There are uncertainties associated with the box model (e.g., Brune et al., 2022; Zhang et al., 2021; Lee et al., 2021), which can be attributed to (i) the lack of inclusion of physical processes such as entrainment/detrainment and diffusion, (ii) discounting the heterogeneous chemistry, (iii) invalid assumption of the diel steady state in areas close to large emission sources or in photochemically less active environments (Thornton et al., 2002; Souri et al., 2021), (iv) errors in the chemical mechanism, and (v) errors in the measurements. These limitations necessitate a thorough validation of the model using unconstrained observations. While models have been known for a long time not to be 100 % accurate (Box, 1976), it is important to characterize whether the model can effectively represent reality. For instance, if the simulated HCHO is poorly correlated with observations and/or displayed large magnitude biases, it will be erroneous to assume that the sources of HCHO, along with the relevant chemical pathways, are appropriate. It is important to acknowledge that the VOC constraints for these model calculations are incomplete, especially for the DISCOVER-AQ campaigns, which lacked comprehensive VOC observations. Nevertheless, we will show that the selected VOCs are sufficient to reproduce a large variance (> 70 %) in observed HCHO.

We diagnose the performance of the box model by comparing the simulated values of six compounds to observations: HCHO, NO, NO2, PAN, hydroperoxyl radical (HO2), and OH. Figure 2 depicts the scatterplot of the comparisons along with several statistics. HCHO observations are usually constrained in box models to improve the representation of HO2 (Schroeder et al., 2017; Souri et al., 2020; Brune et al., 2022); however, this constraint may mask the realistic characterization of the chemical mechanism with respect to the treatment of VOCs. Additionally, it is important to know whether the sources of HCHO are adequate. Therefore, we detach the model from this constraint to perform a fairer and more stringent validation.

Concerning HCHO, our model does have considerable skill in reproducing the variability of observed HCHO (R2=0.73). To evaluate whether this agreement is accidentally caused by the choice of the dilution factor and to identify whether our VOC treatment is inferior compared to the one adopted in the NASA LaRC (Schroeder et al., 2020), we conducted three sets of sensitivity tests for the KORUS-AQ campaign, including ones with and without considering a dilution factor and another one without HNO3 and H2O2 constraints (Fig. S2). The lack of consideration of a dilution factor results in no difference in the variance in HCHO captured by our model (R2=0.81). Our model without the dilution factor is still skillful in replicating the magnitude of HCHO with less than 12 % bias. This is why the optimal dilution factor for each campaign is within 12 to 24 h, which is not different from other box modeling studies (e.g., Brune et al., 2022; Miller and Brune, 2022). We observe no difference in the simulated HCHO when HNO3 and H2O2 values are not constrained. The unconstrained NASA LaRC setup oversampled at 10 s frequency captures 86 % variance in the measurements, only slightly (6 %) outperforming our result. However, the unconstrained NASA LaRC setup greatly underestimates the magnitude of HCHO compared to our model results.

