TROPOMI is the single payload on the Sentinel 5 Precursor (S5P) satellite, which has a sun-synchronous orbit with an overpass time of around 13:30 local solar time (Veefkind et al., 2012). TROPOMI has near-full earth-surface coverage on a daily basis. The instrument contains three spectrometers that cover the ultraviolet-near infrared (UVN) with two spectral bands at 270–500 and 675–775 nm and one spectrometer that covers the shortwave infrared. The UVN detector developed for TROPOMI is a back-illuminated 1024×1024 pixel frame transfer charge-coupled device (CCD) (Kleipool et al., 2018). The instrument has a high spatial resolution of 7km×3.5km (along-track × across-track) at nadir for bands 2–6 (UVN module) (Eskes et al., 2019) (note that since 6 August 2019, the resolution has improved to 5.5km×3.5km). The high spatial resolution of TROPOMI is a major improvement compared to its predecessor, OMI, which has a ground footprint of roughly 13km×24km at nadir (de Graaf et al., 2016).

OMI is a Dutch–Finnish nadir-viewing UV-visible spectrometer aboard NASA's Earth Observing System (EOS) Aura satellite that was launched in July 2004. It measures the solar radiation backscattered by the earth's atmosphere and surface between 270 and 500 nm with a spectral resolution of 0.5 nm (Levelt et al., 2006, 2018). OMI has a 780×576 CCD detector that measures at 60 across-track positions simultaneously and thus does not require across-track scanning. Due to this approach, the spatial resolution of the CCD pixels varies significantly along the across-track direction: the pixels near the track centre have a ground footprint of 13km×24km, whereas the pixels close to the track edge (e.g., view zenith angle =56∘) have a ground footprint of roughly 23km×126km (de Graaf et al., 2016). Note that from 2012 onwards, the smallest pixels (across-track positions) can no longer be used and are excluded from the analysis (known as the “row anomaly”, i.e., Levelt et al., 2018). This means the “smallest” pixels available for OMI are larger than 13km×24km.

The Pandora instrument records spectra between 280 and 530 nm with a resolution of 0.6 nm (Herman et al., 2009, 2015; Tzortziou et al., 2012). It uses a temperature-stabilized Czerny–Turner spectrometer, with a 50 µm entrance slit, 1200 groove mm-1 grating, and a 2048×64 back-thinned Hamamatsu CCD detector. The spectra are analyzed using a total optical absorption spectroscopy (TOAS) technique (Cede, 2019), in which absorption cross sections for multiple atmospheric absorbers, such as ozone, NO2, and sulfur dioxide (SO2), are fitted to the spectra.

The Pandora direct-sun total column NO2 data are produced using Pandora's standard NO2 algorithm implemented in the BlickP software (Cede, 2019). The measured direct-sun spectra from 400 to 440 nm are used in the TOAS analysis. A synthetic reference spectrum is produced by averaging multiple measured spectra which get corrected for the estimated total optical depth included in it. Cross sections of NO2 at an effective temperature of 254.5 K (Vandaele et al., 1998), ozone at an effective temperature of 225 K (Brion et al., 1993, 1998; Daumont et al., 1992), and a fourth-order polynomial are all fitted. The resulting NO2 SCDs are then converted to total column by using direct-sun geometry AMFs. Herman et al. (2009) showed that Pandora direct-sun total column NO2 has a clear-sky precision of 0.01 DU (in the slant column) and a nominal accuracy of 0.1 DU (in the vertical column). Additional information on Pandora calibrations, operation, and retrieval algorithms can be found in Herman et al. (2009) and Cede (2019).

Pandora instrument nos. 103 and 104 have been deployed in Downsview, Toronto (43.781∘ N, -79.468∘ W; suburban), since 2013 to perform direct-sun measurements (Zhao et al., 2016). The instruments are installed on the roof of the ECCC Downsview building at an altitude of 187 m a.s.l. The building is located in a suburban area with multiple roads nearby. Since February 2018, the instruments have employed an alternating direct-sun, zenith-sky, and multi-axis observation schedule, which includes direct-sun measurements every 5 min during the sunlit period.

