A Classification of Bias Correction Methodologies used in Air Quality Scenarios Modelling Applications

A Classification of Bias Correction Methodologies used in Air Quality Scenarios Modelling Applications version 1.0 G. Bonafè1, L. Colombo2, C. Collaveri3, F. Guarnieri4, S. Bande5, R. Amorati6, A. Dalla Fontana7, B. Intini7, L. Vitali8, and M. Mircea8 1ARPA FVG, 2Arpa Lombardia, 3ARPAT, 4LaMMA, 5Arpa Piemonte, 6Arpae, 7ARPAV, 8ENEA November 3, 2025 Air quality modelling systems (AQMS) cannot always reproduce observed pollutant concentrations due to limited input data and incomplete knowledge of atmospheric processes. Bias correction methodologies (BCM) are continuously developed to address these limitations when predicting air concentrations for emission scenarios. We propose a consistent framework for classifying BCMs along four key dimensions: (i) the sequence of operations (spatialization, calibration, and application of adjustment), (ii) the type of adjustment (additive, multiplicative, linear, etc.), (iii) the calibration strategy (per station, per grid cell, or global), and (iv) the spatialization approach (e.g. thin-plate spline, ordinary kriging, kriging with external drift). Examples presented in this report show how existing BCMs can be mapped into this framework. The proposed classification aims to improve comparability and reproducibility within the scientific community and to support clear communication of results to stakeholders. I sistemi di modellistica della qualità dell’aria (AQMS) non riescono sempre a riprodurre le concentrazioni osservate degli inquinanti, a causa della limitata disponibilità di dati di input e della conoscenza incompleta dei processi atmosferici. Le metodologie di correzione del bias (BCM) vengono continuamente sviluppate per affrontare tali limitazioni nella previsione delle concentrazioni in scenari emissivi. In questo rapporto proponiamo un quadro coerente di classificazione delle BCM articolato lungo quattro dimensioni principali: (i) la sequenza delle operazioni (spazializzazione, calibrazione e applicazione dell’aggiustamento), (ii) il tipo di aggiustamento (additivo, moltiplicativo, lineare, ecc.), (iii) la strategia di calibrazione (per stazione, per cella o globale) e (iv) l’approccio di spazializzazione (ad esempio thin-plate spline, kriging ordinario, kriging con deriva esterna). Gli esempi presentati mostrano come le BCM esistenti possano essere ricondotte a questo schema. La classificazione proposta mira a migliorare la confrontabilità e la riproducibilità all’interno della comunità scientifica e a favorire una comunicazione più chiara dei risultati verso gli stakeholder. 1 Contents 1 Introduction 3 2 Key elements of the classification scheme 4 2.1 Unbiasingsequences .................................. 4 2.2 Adjustmentmethods.................................. 5 2.3 Calibrationstrategy .................................. 5 2.4 Spatialization algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Constraints, extensions, and limitations of the classification scheme 6 3.1 Constraints on valid combinations . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 Extensionmechanisms................................. 6 3.3 Hybridapproaches ................................... 7 3.4 Limitations and complementary documentation . . . . . . . . . . . . . . . . . . . 7 4 Examples 8 5 Discussion and conclusions 9 2 1 Introduction The use of atmospheric modelling systems to assess air quality and to inform the development of air quality plans plays a central role in the new Air Quality Directive (EU) 2024/2881. However, the pollutant concentrations produced by atmospheric modelling applications often differ from those observed. These differences, hereafter referred to as biases, originate from multiple sources: uncertainties in emission inventories, inaccuracies in meteorological inputs, limited knowledge or simplifications of atmospheric physical and chemical processes, surfaceatmosphere interactions, and the spatial resolution adopted. Bias correction algorithms are needed to relate the outputs of atmospheric modelling applications to measurements, and to further apply them for improving the modelling of air quality scenarios. In scenario modelling applications, direct observations are not available, and the correction relies on relationships previously established through the comparison between an atmospheric modelling application (referred to as the reference or baseline case, which uses a baseline emission scenario) and observations. Bias correction methodologies (BCM) range from simple statistical adjustments at station level to complex combinations of spatial or temporal adjustments. This variety complicates the comparison of results, the interpretation of differences between studies, and the reproducibility of methods. A shared terminology and a structured framework help to describe BCMs in a consistent way. In this report, a scenario denotes a model simulation that uses input data different from the baseline scenario, representing a specific condition or assumption, such as a reference year, a future projection, or the implementation of an emission reduction policy. The base case (or reference case) refers to the modelling application for the baseline scenario. The output of this application is one of the key inputs in developing bias correction methodologies for the other scenario modelling applications. The air quality modelling systems (AQMS) considered here operate on a regular grid and are typically chemical transport models (CTMs). They estimate pollutant concentrations as averages over spatial cells and time intervals, usually on an hourly basis. In contrast, observations refer to point measurements at monitoring stations. The classification proposed here mainly applies to 2D gridded CTMs but can be extended to other types of numerical models with similar spatial structures. In general, a BCM aims to characterise the systematic model error by comparing the base case with available observations. The resulting relationships are then applied to the scenario simulation, for which direct measurements are not available. The underlying assumption is that the structure of the model errors in the base case can be transferred to the scenario through suitable mathematical formulations and auxiliary information. This