When Does Geostatistical Interpolation Work? Monthly and Hourly Sensitivity of Ordinary Kriging for Urban Air Pollutant Mapping in Mexico City

Abstract

Urban air quality assessment increasingly relies on spatial interpolation to complement fixed monitoring networks; however, the reliability of geostatistical methods depends strongly on temporal conditions and pollutant characteristics. Despite extensive application, limited attention has been paid to how kriging performance varies across hours of the day and months of the year, particularly when contrasting primary pollutants driven by local emissions with secondary pollutants formed through atmospheric chemistry. This study evaluates the temporal sensitivity of Ordinary Kriging (OK) for mapping urban air pollutants in the Mexico City Metropolitan Area. Using hourly observations from the official air quality monitoring network (2021), we analyze ozone (O₃), a secondary pollutant, and sulfur dioxide (SO₂), a primary pollutant, under representative diurnal and monthly scenarios. Variogram model selection and predictive performance are assessed through leave-one-out cross-validation and external hold-out validation across multiple temporal blocks and months. Results indicate that kriging performance is highly sensitive to both hour of day and month. For O₃, smoother Gaussian variogram structures perform best during peak photochemical conditions, producing coherent regional concentration fields with gradual spatial gradients. In contrast, SO₂ exhibits stronger local variability and sharper spatial gradients, favoring exponential variogram models, particularly under stable morning atmospheric conditions associated with primary emission accumulation. Sensitivity analyses further reveal that no single variogram model is universally optimal and that interpolation accuracy depends more on temporal stratification and pollutant behavior than on variogram form alone. These findings demonstrate that geostatistical interpolation is a valuable tool for urban air quality assessment only when temporal sensitivity and pollutant-specific dynamics are explicitly incorporated. The proposed framework provides practical guidance for the responsible use of interpolated air quality maps, supports sustainable urban monitoring strategies, and contributes to more reliable exposure assessment in megacities with limited sensor coverage.

1. Introduction

1.1. Urban Air Quality Monitoring and Sustainability Challenges

Air pollution is one of the most significant environmental health risks worldwide [1]. According to the World Health Organization (WHO), 99% of the global population lives in areas where air pollution levels exceed recommended guideline limits, and only 17% of cities worldwide complied with these guidelines [2]. This situation poses a major challenge for urban sustainability, particularly in megacities characterized by high population density, complex topography, and limited atmospheric ventilation.

In Mexico, air quality remains a critical concern, driven by high emissions, adverse topographic conditions, and limited atmospheric dispersion [3]. Mexico City, in particular, exhibits pronounced spatial heterogeneity in pollutant concentrations, with several zones presenting moderate-to-high risk levels for sensitive populations. This variability underscores the need for continuous and spatially resolved air quality monitoring. Despite more than three decades of sustained regulatory and environmental management efforts, and despite improvements in some indicators, pollutants such as ozone and particulate matter continue to exceed air quality standards during several periods of the year [4].

Air quality monitoring in megacities plays a fundamental role in protecting public health, mitigating climate change impacts, and supporting evidence-based urban planning and policy-making [5]. Moreover, although difficult to quantify, previous studies suggest that effective air quality management can generate substantial long-term economic benefits by reducing health-related costs and productivity losses [6].

However, conventional air quality monitoring networks face important limitations related to cost, spatial coverage, and their ability to capture fine-scale spatial–temporal variability. Factors such as traffic intensity, industrial activity, and population density can lead to significant concentration differences over very short distances (≈0.5 km) [4,7]. While complementary approaches, such as satellite-based observations and atmospheric models, are increasingly used, they require complex processing and integration to provide reliable information at the urban scale [6]. Similarly, low-cost sensor deployments have demonstrated potential for short-term urban studies [1,8,9], but their application is often constrained by limited deployment periods and challenges related to long-term data stability and sensor maintenance.

In this context, spatial interpolation emerges as a complementary and cost-effective approach to estimate pollutant concentrations in areas without fixed monitoring stations by exploiting short- to medium-range spatial dependence inherent in urban environmental data. Geostatistical methods such as Ordinary Kriging (OK) are particularly attractive because they generate continuous spatial estimates while explicitly quantifying prediction uncertainty, a feature essential for decision-making in complex urban environments characterized by heterogeneous emission sources and variable atmospheric mixing [10].

1.2. Spatial Interpolation in Air Quality Studies

The application of geostatistical methods, particularly kriging, in air quality research can be broadly grouped into three main strands: (1) spatial analysis and classification of polluted areas, (2) methodological developments and comparisons of interpolation techniques, and (3) health risk assessment associated with air pollutant exposure. Together, these studies highlight the versatility of kriging as a spatial analysis tool, while also revealing important limitations regarding its temporal sensitivity and pollutant-specific behavior.

The first group of studies focuses on the spatial characterization and classification of air pollution across urban or regional scales. For example, Hopkins et al. [11] applied geospatial techniques to air quality data collected in India between 2018 and 2023. In this study, the concentrations of PM_2.5 (Particulate matter < 2.5 um), PM₁₀ (particulate matter < 10 um), NO₂, and SO₂ decreased during the COVID-19 lockdown period, and special emphasis was placed on classifying zones according to dominant pollutants. Similarly, Thakur et al. [12] investigated the spatial distribution of O₃ in China using hourly monitoring data from 2020 and kriging interpolation, identifying peak and low concentration periods and showing relationships between O₃ levels, rainfall, and co-pollutants such as SO₂ and NO₂. Long-term spatial analyses have also been conducted in major Chinese cities. Li et al. [13] analyzed air quality trends in Beijing from 2013 to 2019, reporting lower levels of SO₂, CO, PM_2.5, PM₁₀, and NO₂, while ozone exhibited a more variable behavior. Zhang et al. [14] focused on PM_2.5 in Beijing, finding a decreasing trend over time influenced by humidity and wind speed, whereas [15] examined multiple pollutants in Hangzhou and identified strong relationships between air pollution, land-use patterns, seasonal variability, and spatial distribution. At a broader scale, Denby et al. [16] mapped spatial trends in air quality across Europe using kriging combined with multiple linear regression. In the Mexican context, Ramirez et al. [17] described the spatial and temporal distribution of CO, NO₂, SO₂, PM₁₀, and O₃ in the city of Guadalajara from 2000 to 2005 using ordinary kriging, while [18] evaluated whether monitoring networks were adequately representative and sufficient in the Rhön region of Germany through data analysis and geostatistical methods.

A second body of literature focuses on methodological developments and comparative evaluations of air pollution interpolation techniques. Dai et al. [19] proposed a hybrid modeling framework that integrates vector autoregression (VAR), ordinary kriging, and XGBoost to predict ozone (O₃) concentrations across multiple seasons in China, demonstrating superior performance relative to traditional machine learning approaches such as decision trees. Dharmalingam et al. [20] conducted a comprehensive comparison of nine prediction methods applied to eight air pollutants, including Site Average, Inverse Distance Weighting, Kriging, and Random Forest. Their findings indicate that predictive efficiency varies substantially depending on pollutant type, highlighting the importance of pollutant-specific methodological selection.