Pandora instrument nos. 108 and 145 have been deployed in Egbert (44.230∘ N, -79.780∘ W; rural) and the University of Toronto St. George Campus (43.661∘ N, -79.399∘ W, referred to as UTSG; urban), respectively, since May 2018. The same alternating observation schedule is implemented. Pandora no. 108 is located on the roof of the ECCC Center for Atmospheric Research Experiments (CARE) building in Egbert at an altitude of 251 m a.s.l. The building is in a rural area, which is surrounded by farmlands. Pandora no. 145 is located in the University of Toronto Atmospheric Observatory (TAO) in downtown Toronto at an altitude of 174 m a.s.l. A map of the GTA and surrounding areas is shown in Fig. 1, overlaid with a colour map of TROPOMI KNMI NO2 tropospheric columns averaged over the March 2018 to March 2019 period utilizing the pixel-averaging technique (Fioletov et al., 2011; Sun et al., 2018). It is clear that the three sites (Downsview, Egbert, and UTSG) represent three different NO2 pollution levels.

To validate the satellite measurements, coincident ground-based data are required. The coincidence criteria are normally composed of spatial, temporal and quality control criteria (e.g., Boersma et al., 2018; Drosoglou et al., 2017; Griffin et al., 2019; Irie et al., 2008; Toohey and Strong, 2007). For example, in Zhao et al. (2019b), the coincidence criteria used to pair ground-based observations (Pandora) and OMI data include (1) the nearest (in time) measurement that was within ±30 min of the OMI overpass time, (2) the closest OMI ground pixel (having a distance of less than 20 km from the ground pixel centre to the location of the Pandora instrument), and (3) cloud fraction < =0.3 (the effective geometric cloud fraction as determined by the OMCLDO2 algorithm), and only high-quality OMI data are used (VcdQualityFlags =0) (Celarier et al., 2016). This simple coincident measurement selection scheme is referred to here as the “standard” method.

In this work, similar criteria are used with some adjustments. The temporal criterion is changed from ±30 to ±10 min of TROPOMI overpass time (this is to ensure the standard method can be fairly compared with the new wind-based method; see Sect. 3.2). Since TROPOMI has better spatial resolution than that of OMI, the selected spatial criterion is set to 10 km for TROPOMI. Similarly to OMI, only high-quality TROPOMI data are used (qa_value > 0.75) (Eskes et al., 2019). Pandora direct-sun NO2 total column data of assured high quality (L2 data quality flag =0) are used in the validation (Cede, 2019). Note that the TROPOMI quality assurance value filter (qa_value > 0.75) removes cloud-covered scenes with cloud radiance fraction > 0.5. In this study, to make a straightforward comparison with OMI, an additional cloud fraction filter is used (cloud fraction < =0.3) for TROPOMI data.

To make more use of the high-resolution measurements made by TROPOMI and to improve their validation, a wind-based method is tested, which can increase the number of coincident measurements. In addition to coincident and co-located data, this method looks at upwind (downwind) TROPOMI pixels that will arrive at (have passed over) the Pandora site within a short time window. Technically, this is done by using wind rotation and aligning all wind directions to the preferred direction. After the rotation, all ground pixels have a common effective wind direction and can be analyzed together regardless of the true wind direction. Any NO2 source located between the satellite pixel and the Pandora site will affect the performance, and we need to look at the TROPOMI–Pandora differences as a function of the wind direction.

In general, the wind-based method adapts a pixel-rotation technique developed and used in several previous studies (e.g., Fioletov et al., 2015; Pommier et al., 2013). Previously, the pixel rotation involved a rotation of each satellite ground pixel around the SO2 source location (e.g., smelters or mining areas). In this work, we adapted the pixel-rotation technique, where the satellite observations are rotated around a point, which in this case is the location of the ground-based instrument.