The model performs well with regard to the simulation of NO (R2=0.89) and NO2 (R2=0.99) on the logarithmic scale. Immediately evident is the underestimation of NO in highly polluted regions, in contrast to an overestimation in clean ones. This discrepancy leads to an underestimation (overestimation) of NO/NO2 in polluted (clean) regions. The primary drivers of NO/NO2 are jNO2 and O3, both of which are constrained in the model. What can essentially deviate the partitioning between NO and NO2 from that of observations in polluted areas is the assumption of the diel steady state, which is rarely strictly valid where measurements are close to large emitters. The overestimation of NO in low-NOx areas is often blamed on the lack of chemical sink pathways of NO in chemical mechanisms (e.g., Newland et al., 2021). The relatively reasonable performance of PAN (R2=0.63) is possibly due to constraining some of the oxygenated VOCs, such as acetaldehyde. Xu et al. (2021) observed a strong dependency of PAN concentrations on NO/NO2 ratios. Smaller NO/NO2 ratios are usually associated with larger PAN mixing ratios because NO can effectively remove peroxyacetyl radicals. We observe an overestimated PAN (0.27 ppbv), possibly due to an underestimation of NO/NO2. Moreover, we should not rule out the impact of the first-order dilution factor, which was only empirically set in this study. For instance, if we ignore the dilution process for the KORUS-AQ campaign, the bias of the model in terms of PAN will increase by 33 %, resulting in poor performance (R2=0.40) (Fig. S3). We notice that this poor performance primarily occurs for high-altitude measurements where PAN is thermally stable (Fig. S4); therefore, this does not impact the majority of rapid atmospheric chemistry occurring in the lower troposphere, such as the formation of HCHO. Schroeder et al. (2020) found that proper simulation of PAN in the polluted PBL during KORUS-AQ required a first-order loss rate based on thermal decomposition at the average PBL temperature, which was more realistic than the widely varying local PAN lifetimes associated with temperature gradients between the surface and the top of the PBL. This solution is computationally equivalent to the dilution rate used in this study.

KORUS-AQ was the only field campaign providing OH and HO2 measurements. Concerning HO2, former studies such as Schroeder et al. (2017), Souri et al. (2020), and Brune et al. (2022) managed to reproduce HO2 mixing ratios with R2 ranging from 0.6 to 0.7. The performance of our model (R2=0.66) is similar to these past studies, with nearly negligible biases (< 1 %). One may argue that the absence of the HO2 uptake by aerosols is contributing to some of the discrepancies we observe in the HO2 comparison. Brune et al. (2022) provided compelling evidence showing that considering the HO2 uptake made their results significantly inconsistent with the observations, suggesting that the HO2 uptake might have been inconsequential during the campaign. Our model manages to reproduce 64 % of the variance of observed OH, outperforming the simulations presented in Souri et al. (2020) and Brune et al. (2022) by > 10 %. The slope (= 1.03) is not too far from the identity line, indicating that our box model systematically overestimates OH by 0.62×106 cm-3. This may be attributed to a missing OH sink in the mechanism or the lack of inclusion of some VOCs. A sensitivity test involving removing the first-order dilution process demonstrates that the simulation of HOx is rather insensitive to this parameter (Fig. S5). In general, the model performance is consistent, or outperforms, results from recent box modeling studies, indicating that it is at least roughly representative of the real-world ozone chemistry and sensitivity regimes.

Kleinman et al. (2001) demonstrated that LROx/LNOx is the most robust ozone regime indicator. Thus, the predictive power of FNRs in detecting the underlying chemical conditions can be challenged by comparing FNRs to LROx/LNOx. Ideally, if they show a strong degree of correspondence (i.e., R2=1.0), we can confidently say that FNRs can realistically portray the chemical regimes. Any divergence of these two quantities indicates the inadequacy of the FNR indicator. Souri et al. (2020) observed a strong linear relationship between the logarithmic-transformed FNRs and those of LROx/LNOx. Our analysis in this study will be based on the simulated values to ensure that the relationship is coherent based on a realization from the well-characterized box model. As pointed out by Schroeder et al. (2017) and Souri et al. (2020), a natural logarithm of LROx/LNOx roughly equal to -1.0 (i.e., LROx/LNOx = 0.35–0.37) perceptibly separates VOC-sensitive from NOx-sensitive regimes, which would make this threshold the baseline of our analysis.

