Introduction

Deteriorating air quality is responsible for more than 7 million premature deaths per year globally, as identified by the World Health Organization1,2,3. Monitoring air pollutant distribution at high spatial resolution provides essential insights into understanding pollution patterns4, assessing public health risks5, and developing and implementing targeted policies to safeguard both environment and public health6,7,8,9,10.

Current air pollution measurement methods are subject to limitations and uncertainties, which hinder our understanding of air pollution distributions at high spatial resolutions4,11,12. Ground station-based measurements offer precise measurements. However, their implementation is constrained by substantial construction expenses, resulting in a restricted deployment of monitoring stations over extensive spatial coverage, especially in suburban areas13,14,15. On the other hand, satellite-based measurements provide extensive spatial coverage and cost-effective measurement, with substantial benefits in the areas where ground-based stations are impractical, thereby bridging the measurement gaps16,17. However, satellite-based measurement encounters limitations in accuracy and spatial resolution (e.g., several to tens of kilometers)11. Additionally, satellite observations are susceptible to interference from cloud cover and other environmental factors, leading to a lower-than-expected availability of data18. To conclude, the present measurements are limited by low-quality labeled data and uneven spatial coverage, posing significant challenges to high-resolution air pollution monitoring.

Currently, two kinds of approaches have been developed to monitor and reconstruct air pollution concentrations at fine scales. The physical knowledge-based methods19,20,21 assess the connection between geographic predictors (e.g., nightlight, road network, land use map) and air pollutants based on the available measurement data. The air pollution concentrations at locations lacking actual measurements can be predicted based on regression analysis22, enabling the derivation of a detailed spatial distribution of air pollution. The complex interplay among different predictors poses a challenge for physical knowledge-based models to map air pollution distributions accurately. On the other hand, AI-based methods are gaining popularity by offering a compelling avenue for capturing the nonlinear relationship between handcrafted predictors and regional air pollution distribution. Some approaches23,24 integrate satellite data with deep learning and physical laws to investigate the connection between air pollution concentration and regional predictors, generating detailed concentration maps. However, these solutions rely on high-quality labeled data and lack transferability—the ability to adapt learned information to unseen regions and related air pollutants—forcing researchers to focus on specific regions and tailor predictors to local conditions, as patterns may vary significantly across spatial contexts. These limitations restrict their applicability to a narrow range of scenarios.

Confronting the above challenges from data and transferability perspectives, to our knowledge for the first time, we introduce a powerful and transferable implicit representation, conceptualized as Height-Field Signed Distance Function (HF-SDF), to monitor air pollution with high spatial adaptability, fine-scale resolution, and comprehensive pollutant coverage even with noisy and low-resolution data (e.g., satellite observations). The key insight is that the continuous spatial distribution of air pollution can be viewed as an unfolded, irregular 3D surface by interpreting pollutant concentration as “height” in an additional dimension. The shape of this surface is then reconstructed by learning HF-SDF functions. Instead of struggling with intricate mapping of air pollution based on various hand-crafted geographic predictors, this new implicit representation, implemented through a self-supervised model, offers a powerful paradigm for integrating 3D geometric features to effectively reconstruct the continuous spatial distribution of air pollutants from low-quality labeled and heterogeneously scattered observations. By avoiding interference from redundancy and uncertainty in various localized predictors, HF-SDF achieves strong transferability and maximizes the network’s ability to capture potential correlations within available observations at minimal cost while producing reliable estimations for unmeasured areas. In this paper, we present the formulation of HF-SDF and further integrate it into an auto-decoder network to support the continuous modeling of air pollution. To validate our new implicit representation, we examined the proposed approach on the reanalysis dataset (TAP: Tracking Air Pollution in China) and raw satellite observations (TROPOMI: TROPOspheric Monitoring Instrument), focusing on its ability to generate high-resolution maps, robustness to sparse or regionally missing data, transferability to related pollutant species, and applicability to real-world observations.