First, the initial coordinates of each satellite pixel (geographic coordinate; Ginitial (lat, lon)) are interpolated to the horizontal distance from the selected centre (local tangent plane coordinate; Pinital (x, y), where x is east–west distance and y is north–south distance). Figure 2a shows the initial positions of pixels in the local tangent plane coordinate, where the ground-based instrument site is at P (0, 0). Then a rotation matrix R is applied to satellite ground pixels, with the rotation angle equal to the negative of the wind direction (θ): 1Protate=R-θPinital,2R-θ=cos⁡-θsin⁡-θ-sin⁡-θcos⁡-θ. After the rotation, each satellite ground pixel maintains its upwind–downwind character. In other words, after rotation, the new ground pixel, Protate (x, y), can be analyzed assuming that the wind always has a constant direction (from “north” to “south”) as shown in Fig. 2. All the pixels in Fig. 2 share the same wind direction, but only three colour-coded pixels are selected to show with the wind arrow. Thus, for a given rotated pixel, the closest distance it can reach to the site, P (0, 0), is xrotate, at a time given by 3tcoincident=tmeas+yrotateν, where tmeas is the measurement time of the pixel, yrotate is the y value of Protate, and ν is the wind speed. Next, the boundaries of coincident measurement selection are defined as 4xrotated≤ρ, where ρ is an arbitrary distance, referred to as rotational-coincident distance. For example, for TROPOMI, we find the optimized ρ value equal to 5 km. Use of a larger ρ value will increase the number of coincident measurements, while the representativeness of coincident measurements (i.e., whether or not the selected satellite pixel can represent the ground-based measurement at a given time) will decrease. Based on sensitivity tests using various ρ values, a balance between number of coincident measurements and representativeness can be achieved. For other satellites with coarse spatial resolution, the rotational-coincident distance value has to be increased (e.g., approximately equal to the satellite's ground-pixel size).

This method is valid if the trace gas concentrations do not change rapidly over the pixel travel time, ttravel. The assumptions made here are that (1) the local emission patterns (strength and spatial distribution), (2) chemical reactions, and (3) vertical atmospheric dynamics (i.e., boundary layer variation) do not change rapidly during this time period. All these assumptions are likely to be close to reality for tropospheric NO2 in most areas around local noon. Even for urban areas, the local emissions are relatively stable around local noon (when sun-synchronous orbiting satellites such as OMI and TROPOMI pass over a given site) compared to morning or evening rush hours. In addition, the NO2 photolysis rate (Dickerson et al., 1982) and boundary layer height (Garratt, 1994) around local noon are also relatively stable compared to morning or evening conditions.

For example, using the rotated plane coordinates (Fig. 2b), any pixel within the rotational-coincident boundaries ±ρ (indicated by the dashed lines) will “overpass” the site at tcoincident, having an “overpass” distance from the ground pixel centre to the location of the ground-based instrument less than or equal to 5 km. In the application, we give a cutoff value to the pixel travel time, ttravel, as 5ttravel=yrotateν≤1h, which ensures that the assumptions we made (i.e., emission, chemical, and dynamic changes are not significant in this period) are valid. The colour-coded pixels in Fig. 2 are examples of upwind pixel (green), downwind pixel (yellow), and out-of-field pixel (red). In general, the wind-based method can identify any pixel that has its simulated “trajectory” passing over the ground-based site. However, if there is a major emission source between the TROPOMI pixel and the ground-based instrument, the difference between satellite and ground-based measurements will increase. This feature is observed and can be used to distinguish local and transported pollution (discussed in Sect. 5.1).