Earlier comparative work by [21] evaluated different air quality prediction models used by the U.S. Environmental Protection Agency and showed that model performance varies substantially by pollutant. Beelen et al. [22] compared ordinary kriging, universal kriging, and regression-based approaches for developing air pollution maps, concluding that the relative efficiency of these methods depends not only on pollutant type but also on key characteristics of the study area, such as urban density, topography, emission source distribution, meteorological conditions, and monitoring network coverage. While these studies contribute to methodological advancement, they generally emphasize predictive accuracy rather than systematically assessing how kriging performance changes across different temporal conditions (e.g., hourly and monthly variability).

The third group of studies employs kriging as a supporting tool in health risk. Xue et al. [23] predicted the occurrence of fibrotic interstitial lung diseases using pollutants such as CO, O₃, SO₂, and NO₂, applying different methods including nearest-neighbor weighting and kriging, and finding that SO₂ and NO₂ were better estimated using kriging. Farhi et al. [24] analyzed cancer risk in newborns associated with prenatal exposure to SO₂, PM₁₀, NOx, and O₃ using GIS (Geographic Information System) and kriging, concluding that no direct relationship was detected. Malek et al. [25] examined the association between exposure to gaseous pollutants (NOx, NO₂, SO₂, O₃) and particulate matter (PM_2.5 and PM₁₀) with neurodegenerative disorders in menopausal women, finding that long-term risks were mainly associated with particulate matter. Additional studies have reported increased risks of chronic inflammatory airway diseases associated with NO₂ and SO₂ exposure [26], potential associations between O₃ and SO₂ exposure and type 2 diabetes [27], and mixed or inconclusive results regarding tuberculosis [28]. Other research has highlighted negative impacts of PM₁₀ on children’s lung function [29], associations between multiple pollutants and chronic obstructive pulmonary disease [30], and increased risks of hypertension, hypercholesterolemia, and diabetes under long-term exposure conditions [31], it can also be associated with kidney failure [32]. Cohort studies have further linked increased PM₁₀ and NOx concentrations during pregnancy with congenital malformations [24].

Although these studies demonstrate the widespread use of kriging as a spatial interpolation tool, it is typically employed as part of a broader analytical framework to characterize polluted areas or estimate exposure for health assessments. Across the three strands of literature, there is limited evaluation of the temporal sensitivity of kriging performance itself, particularly with respect to hourly and seasonal variability. Moreover, explicit comparisons between primary pollutants, which are directly emitted from local sources such as vehicles and industrial activities (e.g., SO₂), and secondary pollutants, which are formed through atmospheric chemical reactions and tend to exhibit broader regional patterns (e.g., O₃), remain scarce under a unified geostatistical validation framework. This gap is especially evident in the Mexican context, where relatively few studies have applied kriging to interpolate urban air pollution, underscoring the need for systematic analyses that account for both temporal dynamics and pollutant-specific spatial behavior.

1.3. Research Gap and Contributions

Although geostatistical interpolation has been widely applied in urban air quality studies, most existing research relies on fixed temporal snapshots or temporally aggregated data, with limited attention to how predictive performance varies across different hours of the day and months of the year. Moreover, many studies focus on a single pollutant or analyze multiple pollutants without explicitly accounting for their distinct physical behavior, particularly the contrast between primary pollutants dominated by local emissions and secondary pollutants formed through atmospheric chemical processes. As a result, the temporal sensitivity of kriging performance and its dependence on pollutant type remain insufficiently explored, especially in the context of megacities in Latin America. To address these gaps, this study evaluates the temporal sensitivity of Ordinary Kriging (OK) for urban air quality mapping in the Mexico City Metropolitan Area. Kriging performance is assessed across representative hourly blocks (09:00, 15:00, and 21:00) and seasonal periods (February, April, May, and August), explicitly comparing a secondary pollutant (O₃) and a primary pollutant (SO₂) under a unified geostatistical validation framework. The analysis combines leave-one-out cross-validation and external hold-out validation to quantify prediction accuracy, bias, and uncertainty.

Accordingly, this study addresses the following research question: How do hourly and monthly temporal scales influence the performance and reliability of Ordinary Kriging for primary and secondary air pollutants in a megacity? By systematically identifying when and for which pollutants kriging provides more robust spatial estimates, this work offers practical insights for the sustainable and physically consistent use of spatial interpolation as a complement to urban air quality monitoring networks.

2. Study Area and Data

2.1. Mexico City Metropolitan Area

Mexico City is located within a high-altitude basin at approximately 2240 m above sea level, surrounded by mountain ranges on three sides, with limited openings to the north, south, and southwest. The surrounding terrain reaches average elevations of about 3200 m, with peaks exceeding 5000 m, including major volcanic structures [33]. This semi-closed, horseshoe-shaped basin functions as a natural barrier to atmospheric dispersion, restricting self-purification processes and favoring pollutant accumulation [34]. Additional topographic features, such as the Chalco passage, Cuautla Valley, Ajusco Mountain, and the Tres Marías pass, further modulate airflow and contribute to spatial heterogeneity in pollutant concentrations [35].

The basin’s complex terrain strongly influences local and regional wind patterns. Thermal forcing generates daytime upslope flows that draw air into the basin, while nocturnal drainage flows and topographic blocking restrict ventilation, particularly around volcanic barriers [33]. Density currents transporting cooler air from the Gulf of Mexico and pressure gradients between the warm basin and surrounding highlands induce gap winds, especially near Chalco, producing strong but highly localized circulation patterns [36].

Meteorological processes further exacerbate dispersion challenges. Temperature inversions occur frequently, up to 25 days per month during winter, trapping pollutants near the surface and limiting vertical mixing [37]. Under weak synoptic forcing, winds are generally light and variable, leading to complex circulation regimes with strong diurnal and seasonal variability. The dry hot season is characterized by atmospheric blocking, low humidity, and minimal wind, conditions that promote pollutant accumulation and ozone episodes. In contrast, spring typically exhibits deeper mixing depths and more effective ventilation [38].

As a result of these interacting topographic and meteorological controls, pollutant transport and dispersion in the basin are highly heterogeneous in space and time. Studies have documented both rapid daily venting under favorable synoptic conditions and multi-day pollutant accumulation during stagnant episodes, with residence times occasionally exceeding several days [35,39]. This context underscores the importance of accounting for strong temporal variability and spatial complexity when analyzing and interpolating air pollutant concentrations in the Mexico City Metropolitan Area.

2.2. Air Quality Monitoring Data

The Mexico City Atmospheric Monitoring Directorate (SIMAT) [40] is managed by the Secretariat of the Environment of Mexico City (SEDEMA) and operates under the Ge-neral Directorate of Air Quality. The monitoring network consists of fixed stations distributed across the Mexico City Metropolitan Area and is operated by the local environmental authority.

Of the total registered stations, data from 21 monitoring stations were consistently available for the period 2021 to 2024. In total, the network comprises 34 air quality monitoring stations; however, the remaining stations did not report valid measurements during the study period and were therefore considered temporarily out of service.

Figure 1 shows the geographic distribution of the monitoring stations across the metropolitan area. Station names and geographic coordinates are provided in

Table A1. Nomenclature associated with air quality monitoring stations.