Results

Reconstructing air pollution map continuously

To showcase the capability of high-resolution mapping, we reconstruct PM2.5 data across China based on TAP data at 10 km resolution. For comparison, both the TAP maps and our results are presented in Fig. 1. The experimental results demonstrate that our approach enables an extremely fine-scale mapping of PM2.5, zooming in from the regional scale to the detailed level of cities or towns. The experiment reports an R value of 0.992, an IOA of 0.996, a MAB of 1.181 μg/m3, and an RMSE of 2.117 μg/m3. Further, more details of the experimental setting and a visualization of the absolute bias are provided in Appendix B.1.

Fig. 1: Reconstruction of PM2.5 distribution in central China.
Fig. 1: Reconstruction of PM2.5 distribution in central China.The alternative text for this image may have been generated using AI.
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In (a) the TAP PM2.5 distribution within China is illustrated at a resolution of 10 km on November 3rd, 2023. For panels (b) (c) and (d) the upper images depict the TAP PM2.5 distribution in Central China, Hubei Province, and Wuhan City, respectively, at a 10 km resolution. The lower images provide detailed patterns of the estimated PM2.5 distribution in the corresponding areas of Central China, Hubei Province, and Wuhan City.

Reconstructing air pollution maps based on sparse observations

In this section, reconstructions are performed based on low-resolution inputs downsampled at various scales. Specifically, the tested input resolutions include 1 km, 5 km, 10 km, 20 km, 30 km, and 40 km PM2.5 data from TAP. Figure 2 shows that our reconstructions exhibit detailed spatial distribution patterns that consistently align with the diverse resolution inputs. The R and IOA values range from 0.97 to 0.99 and 0.98 to 0.99, respectively, while the MAB and RMSE remain lower than 2.63 μg/m3 and 3.93 μg/m3, respectively. Despite these promising qualitative results, an increasing disparity between our reconstructions and the ground truth (TAP 1 km) can be seen in the associated line chart and boxplot in Fig. 2. The assessment of reconstructed results provides insights into the robustness of HF-SDF for sparse data reconstruction. Additional details on the experimental settings and comparison results with Kriging interpolation25 are provided in Appendix B.2.

Fig. 2: Assessment of HF-SDF for ultra-high resolution mapping from sparse observations.
Fig. 2: Assessment of HF-SDF for ultra-high resolution mapping from sparse observations.The alternative text for this image may have been generated using AI.
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In (a) the PM2.5 distribution around the Sichuan Basin from the TAP 1 km dataset was downscaled at various scales, where DS 5 km, DS 10 km, DS 20 km, DS 30 km, DS 40 km illustrate downsampled inputs at resolutions of 5 km, 10 km, 20 km, 30 km, and 40 km, respectively. Panel (b) illustrates our reconstruction results at a resolution of 1 km, inferred from TAP 1 km, DS 5 km, DS 10 km, DS 20 km, DS 30 km, and DS 40 km, respectively. Panel (c) illustrates the difference between our 1 km resolution reconstructions and the data from TAP 1 km. Chart (d) shows two groups of R and IOA: Ours 1 km vs. TAP 1 km and Ours vs. Downsampled Input. Chart (e) illustrates two groups of distribution of reconstruction error: Ours 1 km vs. TAP 1 km and Ours vs. Downsampled Input. See Appendix B.2 for more details.

Reconstructing missing TAP data

To validate the robustness of our reconstruction method in handling unmeasured areas, we evaluate its performance in air pollution reconstruction with varying levels of missing data. In this experiment, we randomly mask five locations (circles in Fig. 3) on an air pollution map to simulate missing data. These masked maps are then used as inputs for high-resolution reconstruction. To assess the effectiveness of our method, we set five different levels of data loss, indicated by the size of the masked circles.

Fig. 3: High-resolution air pollution mapping with varying unmeasured areas.
Fig. 3: High-resolution air pollution mapping with varying unmeasured areas.The alternative text for this image may have been generated using AI.
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Panels (a–e) show reconstruction outcomes for maps with unmeasured areas of radii 0.05°, 0.15°, 0.25°, 0.35°, and 0.45°. GAP1 to GAP5 denote these unmeasured areas. Each panel displays (left to right): the input map, high-resolution output, complete map (GT), and bias visualization (CDF). See Supporting Information Appendix B.3 for more details.

