UISH

NO₂/O₃ photochemical anticorrelation

The NO₂/O₃ cross-correlation (photochemical anticorrelation) is a classic diagnostic plot for data quality assessment.

The NO₂/O₃ cycle is an intraday phenomenon driven by solar radiation. During night-time hours (e.g. 22:00–06:00) photochemistry is inactive and the signal reflects only deposition/emission.

What to look for in the plot: the photochemical anticorrelation manifests as a cloud oriented diagonally — NO₂↑/O₃↓ in the morning (traffic, emissions) and NO₂↓/O₃↑ during the central hours (photolysis). With the legend coloured by hour of day, the hourly cycle should appear as a loop, typically elliptical or horseshoe-shaped.

With the legend coloured by month instead of hour, seasonality emerges: high O₃ in summer with relatively low NO₂; low O₃ in winter with high NO₂.

Pearson correlation coefficient

Interpretation:

• r ≈ −0.7 / −0.9 in the 22:00–06:00 hours → well-defined photochemical anticorrelation
• r close to 0 → hour at which the cycle is perturbed (local emissions, transport)
• positive r → anomalous signal, possible volcanic episode (SO₂ interfering with O₃)

The Pearson coefficient measures only the strength of the linear relationship between two variables. By itself it neither validates nor invalidates a model. What it can indicate:

• If r(NO₂, O₃) from the model differs significantly from r computed on real observed data → the model does not correctly reproduce the photochemical dynamics
• If r varies by hour differently between model and observations → the model mis-times the cycle

In the specific NO₂/O₃ context, Pearson is useful as an internal model diagnostic: it verifies that the model reproduces the anticorrelation expected from Leighton chemistry. If the model produces a positive r where photochemistry predicts a negative r, there is a structural error in the chemical mechanism, independently of absolute values.

A complete validation requires observed data (ARPA/EEA stations) to be compared with model output over the same spatio-temporal domain.

What to expect: depends on the dominant mechanism.

• Maximum anticorrelation in winter, if the signal is dominated by the Leighton reaction:
  
  
  Direct, linear, non-photochemical reaction. At night in winter this is the only active mechanism → very negative and stable r.
• Maximum anticorrelation in summer — if the signal is dominated by photolysis:
  
  
  High UV radiation → complete and rapid photochemical cycle → marked anticorrelation during daytime/evening hours.

During night-time hours (22:00–06:00) specifically, photolysis is absent or negligible → the dominant mechanism is Leighton → the anticorrelation is more pronounced in winter, consistent with the Catania data analysed.

Why the anticorrelation is strong in winter:

• Long nights → the 22:00–06:00 window covers a complete chemical cycle
• Low temperature → lower VOC volatility, more linear chemistry
• The NO₂/O₃ cycle is dominated by the Leighton reaction: NO + O₃ → NO₂ + O₂, well defined and predictable

Why it weakens or reverses in summer:

• Secondary photochemistry: during the day, high UV radiation produces O₃ through VOC and CH₄ oxidation, which recycles NO back to NO₂ without consuming ozone and breaks the linear NO₂/O₃ relationship of the photostationary state alone. At night photochemical production stops (it requires NO₂ photolysis), but the ozone produced in the afternoon survives aloft in the residual layer and is mixed down in the morning: what persists overnight are high concentrations, not night-time production
• Long-range transport: air masses rich in tropospheric O₃ advected from the south/east (Mediterranean, North Africa) add O₃ independently of local NO₂
• In the specific case (Etna): summer is the season of greatest degassing activity: SO₂ and volcanic particulate matter perturb ozone chemistry in a non-linear manner

The p-value

The p-value is the probability of observing a result equal to or more extreme than the measured one, assuming the null hypothesis is true.

In the context of the Pearson coefficient:

• Null hypothesis (H₀): no correlation exists between NO₂ and O₃ in the population (r = 0)
• Alternative hypothesis (H₁): a correlation exists (r ≠ 0)

p = 0.003 means: "if NO₂ and O₃ were truly uncorrelated, there would be only a 0.3% probability of obtaining by chance an r this far from zero with these data"

Conventional thresholds:

From the t-statistic:

where:

• r = Pearson coefficient
• n = number of data pairs
• t follows a Student's t-distribution with ν = n − 2 degrees of freedom

The two-tailed p-value:

where F_t(∣t∣,ν) is the CDF (Cumulative Distribution Function) of the Student's t-distribution — it has no elementary closed form and is expressed via the regularised incomplete beta function:

The code uses the Abramowitz & Stegun approximation of the incomplete beta function, valid for n > 10 and |r| not too close to 1.

Interactive example animation

In this interactive animation the user modifies the value of r or the number of points and instantly sees the point cloud deform, the t-value update, and the tails of the distribution widen/narrow:

• Left panel: scatter plot with synthetic points generated with target correlation r (Cholesky method: )
• Right panel: Student's t curve computed numerically, with coloured tails as separate area series (left tail, centre, right tail = three overlapping series)

How it works:

• r slider: generates synthetic points with target correlation via (Cholesky decomposition for two variables)
• n slider: changes the number of points; with small n and moderate r the p-value will be non-significant even when the correlation is visually apparent
• ↻ Regenerate: new random points with the same parameters, showing the sampling variability of r
• Red tail: the coloured area is exactly p/2 per tail; it widens/narrows in real time
• Dashed lines: position of ±t_obs on the distribution

References

• Wasserstein, R. L., & Lazar, N. A. (2016). The ASA statement on p-values: Context, process, and purpose. The American Statistician, 70(2), 129–133. https://doi.org/10.1080/00031305.2016.1154108
• Núñez-Alonso, D., Pérez-Arribas, L.V., Manzoor, S. & Cáceres, J.O. (2019). Statistical Tools for Air Pollution Assessment: Multivariate and Spatial Analysis Studies in the Madrid Region. Journal of Analytical Methods in Chemistry, 2019, 9753927. DOI: 10.1155/2019/9753927 — PMC6387705 — PMID: 30881728