Figure 3 demonstrates the log–log relationship of LROx/LNOx, FNRs, and P(O3) from all four air quality campaigns. The log–log relationships from each individual campaign are shown in Figs. S6–S9. We overlay the LROx/LNOx baseline threshold along with two commonly used thresholds for FNRs suggested by Duncan et al. (2010); they defined VOC-sensitive regimes if FNR < 1 and NOx-sensitive ones if FNR > 2. Any region undergoing a value between these thresholds is unlabeled and considered to be in a transitional regime. The size of each data point is proportional to the HCHO × NO2 concentration magnitude. One striking finding from this plot is that there is indeed a strong linear relationship between the logarithmic-transformed LROx/LNOx and the FNR (R2=0.91). A strong linear relationship between the two quantities in the log–log scale indicates a power law dependence (i.e., y=axb). A strong power law dependence means that these two quantities have a poor correlation at their low and high values. This is mainly caused by the fact that HCHO does not fully describe VOC reactivity rates in environments with high and low VOC concentrations (Souri et al., 2020). The question is what range of FNRs will fall in ln(LROx/LNOx) = -1.0 ± 0.2. Following the baseline, the transitioning ratios follow a normal distribution with a mean of 1.8, a standard deviation of 0.4, and a range from 1 to 4 (Fig. S10). We define the chemical error in the application of FNRs to separate the chemical regimes as the relative error standard deviation (i.e., σ/μ) of the transitioning ratios leading to ∼ 20 %. These numbers are based on a single model realization and can change if a different mechanism is used; nonetheless, the model has considerable skill in reproducing many different unconstrained compounds, especially OH, suggesting that it is a rather reliable realization. Comparing the transitioning FNRs to the NO2 concentrations suggests no correlation (r=0.02), whereas there is a linear correlation between the transitioning ratios and the HCHO concentrations (r=0.56). This tendency reinforces the study of Souri et al. (2020), who, primarily due to the HCHO–NO2 feedback, observed a larger FNR threshold in VOC-rich environments to be able to detect the chemical regimes.

A striking and perhaps intuitive tendency observed from Fig. 3 is that large PO3 rates are mostly tied to higher HCHO × NO2. However, this relationship gradually weakens as we move towards VOC-sensitive regions (smaller LROx/LNOx ratios). This is a textbook example of nonlinear ozone chemistry. In VOC-sensitive areas, PO3 can be strongly inhibited by NO2+ OH and the formation of organic nitrates despite the abundance of the precursors. In the application of remote sensing of ozone precursors, the greatest unused metric describing the mass of the ozone precursors is HCHO × NO2. However, this metric should only be used in conjunction with FNRs. To demonstrate this, based on what the baseline (LROx/LNOx) suggests against thresholds on FNRs defined by Duncan et al. (2010), we group the data into four regions: NOx-sensitive–NOx-sensitive, NOx-sensitive–transitional, VOC-sensitive–transitional, and VOC-sensitive–VOC-sensitive. A different perspective on this categorization is that the transitional regimes are a weaker characterization of the main regime; for instance, NOx-sensitive–transitional regions are less NOx-sensitive than NOx-sensitive–NOx-sensitive. Subsequently, the cumulative distribution functions (CDFs) of PO3 and HCHO × NO2 with respect to the aforementioned groups are calculated, which is shown in Fig. 4. Regarding NOx-sensitive–NOx-sensitive regions, we see the PO3 CDF very quickly converging to the probability of 100 %, indicating that the distribution of PO3 is skewed towards very low values. The median of PO3 for this particular regime (where CDF = 50 %) is only 0.25 ppbv h-1. This agrees with previous studies such as Martin et al. (2002), Choi et al. (2012), Jin et al. (2017), and Souri et al. (2017) reporting that NOx-sensitive regimes dominate in pristine areas. The PO3 CDFs between NOx-sensitive–transitional and VOC-sensitive–VOC-sensitive are not too distinct, whereas their HCHO × NO2 CDFs are substantially different. The nonlinear ozone chemistry suppresses PO3 in highly VOC-sensitive areas such that those values are not too different from those in mildly polluted areas (NOx-sensitive–transitional). Perhaps the most interesting conclusion from this figure is that elevated PO3 values (median = 4.6 ppbv h-1), a factor of 2 larger than two previous regimes, are mostly found in VOC-sensitive–transitional. This is primarily due to two causes: (i) this particular regime is not strongly inhibited by the nonlinear chemistry, particularly NO2+ OH, and (ii) it is associated with abundant precursors evident in the median of HCHO × NO2 being 3 times as large of those in NOx-sensitive–transitional. This tendency illustrates the notion of nonlinear chemistry and how this may affect regulations. Simply knowing where the regimes are might not suffice to pinpoint the peak of PO3, as this analysis suggests that we need to consider both the FNR and HCHO × NO2; both metrics are readily accessible from satellite remote-sensing sensors.