Figure 3 summarizes the results of our method in addressing missing data, influenced by the scale of the unmeasured area and the connectivity of surrounding observations. Regarding the scale, for each unmeasured area, the CDF visualizations in Fig. 3 show that our reconstructions closely match the original distribution for smaller unmeasured areas (radius of 0.05°, 0.15°, and 0.25°), reporting bias lower than 7.5 μg/m3. However, the maximal bias of reconstruction increases to 15 μg/m3 as the scale of the unmeasured areas expands to 0.25° and grows further with larger areas. In terms of surrounding observations, analyzing the CDF line charts in Fig. 3 vertically reveals the following insights: 1. Similarity of Surrounding Concentration Patterns: If the surrounding concentration patterns share high similarity with the unmeasured area, the reconstructions are likely to have better quality. Specifically, GAP2, a moderately polluted area, and GAP4, a low-pollution area, both provide well-reconstructed results regardless of the scale of missing data. In contrast, GAP3, representing a heavily polluted spot, shows rapid degradation in reconstruction quality as the scale of the unmeasured area expands, with decreasing similarity. Essentially, the reconstruction quality depends on the specific relevance between the unmeasured distribution and accessible surrounding measurements. 2. Variability of Surrounding Observations: If the surrounding observations vary greatly in different directions, the reconstruction tends to yield more acceptable results. GAP1 and GAP5 consistently provide high-quality reconstructions, although the quality may slightly degrade with increasing unmeasured areas. Additionally, the reconstructions of regions without missing data consistently exhibit high quality. Further details of the experimental settings and comparison results are provided in Appendix B.3.

Reconstructing air pollution maps on unseen data

In this section, we utilize the coded map network to evaluate the generalization capability of HF-SDF within trained regions, as well as its transferability to entirely unseen locations and different air pollutants. Details regarding the dataset used for model training and testing, as well as model performance, are summarized in Table 1. The training set comprises PM2.5 data (TAP) collected from 2021 to 2022 in six locations: Chengdu (CD), Bazhou (BZ), Taiyuan (TY), Changsha (CS), Xining (XN), and Longnan (LN), each covering a geographic area of 10° × 10°. For additional details, refer to the Supporting Information (Table A7). To address concerns of homogeneity, the training dataset includes 1080 instances of concentration maps, randomly sampled from 180 days between 2021 and 2022 for each location. Subsequently, test sets, which include different regions, times, and air pollutants, were utilized to assess the model’s generalization capability and transferability. For more details of the experimental settings, see Appendix B.4.

Table 1 Performance metrics for training and testing datasets
Full size table

As summarized in Table 1, the model’s performance on the training set demonstrates its ability to effectively learn and fit the data, exhibiting high consistency with an R value of 0.96 and an IOA of 0.98. Regarding generalization, the model achieved a remarkable level of agreement when tested across different periods (2023) and regions (Yinchuan (YC)), with R values of 0.95 and 0.93, and IOA values of 0.97 and 0.95, respectively. This strongly indicates that HF-SDF is capable of producing fine-scale reconstructions of PM2.5 in previously unseen scenarios. In terms of transferability, the validation conducted on test sets that encompass various air pollutants related to PM2.5, including \({{\rm{SO}}}_{4}^{2-}\), \({{\rm{NO}}}_{3}^{-}\), and \({{\rm{NH}}}_{4}^{+}\), confirms the model’s superior capability in transferring knowledge across different types of pollutants. The R and IOA values for these pollutants range from 0.62 to 0.72. In particular, this variation in performance likely arises from the model, which was trained on PM2.5, not fully capturing pollutant-specific patterns present in these unseen datasets. Nonetheless, our findings suggest that with appropriate fine-tuning tailored to the characteristics of different pollutants, the model’s reconstruction accuracy could be significantly enhanced.