Code	Name	Municipality	State	Latitude	Longitude
AJU	Ajusco	Tlalpan	CDMX	19.1542	−99.1626
AJM	Ajusco Medio	Tlalpan	CDMX	19.2721	−99.2077
ATI	Atizapán	Atizapán de Zaragoza	Estado de México	19.5769	−99.2541
BJU	Benito Juarez	Benito Juárez	CDMX	19.3716	−99.1590
CAM	Camarones	Azcapotzalco	CDMX	19.4684	−99.1697
CHO	Chalco	Chalco	Estado de México	9.487227	−99.1142
CUA	Cuajimalpa	Cuajimalpa de Morelos	CDMX	19.365	−99.2917
CUT	Cuautitlán	Cuautitlán Izcalli	Estado de México	19.6467	99.2102
FAC	FES Acatlán	Naucalpan de Juárez	Estado de México	19.4824	−99.2435
GAM	Gustavo A. Madero	Gustavo A. Madero	CDMX	19.4827	−99.0946
HGM	Hospital General de México	Cuauhtémoc	CDMX	19.41161	−99.1522
INN	Investigaciones Nucleares	Ocoyoacac	Estado de México	19.2919	−99.3805
IZT	Iztacalco	Iztacalco	CDMX	19.3844	−99.1176
MGH	Miguel Hidalgo	Miguel Hidalgo	CDMX	19.4246	−99.1195
MPA	Milpa Alta	Milpa Alta	CDMX	19.2007	99.0113
PED	Pedregal	Álvaro Obregón	CDMX	19.3251	−99.2041
SAG	San Agustín	Ecatepec de Morelos	Estado de México	19.5329	−99.0303
TLA	Tlalnepantla	Tlalnepantla de Baz	Estado de México	19.5290	−99.2045
TLI	Tultitlán	Tultitlán	Estado de México	19.6025	−99.1771
UIZ	UAM Iztapalapa	Iztapalapa	CDMX	19.3607	−99.0738
VIF	Villa de las Flores	Coacalco de Berriozábal	Estado de México	19.6582	−99.0965

in Appendix A.

This study focuses on monitoring O₃ and SO₂, two pollutants with contrasting atmospheric behavior and relevance to urban air quality assessment. O₃ is a secondary pollutant formed through photochemical reactions and exhibits strong diurnal and seasonal variability, with concentration peaks typically occurring in the afternoon. In contrast, SO₂ is a predominantly primary pollutant, closely associated with combustion and industrial emissions, and is strongly influenced by local emission patterns and atmospheric dispersion conditions.

The selection of O₃ and SO₂ enables a comparative evaluation of the temporal sensitivity of spatial interpolation methods for pollutants governed by different physical and chemical processes. Descriptive statistics by month and hour for 2024 were used to characterize peak periods and temporal variability before geostatistical modeling. For O₃, May was the average peak month across all monitoring stations, with peak concentrations occurring between 14:00 and 16:00. For SO₂, February was the predominant peak month, with peak hours typically observed between 09:00 and 11:00, though some station-specific deviations were observed. Figure 2 illustrates an example of the temporal behavior of SO₂ at the UAM Iztapalapa (UIZ) station.

Other criteria pollutants available in the monitoring network, including PM₁₀, PM_2.5, NO₂, NOx, and CO, are not included in the present analysis and are considered as potential extensions for future research.

3. Data Preprocessing

3.1. Data Cleaning and Quality Control

Hourly air quality data for ozone (O₃) and sulfur dioxide (SO₂) from 2024 were obtained from SIMAT [40]. The dataset includes station identifiers, geographic coordinates (latitude and longitude), hourly averaged pollutant concentrations, date, and time.

A systematic data-cleaning and quality-control process was applied before geostatistical modeling. First, station coordinates were reviewed and converted to numeric format, correcting textual inconsistencies and decimal separators. Records with missing, inconsistent, or physically implausible concentration values were removed. All variables were homogenized to ensure a consistent format across stations and time periods. Stations with insufficient data coverage during the selected periods were excluded from the analysis.

Given the relatively compact spatial extent of the monitoring network (maximum inter-station separation ≈ 56 km), geographic coordinates were transformed into local Cartesian coordinates (x, y) expressed in kilometers to allow the use of Euclidean distances in variogram modeling. The conversion was performed using local approximations of 1° latitude ≈ 110.6 km and 1° longitude ≈ 111.3 km × cos (mean latitude). To ensure temporal stationarity for geostatistical analysis, representative months and hours were selected based on descriptive statistics of pollutant concentrations. For O₃, May and 15:00 were identified as the peak month and hour, corresponding to conditions of maximum photochemical activity. For SO₂, February and morning hours were identified as representative peak periods. The same preprocessing procedure was applied to both pollutants.

All data preprocessing, statistical analyses, and geostatistical modeling were performed using Python 3.10 in Google Colab (cloud-based Jupyter Notebook environment). The main libraries employed included NumPy, Pandas, SciPy, Matplotlib, and PyKrige for Ordinary Kriging implementation and variogram modeling.

Google Colab provides a Linux-based cloud execution environment with access to virtual CPUs and high-memory configurations, ensuring computational reproducibility and stability. No commercial or proprietary geostatistical software was used.

All simulations and code development were conducted on a MacBook Pro 16-inch equipped with Apple Silicon (M-series processor), 16 GB RAM, and macOS Sonoma, although all computational processing was performed in the Google Colab cloud environment rather than locally.

3.2. Geostatistical Modeling Using Ordinary Kriging

Spatial interpolation was performed using OK, a geostatistical method widely applied in environmental and air quality studies to estimate pollutant concentrations at unsampled locations based on spatial autocorrelation among observations [10,22]. Kriging has a long-standing theoretical foundation and has been extensively developed and applied over the past five decades, becoming a cornerstone of spatial statistics in geosciences Chilès & Desassis [41]. OK assumes an unknown but locally constant mean within the neighborhood of estimation, making it appropriate for environmental variables when no explicit deterministic spatial trend is specified.

Under this framework, the pollutant concentration at an unsampled location $x_0$ is estimated as a weighted linear combination of observed values at surrounding monitoring stations:

$$\widehat{Z}(x_0)={\sum }_{i=1}^n{\lambda }_{i}Z(x_i)$$

(1)

where $Z\left(x_i\right)$ denotes the observed concentration at the location $x_i$, ${\lambda }_i$ are the kriging weights [41], and $n$ is the number of neighboring observations used in the estimation. The kriging weights are obtained by solving a system of linear equations derived using Lagrange multipliers, ensuring predictions while minimizing estimation variance, subject to the spatial autocorrelation structure described by the fitted variogram model.

Spatial dependence among observations was characterized using the experimental semi-variogram, which describes how variance between observation pairs changes as a function of separation distance. The experimental semi-variogram was computed as [42]:

$$\gamma \left(h\right)={\displaystyle \frac{1}{2N\left(h\right)}}\sum _{i=1}^{N\left(h\right)}{\left[Z\left(x_i\right)-Z\left(x_i+h\right)\right]}^2$$

(2)

where $\gamma \left(h\right)$ represents the semivariance at lag distance $h$, and $N\left(h\right)$ is the number of data pairs separated by that distance. To obtain a continuous representation of spatial dependence suitable for kriging interpolation, the experimental semi-variogram was fitted using three commonly adopted theoretical models in air quality applications: spherical, exponential, and Gaussian. Although recent methodological advances have proposed regularized and hybrid kriging formulations to improve estimation under complex conditions [43], this study adopts classical variogram models to maintain interpretability and comparability with previous air pollution studies.

