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Air Quality Is a Dilution Ratio

Supplied working report; original verdicts not validated.

Review status: Needs review. This is an archival source note, not a passed proof.

README.md

# Air Quality Is a Dilution Ratio

**Verdict: supported.** Across 25 German urban-background stations, daily PM2.5 rises when the boundary layer shrinks. The elasticity is −0.50 (95% CI −0.56 to −0.44): halving the depth of the mixed layer raises particle concentration by about 40 per cent. The same data show that the traffic cycle, the usual proxy for emissions, is almost irrelevant by comparison.

## What was claimed

The posted hypothesis (#251, Volume VIII) says that "urban air pollution is set by the ratio of emission flux to mixing height rather than by emissions alone, so a city with high emissions and deep mixing can be cleaner than one with lower emissions and a shallow boundary layer". Its first prediction is that concentration should track the flux-to-mixing ratio; its third is that mixing height should predict episodes better than emissions.

That is a sharper claim than "weather affects air quality". It says the denominator — the volume of air into which a city's emissions are mixed — does most of the work, and that emissions inventories are a poor explanation without it.

## How it was tested

Particulate measurements come from the German environment agency's public air-data API: hourly PM2.5 at 25 urban background stations, 2023–2024, aggregated to 12,814 station-days. Meteorology comes from ERA5 reanalysis through the Open-Meteo archive: daily mean boundary-layer height, 10-metre wind, temperature and precipitation, interpolated to each station's coordinates.

The test is a within-city comparison. For every station-day I take the deviation of log PM2.5 from its city-month mean and regress it on the deviation of log boundary-layer height, with wind, precipitation and day-of-week controls. City and year-month fixed effects absorb the level differences that make cross-city comparisons so treacherous. Standard errors are clustered by station. Because the emissions themselves are not observed, day-of-week serves as a crude emissions instrument: if traffic dominates, weekends should be measurably cleaner.

The decisive comparison was fixed before the number was computed. Support required four things: a negative elasticity significant at p < 10⁻⁶, a partial R² for mixing height of at least 0.05, an episode odds ratio of at least 1.5 per standard deviation of lower mixing depth, and a mixing-height partial R² more than five times the day-of-week partial R².

## What happened

The elasticity is −0.499, with a confidence interval that excludes any value above −0.44. Within a station, the median correlation between log PM2.5 and log mixing depth is −0.47. Mixing height alone adds 9.8 per cent of explained variance after season, city, wind and precipitation; day-of-week adds 0.1 per cent. The ratio is 96 to one.

The episode analysis points the same way. Days in the shallowest mixing decile are three times more likely to sit in a station's dirtiest PM2.5 decile than days in the deepest decile, and the odds ratio per standard deviation of lower mixing depth is 2.03 after month and station controls.

| Check | Value |
|---|---:|
| Elasticity to boundary-layer height | −0.499 (CI −0.555 to −0.444) |
| Partial R², mixing height | 0.098 |
| Partial R², day-of-week | 0.001 |
| Median within-station correlation | −0.47 |
| Episode odds ratio per SD | 2.03 |
| Elasticity to wind speed | +0.02 (p = 0.60) |

The wind coefficient is worth a remark. In a model that already contains mixing height, wind adds nothing and its sign is slightly positive. That is not evidence that wind fails to disperse pollution; it is evidence that the two variables are collinear and that the boundary layer carries the information in this sample. A test that dropped mixing height would find a strong wind effect, as the earlier literature does.

## The sensitivity checks

The result does not depend on the awkward choices. Flooring PM2.5 at 1 instead of dropping zero-concentration days gives −0.497. Running 2023 alone gives −0.500; 2024 alone gives −0.516. Restricting to the six months when the boundary layer is most variable (summer and winter) gives −0.489. All four are inside the headline interval.

One check does not support the stronger, cross-city version of the claim. Across the 25 cities, annual mean PM2.5 and annual mean mixing height correlate at ρ = +0.34 (p = 0.10) — the wrong sign, and not distinguishable from zero. This is expected if emissions differ by city: a deep-mixing city with heavy industry can be dirtier than a shallow-mixing city with none. The posted claim's second prediction, that high-emission deep-mixing cities should be cleaner, needs an emissions inventory that this dataset does not contain, so it was not tested. The within-city result is the part the data can decide.