Figure 4 shows the comparison of results obtained using the standard and new wind-based methods for defining coincident measurements. The standard and wind-based methods show similar performance in terms of correlation coefficient. Although the correlation coefficients (R) decreased slightly for two of the three sites (for the Downsview site, it decreased from 0.75 to 0.71; for the UTSG site, it decreased from 0.71 and 0.65), the R increased for the Egbert site (from 0.78 to 0.89). Egbert, as a rural area, has much lower NO2 total columns than the values in urban and suburban areas. Compared to Downsview and UTSG, the correlation coefficients between TROPOMI and Pandora data are improved by the wind-based method due to increased observations of transported pollution events. In general, the number of coincident measurements (N) increased for all sites by about a factor of 5 (e.g., from 174 to 939 for Downsview) when using the wind-based method.

For Downsview, Fig. 4a and b show that the multiplicative biases between TROPOMI and Pandora total column NO2 data (indicated by the slopes of the fitted lines, with fixed zero intercept) are -30% and -23% for the standard and wind-based methods, respectively. The results for UTSG and Egbert are shown in Fig. 4c to f. Similarly to Downsview, TROPOMI data show -28% (standard) and -24% (wind-based) multiplicative biases relative to the Pandora at UTSG. However, in contrast to Downsview, where TROPOMI data show negative bias relative to Pandora data, TROPOMI NO2 observations have 10 % (standard) and 4 % (wind-based) positive multiplicative biases at Egbert. The positive bias at Egbert might be due to TROPOMI overestimating stratospheric NO2 (e.g., Wang et al., 2020). Other typical satellite validation metrics, including absolute differences, relative mean differences, and regression slopes, are provided in Appendix A.

In general, Fig. 4 shows that the larger number of coincident measurements paired by the wind-based method maintained similar good quality to the ones paired by the standard method. Further sensitivity tests on the parameters used in the wind-based method are shown in Appendix B. It is expected that the local emissions will be more significant than transported NO2 in downtown Toronto, whereas the transported NO2 is more significant than local emissions in Egbert. Thus, the wind-based method's sensitivity is dependent on local pollution patterns. Details about the sensitivity distance are discussed in Appendix B. In general, for individual sites, a unique sensitivity distance should be evaluated and applied to achieve the best balance between the number of coincident measurements and representativeness of the enlarged dataset (i.e., whether or not the expanded dataset can represent the real local or regional conditions).

The ECCC TROPOMI NO2 data product was also compared to the Pandora measurements. The results from the three sites are shown in Fig. 5. A clear difference between the KNMI version and the ECCC version of TROPOMI data is the multiplicative bias. In general, ECCC data, which are based on a model with much higher spatial resolution, show a positive shift of the fitted slopes for all three sites by roughly 5 % to 15 %. For Downsview, ECCC data decreased the multiplicative bias between satellite and Pandora data from 24 %–28 % to 15 %–24 %. For UTSG, similar improvement was found as the multiplicative bias decreased from 24 %–28 % to 13 %–20 %. However, for the clean background (rural) site in Egbert, ECCC data increased the multiplicative bias from 4 %–10 % to 14 %–15 %. Thus, the results show that ECCC data have a positive shift in the total column values compared to KNMI data when compared to Pandora measurements. The ECCC NO2 product has a better representation of the albedo for snow-covered areas (Griffin et al., 2019). However, for this period of measurements, the snow-covered satellite ground pixels are too sparse. Future studies will be performed to evaluate the performance of the ECCC data in snow-covered conditions, after accumulating a sufficient number of snow-covered pixels. More comparison results, such as the absolute difference and relative difference between TROPOMI ECCC data and Pandora data, are shown in Appendix A. In general, TROPOMI ECCC data have smaller absolute and relative differences (compared with Pandora data) at Downsview and Egbert but slightly larger differences at UTSG.

