A PM_2.5 magnitude-rank index for summarizing annual concentration-frequency structure

Fine particulate matter (PM_2.5) is routinely monitored worldwide to support environmental regulation and public health policy. Commonly reported metrics, including annual mean concentrations and short-term regulatory exceedance counts, provide essential summaries of long-term exposure and episodic events but do not fully characterize the distribution of daily pollution levels; exceedance counts also depend on policy-specific thresholds. To jointly characterize the magnitude and frequency of elevated PM_2.5 concentrations, we introduce the PM_2.5 Magnitude-Rank Index (PMRI), defined as the largest integer k such that at least k days in a year have daily mean PM_2.5 concentrations ≥ k µg/m³. For example, a PMRI of 23 indicates that at least 23 days had concentrations of 23 µg/m³ or higher. Applied to 918 qualifying Metropolitan Statistical Area-years (MSA-years) from 96 U.S. metropolitan areas during 2013-2022, PMRI ranged from 6 to 38 (median, 16). It correlated strongly with the annual mean (Spearman ρ = 0.83), annual 98th percentile (ρ = 0.95), and number of days ≥ 15 µg/m³ (ρ = 0.96), while distinguishing MSA-years with similar annual means but different concentration-frequency profiles. By formalizing the magnitude-frequency relationship in an intuitive, threshold-free form, PMRI offers a transparent complement to conventional metrics for characterizing particulate pollution.