Reconstructing air pollution maps from satellite data: take the TROPOMI NO2 column as an example

Satellites provide a global perspective on air quality monitoring; however, limitations such as low spatial resolution and unexpected data gaps reduce data availability. As discussed above, HF-SDF facilitates the continuous reconstruction of air pollution data despite these challenges, enabling effective reconstruction of satellite data. Here, our method is employed to estimate NO2 maps for both a local area (35° N to 45° N, 115° E to 125° E) and a regional area (18° N to 54° N, 72° E to 136° E) on daily and 10-day scales using TROPOMI data, addressing the challenges of sparse, noisy, and unexpectedly missing satellite data. For local reconstruction (Fig. 4a), our continuous reconstruction is performed at various resolutions (10 km, 2 km, and 1 km), achieving an R value of 0.983 and an IOA of 0.991. These results highlight the effectiveness of HF-SDF in overcoming challenges related to real-world air pollution mapping. For the daily regional reconstruction (Fig. 4b), the experimental results demonstrate that our method successfully captures detailed patterns of air pollutants, significantly improving resolution to a fine degree, with an R value of 0.978 and an IOA of 0.989. While HF-SDF shows strong promise for high-resolution satellite mapping, we acknowledge that extensive missing observations can lead to reconstruction challenges. For example, in experiments using single-day TROPOMI data (see Appendix B.5), the reliability of reconstructions in areas with extremely limited data is compromised. In such cases, additional information, such as temporal features or data from other sensors, would be necessary to enhance inference and support robust reconstructions. Additional details of the experimental settings are provided in Appendix B.5.

Fig. 4: Experiments on TROPOMI.
Fig. 4: Experiments on TROPOMI.The alternative text for this image may have been generated using AI.
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The top row illustrates the projection of satellite data alongside our continuous reconstruction at various resolutions: a.1 presents raw NO2 data from TROPOMI; a.2 showcases our reconstruction projected at a resolution of 10 km; a.3 displays our reconstruction projected at a resolution of 2 km; a.4 exhibits our reconstruction projected at a resolution of 1 km; and a.5 visualizes the reconstructed 3D concentration surface. The bottom row depicts NO2 concentration maps over China on the 10-day scale. b.1 Features averaged NO2 concentration maps over 10 continuous days from May 1st to 10th, 2018, while b.2 presents the corresponding reconstruction. Reconstruction on a single day can be found in Supporting Information Fig. A7.

Discussion

In summary, the proposed implicit representation frameworks thus hold several advantages regarding the high-resolution mapping of air pollution. First, HF-SDF provides an advanced paradigm for end-to-end high-resolution mapping of air pollution from low-quality labeled or heterogeneous-distributed data to maps at expected resolution, significantly improving the quality of those data that suffer from lower-than-expected availability. Furthermore. the ability to be independent of other geographic predictors, such as road networks and land use maps, allows the structure of the proposed framework to be tailored for a broad range of air pollution-related applications, substantially widening the applicable scenarios with HF-SDF’s strong transferability (easily to be applied to unseen locations or species). The feasibility of new implicit representation would further permit direct data recovery from satellite observations, driving progress in air pollution monitoring by enriching perceptual capabilities.

Nevertheless, the current implementation of HF-SDF has some limitations. For one, the use of the Marching Cubes algorithm can introduce irregular spatial gradients where sharp transitions or noise in the data may cause discontinuities or artifacts. For another, when measurements are overly sparse, our method tends to produce smoother estimations, potentially underestimating pollution levels at hotspots while slightly overestimating them in cleaner areas. In future studies, incorporating more air pollution data from different periods and regions will be necessary to further refine the model, enhancing its accuracy and robustness.

This continuous model not only significantly aids in facilitating specific downstream tasks, such as air-quality-related health research and the development of air pollutant disposal strategies, but also may offer a new approach for addressing broader data assimilation challenges in the field of Earth Science. This is made possible by the growing demand for a unified representation, which is likely to extend to Earth System Models (ESMs) that integrate diverse environmental factors beyond the scope of specific pollutants across varying resolutions. Additionally, we anticipate the potential to use the proposed representation to learn spatial patterns of relevant geophysical parameters with higher resolution or fewer gaps, further enhancing the richness of air pollution monitoring.