The parameters of the theoretical variogram models (nugget, sill, and range) were estimated using weighted least squares (WLS) fitting to the experimental semivariogram. The fitting procedure minimized the squared difference between experimental and modeled semivariance values across lag distances, assigning greater weight to bins with more point pairs. Automatic fitting routines implemented in the PyKrige library were used to ensure objective, reproducible, and algorithmically consistent parameter estimation. No manual or purely visual fitting was applied.

In addition to point estimates, OK provides a quantitative measure of prediction uncertainty through the kriging variance, which reflects the reliability of spatial estimates given the spatial configuration of monitoring stations:

$${\sigma }_K^2\left(x_0\right)=\sum _{i=1}^n{\lambda }_i\gamma \left(x_i-x_0\right)+\mu$$

(3)

where ${\sigma }_K^2\left(x_0\right)$ denotes the kriging variance at location $x_0$, $\gamma (x_i-x_0)$ is the modeled semivariance between observation and prediction locations, and $\mu$ is the Lagrange multiplier enforcing the unbiasedness constraint. The kriging weights are obtained by solving a system of linear equations derived using Lagrange multipliers, ensuring unbiased predictions while minimizing estimation variance, subject to the spatial autocorrelation structure described by the fitted variogram model.

To ensure temporal stationarity in the geostatistical modeling, kriging was applied to datasets corresponding to specific combinations of month and hour, as defined in Section 3.1. For O₃, spatial interpolation focused on May at 15:00, representing conditions of maximum photochemical activity and peak concentrations. For SO₂, representative peak conditions were selected in February and during the morning hours. This temporal stratification enables consistent comparison of spatial patterns across different atmospheric regimes while avoiding mixing heterogeneous temporal processes. The same geostatistical framework was applied to both pollutants, allowing a direct comparison of kriging performance for a secondary pollutant dominated by photochemical processes (O₃) and a primary pollutant primarily driven by local emissions and dispersion conditions (SO₂).

3.3. Model Validation and Performance Assessment

The performance of the geostatistical models was evaluated using a combination of internal and external validation strategies [44], thereby enabling a robust assessment of prediction accuracy and uncertainty under varying temporal conditions. Two complementary approaches were adopted: leave-one-out cross-validation (LOOCV) and hold-out validation using independent monitoring.

LOOCV was employed as an internal validation method to assess model robustness using the full dataset. In this procedure, each monitoring station was sequentially removed from the dataset, and its pollutant concentration was estimated using OK based on the remaining stations [44,45]. The predicted values were then compared with the observed concentrations at the omitted locations. This approach allows evaluation of model sensitivity to individual observations while preserving the spatial configuration of the monitoring network. For each variogram model (spherical, exponential, and Gaussian), prediction errors were quantified using three standard performance metrics: root-mean-square error (RMSE), mean absolute error (MAE), and mean bias error (bias). RMSE was used to emphasize larger deviations, MAE to provide a robust measure of average error magnitude, and bias to identify systematic over- or underestimation by the model.

To complement LOOCV and evaluate the models’ predictive capability at truly unobserved locations, an external hold-out validation was conducted. A subset of monitoring stations was randomly selected and excluded from the model calibration process, while the remaining stations were used to fit the variogram and perform kriging interpolation [42,44]. Pollutant concentrations at the hold-out stations were then predicted and compared against observed values. This validation strategy is particularly relevant for assessing the applicability of spatial interpolation in areas with limited monitoring coverage, as it simulates the practical scenario of estimating pollutant concentrations at locations where no measurements are available. The same performance metrics (RMSE, MAE, and bias) were computed to ensure consistency with the LOOCV evaluation.

Approximately 20% of monitoring stations were randomly selected using a fixed random seed to ensure reproducibility of the validation procedure. A single hold-out split was applied per temporal scenario (month–hour combination) to maintain methodological consistency and comparability across pollutants and time periods.

Model performance was systematically compared across the three theoretical variogram models considered in this study. The selection of the most appropriate variogram model for each pollutant and temporal scenario was based on a combined evaluation of predictive accuracy (RMSE and MAE), bias minimization, and the spatial plausibility of the resulting interpolated surfaces. Rather than relying on a single metric, this multi-criteria evaluation allowed identification of models that balance accuracy and stability across different validation approaches.

In addition to accuracy metrics, kriging variance was analyzed to assess spatial patterns of prediction uncertainty. Areas with higher kriging variance were interpreted as locations where estimates are less reliable due to sparse monitoring coverage or increased spatial variability. This uncertainty information complements point predictions and provides valuable insights for identifying regions where additional monitoring could improve air quality assessments.

The validation framework was applied consistently across different temporal scenarios, including selected hours and months representative of peak and non-peak conditions for O₃ and SO₂. This approach enables a direct comparison of model performance across temporal regimes and supports the evaluation of how spatial interpolation accuracy varies with atmospheric conditions and pollutant type. All validation procedures were implemented within a fully reproducible computational workflow, ensuring consistent parameter estimation, station selection, and performance metric calculation across all temporal scenarios.

The methodological flowchart of the Ordinary Kriging framework applied in this study is summarized in Figure 3. The flowchart illustrates the sequential process from data preprocessing and temporal stratification to geostatistical modeling, validation, and uncertainty assessment, ensuring transparency and reproducibility of the proposed framework.

4. Results

4.1. Descriptive Temporal Patterns of O₃ and SO₂

Descriptive statistics for ozone (O₃) concentrations in 2024 are presented in

Table 1. Monthly descriptive statistics at UIZ station, O₃, 2024.

Month	Mean	Median	Std *	Min	Max	n	p25 **	p75 ***
1	46.64188	42.52162	19.99761	10.51613	135.1667	460	33.59821	57.14259
2	51.96884	47	22.83287	16.24138	171.0769	483	37.20761	63.97222
3	55.55699	51.72727	20.98059	15.03226	166.3333	483	42.01613	64.73656
4	54.50459	52.92	17.30289	15.03704	112.1429	483	42.52013	63.89286
5	58.03709	55.03571	17.71254	19.53846	128.3333	483	46.73157	65.48746
6	37.87682	35	12.60866	14	90.2069	483	29.70417	42.94
7	30.3972	29.03704	9.253548	14.19048	70.78571	437	24.23077	35.86364
8	27.1636	24.90476	11.89331	8.157895	90.32258	435	19.425	33.69587

* Std = standard deviation; ** p25 = 25th percentile; *** p75 = 75th percentile, n = data number.

and

Table 2. Descriptive statistics by hour at the UIZ station during May, O₃, 2024.