One motivation to increase the number of coincident measurements is to study NO2 spatial distribution and local air quality conditions. In Fig. 6, the coincident TROPOMI and Pandora data are grouped by wind direction, and the mean values of each group are shown as a function of wind direction. For example, Fig. 6a shows the NO2 results from the Downsview site; the purple line with error bars is Pandora no. 104 total columns, the blue line is TROPOMI KNMI total columns, and the red and yellow lines are the TROPOMI tropospheric and stratospheric components. Although there is a clear offset between the purple and blue lines, indicating an offset between TROPOMI and Pandora NO2 total columns, the general pattern between two datasets is similar. Figure 6a reveals several peaks in the mean NO2 total columns at Downsview, such as at wind directions 180 and 240∘, which correspond to the directions from downtown Toronto and the major city airport (Toronto Pearson International Airport, referred to here as YYZ by its airport code, the largest and busiest airport in Canada; in the city of Mississauga, 0.72 million population), respectively. In addition, high Pandora NO2 values for the wind direction of 240∘ may be related to vehicle emissions from a busy local street located about 100 m from the site. Meanwhile, low mean NO2 values are found in the 270–360∘ range, which corresponds to the direction of suburban Downsview, a relatively clean area (in the northern part of the city of Toronto). Here we define the clean wind direction by using TROPOMI stratospheric and tropospheric NO2. For a given site, any direction that has TROPOMI VCDtrop≤ VCDstrat is considered a clean wind direction. In general, for clean wind directions, the mean difference between TROPOMI and Pandora total columns is within ±0.05 DU and the mean relative differences are typically within ±20 %. Here, the mean relative difference is defined as 6Δrel=100×1N∑i=1NM1i-M2iM1i+M2i/2, where N is the number of measurements. We select M1 to be TROPOMI measurements and M2 to be Pandora measurements. Figure 6b shows the results from TROPOMI ECCC data. It is clear that the offset between TROPOMI ECCC data and Pandora data has decreased for most polluted wind directions.

The results for the UTSG and Egbert sites are shown in Fig. 6c to f. For Egbert, almost all wind directions are considered clean air directions, except for 180∘. These results highlight that, compared to the GTA, other nearby small cities, such as Barrie (see Fig. 1, 0.14 million population; 15 km away from Egbert, within the 30∘ wind direction bin), are not significant NO2 sources to Egbert. The UTSG site experiences relatively clean air from 60 to 120∘. The major NO2 peak at 150∘ is linked to the direction from the city's central business district (2 km away from the measurement site). The second peak at 240 to 270∘ is likely linked to the direction of a large diesel train yard for the local train service and the YYZ airport (18 km away from the measurement site).

The similarity of the NO2 horizontal distribution patterns observed by TROPOMI and Pandora is also evaluated. The correlation coefficients between TROPOMI (blue lines) and Pandora (purple lines) angular total column NO2 data (as a function of wind direction) are shown in Fig. 6, referred to here as Rangle. In general, the patterns show high similarity between satellite and ground-based results, with Rangle larger than 0.8 for all three sites, and TROPOMI ECCC data have equal or higher correlation coefficients with Pandora data compared to TROPOMI KNMI data.

To further evaluate the agreement between sets of coincident measurements, the mean difference and mean relative difference between satellite and ground-based results are shown in Fig. 7. The mean differences between TROPOMI and Pandora are within ±0.1 DU, except for Downsview data from the 240∘ wind direction. For Downsview, the highest relative difference is found to be -36% for the 240∘ wind direction. Similarly, the largest discrepancies between Pandora and TROPOMI at Egbert and UTSG are found at the wind directions with the highest NO2 values, such as 240∘ for UTSG and 180∘ for Egbert. For the clean air direction, such as 270–360∘ for Downsview, the mean relative differences are typically within ±30 %. TROPOMI ECCC data performed better for the Downsview and UTSG sites, whereas TROPOMI KNMI data performed slightly better for the Egbert site.

