The use of aircraft observations is twofold: first, they provide a vast number of measured geophysical variables suitable to simulate our observationally-constrained PO₃ training dataset (Sect. 3.1); second, they enable a rigorous validation of column-to-planetary boundary layer (PBL) conversion factors derived from a chemical transport model (Appendix B). We use the dataset compiled by Souri et al. (2025), who curated various aircraft campaigns measuring photolysis rates, meteorological variables, and atmospheric composition from varying atmospheric conditions, including urban/suburban settings (DISCOVER-AQs, and KORUS-AQ), high-vegetated regions (SENEX), and remote areas (INTEX-B and AToms). The sampling frequency varies from 10 to 30 s. More detailed information regarding the choice of instrument, gap filling, and data exclusion can be found in Souri et al. (2025).

We use a global chemical transport model simulation designed to support trace gas retrievals. The simulation, called Multi-Decadal Nitrogen Dioxide and Derived Products from Satellites (MINDS) (Fisher et al., 2024), was generated using the Goddard Earth Observing System (GEOS) Earth system model (Molod et al., 2015; Nielsen et al., 2017) equipped with the full chemistry Global Modeling Initiative (GMI) mechanism (Duncan et al., 2007; Strahan et al., 2007) and coupled with the Goddard Chemistry Aerosol Radiation and Transport (GOCART) aerosol module (Chin et al., 2002). The rapid radiative transfer model, which was designed for global climate models (GCMs) and is known as the Radiative Transfer Module for GCM (RRTMG), calculates the longwave and shortwave radiation influenced by aerosols simulated by GOCART, enabling the incorporation of the direct effects of aerosols on meteorological conditions (Nielsen et al., 2017). Meteorology is resolved using GEOS with several prognostic inputs, including water vapor, being constrained by MERRA-2 reanalysis using “replay” mode at 3-hourly basis (Orbe et al., 2017). The model is setup at c360 grid (0.25° × 0.25°) and covers the period of 1993 until the end of 2023. The model follows 72 hybrid sigma values ranging from the surface to 0.01 hPa.

Lightning production of NO is parametrized based on the simulated convection. The model uses the Monitoring Atmospheric Chemistry and Climate and CityZen (MACCity) inventory (Granier et al., 2011) of anthropogenic emissions downscaled to 0.1° × 0.1° using the Emissions Database for Global Atmospheric Research version 4.2 (EDGAR 4.2). These anthropogenic emissions change by year and month. Biomass burning emissions rely on the Fire Energetics and Emissions Research (FEER) dataset (Ichoku and Ellison, 2014). Biogenic emissions are modeled interactively by the Model of Emissions of Gases and Aerosols from Nature (MEGAN) v2.1 (Guenther et al., 2012). It is known that isoprene emissions in MEGANv2.1 are largely overestimated (Bauwens et al., 2016; Souri et al., 2020b), therefore they are scaled down by a factor of two.

To estimate jNO₂ and jO¹D, we refer to a detailed look-up table provided by the Framework for 0-D Atmospheric Modeling (F0AM) model (Wolfe et al. 2016). This table is developed for clear-sky conditions based on over 20 064 solar spectra calculations. The data encompasses a broad spectrum of solar zenith angles (SZA) from 0 to 90° in 5° increments, altitudes ranging from 0 to 15 km in 1 km steps, overhead total ozone columns from 100 to 600 DU in increments of 50 DU, and surface UV albedo values from 0 to 1 in 0.2 increments. These calculations were carried out using NCAR's Tropospheric Ultraviolet and Visible radiation model (TUV v5.2), along with cross sections and quantum yields from IUPAC and JPL (Wolfe et al., 2016). Information on SZA and surface elevation is obtained from the L2 TROPOMI/OMI granule data. Surface albedo is based on the TROPOMI DLER climatology (Sect. 2.1.4). The overhead total ozone columns are derived from MINDS simulations (Sect. 2.3). For any values that fall between the entries in the tables, we apply a linear interpolation method.

We will compare our new product (Sect. 3.2) to an empirical method developed by Souri et al. (2025), who took advantage of simulated PO₃ data constrained by aircraft measurements to parameterize PO₃ using four geophysical variables: NO₂, HCHO, jNO₂, and jO¹D. Their algorithm used a piecewise L1-regularized linear regression model known as Least Absolute Shrinkage and Selection Operator (LASSO). Since the algorithm was based on a linear model which was ill-suited for the non-linear ozone chemistry, it was necessary to linearize the parameterization using various thresholds for FNRs. Despite the method's simplicity, Souri et al. (2025) were able to reproduce approximately 88 % of the variance with low biases (less than 20 %) in observationally-constrained PO₃. Using the empirical method, they generated the first maps of PO₃ by combining bias-corrected TROPOMI HCHO and NO₂ columns, simulated photolysis rates, and a global transport model designed for the conversion from column measurements to the PBL.