report introduces a practical classification scheme for BCMs. Each methodology can be represented as a combination of defined operations and algorithms, as shown in Section 2. In addition to this theoretical classification of BCMs, Section 3 show their potential constraints, extensions, and limitations which may be experiences in their application. Finally, Section 4 provides practical examples of how existing BCMs can be mapped into this classification framework. The classification scheme was developed and refined in the framework of two collaborative activities: the FAIRMODE WG5 Unbias Exercise1and the Italian Unbias Exercise2. The FAIRMODE WG5 Unbias Exercise benchmarks bias correction methodologies applied to air quality models across Europe, focusing on policy-oriented scenarios. The Italian Unbias Exercise adapts this framework to national-scale applications, where complex orography and diverse 1Forum for Air Quality Modelling in Europe, Working Group 5, https://fairmode.jrc.ec.europa.eu/ Activity/CT5 2Working Group for Air Quality Model Application established by Italian Ministry of the Environment and Energy Security (MASE) 3 emission patterns test the robustness of BCMs. The datasets used include annual concentrations of PM2.5, NO2, and O3for both exercises, and PM10 for the Italian exercise only, which were used for testing this classification. 2 Key elements of the classification scheme A BCM can be described by four key elements. The first is the sequence, which specifies the order of applying the fundamental operations: calibrate the adjustment algorithm (C), apply the adjustment (A), and spatialize the data (S). The second element is the adjustment algorithm, the mathematical function used to adjust the model values. The third is the calibration method, which defines the strategy for estimating the parameters of the adjustment algorithm from the data. The fourth is the spatialization algorithm, the method used to transform point-based information into a continuous field. These elements are combined into a standardized code that uniquely identifies each method according to the classification pattern: <sequence>.<adjustment>.<calibration>.<spatialization> For example, the code CAS.Lin.Each.ked describes a method that: calibrates a linear adjustment algorithm at each station, applies it to obtain adjusted point values, then spatializes these using kriging with external drift. Possible values for the key elements are listed in Table 1. The following subsections describe each element in detail to clarify the meaning of the entries in the table. Table 1: Possible values for the elements of the unbiasing method classification scheme, in the pattern <sequence>.<adjustment>.<calibration>.<spatialization> <sequence> <adjustment> <calibration> <spatialization> SCA Add All tps CSA Mult Each idw CAS Lin Grid ok CA Resc Cell ked Quant Neigh mlr ebk . . . 2.1 Unbiasing sequences The unbiasing sequence defines the workflow by specifying the order in which the fundamental actions – Calibrate (C), Apply (A), and Spatialize (S) – are performed. We define four main sequences. SCA (Spatialize →Calibrate →Apply) This sequence first spatializes the observed data to the entire grid. Then, the adjustment algorithm is calibrated by comparing the spatialized observations to the model base case, and finally applied to the scenario. CSA (Calibrate →Spatialize →Apply) This sequence starts by calibrating the adjustment algorithm at each monitoring site, generating point-based adjustment parameters (e.g., bias offsets). These parameters are then spatialized across the domain to create a continuous adjustment field, which is applied to the model scenario. 4 CAS (Calibrate →Apply →Spatialize) In this sequence, the adjustment algorithm is calibrated and applied directly at each monitoring station, creating unbiased point values. These corrected values are then spatialized over the entire domain to produce the final unbiased grid. CA (Calibrate →Apply) This sequence involves applying an adjustment after calibration, either globally over the entire grid or locally at monitoring sites. This sequence bypasses the need for a separate spatialization step from sparse points, as the adjustment is applied directly to the grid cells. 2.2 Adjustment methods The adjustment method defines the mathematical transformation used to adjust the model value. Several methods are commonly used. Add The additive method adjusts by adding a constant value: Ccorr =Cmodel +∆. It is suitable for biases that are independent of the concentration level. Mult The multiplicative method adjusts by multiplying by a factor: Ccorr =α·Cmodel, addressing proportional errors. Lin The linear method applies a linear transformation: Ccorr =α·Cmodel + ∆. Resc The rescaled additive method adds an adjustment term that is scaled by the ratio between base case and scenario simulations: Ccorr =Cmodel + ∆ ·Cmodel Cbase . Quant Finally, the quantile-based method adjusts values based on quantile-specific factors derived from matching the cumulative distribution functions of model and observations, which corrects biases across the entire value range. 2.3 Calibration strategy The calibration strategy specifies how the parameters of the adjustment algorithm (e.g., ∆,α) are estimated from the data. Different strategies can be used. All In this global, point-based method, a single set of adjustment parameters is derived using data from all available monitoring stations combined. Each This local, point-based method, calibrates a distinct set of parameters locally for each monitoring site. Grid This global, grid-based method, performs calibration over the entire grid, implying a single, globally applied set of parameters. Cell This local, cell-by-cell method, considers each model grid cell individually, adjusting parameters to remove bias at the cell level. Neigh The cell neighborhood method, uses information from surrounding cells to adjust parameters for a target cell. 5 2.4 Spatialization algorithms The spatialization algorithm determines how point-based information is propagated to the entire grid. The nature of the information being spatialized depends on the unbiasing sequence: it can be air quality concentrations (for sequences SCA, CAS) or the parameters of the adjustment algorithm (for sequence CSA). Common spatialization algorithms include tps thin-plate spline, idw inverse distance weighted, ok ordinary kriging, ked kriging with external drift, mlr multiple linear regression with residual interpolation, oi optimal interpolation, mwk moving window kriging. 