The discrepancies between TROPOMI and Pandora mean differences also indicate the types of NO2 sources. A NO2 peak value is more likely from a regionally transported NO2 source (e.g., a few ground pixels away) if the mean difference between Pandora and TROPOMI is small (i.e., both Pandora and TROPOMI measured the peak). If the mean difference is large (i.e., Pandora measured the peak, whereas TROPOMI did not), then the measured NO2 peak is likely from a localized source (e.g., within or around one ground pixel). For example, in Fig. 7a for Downsview, the 180∘ peak shows -0.06 DU mean difference, whereas the 240∘ peak shows -0.13 DU mean difference. Thus, the 240∘ peak is more influenced by some near-local NO2 source (e.g., nearby heavy traffic roads). Similarly, in Fig. 6b at Egbert, the 180∘ peak shows only -0.005 DU mean difference. Thus this peak is more weighted by a far-transported NO2 source. In general, this NO2 distribution study shows that combining Pandora and satellite measurements can be a powerful tool to study local or regional air quality.

The number of coincident pairs and the number of unique days for each wind bin are shown in Fig. A1. In general, due to the uneven distribution of wind direction, some binned wind directions have a limited number of coincident pairs between TROPOMI and Pandora, e.g., 60∘. Thus, the interpretation of results from these bins is difficult. However, for other bins, such as the 180∘ bin for Egbert, there are 46 coincident measurements from 7 d. Thus, we can be confident about the sharp peak signal observed in Fig. 6e and f and conclude that for Egbert, the main pollution events are transported from the Toronto area.

The standard and wind-based methods for determining coincidences were also applied to OMI NO2 observations. In this study, we used OMI and Pandora no. 104 measurements from 2015 to 2018 at Downsview. By extending the observation period by a factor of 4 (i.e., only 1 year of TROPOMI observations were used in Sects. 4 and 5.1, while 4 years of OMI observations were used here), OMI measurements can reveal similar NO2 spatial distributions to those of TROPOMI (i.e., results in Sect. 5.1).

Figure 9a to d show the results of applying the standard and wind-based methods to OMI data. Figure 9a and b show the results using the standard method and the OMI SPv3 NO2 and OMI ECCC NO2 data products (see Sect. 2.2), respectively. The general performances of OMI SPv3 and OMI ECCC data are similar, and OMI ECCC data have a slightly lower multiplicative bias (28 %) than OMI SPv3 (34 %). Due to the lower spatial resolution of OMI, we modified the coincident criteria used above for TROPOMI. For the wind-based method, the ρ value criterion was changed from 5 to 20 km, and ttravel was changed from 1 to 3 h, compared to the criteria used for TROPOMI (see Sect. 3.3). The correlations of OMI and Pandora total columns are smaller than those found in Sect. 4 (using TROPOMI and the ERA-5 wind field). To make the comparison between OMI and TROPOMI consistent, these modified wind-based criteria (ρ and ttravel value) are also applied to TROPOMI data (see Fig. 8e). Similarly to other studies (e.g., Eskes and Eichmann, 2019), OMI data show a larger multiplicative bias relative to Pandora than TROPOMI. Although TROPOMI data used in this work only cover 1 year and OMI data cover 4 years, TROPOMI data have about 5 times the number of coincident measurements compared to OMI data (see Fig. 9c and e). In general, the proposed wind-based method is more powerful for high-spatial-resolution satellite instruments than medium-resolution instruments. The same tests were performed on the OMI ECCC data products, as shown in Fig. 9b, d, and f. Similarly to the results in Sect. 4, OMI ECCC and TROPOMI ECCC data have lower multiplicative biases relative to Pandora NO2 total columns than do OMI SPv3 or TROPOMI KNMI data.

Another motivation for applying the wind-based method to OMI is to assess whether the spatial distribution of NO2 in this area (Downsview) has had any significant changes over this 4-year period. Binning the data by wind direction (see Fig. 9) shows that the NO2 spatial distribution patterns revealed by OMI and TROPOMI are similar; i.e., the main pollution sources are from the south (from downtown Toronto) and southwest (from the YYZ airport), and clean air from the north (from the suburban area).

For 2018, depending on the wind direction, absolute differences of up to 0.13–0.15 DU can be observed by Pandora and TROPOMI (indicated by Fig. 10b and c). From 2015 to 2018, absolute differences of up to 0.17 DU can be seen for Pandora and 0.12 and 0.13 DU for OMI SPv3.1 and OMI ECCC, respectively (indicated by Fig. 10c and d). In general, OMI data for 2015–2018 and TROPOMI data for 2018 demonstrate a similar dependence on the wind direction.