It may be advantageous to construct an empirical function fitted to these two quantities and elucidate the maximum variance (information) we can potentially gain to recreate PO3. After several attempts, we found a bilinear function (z=a0+a1x+a2y+a3xy) to be a good fit without overparameterization. Due to the presence of extreme values in both the FNR and HCHO × NO2, we use a weighted least-squares method for the curve fitting based on the distance of the fitted curve to the data points (known as bi-squares weighting). The best fit with R2 equals 0.94, and an RMSE of 0.60 ppbv h-1 is 4PO3=0.74-0.09x-0.02y+0.25xy, where x and y are the FNR (unitless) and HCHO × NO2 (ppbv2), respectively. The residual of the fit is shown in Fig. S11. The gradients of PO3 with respect to x and y are 5dPO3dx=0.25y-0.09,6dPO3dy=0.25x-0.02. An apparent observation arises from these equations: i.e., the derivative of PO3 to each metric depends on the other one underscoring their interconnectedness. For instance, Eq. (6) suggests that larger FNRs (x) result in a larger gradient of PO3 to the abundance of HCHO × NO2(y). In very low FNRs, this gradient can become very small, rendering PO3 insensitive to (or in extreme cases, negatively correlated with) HCHO × NO2. This analysis provides encouraging results about the future application of the satellite-derived HCHO × NO2; however, the wide class of problems relating to the application of satellite-derived FNR columns, such as satellite errors in columns or the translation between columns to the PBL, is also present in Eq. (4), even in a more pronounced way due to HCHO × NO2 and HCHO2 (=xy). This new perspective on PO3 estimation deserves a separate study.

A lingering concern over the application of satellite-based FNR tropospheric columns is that the vertical distributions of HCHO and NO2 are integrated into columns; thus, this vertical information is permanently lost. Here, we provide insights into the vertical distribution of FNRs within the tropospheric column. This task requires information about the differences between (i) the vertical shape of HCHO and that of NO2 and (ii) the vertical shape in the sensitivity of the retrievals to the different altitude layers (described as scattering weights). Ideally, if both compounds show an identical relative shape, the FNR columns will be valid for every air parcel along the vertical path (i.e., a straight line). Previous studies such as Jin et al. (2017) and Schroeder et al. (2017) observed a large degree of vertical inhomogeneity in both HCHO and NO2 concentrations, suggesting that this ideal condition cannot be met. We do not always have precise observations of HCHO and NO2 vertical distributions, but we can constitute some degree of generalization by leveraging the measurements made during the aircraft campaigns. As for the differences in the vertical shapes (i.e., the curvature) of the sensitivity of the retrievals between HCHO and NO2 channels (i.e., ∼ 340 and ∼ 440 nm), under normal atmospheric and viewing geometry conditions, several studies such as Nowlan et al. (2018) and Lorente et al. (2017) showed small differences in the vertical shapes of the scattering weights in the first few kilometers in altitude above the surface, where the significant fluctuations in FNRs usually take place. Therefore, our analysis does not consider the varying vertical shapes in the scattering weights. However, this assumption might not hold for excessive aerosol loading with variable extinction efficiency between ∼ 340 and ∼ 440 nm wavelengths or extreme solar zenith angles.