Methods

The satellite data for air pollution monitoring has been limited by its inadequate availability, typically characterized by low resolution or incomplete observations. To address these challenges, we propose a novel implicit representation, called the Height-Field Signed Distance Function (HF-SDF), which models air pollutant distribution as a stretched surface within the abstract 3D space defined by latitude, longitude, and air pollutant density. Built upon HF-SDF representation, our general framework for high-resolution mapping of air pollution includes measurement, data processing, model training, and map inference (Fig. 5a–d). Next, we first introduce the design of our implicit representation, followed by the implementation of the pipeline, and finally, the model configuration and validation. For further details on the self-supervised model settings, please refer to Appendix A.

Fig. 5: Model framework.
Fig. 5: Model framework.The alternative text for this image may have been generated using AI.
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a–d illustrate the stepwise modeling approach, including measurement, data processing, model training, and map inference. Specifically, (a.1), (b), (c.1), and (d) represent the high-resolution mapping pipeline for single-day measurements, while (a.2), (b), (c.2), and (d) correspond to the high-resolution mapping pipeline for multi-day measurements. Notably, in (d), the zero iso-surface of the HF-SDF field is shown in white, with points having positive HF-SDF values colored in red to indicate positive discrepancies and points with negative values in blue to indicate negative discrepancies.

Implicit representation: Height-Field Signed Distance Function (HF-SDF)

Implicit representation refers to a class of neural networks26,27,28,29 that can learn to represent complex, high-dimensional functions without explicit parameterization, where the signal can be parameterized into a continuous field. In this study, we propose a novel implicit representation, called Height-Field Signed Distance Function (HF-SDF), to reconstruct continuous concentration maps for air pollutants. Our approach models the distribution of air pollution as an unfolded surface within a customized 3D space constructed by latitude, longitude, and air pollutant density.

In the context of HF-SDF shown in Fig. 6c, we assume that there is a manifold traversing the customized 3D space, dividing it into two disjoint portions (the upper and the lower). This manifold is referred to as the reference surface (or concentration surface) S. For each 3D point, we compute the HF-SDF value that measures the distance between the current position and the reference surface S along the height axis, with the sign indicating whether the point is above or below the reference surface S. To clarify, for an arbitrary point in the space represented as p(x, y, z) ∈ R3, we define the HF-SDF as a mapping: R3 ↦ R, denoted as SDF*:

$$SD{F}^{* }(x,y,z)=z-{z}^{{\prime} },\quad (x,y,{z}^{{\prime} })\in S,$$
(1)

where S is the 3D reference surface, the height (or z-axis) indicates air pollutant density, and \({p}^{{\prime} }:= (x,y,{z}^{{\prime} })\in S\) is a point sampled from the reference surface S. Particularly, the point \({p}^{{\prime} }\) shares the same (x, y) coordinates with point p, as it is the projection of p onto the reference surface S with respect to the height. It is straightforward to show that HF-SDF has an inherent differential property:

$$\frac{\partial SD{F}^{* }(x,y,z)}{\partial z}=1,\quad \forall (x,y,z)\in {{\mathbb{R}}}^{3}.$$
(2)
Fig. 6: Concept Illustration of HF-SDF.
Fig. 6: Concept Illustration of HF-SDF.The alternative text for this image may have been generated using AI.
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a Decision boundary of the implicit reference surface (the curved line) represents the zero iso-surface in the HF-SDF formulation. Red points indicate samples with positive HF-SDF values (SDF* > 0), while blue points indicate samples with negative values (SDF* < 0). b Zero iso-surface from a vertical cross-sectional perspective. c HF-SDF field modeling for air pollutants, where the space is defined by latitude, longitude, and estimated concentration. The reference surface S is the target of interest, corresponding to the decision boundary in (a); the upper and lower regions represent the positive and negative HF-SDF subspaces, respectively.

HF-SDF constitutes a continuous representation that may be intuitively understood as a learnable shape-conditioned classifier, for which the decision boundary is the concentration surface S (Fig. 6). In the setting of HF-SDF, the coordinates of each 3D point hold tailored physical meaning: The height represents the estimated density value at the geographic location (x, y), where x and y indicate latitude and longitude, respectively. Since the concentration surface is a 3D representation of an air pollution map, the HF-SDF value at each point p(x, y, z) can be understood as the discrepancy between estimated density and actual density at location (x, y). Intuitively, points above the concentration surface have positive discrepancies, leading to positive HF-SDF values (Fig. 6a, red points). Conversely, points below the concentration surface have negative discrepancies, resulting in negative HF-SDF values (Fig. 6a, blue points). Points lying on the concentration surface have HF-SDF values of zero (Fig. 6b, the white boundary between red and blue points).

