Interpreting exposure: dose, TWA and Lex,8

Lesson 4 described how noise is measured. This lesson is about what the numbers mean. A worker's day is rarely spent at one steady level; it is a mixture of loud tasks, quieter periods, breaks and sometimes overtime. Regulations compare that mixture with a single limit, so the measurements have to be combined into one figure: a dose expressed as a percentage, a time-weighted average or an 8-hour normalized level. Understanding how those figures are calculated lets you check reports, see which tasks drive the dose and test what-if scenarios before spending money on controls.

Noise dose

The noise dose expresses a worker's exposure as a percentage of the daily allowance. A dose of 100 percent means the worker received exactly the amount allowed by the limit; 200 percent means twice the allowance. For a day made up of periods at different levels, each period uses up a fraction of the allowance equal to the time spent divided by the time allowed at that level:

Here C is the time actually spent at each level and T is the allowable time at that level from the formula in Lesson 3. Because the allowable time depends on the criterion level and exchange rate, the same day can give quite different doses under different rules.

Converting dose to a level

A percentage is useful but people more often want a level that can be compared directly with the limit. With a 3 dB exchange rate, the 8-hour equivalent level, called Lex,8 or LEX,8h in Canada and ISO standards, is:

So a dose of 258 percent with an 85 dBA criterion gives 85 + 10 log10(2.58) ≈ 89.1 dBA. Lex,8 can also be calculated directly from the levels and durations, by summing the energy of each period and normalizing to 8 hours:

with each duration t in hours. Both formulas give the same answer for a 3 dB exchange rate.

For OSHA's 5 dB exchange rate, the corresponding figure is the time-weighted average, TWA, given in Appendix A of 1910.95:

A dose of 100 percent is a TWA of 90 dBA, the PEL, and a dose of 50 percent is a TWA of 85 dBA, the action level. The constant 16.61 is 5 divided by log10(2). For comparisons with the NIOSH REL, use the 3 dB form, Lex,8 = 85 + 10 log10(D ÷ 100).

Doing the calculation in code

A short script makes it easy to compare rules, check a report or test the effect of a control. The function below calculates dose and the equivalent level for any criterion, exchange rate and threshold.

import math

def dose(segments, criterion=85, exchange=3, threshold=None):
    """segments: list of (hours, dBA). Returns the dose as a percentage."""
    d = 0.0
    for hours, level in segments:
        if threshold is not None and level < threshold:
            continue
        allowed = 8 / 2 ** ((level - criterion) / exchange)
        d += hours / allowed
    return 100 * d

def level_from_dose(d, criterion=85, exchange=3):
    q = exchange / math.log10(2)
    return criterion + q * math.log10(d / 100)

day = [(3, 92), (2, 88), (3, 82)]

d_ca = dose(day, 85, 3)
print(f"85/3 rule:   dose {d_ca:5.0f} %, Lex,8 {level_from_dose(d_ca, 85, 3):.1f} dBA")
d_fed = dose(day, 87, 3)
print(f"87/3 rule:   dose {d_fed:5.0f} %, level {level_from_dose(d_fed, 87, 3):.1f} dBA")
d_pel = dose(day, 90, 5, threshold=90)
print(f"OSHA PEL:    dose {d_pel:5.0f} %, TWA {level_from_dose(d_pel, 90, 5):.1f} dBA")
d_al = dose(day, 90, 5, threshold=80)
print(f"OSHA action: dose {d_al:5.0f} %, TWA {level_from_dose(d_al, 90, 5):.1f} dBA")

85/3 rule:   dose   258 %, Lex,8 89.1 dBA
87/3 rule:   dose   162 %, level 89.1 dBA
OSHA PEL:    dose    49 %, TWA 84.9 dBA
OSHA action: dose    81 %, TWA 88.5 dBA

The same day is well over the limit under the Canadian provincial rule, over the federal Canadian rule, under the OSHA PEL and over the OSHA action level. That is not a contradiction; it shows how much the rule chosen matters.

Which tasks drive the dose?

Dose calculations are most useful when they show where the dose comes from. Because allowable time halves with each 3 dB, short periods at high levels can dominate. A worker who spends 7 hours at a comfortable 82 dBA and only 30 minutes using a cut-off saw at 105 dBA receives most of the day's dose from the saw.

import math

def allowed(level, criterion=85, exchange=3):
    return 8 / 2 ** ((level - criterion) / exchange)

tasks = [
    ("bench assembly", 7.0, 82),
    ("cut-off saw", 0.5, 105),
    ("break room", 0.5, 65),
]
total = sum(h / allowed(L) for _, h, L in tasks)
print(f"Total dose: {100 * total:.0f} %  (Lex,8 {85 + 10 * math.log10(total):.1f} dBA)")
for name, h, L in tasks:
    share = (h / allowed(L)) / total
    print(f"  {name:<15} {h:4.1f} h at {L} dBA -> {share:6.1%} of dose")

