB, which would print 80,0; the result keeps that chain as full_precision_index_db. The lines alone cannot tell the two chains apart, because their intercepts are printed to one decimal and their slopes to two: over every line that agrees with the intercept, slope, mean level, mean speed and vehicle sound level Annex E prints, to their last digit, the unrounded index runs from 79,948 dB to 79,996 dB. What the page does show is that the index is rounded once: 79,946 dB taken first to two decimals, 79,95, and then to one would print 80,0 as well. The two chains never differ by more than 0,05 dB, the most a level moves when it is rounded, which is below any precision the method claims but can move the last printed digit. The conformance row that pins 79,9 dB reads index_db of the run from pass-bys.

Two more things about Annex E are worth knowing before it is used as a check. Its reference index of 77,3 dB is described as a normalized reference case of seven surfaces, three stone mastic asphalt and four asphalt concrete, which is the composition of Annex D; but the Annex D levels give an index of 78,92 dB with the same weights, not 77,3, and the reference box gives a maximum chipping size of 12 mm to 16 mm where Annex D lists a surface with 11 mm chippings. The standard does not say the two are the same database, so the library treats 77,3 dB as the input it is printed as. And its speed spreads (13,3, 7,5 and 6,4 km/h) do not fit the slopes, correlations and level spreads printed beside them; the errata registry has the arithmetic. The vehicle sound levels depend on neither.

• Covered

  The computation of ISO 11819-1:1997 from a list of measured pass-bys: pass_by_regression (9.1, 9.2 and the 9.3 window), statistical_pass_by (the three categories, the 7.3 counts, the index of 9.5 from the levels of 9.2 to one decimal, temperature-corrected levels as an input, and the difference from a reference surface given as an index or as levels), statistical_pass_by_index for levels already reported, and normalized_reference_levels for a reference built as in 10.2 and Annex D. Tables 1 and 2, the 7.3 counts and the 9.3 factors are published as read-only tables, and both results draw themselves with .plot().

• Not covered

  The measurement itself: telling a vehicle’s category, screening pass-bys for the 6 dB dips of 7.2, the site and weather requirements of clauses 6, 11 and 12, which are documented above as conditions and are the tester’s. A temperature correction, which the standard does not give. The Close-Proximity method of ISO 11819-2, which works with reference tyres rather than with the traffic (Annex C, NOTE 3). Later editions of ISO 11819-1 were not consulted; this page implements the 1997 text.