Three-staged modeling

In practice, we incorporate the proposed implicit representation (HF-SDF) into an MLP-based deep framework for high-resolution mapping. Given the measurement (Fig. 5a), the pipeline consists of three stages: data processing (Fig. 5b), model training (Fig. 5c), and map inference (Fig. 5d).

The first stage, called dimension lifting, transforms raw air pollution measurements by interpreting pollutant concentration as “height” in an additional dimension. This allows the data to be mapped into a 3D concentration space defined by {(latitude, longitude, air pollutant density)}. The transformed measurements correspond to those scattered 3D points sampled on the concentration surface. In doing so, we convert the goal of high-resolution mapping to the reconstruction of a continuous concentration surface within the 3D space. For the sake of training ease, we normalize the points to be within a regular cube. To train a neural network to fit the underlying surface, we further sample points with positive and negative HF-SDF values around the transformed measurements.

In the second stage, two networks are tailored for HF-SDF to model the geometric structure of the concentration surface, distinguishing between single-day and multi-day measurements. These networks share the same backbone, as shown in Fig. 5c. If only single-day measurement is available, then we employ an MLP-based network (Fig. 5c.1) to regress a continuous HF-SDF Field for its concentration surface with the pre-processed data from the last stage, which is regularized by a certain geometric constraint of HF-SDF. Specifically, it inputs the 3D points from single-day observations and outputs their corresponding HF-SDF values. We can further enhance our model given access to multi-day measurements through an auto-decoder network. In this case, we introduce a learnable low-dimensional vector for coded map HF-SDF, known as a latent code, which encapsulates most of the prior knowledge. As illustrated in Fig. 5c.2, the latent code, concatenated with the 3D point, serves as an additional input to predict HF-SDF values, further refining the captured patterns through iterative updates. With the well-trained network, a continuous HF-SDF field can be approximated for a set of observations.

The third stage is a procedure called high-resolution map inference, which involves identifying the concentration surface from an approximated continuous HF-SDF field and then projecting it onto a 2D map. Under the assumption of the HF-SDF field, the concentration surface is implicitly represented as the zero iso-surface of the HF-SDF field. To identify this concentration surface, we first sample n3 grid points at expected resolution in the unit cube (xi, yj, zk), i, j, k ∈ {1, 2, ⋯, n}. Here, n can be tuned so that the query resolution is sufficient for the user. With a trained neural network, we evaluate the HF-SDF value for each grid point, which indicates the distance from this point to the underlying concentration surface (e.g., corresponding to the white part of the HF-SDF field in Fig. 5d). The Marching Cubes algorithm30 is then applied to identify the concentration surface of interest with a triangle mesh based on the grid points and their HF-SDF values. Once the surface points have been identified, it is straightforward to project the result vertically onto a 2D map with respect to ‘height’—the dimension indicating air pollutant density.

Model configuration and validation

The self-supervised model, an MLP-based auto-decoder network defined by Eq. (3), is constructed to reconstruct the concentration surface from available 3D points. Regarding the single map network, the MLP with 8 layers is taken to estimate the HF-SDF field with a customized loss function (Eq. (4)).

$${f}_{\theta }(x)\approx SD{F}^{* }(x),\forall x\in \Omega$$
(3)
$${\mathcal{L}}(\theta )={{\mathcal{L}}}_{{\mathcal{I}}}(\theta )+\lambda {{\mathcal{L}}}_{{\mathcal{J}}}^{{\prime} }(\theta )$$
(4)

Here, λ = 0.1 is a hyperparameter. The first term \({{\mathcal{L}}}_{{\mathcal{I}}}(\theta )\) is given by:

$${{\mathcal{L}}}_{{\mathcal{I}}}(\theta )=\frac{1}{| {\mathcal{I}}| }\sum _{{x}_{i}\in {\mathcal{I}}}\left\vert {f}_{\theta }({x}_{i})\right\vert .$$