Figure 5 demonstrates the violin plot of the afternoon (> 12:00 LT) vertical distribution of HCHO, NO2, and FNRs observed by NASA's aircraft during the four field campaigns analyzed in this study superimposed by the simulated PO3 rates. The vertical layers are grouped into 16 altitudes ranging from 0.25 to 7.75 km. Each vertical layer incorporates measurements ±0.25 km of the mid-layer height. The observations do not follow a normal distribution, particularly in the lower parts of the atmosphere; thus, medians are preferred to represent the central tendency. While the largest PO3 rates tend to occur in areas close to the surface (< 2 km a.g.l.), non-negligible fractions of the elevated PO3 rates are also observed in other parts of the atmosphere, such as in the free troposphere.

Several intriguing features are observed in Fig. 5. First, up to the 5.75 km range, which encompasses the PBL area and a large portion of the free troposphere, NO2 concentrations tend to decrease more quickly than those of HCHO, in line with previous studies such as Schroeder et al. (2017), Jin et al. (2017), Chan et al. (2019), and Ren et al. (2022). Second, above 5.75 km, HCHO levels off, whereas NO2 shows an increasing trend. Finally, due to their different vertical shapes, we observe non-uniformities in the vertical distribution of FNRs: they become more NOx-sensitive with altitude up to a turning point at 5.75 km and then shift back to the VOC-sensitive direction.

It is attractive to model these shapes and apply parameterizations to understand how their shapes will complicate the use of tropospheric column retrieval from satellites. First-order rational functions are a good candidate to use. Concerning the vertical dependency of HCHO and NO2, we find a reasonable fit (R2=0.73) as 7HCHO,NO2=a0z+a1z+a2, where z is altitude in kilometers. ai (i=0,1,2) are fitting parameters. From this equation it is determined that FNRs follow a second-order rational function: 8fz=HCHONO2=b0z2+b1z+b2b3z2+b4z+b5, where bi (i=0,…,5) are fitting parameters. One can effortlessly fit this function to different bounds of the vertical distribution of FNRs such as the 25th and 75th percentiles and subsequently estimate the first moment of the resultant polygon along z divided by the total area bounded to the polygon (the centroid G) via 9G(z1z2)=12A∫z1z2f2(z)75th-f2(z)25thdz, where A is the area of the polygon bounded by the 75th percentiles, f(z)75th, and 25th percentiles (f(z)25th) of FNRs (shown in Fig. 5 as solid black lines). We define an altitude adjustment factor (fadj) such that one can translate observed FNR tropospheric column ratios, such as those retrieved from satellites, to a defined altitude and below that point (zt) through 10fadj=G(0,zt)G(0,8km), where zt can be interchanged to match the planetary boundary layer height (PBLH). This definition is more beneficial than using the entire tropospheric column to surface conversion (e.g., Jin et al., 2017) because ozone can form in various vertical layers. Using the observations collected during the campaign, we estimate Eq. (10) along with ±1σ boundaries shown in Fig. 6. To determine the adjustment factor error, we reestimate Eq. (9) with the ±1σ level in the coefficients obtained from Eq. (8). The resultant error is shown in the dashed red line in Fig. 6. This error results from uncertainties associated with assuming that the second-order rational function can explain the vertical distribution of FNRs. The shape of the resulting adjustment factor is in line with the vertical distribution of FNRs (see Fig. 5): the adjustment factor curve closer to the surface has values smaller than 1, increases to values larger than 1 in the mid-troposphere, and finally converges to 1 near the top of measured concentrations. If one picks out an altitude pertaining to a PBLH, one can easily apply fadj to the observed FNR columns to estimate the corresponding ratio for that specific PBLH. A more evolved PBLH (i.e., a large zt) results in stronger vertical mixing, rendering fadj closer to 1. The standard error deviation of this conversion is around 19 %. The relatively low fluctuations in the adjustment factor around 1 suggest that under the observed atmospheric conditions (clear-sky afternoon summers), the columnar tropospheric ratios do not poorly represent the chemical conditions in the PBL region.