To isolate the performance of the PO₃ estimator used in Souri et al. (2025) in comparison to the proposed algorithm in this study, we will ensure that the input variables, including the mixing ratios of HCHO and NO₂ within the PBL as well as the photolysis rates, remain identical for both the empirical product and our new algorithm. Hereafter, we will refer to this empirical product as “PO₃LASSO”.

To establish a relationship between several geophysical variables related to PO₃, we use F0AM version 4 box model (Wolfe et al., 2016). This model is capable of simulating detailed chemical kinetics based on user inputs regarding meteorological variables, atmospheric compositions, and photolysis rates. F0AM uses a solver for ordinary differential equations (ODEs) designed for stiff systems, which allows it to determine the chemical evolution of all species included in the selected chemical mechanism. We adhere to previous configurations that apply the Carbon Bond 6 (CB06, r2) chemical mechanism within F0AM (Souri et al., 2020a; Souri et al., 2023a; Souri et al., 2025). The model is constrained by data collected during aircraft campaigns, including meteorological data, photolysis rates, and various trace gas concentrations. Additional details regarding the selection of instruments, bias corrections for photolysis, choices of dilution factors, and other configurations can be found in Souri et al. (2025). We incorporate data from seven aircraft campaigns, including DISCOVER-AQ (Texas, Washington D.C., Colorado), KORUS-AQ, ATOMs, INTEX-B, and SENEX, to further constrain the model. Souri et al. (2025) demonstrated that this setup effectively reproduces several unconstrained yet measured compounds, such as HCHO, HO₂, OH, and PAN; moreover, the performance of the model was on par with other studies (e.g., Brune et al., 2020; Brune et al., 2022; Miller and Brune, 2022), indicating that it is a suitable model setup for understanding local ozone chemistry. This model-derived dataset consists of ∼ 134 000 points.

A limitation to the training dataset prepared by Souri et al. (2025) originates from the fact that only a subset of atmospheric conditions could be observed by the suborbital missions. A remedy for this limitation is to synthetically regenerate data by systematically perturbing several of the inputs used in the F0AM model. As a result, we apply a scaling factor, ranging from 0.1 up to 10 in 12 evenly-spaced steps, separately to NO_x, VOCs, H₂O(v), and photolysis rates. This expands the dataset to ∼ 6.4 million datapoints, covering a much wider range of atmospheric states.

Once the simulations are done, we determine simulated PO₃ by: 1PO3=FO3-LO3 where LO₃ is all possible chemical loss pathways of ozone (negative stoichiometric multiplier matrix) and FO₃ is all possible chemical pathways producing ozone molecules (positive stoichiometric multiplier matrix). This equation is also known as ozone tendency. This definition simplifies intercomparison with estimates derived from different chemical mechanisms by eliminating the requirement to explicitly match individual production and loss terms, which often exhibit inconsistencies across mechanisms, especially in their treatment of peroxy radicals. The calculation of PO₃ is under a steady-state assumption.

The overall architecture of the DNN model is portrayed in Fig. 2. The design consists of three fully-connected hidden layers each having 32 neurons. The neurons are equipped with rectified linear unit (ReLU) activation functions. The training dataset (∼ 6.4 millions) is split into 20 % test, 24 % validation, and 56 % training. Training inputs to the parametrization consists of HCHO, NO₂, jO¹D, jNO₂, and H₂O(v). Prior to the training, we normalize them, such that each feature (x) is rescaled according to x′=x-μσx, where μ and σ represent the mean and standard deviation of the feature, respectively, ensuring a mean of zero and a variance of one. The optimization (training) of the DNN follows the backpropagation rule armed with Adaptive Moment Estimation (ADAM) optimizer which is known to perform well with noisy data (Kingma and Ba, 2017). The initial learning rate is set to 10⁻⁵. We use 500 epochs. The loss function (L) of the optimalization problem is: 2L=12∑k=1N(yk-ok)2+λ∑i=1pwi2 where the first term on the right side represents the mean squares error (MSE) of the prediction derived from difference between the target PO₃ (y) and the predicted PO₃ (o). N represents the number of training datapoints. The second term is L2-regularization with a factor of λ to reduce the squares of p number of neuron weights (w).