## What this does not show

The boundary-layer height is a reanalysis grid value, not a measurement over each station; ERA5's mixing depth can be wrong by tens of per cent on stable nights, which is exactly when the elasticity matters most. German urban background sites are a narrow sample: 25 cities in one temperate country, none of them dominated by the extreme emission regimes the hypothesis is really about. And the day-of-week instrument is weak: much of the particle load is secondary aerosol and long-range transport, not traffic, so a null day-of-week effect is weaker evidence about emissions than it first appears.

None of this establishes causation. The design shows that concentration co-moves with mixing depth within cities, which is what a dilution mechanism predicts; it does not isolate the mechanism from correlated meteorology such as temperature inversions, humidity and stagnation, all of which arrive with a shallow boundary layer.

## Why the result matters for the notebook

The claim was posted as a correction to emissions-only thinking, and in one narrow but important sense it holds: for a fixed city, the day-to-day variation in particle concentration is mostly a story about the depth of the air, not about how much is emitted that day. The practical implication is narrow — episode forecasting is a meteorological problem before it is an inventory problem — and the political implication is not implied by this test. A city can be clean today because the air is deep and dirty next week with unchanged emissions. That is a statement about variability, not about annual averages or about whether emissions control works.

## Reproduce

`run.py` reads `data/daily.csv` (committed) and writes `results.json` and both figures. `fetch.py` rebuilds the raw table from the UBA API and the Open-Meteo archive; `SOURCE.md` records the URLs, the query strings, the fetch date and the hash of every raw file.

hypothesis.md

# Posted hypothesis, quoted verbatim

**Entry 251, Volume VIII — Energy & Environment.** Published in the theory notebook.

> **Air Quality Is a Dilution Ratio.** Urban air pollution is set by the ratio of
> emission flux to mixing height rather than by emissions alone, so a city with
> high emissions and deep mixing can be cleaner than one with lower emissions and
> a shallow boundary layer.

Posted predictions:

- **P1.** Concentration should track the flux-to-mixing ratio.
- **P2.** High-emission deep-mixing cities should be cleaner.
- **P3.** Mixing height should predict episodes better than emissions.

Posted refutation: *concentration tracking emissions at controlled mixing would
restore inventory management.*

# The decisive comparison, fixed before running

The observable quantity is the elasticity of daily PM2.5 to daily boundary-layer
height, estimated within cities so that emission levels are held roughly
constant. A dilution mechanism implies a negative elasticity of order one;
emissions-dominated variation implies an elasticity near zero.

Because emission flux is not observed, the test uses two proxies for it: the
day-of-week cycle (a traffic instrument) and the between-city variation in
annual means. These are weak instruments, and the second is used descriptively
only.

Support requires all four:

1. elasticity < 0 with cluster-robust p < 10⁻⁶ (25 station clusters);
2. partial R² of mixing height ≥ 0.05 after city, year-month, wind and precipitation;
3. episode odds ratio ≥ 1.5 per standard deviation of lower mixing depth;
4. mixing-height partial R² more than five times the day-of-week partial R².

Refutation: elasticity ≥ 0, or mixing height adding less than 0.01 of R² after
controls, or the day-of-week proxy explaining more than mixing height.

Partial support is recorded if the within-city relation holds but the
cross-city prediction (P2) cannot be evaluated, which is the expected outcome
because no emissions inventory is used.

# What the test cannot see

Boundary-layer height is a reanalysis grid quantity, not a measurement over the
station. It is least reliable in the stable nocturnal cases that dominate high
concentrations. Day-of-week is a poor emissions instrument for secondary
aerosol. The sample is one temperate country, so the cross-city clause of the
claim — the one about high-emission cities with deep mixing — is outside the
design.

# Outcome

Supported on P1 and P3 within cities; P2 not tested. See `results.json` and
`README.md`.

Included code and results