It is beneficial to model this curve to make this data-driven conversion easier for future applications. A second-order polynomial can well describe (R2=0.97) this curve: 11fadj=azt2+bzt+c,a=-0.01,b=0.15,c=0.78.

Although Eq. (11) does not include observations above 8 km, the area bounded between f(z)75th and f(z)25th at higher altitudes is too small to make a noticeable impact on this adjustment factor.

One may object that since we estimated the adjustment factor based on two boundaries (25th and 75th percentiles) of the data, we are no longer really dealing with 50 % of features observed in the vertical shapes of FNRs. This valid critique can be overcome by gradually relaxing the lower and upper limits and examining the resulting change in fadj. When we reduce the lower limit in Eq. (9) from the 25th to 1st percentiles, the optimal curve is similar to the one shown in Fig. 6 (Fig. S12). However, when we extend the upper limit from the 75th percentile to greater values, we see the fit becoming less robust above the 80th percentile, indicating that the formulation applies to ∼ 80 % of the data. The reason behind the poor representation of the adjustment factor for the upper tail of the population is the extremely steep turning point between 5.5 and 6.0 km, necessitating a higher-order rational function to be used for Eqs. (7) and (8). We prefer to limit this analysis to both boundaries and the order defined in Eqs. (8) and (9) because extreme value predictions usually lack robustness.

A caveat with these results is that our analysis is limited to afternoon observations because we focus on afternoon low-orbiting sensors such as OMI and TROPOMI. Nonetheless, Schroeder et al. (2017) and Crawford et al. (2021) observed large diurnal variability in these profiles due to diurnal variability in sinks and sources of NO2 and HCHO and atmospheric dynamics. The diurnal cycle has an important implication for geostationary satellites such as Tropospheric Emissions indeed: Monitoring of Pollution (TEMPO) (Chance et al., 2019). Limiting the observations to morning time results in a smaller adjustment factor for altitudes close to the surface resulting from steeper vertical gradients of HCHO/NO2 (Figs. S13 and S14). This tendency agrees with Jin et al. (2017), who observed a larger deviation from 1 in an adjustment factor used for the column–surface conversion in winter.

Another important caveat with our analysis is that it is based upon four air quality campaigns in warm seasons that avoid times/areas with convective transport; as such, our analysis needs to be made aware of the vertical shapes of FNRs during convective activities and cold seasons. However, a few compelling assumptions can minimize these oversights: first, it is very atypical to encounter elevated ozone production rates during cold seasons, with few exceptions (Ahmadov et al., 2015; Rappenglück et al., 2014); second, the notion of ozone regimes is only appropriate in photochemically active environments where the ROx–HOx cycle is active. An example of this can be found in Souri et al. (2021), who observed an enhancement of surface ozone in central Europe during a lockdown in April 2020 (up to 5 ppbv) compared to a baseline which was explainable by the reduced O3 titration through NO in place of the photochemically induced production. An exaggerated extension to this example is the nighttime chemistry where NO–O3–NO2 partitioning is the primary driver of negative ozone production rates; at night, the definition of NOx-sensitive or VOC-sensitive is meaningless, so it is in photochemically less active environments. Third, it is rarely advisable to use cloudy scenes in satellite UV–Vis gas retrievals due to the arguable assumption about Lambertian clouds and a highly uncertain cloud optical centroid and albedo. Accordingly, atmospheric convection occurring during storms or fires is commonly masked in satellite-based studies. Therefore, the limitations associated with the adjustment factor are mild compared to the advantages.

The spatial representation error resulting from unresolved processes and scales (Janjić et al., 2018; Valin et al., 2011; Souri et al., 2022) refers to the amount of information lost due to satellite footprint or unresolved inputs used in satellite retrieval algorithms. Unfortunately, this source of error cannot be determined when we do not know the true state of the spatial variability. There is, however, a practical way of resolving this by conducting multi-scale intercomparisons of a coarse spatial resolution output against a finer one. Yet despite the absence of the truth in this approach, we tend to find their comparisons useful in giving us an appreciation of the error.