An important aspect of this optimization is the use of L2 regularization, which effectively helped us determine the optimal number of hidden layers and neurons. L2 regularization penalizes the cost function if an illusion of high prediction accuracy (the first term) is achieved with excessive variance in the solution (weights). Failing to balance the prediction error and the solution variance can lead to overfitting, which harms model performance in two ways: (i) it results in erroneous predictions for atmospheric conditions that fall outside the training dataset; (ii) it diminishes the physical interpretability of the statistical model because of large fluctuations in the weights, a common issue in regression models known as collinearity. When we used too many neurons or layers, the regularization penalized the weights, causing a substantial proportion to approach zero (not shown), indicating that those neurons were unnecessary. However, incorporating regularization does have some drawbacks: (i) it requires a smaller initial learning rate (set to 10⁻⁵) to avoid falling into local minima, which demands more computational resources; and (ii) the regularization factor also needs to be optimized. We find that a value of λ= 10⁻⁵ provides the best results among the set of values [10⁻⁴, 10⁻⁵, and 10⁻⁶], based on the symmetry in the statistical distributions of the test residuals, MSE, and the overall level of physical interpretability observed in the sensitivity tests.

The implementation of the DNN model is done using the open-source TensorFlow application programming interface (API) package in Python (Abadi et al., 2016). To thoroughly validate the performance of this model from various angles we (i) compare the DNN prediction with the test data using various standard metrics, (ii) investigate the evolution of the loss function derived from both the training set and the validation one over epochs, (iii) study the physical explanation of the response of PO₃ to NO₂ and HCHO, water vapor, and photolysis rates, and (iv) finally compare the DNN results to PO₃LASSO. We will use a number of statistical metrics, including the coefficient of the determination (R2), mean bias, mean square error, mean absolute error, and root mean square error (RMSE), to carry out the quantitative assessment (Sect. 4.1).

To elucidate the response of PO₃ to its inputs, we calculate the semi-normalized sensitivities through the finite difference method: 3SPO3i=[PO3]i110%-[PO3]i90%0.2 where [PO3]i110% and [PO3]i90% are PO₃ from perturbing input parameters (i= 1 for NO₂, and i= 2 for HCHO) by 1.1 and 0.9 scaling factors. A mathematical proof showing that these sensitivity calculations are equivalent to the directional derivative is provided in Appendix A.

Since the PO₃DNN integrates atmospheric models, satellite trace gas retrievals, ground remote sensing, and a machine learning approach, it contains various sources of errors, some of which will be formulated in this section. Spatially and temporally averaging satellite-based products is a common practice to reduce noise and fill gaps; therefore, we attempt to separate systematic errors (irreducible by averaging) from random ones (reducible by averaging). We assign the total PO₃ within PBL region error (etotal) based on the following equation: 4etotal=esyst2+erand2 where esyst and erand are systematic and random errors associated with PO₃ estimates. Systematic errors account for the errors associated with the bias correction of OMI and TROPOMI against ground remote sensing retrievals (eHCHO_bias_c and eNO2_bias_c), the model-based conversion of columns to the PBL mixing ratios (eHCHO_conversion,eNO2_conversion), and the DNN estimator error (eDDN), and are given by: 5esyst=eHCHO_bias_c2+eNO2_bias_c2+eHCHO_conversion2+eNO2_conversion2+eDNN26eHCHO_bias_c2=∂PO3∂HCHO⋅γ⋅ebc-HCHO27eNO2_bias_c2=∂PO3∂NO2⋅γ⋅ebc-NO228eHCHO_conversion2=∂PO3∂HCHO⋅VCDHCHO⋅econv-HCHO29eNO2_conversion2=∂PO3∂NO2⋅VCDNO2⋅econv-NO22 where γ is the conversion factor of the satellite total to the PBL columns translation based on MINDS and the formulation by Souri et al. (2025); ebc-HCHO and ebc-NO2, in column units, are calculated following the formulation from Souri et al. (2025) who used the errors of slope and offset obtained from the comparison of satellite VCDs to ground remote sensing benchmarks; econv-HCHO and econv-NO2 are quantified by validating the simulated conversion factors compared to those of aircraft vertical spirals (Appendix B). The unit for these two errors is ppbv per the column unit; accordingly, we multiply these terms to satellite VCDs. The last term in Eq. (5) is a fixed systematic error associated with the DNN estimates which will be quantified based on the MSE of the DNN prediction. Both ∂PO3∂HCHO and ∂PO3∂NO2 are derived from the sensitivity calculations from Eq. (3) divided by the satellite columns. All error terms in Eqs. (6)–(9) are spatially and temporally invariant, but the derivatives vary from pixel to pixel resulting in spatiotemporally-varying systematic errors.