Hour	Mean	Median	Std	Min	Max	n	p25	p75
1	38.2619	35.04348	16.57394	10.5	103	163	26.42955	46.21667
2	36.01109	33.375	15.10307	9.411765	88.5	163	24.875	44.82923
3	34.54254	33.28571	14.65417	9.176471	83.5	163	23.24815	42.76157
4	34.3171	32.22222	13.91843	8.157895	72.84615	163	22.80645	43.53784
5	34.39122	32.96	14.73063	8.730769	82	163	22.38889	43.86
6	37.21212	34.84	16.71861	9.423077	89.83871	163	24.79755	47.75714
7	43.56984	39.5	21.13661	8.962963	116	163	27.26154	55.69355
8	52.07109	45.55556	27.9995	10.38462	153.25	163	29.72989	68.98103
9	55.46481	48	28.98378	11.5	171.0769	163	33.85809	72.21667
10	54.58491	53.20833	24.2602	12.37037	135.1923	163	35.7577	68.81167
11	51.65196	51.83871	17.88423	13.40741	103.6	163	37.54839	62.25403
12	48.17588	47.44	14.11244	16.11538	96.93333	163	38.04849	57.81731
13	46.07734	45.43333	15.03253	19.5	110.3667	163	35.27381	51.98
14	46.85535	43.37931	17.8491	20.85714	116.1333	163	34.1523	53.17324
15	47.69461	45.98601	18.34663	14.25	114.0968	162	33.57955	56.18407
16	49.91696	48.20172	18.34619	21.3913	115.7895	162	36.09596	58.99107
17	52.03073	51.75	19.25995	17.64	111.2105	163	37.24702	60.63333
18	54.46105	52.55556	23.43304	17.11111	166.3333	163	37.03448	66.50862
19	53.42305	51.3	23.03444	16.33333	128.3333	163	38.18122	66.2963
20	49.52446	47.43478	20.56966	13.07407	109	163	35.39091	61.58796
21	46.4017	44.47368	19.08623	13.10714	105.9091	163	32.97222	55.16071
22	42.76662	40.72414	16.94158	13.82143	112.6	163	30.50278	50.79048
23	41.14418	39.30769	17.08175	12.36	120.2	163	29.43111	48.96552

.

Table 1 summarizes monthly statistics, indicating that May exhibits the highest mean O₃ concentrations, while February shows the largest variability, as reflected by the standard deviation. Regarding diurnal behavior, Table 2 shows that ozone (O₃) concentrations increase during the late morning and reach their highest levels in the afternoon, with peak mean values typically occurring between 14:00 and 18:00. At the station level, May is the most frequently identified peak month across the monitoring network. Figure 4 illustrates the hourly variation of ozone concentrations at the UIZ station as a representative example.

For SO₂, February exhibits the highest mean concentrations across the monitoring network, with greater variability observed during the winter months (January to March). As shown in Figure 4, the morning hours between 09:00 and 11:00 consistently exhibit the highest SO₂ levels, reflecting the combined influence of primary emissions and reduced atmospheric mixing under stable boundary-layer conditions. Across all stations in the network, February is the most frequent peak month for SO₂ (

Table 3. Monthly descriptive statistics at the CUA station, SO₂, 2024.

Month	Mean	Median	Std	Min	Max	n	p25	p75
1	3.479474	2.741338	2.426745	0.142857	13.28571	598	1.730769	4.41916
2	4.085112	3.285714	2.833703	0.26087	17.89286	621	2.125	5.285714
3	3.041043	2.101724	2.546145	0.413793	15.6	598	1.286866	3.967742
4	2.606915	2	1.847152	0.033333	10.68966	575	1.327381	3.25
5	1.716499	1.6	0.959801	0	7.166667	621	1.096774	2.24
6	1.46151	1.333333	0.797613	0	5.166667	598	1	1.857143
7	1.549131	1.381963	0.889087	0	6.1	598	1.034483	1.831897
8	1.610458	1.4	0.946928	0.068966	7.166667	529	1.086957	1.84

), although localized deviations in peak timing and magnitude were observed. To illustrate the typical diurnal pattern,

Table 4. Descriptive statistics by hour is this at CUA station during February, SO_2, 2024.

Hour	Mean	Median	Std	Min	Max	n	p25	p75
1	2.353397	1.375	2.410001	0	16.33333	206	1	2.899209
2	2.498405	1.384615	2.580064	0.04	15.70833	206	1.01	2.990385
3	2.471999	1.583333	2.380296	0	16	206	1.069231	2.875
4	2.598325	1.833333	2.393519	0	17.89286	206	1.161667	3.089822
5	2.663304	1.833333	2.366554	0	15.67857	206	1.202083	3.283251
6	2.79861	1.945906	2.479411	0.033333	16.71429	206	1.29	3.498208
7	3.002298	2.080645	2.511083	0	15.71429	206	1.403448	3.849624
8	3.462918	2.706897	2.455624	0	14.2	206	1.780357	4.729167
9	4.026582	3.196296	2.772314	0.166667	15.6	206	1.925066	5.446154
10	4.109744	3.058574	2.807956	0.307692	12.7619	206	2.046667	5.607184
11	3.78053	2.766667	2.613158	0.142857	11.88	206	1.882222	5.104809
12	2.996753	2.305172	1.835829	0.214286	9.5	206	1.666667	3.915
13	2.453683	2.135632	1.236239	0.5	7.25	206	1.6	3.145594
14	2.160181	1.983871	0.979425	0.3	7.2	206	1.460417	2.66129
15	2.017341	1.850389	0.920984	0.45	7	206	1.368881	2.478736
16	1.944375	1.75431	1.086287	0.44	10.57143	206	1.235345	2.4
17	1.852506	1.650167	0.934229	0.3	6.285714	206	1.252717	2.26859
18	1.757772	1.529825	1.022032	0.08	6.586207	206	1.168103	2.057745
19	1.580284	1.368519	0.959157	0	6	206	1.008065	1.75
20	1.442248	1.22957	0.890044	0.041667	5.666667	206	0.965517	1.621976
21	1.40945	1.196774	0.925054	0.033333	5.222222	206	0.929187	1.62069
22	1.499034	1.22381	1.011435	0	5.966667	206	0.896552	1.783333
23	1.689879	1.303321	1.30947	0.066667	6.75	206	0.931034	1.919167

and Figure 5 presents the hourly variation of SO₂ concentrations at the CUA station as a representative example; however, the reported temporal trends are derived from the full set of monitoring stations.

Descriptive analysis revealed clear and contrasting temporal patterns for both pollutants. O₃ concentrations exhibited strong diurnal variability, with consistent afternoon peaks driven by photochemical activity, whereas SO₂ showed morning maxima associated with primary emissions and limited atmospheric mixing. Across the monitoring network, May was identified as the peak month for O₃, with maximum concentrations occurring between 14:00 and 18:00, while February and morning hours (09:00–11:00) were predominant for SO₂. Although peak periods were generally consistent across stations, some local deviations were observed, reflecting site-specific emission and dispersion conditions. Interquartile ranges (p25–p75) further confirm higher temporal variability in O3 during the warm season and in SO₂ during the morning hours. These temporal patterns were used to define representative scenarios for subsequent geostatistical modeling.

4.2. Variogram Analysis and Model Selection

Variogram-based model selection was conducted using OK for the representative temporal scenarios identified in the descriptive analysis. Three theoretical variogram models, spherical, exponential, and Gaussian, were evaluated to characterize spatial dependence and assess predictive performance. Model evaluation was performed using leave-one-out cross-validation (LOOCV) and an external hold-out validation, with the root-mean-square error (RMSE), mean absolute error (MAE), and bias as performance metrics.

Table 5. LOOCV and Hold-out Results, O₃, Hour 15 h, May.

Model	LOOCV RMSE	LOOCV MAE	LOOCV Bias	Hold-Out RMSE	Hold-Out MAE	Hold-Out Bias
spherical	16.9307	10.8707	−0.1935	9.3855	7.8362	−3.0027
exponential	17.1476	10.8138	−0.2868	9.5189	7.9918	−2.8750
gaussian	14.5492	8.81597	−2.6317	10.3491	8.9841	−3.1821

presents the LOOCV and hold-out results for the representative O₃ scenario (May, 15:00 h). Under LOOCV, the Gaussian model achieved the lowest prediction errors (RMSE = 14.55 ppb; MAE = 8.82 ppb), outperforming the spherical and exponential models. Although the Gaussian variogram exhibited a moderately negative bias (−2.63 ppb), indicating a slight tendency to underestimate O₃ concentrations, its overall predictive accuracy was superior. Consequently, the Gaussian model was selected as the most appropriate variogram for O₃ spatial interpolation and uncertainty assessment.