We build the reference data on qualified pixels (qa_value > 0.75) of the offline TROPOMI tropospheric NO2 version 2.2.0 (van Geffen et al., 2022; Boersma et al., 2018) and total HCHO columns version 2.02.01 (De Smedt et al., 2018) oversampled at 3 × 3 km2 in summer 2021 over the US. Figure 7 shows the map of those tropospheric columns as well as FNRs. Encouragingly, the small footprint and relatively low detection limit of TROPOMI compared to its predecessor satellite sensors (e.g., OMI) enable us to have possibly one of the finest maps of HCHO over the US to date. Large values of HCHO columns are found in the southeast due to strong isoprene emissions (e.g., Zhu et al., 2016; Wells et al., 2020). Cities like Houston (Boeke et al., 2011; Zhu et al., 2014; Pan et al., 2015; Diao et al., 2016), Kansas City, Phoenix (Nunnermacker et al., 2004), and Los Angeles (de Gouw et al., 2018) also show pronounced enhancements of HCHO possibly due to anthropogenic sources. Expectedly, large tropospheric NO2 columns are often confined to cities and some coal-fired power plants along the Ohio River basin. Concerning FNRs, low values dominate cities, whereas high values are found in remote regions. An immediate tendency observed from these maps is that the length scale of HCHO columns is longer than that of NO2. This indicates that NO2 columns are more heterogeneous. Because of this, we observe a large degree of spatial heterogeneity with respect to FNRs.

Here we limit our analysis to Los Angeles due to computational costs imposed by the subsequent experiment. To quantify the spatial representation errors caused by satellite footprint size, we upscale the FNRs by convolving the values with four low-pass box filters with sizes of 13 × 24, 36 × 36, 108 × 108, and 216 × 216 km2, shown in the first column of Fig. 8. Subsequently, to extract the spatial variance (information), we follow the definition of the experimental semivariogram (Matheron, 1963): 12γh=12N(h)∑xi-xj-h≤ε[Zxi-Z(xj)]2, where Zxi and Zxj are discrete pixels of FNRs, and N(h) is the number of paired pixels separated by the vector of h. The |.| operator indicates the length of a vector. The condition of xi-xj-h≤ε is to permit a certain tolerance for differences in the length of the vector. Here, we ignore the directional dependence in γh which makes the vector of h a scalar (h=|h|). Moreover, we bin γ values in 100 evenly spaced intervals ranging from 0 to 5∘. To remove potential outliers (such as noise), it is wise to model the semivariogram using an empirical regression model. To model the semivariogram, we follow the stable Gaussian function used by Souri et al. (2022): 13γh=s1-e-(hr)c0:c0=1.5, where r and s are fitting parameters. For the most part, geophysical quantities become spatially uncorrelated at a certain distance called the range, and the variance associated with that distance is called the sill. The fitting parameters, r, and s, describe these two quantities as long as the stable Gaussian function can well fit the shape of a semivariogram. The semivariograms and the fits associated with each map are depicted in the second column of Fig. 8.

The modeled semivariograms suggest that a coarser field comes with a smaller sill, implying a loss in the spatial information (variance). The length scale (i.e., the range) only sharply increases at coarser footprints (> 36 × 36 km2). This indicates that several coarse-resolution satellite sensors, such as OMI (13 × 24 km2), are rather able to determine the length scales of FNRs over a major city such as Los Angeles. By leveraging the modeled semivariograms, we can effortlessly determine the spatial representation error for a specific scale (e.g., h=10 km) through 14e2h=1-γhγrefh, where γh and γrefh are the modeled semivariogram of the target and the reference fields (3 × 3 km2). This equation articulates the amount of information lost in the target field compared to the reference. Accordingly, the proposed formulation of the spatial representation error is relative. Figure 9 depicts the representation errors for various footprints. For the most part, the OMI nadir pixel (13 × 24 km2) only has a ∼ 12 % loss of spatial variance. By contrast, a grid box with a size of 216 × 216 km2 fails at capturing ∼ 65 % of the spatial information in FNRs with a 50 km length scale comparable to the extent of Los Angeles. The advantage of our method is that we can mathematically describe the spatial representation error as a function of the length of our target. The present method can be easily applied to other atmospheric compounds and locations. We have named this method the SpaTial Representation Error EstimaTor (STREET), which is publicly available as an open-source package (Souri, 2022).