This term encourages the function f to vanish on the point cloud \({\mathcal{I}}\), where \({\mathcal{I}}\) represents the set of observations. The second term \({{\mathcal{L}}}_{{\mathcal{J}}}^{{\prime} }(\theta )\) is defined as:

$${{\mathcal{L}}}_{{\mathcal{J}}}^{{\prime} }(\theta )=\frac{1}{| {\mathcal{J}}| }\sum _{{x}_{j}\in {\mathcal{J}}}\left| \frac{\partial {f}_{\theta }({x}_{j})}{\partial z}-1\right| .$$

This term encourages the partial gradient \(\frac{\partial {f}_{\theta }(x)}{\partial z}\) to be close to 1, as indicated by Eq. (2). Here, \({\mathcal{J}}\) represents a point cloud randomly sampled round about \({\mathcal{I}}\), xj denotes any simple in \({\mathcal{J}}\), and z indicates height.

Regarding the coded map network (Eq. (5)), it takes latent code \({\mathcal{Z}}\) as an additional input to the MLP-based model aside from 3D points, allowing the network to learn domain knowledge of air pollution. At training time, we use the loss function defined as Eq. (6).

$${f}_{\theta }({\mathcal{Z}},x)\approx SD{F}^{* }(x)$$
(5)
$${{\mathcal{L}}}_{coded}(\theta )=\frac{1}{| {\mathcal{B}}| }\sum _{k\in {\mathcal{B}}}{\mathcal{L}}(\theta ,{{\mathcal{Z}}}_{k})+\alpha \left\Vert {{\mathcal{Z}}}_{k}\right\Vert ,$$
(6)

where \({{\mathcal{Z}}}_{k}\in {R}^{256}\), initialized to 0, is a latent vector corresponding to each training example \(k\in {\mathcal{K}}\), \({\mathcal{B}}\subset {\mathcal{K}}\) is a batch, α = 0.01, \({\mathcal{L}}\) defined in Eq. (4), λ = 0.1 is a hyperparameter. Once the network is well-trained, for each reconstruction target, a latent code \({\mathcal{Z}}\) corresponding to its observations \({\mathcal{X}}\) can be estimated via Maximum-a-Posterior (MAP) estimation (Eq. (7)). Information regarding the specified hyperparameters can be found in Supporting Information.

$${\hat{{\mathcal{Z}}}} = \mathop{{\mathrm{arg}}\,{\mathrm{min}}}\limits_{{{{\mathcal{Z}}}}}\quad {{\mathcal{L}}}(\theta,{{\mathcal{Z}}})+\alpha \left\| {{\mathcal{Z}}} \right\|$$
(7)

For model validation, statistical metrics, including correlation coefficient (R), index of agreement (IOA), mean absolute bias (MAB), and root mean squared error (RMSE), were used for model evaluation, according to Eqs. (8)–(11), respectively. In these equations, Oi denotes observation, while Ei represents estimation.

$$R=\frac{\mathop{\sum }\nolimits_{i = 1}^{n}({O}_{i}-\bar{O})({E}_{i}-\bar{E})}{\sqrt{\mathop{\sum }\nolimits_{i = 1}^{n}{({O}_{i}-\bar{O})}^{2}\mathop{\sum }\nolimits_{i = 1}^{n}{({E}_{i}-\bar{E})}^{2}}}$$
(8)
$$IOA=1-\frac{\mathop{\sum }\nolimits_{i = 1}^{n}{({O}_{i}-{E}_{i})}^{2}}{\mathop{\sum }\nolimits_{i = 1}^{n}{(| {O}_{i}-\bar{O}| +| {E}_{i}-\bar{E}| )}^{2}}$$
(9)
$$MAB=\frac{1}{n}\sum | {E}_{i}-{O}_{i}|$$
(10)
$$RMSE=\sqrt{\frac{1}{n}\sum {({E}_{i}-{O}_{i})}^{2}}$$
(11)