To further assess model robustness, external validation was conducted using three randomly selected monitoring stations (CAM, CUT, and CHO) as a hold-out set (Table 5). While the spherical model yielded marginally lower RMSE and MAE values in the hold-out evaluation, differences among models were small, and all exhibited consistent negative bias. These results indicate moderate sensitivity to the validation approach but confirm the stability of the Gaussian model for representing O₃ spatial patterns under peak photochemical conditions.

For comparative purposes, the same validation framework was applied to SO₂. For each pollutant, representative scenarios were defined based on descriptive peak conditions to capture the dominant physical and chemical regimes: May at 15:00 h for O₃ (photochemical peak) and February at 10:00 h for SO₂ (primary emissions under stable morning conditions). The corresponding LOOCV and hold-out results for SO₂ are shown in

Table 6. LOOCV and Hold-out Results, SO₂, Hour 10 h, February.

Model	LOOCV RMSE	LOOCV MAE	LOOCV Bias	Hold-Out RMSE	Hold-Out MAE	Hold-Out Bias
spherical	1.451685	1.248203	0.102410	2.334734	2.175746	2.175746
exponential	1.552509	1.293633	0.123260	2.253626	1.956587	1.956587
gaussian	31.701103	11.507313	−6.369781	9.378000	8.416435	−0.454333

.

In contrast to O₃, SO₂ exhibited higher sensitivity to variogram model selection. The exponential model yielded the lowest RMSE and MAE values for both LOOCV and hold-out validation, indicating superior predictive performance. The spherical model showed comparable accuracy, while the Gaussian model performed poorly, with substantially higher errors and instability. These results are consistent with the predominantly primary nature of SO₂ emissions, which generate sharper spatial gradients and shorter correlation ranges, better captured by exponential-type variogram structures. Although a small bias was observed for the exponential and spherical models, its magnitude was negligible, indicating stable and reliable performance.

Overall, the results demonstrate that optimal variogram selection depends on both temporal conditions and pollutant characteristics. O₃, as a secondary pollutant, is better represented by smoother spatial structures, whereas sulfur SO₂ (a primary pollutant) exhibits stronger local variability requiring models with shorter-range spatial dependence. These findings highlight the importance of selecting pollutant-specific variograms when applying kriging-based interpolation in urban air quality studies.

4.3. Spatial Interpolation and Uncertainty Assessment

Spatial interpolation was performed using OK to estimate pollutant concentrations across the Mexico City Metropolitan Area for the representative temporal scenarios identified in the descriptive analysis. Interpolations were conducted separately for O₃ and SO₂, enabling a comparative assessment of spatial patterns of secondary and primary pollutants within a unified methodological framework.

For O₃, the interpolated concentration fields exhibited relatively smooth spatial gradients across the basin, with higher concentrations generally observed during the afternoon peak period. This spatial structure is consistent with the regional formation of O₃ through photochemical processes and its subsequent transport under prevailing wind conditions. The resulting kriging maps reveal coherent spatial patterns that extend beyond individual monitoring stations, highlighting areas of elevated O₃ exposure that are not directly captured by the fixed monitoring network. Figure 6a presents the spatial interpolation of ozone concentrations using OK with a Gaussian variogram for the representative peak scenario (May, 15:00 h). Higher concentrations are observed predominantly in the southern sector of the basin, whereas lower values are observed toward the northern areas. The absence of abrupt local gradients reflects the secondary nature of O₃ and supports the suitability of smooth variogram structures for this pollutant.

Figure 6b shows the corresponding kriging variance map, which quantifies the spatial uncertainty in predictions. Lower uncertainty is observed in areas with higher station density, whereas uncertainty increases toward the basin’s peripheral regions. Notably, regions of elevated O₃ concentration do not necessarily coincide with areas of highest uncertainty, underscoring the importance of jointly analyzing concentration and uncertainty fields when interpreting interpolated air quality surfaces.

In contrast, SO₂ exhibited more localized spatial patterns characterized by sharper gradients and higher spatial variability. Elevated SO₂ concentrations were primarily confined to specific regions of the basin, reflecting the influence of local combustion sources and industrial emissions combined with limited atmospheric dispersion during morning hours. Figure 7a presents the SO₂ interpolation for the representative scenario (February, 10:00 h), showing spatial patterns dominated by localized hotspots rather than smooth regional gradients. This behavior is consistent with the predominantly primary nature of SO₂ emissions and their stronger dependence on proximity to sources and local meteorological conditions.

The corresponding uncertainty map for SO₂ (Figure 7b) indicates higher spatial variability in prediction confidence, particularly in areas distant from monitoring stations. Compared to O₃, SO₂ uncertainty patterns are more heterogeneous, reflecting both the sharper spatial gradients and the greater sensitivity of primary pollutants to local emission and dispersion processes.

To further illustrate the applicability of the proposed framework, point-based predictions were generated at locations without monitoring stations. As an illustrative example, O₃ concentration was estimated at the Torre Latinoamericana, a central urban landmark. The predicted O₃ concentration was 54.79 ppb, with an associated kriging variance of 47.36, corresponding to an approximate standard deviation of 6.88 ppb. For SO₂, the predicted concentration at the same location was 1.99 ppb with an estimated standard deviation of 0.46 ppb. These results demonstrate the method’s ability to provide both concentration estimates and uncertainty bounds at unmonitored locations, supporting its potential use in exposure assessment and urban air quality analysis.

Overall, the spatial interpolation results highlight clear differences in the spatial behavior of O₃ and SO₂, reflecting their distinct physical and chemical characteristics. While O₃ exhibits broader regional patterns with smoother spatial transitions, SO₂ is dominated by localized hotspots and higher spatial heterogeneity. These contrasting behaviors underscore the importance of pollutant-specific considerations and uncertainty assessment when applying geostatistical methods to urban air-quality monitoring.

4.4. Sensitivity Analysis

To evaluate the robustness and temporal sensitivity of the proposed geostatistical framework, a sensitivity analysis was conducted by varying both the hour of day and the month for each pollutant. This analysis aims to assess how changes in atmospheric conditions affect the spatial dependence structure and the predictive performance of OK, beyond the choice of the variogram model itself.

For O₃, sensitivity analysis focused on representative stages of the diurnal photochemical cycle during May, the month with the highest average concentrations. Three characteristic time blocks were selected:

Morning (09:00 h): Relatively low ozone levels, influenced by residual nocturnal conditions and limited photochemical activity.

Afternoon peak (15:00 h): Maximum photochemical production under strong solar radiation.

Evening (21:00 h): Ozone decay phase, with reduced radiation and changing ventilation conditions.

Table 7. LOOCV cross-validation; 9,15 and 21 h, O_3.