An oversight in the above experiment lies in its lack of appreciation of unresolved physical processes in the satellite measurements: a weak sensitivity of some retrievals to the near-surface pollution due to the choice of spectral windows used for fitting (Yang et al., 2014), using 1-D air mass factor calculation instead of 3-D (Schwaerzel et al., 2020), and neglecting the aerosol effect on the light path are just a few examples to point out. To account for the unresolved processes, one can recalculate Eqs. (12)–(14) using outputs from different retrieval frameworks, which is beyond the scope of this study.

The ultimate task is to compile the aforementioned errors to gauge how each individual source of error contributes to the overall error. Although each error is different in nature, combined they explain the uncertainties of one quantity (FNR) and can be roughly considered independent; therefore, the combined error is given by 16σtotal=σCol2PBL2+σSpatialRep2+σRetreival2. σCol2PBL is the error in the adjustment factor defined in this study. We calculated a 19 % standard error for a wide range of PBLHs. Therefore, σCol2PBL equals 19 % of the observed ratio (i.e., magnitude-dependent). σSpatialRep is more complex. It is a function of the footprint of the satellite (or a model), the spatial variability of the reference field, which varies from environment to environment, and the length scale of our target (e.g., a district, a city, or a state). Equation (14) explicitly quantifies this error. The product of the square root of that value and the observed ratio defines σSpatialRep. The last error depends on the magnitude of HCHO and NO2 tropospheric columns. It can be estimated from Eq. (15) times the observed ratio. We did not include the chemistry error in Eq. (16) because it was suited only for segregating the chemical conditions; it does not describe the level of uncertainty that comes with the observed columnar ratio. Figure 13 shows the total relative error given the observed TROPOMI ratio seen in Fig. 7. We consider the OMI spatial representation error (13 % variance loss) for this case that was computed in a city environment. The retrieval errors are based on TROPOMI sigma values. Areas associated with relatively small errors (< 50 %) are mostly seen in cities due to a stronger signal (smaller σRetreival). Places with low vegetation and anthropogenic sources (i.e., the Rocky Mountains) possess the largest errors (> 100 %).

To produce some examples of the fractional errors and the combined error, we focus on two different environments with two different sets of HCHO and NO2 columns. One represents a heavily polluted area, and the other one is a moderately polluted region. We also include two footprints: OMI (13 × 24 km2) and a 108 × 108 km2 pixel. Finally, we calculate the percentage of each error component for both OMI and TROPOMI sensors. Figure 14 shows the pie charts describing the percentage of each individual error in the total error for TROPOMI. Unless the footprint of the sensor is coarse enough (e.g., 108 km2) to give rise to the spatial representation error dominance, the retrieval error stands out. New satellites are not expected to have very large footprints; as such, retrieval errors appear to be the major obstacle to using FNRs in a robust manner. Figure 15 shows the same calculation but using OMI errors; the retrieval errors massively surpass other errors. This motivates us to do one more experiment: we recalculate the HCHO error distribution in OMI using monthly-averaged data instead of daily data (Fig. S17). This experiment suggests a standard deviation of 9.4 × 1015 molec. cm-2, with which we again observe the retrieval error to be the largest contributor (> 80 %) to the combined error (Fig. S18). A recent study (Johnson et al., 2022) also suggests that retrieval errors can result in considerable disagreement between FNRs from various sensors and retrieval frameworks.