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Abstract Let$$\mathbb {F}_q^d$$ $F_q^d$be thed-dimensional vector space over the finite field withqelements. For a subset$$E\subseteq \mathbb {F}_q^d$$ $E\subseteq F_q^d$and a fixed nonzero$$t\in \mathbb {F}_q$$ $t\in F_q$, let$$\mathcal {H}_t(E)=\{h_y: y\in E\}$$ $H_t\left(E\right)=\{h_y:y\in E\}$, where$$h_y:E\rightarrow \{0,1\}$$ $h_y:E\to \{0,1\}$is the indicator function of the set$$\{x\in E: x\cdot y=t\}$$ $\{x\in E:x·y=t\}$. Two of the authors, with Maxwell Sun, showed in the case$$d=3$$ $d=3$that if$$|E|\ge Cq^{\frac{11}{4}}$$ $\left|E\right|\ge C{q}^{\frac{11}{4}}$andqis sufficiently large, then the VC-dimension of$$\mathcal {H}_t(E)$$ $H_t\left(E\right)$is 3. In this paper, we generalize the result to arbitrary dimension by showing that the VC-dimension of$$\mathcal {H}_t(E)$$ $H_t\left(E\right)$isdwhenever$$E\subseteq \mathbb {F}_q^d$$ $E\subseteq F_q^d$with$$|E|\ge C_d q^{d-\frac{1}{d-1}}$$ $\left|E\right|\ge C_d{q}^{d-\frac{1}{d-1}}$. 

Abstract. High-quality long-term observational records are essential to ensure appropriate and reliable trend detection of tropospheric ozone. However, the necessity of maintaining high sampling frequency, in addition to continuity, is often under-appreciated. A common assumption is that, so long as long-term records (e.g., a span of a few decades) are available, (1) the estimated trends are accurate and precise, and (2) the impact of small-scale variability (e.g., weather) can be eliminated. In this study, we show that the undercoverage bias (e.g., a type of sampling error resulting from statistical inference based on sparse or insufficient samples, such as once-per-week sampling frequency) can persistently reduce the trend accuracy of free tropospheric ozone, even if multi-decadal time series are considered. We use over 40 years of nighttime ozone observations measured at Mauna Loa, Hawaii (representative of the lower free troposphere), to make this demonstration and quantify the bias in monthly means and trends under different sampling strategies. We also show that short-term meteorological variability remains a cause of an inflated long-term trend uncertainty. To improve the trend precision and accuracy due to sampling bias, two remedies are proposed: (1) a data variability attribution of colocated meteorological influence can efficiently reduce estimation uncertainty and moderately reduce the impact of sparse sampling, and (2) an adaptive sampling strategy based on anomaly detection enables us to greatly reduce the sampling bias and produce more accurate trends using fewer samples compared to an intense regular sampling strategy. 

Abstract. Tropical tropospheric ozone (TTO) is important for the global radiation budget because the longwave radiative effect of tropospheric ozone is higher in the tropics than midlatitudes. In recent decades the TTO burden has increased, partly due to the ongoing shift of ozone precursor emissions from midlatitude regions toward the Equator. In this study, we assess the distribution and trends of TTO using ozone profiles measured by high-quality in situ instruments from the IAGOS (In-Service Aircraft for a Global Observing System) commercial aircraft, the SHADOZ (Southern Hemisphere ADditional OZonesondes) network, and the ATom (Atmospheric Tomographic Mission) aircraft campaign, as well as six satellite records reporting tropical tropospheric column ozone (TTCO): TROPOspheric Monitoring Instrument (TROPOMI), Ozone Monitoring Instrument (OMI), OMI/Microwave Limb Sounder (MLS), Ozone Mapping Profiler Suite (OMPS)/Modern-Era Retrospective analysis for Research and Applications version 2 (MERRA-2), Cross-track Infrared Sounder (CrIS), and Infrared Atmospheric Sounding Interferometer (IASI)/Global Ozone Monitoring Experiment 2 (GOME2). With greater availability of ozone profiles across the tropics we can now demonstrate that tropical India is among the most polluted regions (e.g., western Africa, tropical South Atlantic, Southeast Asia, Malaysia and Indonesia), with present-day 95th percentile ozone values reaching 80 nmol mol−1 in the lower free troposphere, comparable to midlatitude regions such as northeastern China and Korea. In situ observations show that TTO increased between 1994 and 2019, with the largest mid- and upper-tropospheric increases above India, Southeast Asia, and Malaysia and Indonesia (from 3.4 ± 0.8 to 6.8 ± 1.8 nmol mol−1 decade−1), reaching 11 ± 2.4 and 8 ± 0.8 nmol mol−1 decade−1 close to the surface (India and Malaysia–Indonesia, respectively). The longest continuous satellite records only span 2004–2019 but also show increasing ozone across the tropics when their full sampling is considered, with maximum trends over Southeast Asia of 2.31 ± 1.34 nmol mol−1 decade−1 (OMI) and 1.69 ± 0.89 nmol mol−1 decade−1 (OMI/MLS). In general, the sparsely sampled aircraft and ozonesonde records do not detect the 2004–2019 ozone increase, which could be due to the genuine trends on this timescale being masked by the additional uncertainty resulting from sparse sampling. The fact that the sign of the trends detected with satellite records changes above three IAGOS regions, when their sampling frequency is limited to that of the in situ observations, demonstrates the limitations of sparse in situ sampling strategies. This study exposes the need to maintain and develop high-frequency continuous observations (in situ and remote sensing) above the tropical Pacific Ocean, the Indian Ocean, western Africa, and South Asia in order to estimate accurate and precise ozone trends for these regions. In contrast, Southeast Asia and Malaysia–Indonesia are regions with such strong increases in ozone that the current in situ sampling frequency is adequate to detect the trends on a relatively short 15-year timescale. 

Volatile chemical products (VCPs) and other non-combustion-related sourceshave become important for urban air quality, and bottom-up calculationsreport emissions of a variety of functionalized compounds that remainunderstudied and uncertain in emissions estimates. Using a new instrumentalconfiguration, we present online measurements of oxygenated organiccompounds in a US megacity over a 10 d wintertime sampling period, whenbiogenic sources and photochemistry were less active. Measurements wereconducted at a rooftop observatory in upper Manhattan, New York City, USAusing a Vocus chemical ionization time-of-flight mass spectrometer, withammonium (NH4+) as the reagent ion operating at 1 Hz. The range ofobservations spanned volatile, intermediate-volatility, and semi-volatileorganic compounds, with targeted analyses of ∼150 ions, whoselikely assignments included a range of functionalized compound classes suchas glycols, glycol ethers, acetates, acids, alcohols, acrylates, esters,ethanolamines, and ketones that are found in various consumer, commercial,and industrial products. Their concentrations varied as a function of winddirection, with enhancements over the highly populated areas of the Bronx,Manhattan, and parts of New Jersey, and included abundant concentrations ofacetates, acrylates, ethylene glycol, and other commonly used oxygenatedcompounds. The results provide top-down constraints on wintertime emissionsof these oxygenated and functionalized compounds, with ratios to commonanthropogenic marker compounds and comparisons of their relative abundancesto two regionally resolved emissions inventories used in urban air qualitymodels.