Model	RMSE 9 h	RMSE 15 h	RMSE 21 h	MAE 9 h	MAE 15 h	MAE 21 h
spherical	8.492602	16.9307	18.917307	6.429629	10.8707	15.973494
exponential	9.147813	17.1476	19.991709	6.685627	10.8138	17.044828
gaussian	139.533149	14.5492	18.285069	68.107795	8.81597	15.263694

presents the leave-one-out cross-validation (LOOCV) results for these three hours. A clear increase in interpolation error from morning to evening was observed across all variogram models. RMSE values increased from approximately 8.4 ppb at 09:00 h to 14.5 ppb at 15:00 h and 18.2 ppb at 21:00 h, indicating increasing spatial heterogeneity during the evening decay period.

Notably, the Gaussian variogram exhibited unstable behavior during the morning period (09:00 h), with unrealistically high RMSE and MAE values. This result reflects the incompatibility of highly smooth spatial assumptions with weakly structured O₃ fields dominated by local variability and low concentrations. In contrast, spherical and exponential models showed more stable performance during the morning hours, suggesting that smoother variogram structures are more appropriate under peak photochemical conditions than under low-concentration regimes.

Seasonal sensitivity for O₃ was evaluated by fixing the hour at 15:00 h and comparing February (winter), April (transition), May (photochemical maximum), and August (rainy season). Results shown in

Table 8. LOOCV cross-validation; February, May and August, O₃.

Model	RMSE Feb	RMSE Apr	RMSE May	RMSE Aug	MAE Feb	MAE Apr	MAE May	MAE Aug
spherical	13.9780	17.3940	16.9307	16.094464	11.2406	11.8240	10.8707	9.906506
exponential	13.5090	17.4597	17.1476	16.391183	10.8734	12.3449	10.8138	10.640398
gaussian	14.4782	17.9417	14.5492	26.260674	11.2225	11.9900	8.8159	16.300298

indicate substantial seasonal variability in spatial dependence. February and May exhibited comparatively lower RMSE values (≈13.5–14.5 ppb), suggesting more coherent spatial fields under winter stagnation and peak photochemical production. In contrast, April and August presented higher interpolation errors (≈16–18 ppb), reflecting increased atmospheric variability associated with transitional conditions and convective mixing. Across all months, differences among variogram models were moderate, indicating that hourly and monthly effects exert a stronger influence on ozone spatial structure than the specific variogram form.

For SO₂, a predominantly primary pollutant, sensitivity analysis was designed to capture conditions of emission accumulation and dispersion. Hourly sensitivity was evaluated during February, the month with the highest SO₂ levels, by selecting four representative periods: 6:00 h (stable morning conditions), 10:00 h (peak emissions), 15:00 h (enhanced mixing), and 21:00 h (evening stabilization).

Table 9. LOOCV cross-validation; 6,10, 15, and 21 h, SO_2.

Model	RMSE 6 h	RMSE 10 h	RMSE 15 h	RMSE 21 h	MAE 6 h	MAE 10 h	MAE 15 h	MAE 21 h
spherical	3.0721	1.4516	0.9739	1.1092	2.1219	1.2482	0.7323	0.9020
exponential	3.0080	1.5525	1.0012	1.1321	2.0817	1.2936	0.7616	0.9104
gaussian	3.1889	31.7011	0.9603	1.0880	2.4605	11.5073	0.7342	0.8950

summarizes the LOOCV results for these hours. The highest prediction errors were consistently observed during early-morning conditions (06:00 h), when stable atmospheric layers favor local accumulation of emissions and the development of sharp spatial gradients. The lowest errors occurred at 15:00 h, coinciding with stronger atmospheric mixing and a more homogeneous spatial field. Evening hours (21:00 h) exhibited intermediate error levels, reflecting partial re-stabilization of the boundary layer.

Across all hours, spherical and exponential variogram models exhibited comparable, stable performance. In contrast, the Gaussian variogram consistently yielded unrealistically high RMSE and MAE values, indicating numerical instability and physical inconsistency in the SO₂ spatial fields. This behavior confirms that highly smooth variogram structures are unsuitable for modeling pollutants with short spatial correlation ranges and strong local emission control.

Seasonal sensitivity for SO₂ was evaluated by fixing the hour at 10:00 h and comparing February, April, and August (

Table 10. LOOCV cross-validation; February, April, and August hours, SO_2.

Model	RMSE Feb	RMSE Apr	RMSE Aug	MAE Feb	MAE Apr	MAE Aug
spherical	1.4516	1.524637	1.093720	1.2482	1.146517	0.792070
exponential	1.5525	1.556291	1.042159	1.2936	1.180675	0.756114
gaussian	31.7011	54.040358	41.609238	11.5073	25.105067	9.331594

). A progressive decrease in prediction error from winter to the rainy season was observed, reflecting enhanced dispersion and wet deposition processes during summer months. As with the hourly analysis, model performance was more strongly governed by atmospheric conditions than by the choice of variogram.

Overall, the sensitivity analysis demonstrates that temporal factors (hour and month) dominate the spatial dependence structure for both pollutants, while the choice of variogram model plays a secondary role. However, the nature of this sensitivity differs substantially between pollutants. O₃ exhibits strong hour and season-dependent variability associated with photochemical production and regional transport, requiring hour and month-specific spatial models. In contrast, SO₂ displays greater sensitivity to atmospheric stability and mixing conditions, with simpler variogram models providing more robust performance. These findings highlight the importance of pollutant-specific temporal stratification when applying geostatistical interpolation to urban air quality data, and caution against the use of static spatial models across heterogeneous atmospheric regimes. In Figure 8, the Leave-one-out cross-validation (LOOCV) RMSE for O₃ in May and SO₂ in February at representative hours of the day is shown. Results highlight the strong temporal sensitivity of spatial interpolation accuracy. For O₃, the Gaussian variogram performs best during peak photochemical conditions (15:00 h) but fails under weakly structured morning fields. For SO₂, primary emission dominance leads to higher errors under stable morning conditions, whereas improved performance is observed under enhanced atmospheric mixing. These results confirm that hour-specific variogram selection is essential for physically consistent geostatistical modeling.

5. Discussion

For O₃, the Gaussian variogram model achieved the lowest prediction errors under the leave-one-out cross-validation (LOOCV) scheme, indicating superior overall predictive performance for the representative peak photochemical scenario. However, this model exhibited a consistent negative bias, reflecting a slight tendency to underestimate O₃ concentrations. Under the hold-out validation scheme, differences in RMSE and MAE across variogram models were relatively small, and all models showed negative bias. These results indicate that, although the Gaussian model provides the best overall fit, uncertainty and bias remain non-negligible and should be explicitly acknowledged when interpreting interpolated O₃ fields.

The O₃ interpolation maps generated using the Gaussian model displayed smooth spatial patterns without abrupt gradients, consistent with the secondary nature of O₃ formation and its regional-scale behavior. Sensitivity analysis further revealed that kriging performance for O₃ is strongly dependent on both the hour of day and season. Smoother variogram structures performed better under conditions of intense photochemical activity (e.g., afternoon peak hours), whereas during periods of lower concentrations and weaker spatial structure (morning and evening), prediction errors increased. This confirms that O₃ spatial dependence is highly dynamic and temporally sensitive, reinforcing the need for hour-specific and month-specific modeling strategies.