# What if the saw time is halved, or the saw is quietened by 6 dB?
for label, saw in [("saw time halved", (0.25, 105)), ("saw 6 dB quieter", (0.5, 99))]:
    t = 7.0 / allowed(82) + saw[0] / allowed(saw[1]) + 0.5 / allowed(65)
    print(f"{label:<17} dose {100 * t:4.0f} %, Lex,8 {85 + 10 * math.log10(t):.1f} dBA")

Total dose: 679 %  (Lex,8 93.3 dBA)
  bench assembly   7.0 h at 82 dBA ->   6.4% of dose
  cut-off saw      0.5 h at 105 dBA ->  93.5% of dose
  break room       0.5 h at 65 dBA ->   0.0% of dose
saw time halved   dose  361 %, Lex,8 90.6 dBA
saw 6 dB quieter  dose  203 %, Lex,8 88.1 dBA

Half an hour on the saw delivers more than nine-tenths of the dose. Quietening the saw by 6 dB, for example with a better blade, a water feed or a shroud, does more than halving the time spent on it. Results like these give supervisors a clear, quantified case for engineering controls.

Shifts longer than 8 hours

Exposure limits are written for an 8-hour day, but many workers do 10- or 12-hour shifts. The calculations above handle this automatically, because the dose adds up all the time actually spent. A worker at a steady 85 dBA for 12 hours has a dose of 150 percent and a Lex,8 of 86.8 dBA, over the limit, even though the level never exceeded 85 dBA. Forgetting to account for the extra hours is a common reason why long-shift workers are underprotected.

OSHA's dose formula covers any shift length, so a 12-hour shift at a steady 90 dBA is a dose of 150 percent and a TWA of about 92.9 dBA, over the PEL. Long shifts also reduce the quiet recovery time between exposures, which is another reason to aim for exposures well below the limit for those workers.

Uncertainty and variation

Noise exposure varies from day to day with production, maintenance, weather (doors open or closed) and the mix of tasks. A single day's measurement is a sample, not the truth. Measurement standards such as ISO 9612 and CSA Z107.56 describe how to estimate uncertainty, which comes from the instrument, microphone position, sampling and day-to-day variation. Typical expanded uncertainties for a careful occupational assessment are in the range of 1.5 to 3 dB.

The practical consequences are straightforward. A result just below the limit does not show compliance with confidence. Many programs treat any result within about 3 dB of the limit as potentially over it, add those workers to the hearing conservation program and measure again. For job groups, several samples on different days give a better estimate, and the group's energy-average level, not the arithmetic average, should be used.

import math, random, statistics

random.seed(11)
# Lex,8 results from 8 full-shift dosimeter samples for one job group (illustrative)
samples = [round(random.gauss(84.5, 2.0), 1) for _ in range(8)]
print("Samples:", samples)
arith = statistics.mean(samples)
energy = 10 * math.log10(sum(10 ** (L / 10) for L in samples) / len(samples))
sd = statistics.stdev(samples)
print(f"Arithmetic mean {arith:.1f} dBA, energy mean {energy:.1f} dBA, SD {sd:.1f} dB")
print(f"Samples above 85 dBA: {sum(L > 85 for L in samples)} of {len(samples)}")
print(f"Energy mean plus 2 SD: {energy + 2 * sd:.1f} dBA")

Samples: [82.1, 85.3, 86.5, 83.5, 81.8, 84.4, 85.5, 86.7]
Arithmetic mean 84.5 dBA, energy mean 84.8 dBA, SD 1.9 dB
Samples above 85 dBA: 4 of 8
Energy mean plus 2 SD: 88.6 dBA

Even though the arithmetic mean is below 85 dBA, the energy mean is close to the limit and several individual samples exceed it. The group should be treated as over the limit and enrolled in the program, with controls considered.

Peak levels

Dose and Lex,8 describe continuous and intermittent noise well, and with a 3 dB exchange rate they also capture the energy of impulse noise. Very intense impulses, though, can cause immediate mechanical damage regardless of the daily dose. That is why jurisdictions that set one have a separate peak limit, typically 140 dBC or 140 dB peak, and why dosimeter reports show the highest peak recorded. Lesson 11 covers impulse noise.

Communicating results to workers

Workers deserve to know their exposure in terms they understand. A dose percentage can be explained as how much of the day's safe allowance was used up; saying that the maintenance crew used up three times their allowance by lunchtime is clearer than quoting 91.8 dBA. Time histories showing the loud spikes from specific tasks help workers see why protectors matter at particular moments. Results should also be shared with the joint health and safety committee or representative, and used to update the hearing conservation program.