To further assess the quality of TROPOMI KNMI and TROPOMI ECCC NO2 data products and to determine whether they meet the TROPOMI design bias and precision requirements (Eskes and Eichmann, 2019), we performed statistical uncertainty and bias estimations for TROPOMI and Pandora data. In general, by comparing the same quantity retrieved from different remote sensing instruments, we can characterize the differences between them, which are a combination of random uncertainties and systematic bias. Theoretically, information about the random uncertainties can be derived from the measurements (Fioletov et al., 2006; Grubbs, 1948; Toohey and Strong, 2007; Zara et al., 2018; Zhao et al., 2016, 2019a).

As an example, we define the two measured NO2 total column data (denoted as M1 and M2, for Pandora nos. 103 and 104 measurements, respectively) as simple linear functions of the true NO2 total column value (X) and instrument random uncertainties (δ1 and δ2) and assume that there is no multiplicative or additive bias between the two Pandora datasets, giving 7M1=X+δ1,M2=X+δ2. Note that these two Pandora instruments are located at the same site, i.e., Downsview. If we assume that the instrument random uncertainties are independent of the measured NO2 total column, the variance of M is the sum of the variances of X (around the mean of the dataset) and δ, 8σM12=σX2+σδ12,σM22=σX2+σδ22. If the difference between two Pandoras does not depend on X (no multiplicative bias) and the random uncertainties of the two instruments are not correlated, then the variance of the difference is equal to the sum of the variance of the random uncertainties, 9σM1–M22=σδ12+σδ22. Then, the variance of the instrument random uncertainties can be solved by 10σδ12=σM12-σM22+σM1-M22/2,σδ22=σM22-σM12+σM1-M22/2. Equation (10) can be used to estimate the standard deviation of instrument random uncertainties (σδ1 and σδ2). The variances σMi2 and σM1–M22 can be estimated from the available measurements (with some uncertainty). The uncertainties of the σδ12 and σδ22 estimates depend on the sum of all three variances σM12, σM22, and σM1–M22 and can be high even if the estimated variance itself is low (but one or more of the variances σM12, σM22, and σM1–M22 are high). Thus, the estimates are only as accurate as the least accurate of these parameters. Following the method in Zhao et al. (2016), the variance estimates can be improved by increasing the number of data points or by reducing variance of X by removing some of its natural variability. Thus, the M1 and M2 used in the statistical uncertainty estimation are replaced by so-called residual NO2, which is defined as the difference between the measured NO2 total column and its daily mean. Figure 11 shows the scatter plots for residual NO2 total columns from Pandora nos. 103 and 104. The model-estimated NO2 total column random uncertainties (UPandora103 and UPandora104) for the two instruments are the same, 0.02 DU, indicating the good consistency between the two co-located instruments. Compared to the TROPOMI NO2 total column random uncertainty requirement (0.032 DU; see Table 1), this result shows Pandora instruments have sufficient precision for the TROPOMI NO2 data product validation work.