The improved performance of the Gaussian variogram during afternoon peak hours can be physically interpreted in terms of boundary-layer dynamics and photochemical production. During midday and early afternoon, enhanced solar radiation promotes photochemical reactions leading to regional-scale O₃ formation, while convective mixing deepens the planetary boundary layer. This vertical mixing reduces localized concentration gradients and produces smoother horizontal spatial structures, which are better captured by variogram models imposing gradual spatial continuity. Conversely, during morning and evening transition periods, boundary-layer stability increases, vertical mixing weakens, and spatial heterogeneity becomes more pronounced, thereby reducing the predictive stability of smooth correlation structures.

In contrast, SO₂ exhibited markedly different behavior. As a predominantly primary pollutant, SO₂ exhibited greater spatial heterogeneity and sharper local gradients, leading to greater sensitivity to variogram model selection. Across both LOOCV and hold-out validation schemes, the exponential variogram consistently yielded lower RMSE and MAE values. In contrast, the Gaussian model exhibited instability and unrealistically high errors, particularly during the morning hours. This behavior reflects the incompatibility of overly smooth variogram structures with pollutants characterized by short correlation ranges and strong local emission influences.

The higher prediction errors observed during early morning hours are consistent with nocturnal boundary-layer stabilization and reduced atmospheric mixing. Under stable stratification, emissions from local combustion sources accumulate near the surface, generating sharp concentration gradients and short correlation ranges. Such conditions amplify spatial discontinuities that are poorly represented by smooth variogram structures. In contrast, during midday periods characterized by increased turbulence and boundary-layer growth, dispersion processes reduce spatial heterogeneity, leading to more stable interpolation performance. Monthly differences, including lower errors during the rainy season, may also reflect enhanced dispersion and wet deposition processes that moderate the contrast in extreme concentrations.

Spatial interpolation maps for SO₂ further support these findings, revealing localized concentration patterns and increased variability, particularly during early morning hours when stable atmospheric conditions favor pollutant accumulation and limit dispersion. Under nocturnal boundary-layer stabilization, reduced vertical mixing enhances short-range spatial dependence and sharp concentration gradients, leading to larger interpolation errors when smooth variogram models are applied. In contrast, during midday and afternoon periods, increased turbulence and boundary-layer growth promote dispersion and produce more homogeneous spatial fields, resulting in improved model stability. Seasonal differences, including lower errors during the rainy season, may reflect enhanced dispersion and wet deposition processes that moderate extreme concentration contrasts.

This behavior indicates that the Gaussian model may overestimate spatial continuity in contexts where the true correlation range is short and highly heterogeneous. In contrast, the exponential model allows sharper changes at short distances and better captures localized emission-driven variability, resulting in more stable and physically consistent predictions. These findings highlight the importance of aligning variogram structure with pollutant-specific spatial behavior, particularly when modeling primary pollutants under stable atmospheric conditions.

Taken together, these results indicate that temporal context (hour of day and month) exerts a stronger influence on kriging performance than the choice of variogram model alone. While Gaussian structures are more appropriate for secondary pollutants, such as O₃ under strong photochemical regimes, exponential models better capture the spatial behavior of primary pollutants such as SO₂. Importantly, no single variogram model can be considered universally optimal across pollutants and temporal conditions.

From an algorithmic perspective, these results demonstrate that variogram model selection is not merely a parametric adjustment but a structural decision that directly affects numerical stability and predictive reliability. The sensitivity of prediction error to variogram functional form underscores the need for temporally stratified and pollutant-aware model specification within geostatistical workflows.

Limitations and Implications

Several limitations of this study should be acknowledged. First, data availability was restricted to a subset of monitoring stations in the Mexico City network, which may limit spatial representativeness in some areas of the metropolitan region. Second, the interpolation framework relied exclusively on concentration measurements and did not explicitly incorporate emission inventories or meteorological variables (e.g., wind speed and direction, humidity, temperature), which are known to influence pollutant dispersion, particularly for primary pollutants.

From a methodological perspective, only three theoretical variogram structures (spherical, exponential, and Gaussian) were evaluated. Although widely used, more flexible covariance formulations (e.g., Matérn models, anisotropic structures, or non-stationary kernels) could better capture complex spatial processes in heterogeneous urban environments. Additionally, the temporal component was handled through discrete hourly and monthly stratification rather than a fully spatial–temporal model, which does not explicitly represent temporal autocorrelation.

The analysis focused exclusively on O₃ and SO₂, leaving other relevant pollutants such as NO_x, PM_2.5, PM₁₀, and CO for future research. Furthermore, the study was limited to a single megacity; extending the framework to a multi-city context would improve the generalizability of the findings. Future work could also evaluate whether similar temporal sensitivity patterns emerge for other secondary pollutants relative to O₃ and for other primary pollutants relative to SO₂.

Despite these limitations, the results demonstrate that Ordinary Kriging, when applied with explicit consideration of pollutant type and temporal context, can serve as a valuable complement to fixed air quality monitoring networks. From a sustainability perspective, this approach supports more informed use of interpolated maps for urban planning and exposure assessment, while emphasizing the importance of responsible interpretation of spatial predictions and associated uncertainty. Moreover, integrating geostatistical methods with emerging low-cost sensors and IoT-based monitoring systems offers promising opportunities to enhance spatial coverage in cities with limited monitoring infrastructure.

6. Conclusions

This study evaluated the temporal sensitivity of Ordinary Kriging (OK) for urban air-quality mapping in the Mexico City Metropolitan Area, explicitly comparing a secondary pollutant (O₃) and a primary pollutant (SO₂) under a unified geostatistical validation framework. By integrating descriptive analysis, leave-one-out cross-validation, external hold-out validation, and sensitivity analysis across hours of the day and months of the year, this work provides a comprehensive assessment of when and for which pollutants spatial interpolation yields reliable results.

The findings demonstrate that interpolation performance is strongly conditioned by both temporal context and pollutant characteristics. For O₃, a secondary pollutant governed by photochemical processes, smoother variogram structures, particularly the Gaussian model, yielded superior predictive performance under peak photochemical conditions, producing spatial fields with gradual gradients and coherent regional patterns. In contrast, SO₂, as a predominantly primary pollutant, exhibited sharper spatial gradients, stronger local variability, and shorter correlation ranges. For this pollutant, the exponential variogram consistently yielded more stable and accurate predictions, whereas overly smooth models led to instability and unrealistic estimates, especially during the morning hours.

Sensitivity analyses revealed that interpolation accuracy varies substantially across both hourly and monthly scales. For O₃, prediction errors increased during evening decay periods, reflecting greater spatial heterogeneity, whereas for SO₂, the largest errors were associated with early morning stable atmospheric conditions. These results confirm that no single variogram model is universally optimal and that geostatistical predictions at unmonitored locations should be performed using hour-specific and month-specific configurations to ensure physical consistency and predictive reliability.

From a sustainability perspective, this study highlights that spatial interpolation can effectively complement fixed air quality monitoring networks only when temporal sensitivity and pollutant-specific behavior are explicitly accounted for. When appropriately applied, geostatistical mapping can support urban planning, exposure assessment, and environmental decision-making by extending information beyond existing monitoring coverage, while requiring responsible interpretation and explicit communication of associated uncertainty.

Finally, the integration of geostatistical methods with emerging low-cost sensors and IoT-based air quality monitoring systems represents a promising pathway to expand spatial coverage in cities with limited monitoring infrastructure. Combined with monthly temporal stratification and pollutant-aware modeling, such approaches can contribute to more resilient, equitable, and sustainable urban air quality management.