Random errors originate from the uncertainty estimates coming with the TROPOMI and OMI L2 products and are somewhat reducible by averaging, and are given by: 10erand=∂PO3∂HCHO⋅γ⋅erand-HCHO2+∂PO3∂NO2⋅γ⋅erand-NO22 where erand-HCHO and erand-NO2 are random retrieval errors. All terms in Eq. (10) vary by time and location.

Table 2 summarizes the numbers used in the above equations and their origin.

It is important to acknowledge that the defined total error budget here is only a good guess and optimistic. Some underlying sources of error, which are difficult to quantify, are not included. For example, errors related to the training dataset derived from the F0AM model are challenging to assess because of the lack of PO₃ measurements. We assume other inputs to the PO₃ parametrization, such as the monthly climatology TROPOMI surface albedo to be error-free. Additionally, all datasets used to estimate PO₃ contain spatial representation errors (Souri et al., 2022), which are difficult to measure without knowing their true state of global spatial variability. Moreover, we do not consider correlated errors among HCHO and NO₂ retrievals. It is worth noting that some of the inputs such as H₂O(v) and the overhead ozone column have minimal biases because of MINDS simulations being observationally constrained (Fisher et al., 2024; Souri et al., 2024).

There are also assumptions regarding the equations mentioned earlier. For instance, it is assumed that the validation of conversion factors can account for all systematic issues related to the vertical distribution of NO₂ and HCHO in MINDS. Furthermore, we presume that the reported retrieval errors are mostly random; however, this is not the case (Eskes and Boersma, 2003; Boersma et al., 2018) and distinguishing between these errors is not straightforward.

Another source of uncertainty arises from partially cloudy pixels and aerosols, which can introduce errors in calculated photolysis rates. While we successfully filtered out cloud cover and strong aerosol loadings (e.g., from wildfires) using effective cloud fraction thresholds, some aerosol or cloud-contaminated pixels may pass cloud screening due to low optical depth or height characteristics. Rigorously quantifying the errors coming from these effects would require running a radiative transfer model with detailed three-dimensional optical properties of clouds and aerosols on a global scale, particularly critical for aerosols, which can have complex effects on photolysis rates depending on their absorption and scattering properties and vertical distribution. Unfortunately, such comprehensive datasets are typically limited to the narrow swaths of spaceborne lidar observations, which themselves carry substantial uncertainties (Thorsen and Fu, 2015). While these complications cannot be entirely avoided, particularly for aerosol effects, users can apply additional quality control measures by filtering pixels using aerosol optical depth retrievals from TROPOMI, OMI, or other sensors to more rigorously identify contaminated observations.

In case of oversampling of the PO₃ product both temporally and spatially, the total error will be given by: 11etotal_oversampled=1m∑esyst2+1m2∑erand2 where m is the total number of samples. Equation (11) suggests that the systematic errors are persistent across all samples and are not reducible by averaging, whereas the random errors become smaller by root square of samples. In this equation, the assumption is that the root-mean-square of the systematic errors is a good approximation of the systematic errors in the oversampled data because they are independent of each other.

We investigate the predictive power of the DNN algorithm against both validation and test data for each air quality campaign or the entire aircraft dataset (Sect. 2.2). All training datasets described in Sect. 3.1 are used in this stage. Except for the early stages of training, both training and validation curves, explaining the evolution of the prediction against the number of epochs corresponding to the number of iterations of training the network for one cycle, closely follow each other, indicating that we possibly do not have overfitting issues (Fig. S11 in the Supplement). The curves are fairly smooth, resulting from using the ADAM optimizer with a strictly small learning rate initially. Both curves converge to RMSE below 0.88 ppbv h⁻¹ which we use to assign the error of PO₃DNN prediction in Eq. (5).

PO₃DNN has promising skill at predicting PO₃ across various atmospheric conditions. Figure 3 presents a comparison of the predicted PO₃ values against observationally-constrained F0AM values for the test data for each suborbital mission. A similar comparison, which includes all data points measured during each mission, can be found in Fig. S12. The primary reason for highlighting the test data is that they have never been used to fine-tune the DNN parameters. There is a strong correlation between the predictions and the benchmarks across most campaigns for both the test data points (Fig. 3) and the complete set of aircraft measurements (Fig. S12). Notably, the slope for the “All” test dataset is close to the unity line. The DNN algorithm can reproduce over 96 % of the variance in the test data. Similar to the approach of Souri et al. (2025), we completely exclude each suborbital mission from the training dataset and use it as an independent benchmark to evaluate the model's performance. The resulting accuracy is comparable to that achieved when 56 % of the data are used for training, indicating that the PO₃ parameterization has reached a high degree of generalization (Fig. S13).