The statistical uncertainty estimation model was also applied to TROPOMI NO2 total column data. Note that the dataset used is the TROPOMI (both KNMI and ECCC products) and Pandora coincident NO2 total column data, paired by the wind-based method. Details of TROPOMI statistical uncertainty calculation are shown in Appendix B. The results are summarized in Fig. 11, which indicate that Pandora NO2 data have lower random uncertainties than TROPOMI NO2 data for all the sites. For example, the first column in Fig. 11a shows that the Pandora NO2 measured at Downsview has 0.03 DU random uncertainty (red cross sign with error bar), which is better than the Pandora total column NO2 nominal accuracy (0.05 DU at 1σ level, e.g., Zhao et al., 2019b). At Downsview, the TROPOMI KNMI and TROPOMI ECCC total column data products have random uncertainties of 0.05 DU (red square with error bar, Fig. 11a) and 0.06 DU (blue square with error bar, Fig. 11b), respectively. The mean of the reported TROPOMI KNMI total column precision is 0.06 DU at this site (black square with error bar, Fig. 11a). The black dashed line shows the TROPOMI total column data product precision requirement. The green dashed line shows the Pandora precision that was estimated using two co-located Pandoras at Downsview (Pandora nos. 103 and 104). Note that the estimates, which use the statistical random uncertainty estimation method, are only as accurate as the least accurate of these two instruments. Thus, the statistical model indicates that Pandora has a 0.03 DU precision when compared with TROPOMI, while Pandora has a 0.02 DU precision when compared with another co-located Pandora. The KNMI-reported precisions show that the satellite data product has better precision at Egbert (0.03 DU) than at Downsview and UTSG. The statistical uncertainty estimation also shows similar results for TROPOMI NO2 total column data (i.e., UDownsview > UUTSG > UEgbert), but about 0.01 DU lower than the reported precisions. The TROPOMI slant column has a reported random uncertainty on the order of about 0.022 DU (Eskes and Eichmann, 2019). The reported random uncertainty for the tropospheric column is then derived by dividing the slant column by the tropospheric AMF. Because the tropospheric AMF at Egbert is larger, the derived vertical column random uncertainty will be smaller. Thus, the changes between the three sites (at least partly) reflect differences in tropospheric AMF at these sites.

In general, this result indicates the good quality of TROPOMI-reported precision. The TROPOMI NO2 total column data products, however, did not meet the design random uncertainty requirement (0.032 DU; see Table 1), except for the clean site (Egbert). On the other hand, although the TROPOMI KNMI data products have higher multiplicative biases than the TROPOMI ECCC data products, their random uncertainties are lower by 0.01 DU at Downsview and UTSG and by 0.003 DU at Egbert (Fig. 11, red squares compared with blue squares). However, this result should be taken with caution since TROPOMI and Pandora do not directly measure the same quantities. Pandora measures NO2 slant columns along the line-of-sight between the instrument and the sun, while TROPOMI measures slant columns from a mixture of scattering optical paths. Then, both are converted into vertical columns. Thus, the statistical uncertainty model-estimated random uncertainties (red and blue symbols in Fig. 11) are upper limits of the TROPOMI total column random uncertainty, since the mismatch of the air masses observed between TROPOMI and Pandora (representativity error) will also produce a random-like signal which adds to the estimate. Moreover, the lower spatial resolution of the parameters used in the KNMI AMF calculations may lead to more uniform retrieved values, i.e., to a lower variability of the retrieved NO2 values and, therefore, lower estimated uncertainties.

Besides precision, the bias of the data is estimated for total column and tropospheric column data products. In Sect. 4, Figs. 4 and 5 show that the TROPOMI KNMI and ECCC total column data have negative multiplicative biases up to 30 % and 25 % (negative relative differences up to 26 % and 19 %; see Appendix A), respectively. These results are slightly better than the finding from the S5P NIDFORVAL (NItrogen Dioxide and FORmaldehyde Validation) project, in which the NO2 total column comparisons with more than 10 Pandora instruments showed TROPOMI has a negative bias (up to 45 % lower) and a lower than expected accuracy (Eskes and Eichmann, 2019). Note that the bias is strongly site dependent and will depend on the local gradients in NO2 around the measurement site and the ability of the coarse global TM5-MP model (used by TROPOMI KNMI data, 1∘ resolution) to produce realistic profiles for individual sites. Apparently the AMF produced by the TM5-MP model has good performance in Egbert, away from the city, but has negative biases inside the city of Toronto. For the tropospheric column data, both the TROPOMI KNMI and TROPOMI ECCC products meet the design bias requirement; KNMI and ECCC tropospheric NO2 columns have a negative multiplicative bias relative to the Pandora measurements of up to 41 % and 34 %, respectively (see Appendix B).