The model performs significantly better than PO₃LASSO over INTEX-B compared to LASSO (as shown in Fig. 7 in Souri et al., 2025). While the DNN's performance over the ATom campaigns is less impressive than in other areas, it still represents a considerable improvement over LASSO, which was unable to reproduce PO₃ in pristine regions (R2<0.05). One key factor contributing to this improvement is the inclusion of H₂O(v) in the input. Various parameters, including HO_x, are known to influence PO₃ in remote regions, but these factors were not included in our parametrization. The method does not artificially inflate results by introducing non-physical relationships in remote regions; the inability of the DNN to fully explain PO₃ during AToms suggests that it does not force unrealistic relationships between PO₃ and the inputs to completely align with the F0AM results, leaving areas for future improvement in parametrization over remote regions.

There are primarily four major benefits of PO₃DNN over PO₃LASSO that make the former parameterization a superior algorithm. The discussion of these advantages is as follows:

Higher predictive power. PO₃LASSO predicted PO₃ for all datapoints collected from the suborbital missions with a R2= 0.88, RMSE = 1.2 ppbv h⁻¹, and a slope of 0.87 (Souri et al., 2025), whereas PO₃DNN reproduced the exact datapoints (Fig. S12) with a R2= 0.96, RMSE = 0.7 ppbv h⁻¹, and a slope of 1.00. Furthermore, as shown in Fig. 3, PO₃DNN has a great degree of generalization for datapoints outside of the training/validation data points. Consequently, these statistics suggest that DNN is a more powerful predictor.

To further elaborate on the capability of PO₃DNN to reasonably respond to variations in its five major parameters in a mathematically continuous fashion, we create six isopleths, each specifically designed to represent a particular atmospheric condition listed in Table 3. These isopleths are based on perturbing HCHO and NO₂ in PO₃DNN and are shown in Fig. 5.

It is immediately apparent that the hyperbolic shape of the PO₃ curve relative to NO₂ and HCHO can be recreated by our algorithm, displaying a positive response to both HCHO and NO₂ on the right and left sides of the ridgelines. This observation underscores the effective parametrization of the non-linearities in ozone photochemistry achieved through the DNN algorithm. In the subplot representing normal conditions, we overlaid three lines indicating FNR values of 1.5 (blue), 2.5 (green), and 3.5 (cyan). Souri et al. (2025) used these lines to determine various coefficients in the PO₃LASSO parameterization. For instance, the derivative of PO₃ with respect to NO₂ was determined to be -0.14 ppbv h⁻¹ for FNR < 1.5 but increased to 6.54 ppbv h⁻¹ for FNR > 3.5. However, in practice, the thickness and curvature of the PO₃ isopleths vary based on the prevailing atmospheric conditions, implying that the derivatives cannot consistently retain the same values across the broad range of conditions.

In bright conditions, not only do we observe a significantly accelerated response of PO₃ compared to the norm at identical NO₂ and HCHO concentrations, but the responses of PO₃ to these two compounds also become more pronounced. Conversely, in dim conditions, both the magnitudes and responses are weaker.

These results underscore the importance of including photolysis rates in ozone sensitivity analysis, rather than relying solely on FNR in former studies. For example, a lower FNR in the morning (∼ 09:30 LST) compared to the afternoon may wrongly suggest that PO₃ would become more sensitive to VOCs earlier in the day. However, decreased light in the morning reduces the sensitivity of PO₃ to VOCs, despite a lower FNR (Sect. S1).

The contrast between dry and humid isopleths suggests that the presence of H₂O(v) enhances PO₃ when abundant NO₂ and HCHO are present. This trend is similarly observed in the F0AM model, as depicted in Figure S4, indicating that an increase in H₂O(v) over polluted regions (arbitrarily defined as HCHO × NO₂ > 10) increases PO₃. Nonetheless, more humidity suppresses PO₃ especially where VOC is limited and NO₂ is elevated possibly because the generated OH molecules from O¹D + H₂O(v) predominantly react with elevated NO₂.

Lastly, we see the highest PO₃ rates recorded among all scenarios under a hypothetical condition characterized by high humidity and photolysis rates. This condition is rare in nature because large amounts of H₂O(v) (0.8 × 10¹⁸ molec. cm⁻³) are confined to marine regions where surface reflectivity is low; nonetheless, an intuitive tendency from PO₃DNN suggests that the algorithm does not create non-physical extrapolation values.