3 Constraints, extensions, and limitations of the classification scheme While the classification scheme provides a comprehensive framework for describing unbiasing methods, several important constraints, extension mechanisms, and limitations must be considered for its proper application. 3.1 Constraints on valid combinations Not all theoretical combinations of the classification elements are methodologically valid or practically meaningful. The main constraints arise from the logical dependencies between the unbiasing sequence and the other elements. First, there are constraints related to the sequence and calibration strategies. For the SCA sequence, the calibration must operate on gridded data because the spatialization step comes first. Therefore, only Grid,Cell, or Neigh calibration strategies are permitted. For the CA,CSA, and CAS sequences, calibration occurs at observation points, so only All or Each calibration strategies are valid. Second, there are constraints between the adjustment algorithm and the calibration strategy. When applied to a single time step (or temporally averaged data), some combinations are invalid because calibration and adjustment can rely only on spatial information. For example, Linear (Lin) and Quantile-based (Quant) algorithms cannot be calibrated using Each or Cell strategies, as these would attempt to fit complex models with insufficient data. Such algorithms require pooled data from multiple locations, so only All or Grid calibration strategies are appropriate. Simple algorithms like Additive (Add) and Multiplicative (Mult) can work with any calibration strategy, including single-point or cell-by-cell calibrations. These constraints can be relaxed when multi-temporal data are available, allowing greater flexibility in combining adjustment methods and calibration strategies. 3.2 Extension mechanisms The classification scheme is designed to be extensible to accommodate methodological innovations. No additional sequences beyond SCA, CSA, CAS, and CA are currently defined, as these 6 cover the fundamental ordering possibilities of the three core operations. If a method uses an adjustment method, calibration method, or spatialization technique not included in the standard classification, new codes can be added. The temporal dimension of calibration or adjustment can be specified using the hyphen operator "-". For example, Mult-day indicates a multiplicative adjustment applied on a daily basis, Add-month denotes an additive adjustment calibrated monthly, and Lin-season specifies a linear adjustment calibrated separately for each season. This allows the classification scheme to describe whether adjustments are applied annually (default) or at higher temporal resolutions before aggregation. 3.3 Hybrid approaches Some methods combine multiple algorithms within the same unbiasing workflow. The classification scheme accommodates these hybrid approaches using the "+" operator. For example, a method might apply different adjustments for different concentration ranges (e.g., additive for low values, multiplicative for high values), which would be coded as Add+Mult. Similarly, an approach might use different interpolation techniques sequentially (e.g., MLR for the background field followed by kriging of residuals), coded as mlr+ked. A sophisticated example of method hybridization can be found in approaches that combine class-based calibration with spatial interpolation. Consider a methodology that: (1) builds a classified frequency distribution from all concentration values of the base case simulation, (2) calculates class-specific average changes as mean differences between scenario and base case concentrations within each class, (3) estimates scenario "observations" by adding class-specific changes to actual measurements, and (4) applies Optimal Interpolation using these adjusted observations and the scenario model as background. This method presents classification challenges due to its hybrid nature. The calibration uses global grid data to define classes (suggesting Grid calibration), but applies class-specific adjustments at station locations (suggesting Each calibration). The extreme cases reveal the method’s dual character: with one class, it is equivalent to CAS.All.Add.oi; with maximum classes, it becomes CAS.Each.Add.oi. The actual implementation operates between these extremes, justifying the hybrid classification CAS.All+Each.Add.oi. This coding captures both the global class calibration and the station-specific application, demonstrating how the classification scheme can represent complex methodological hybrids that don’t fit neatly into single categories. This example illustrates how the classification scheme accommodates sophisticated methods through hybrid coding while maintaining systematic classification principles. 3.4 Limitations and complementary documentation The classification scheme provides a high-level organization but does not capture all methodological details. The codes do not specify algorithm parameters (e.g., variogram parameters for kriging; distance weighting exponents for IDW; specific quantile mappings). These details must be documented separately in method descriptions or metadata. Subtle implementation choices (e.g., handling of negative concentrations, treatment of outlier or missing data) are not captured by the classification and require complementary documentation. The classification scheme is descriptive, not evaluative - it classifies how methods work, not how well they perform. Despite these limitations, the classification scheme serves as a crucial first step in methodological documentation, enabling systematic organization and comparison while pointing to the need for additional, detailed documentation of implementation specifics. 7 4 Examples Modelers often describe their bias correction methods using detailed, narrative explanations that reflect their specific implementation choices and justifications. While informative, these descriptions can be lengthy and difficult to compare directly. The classification presented in this report allows us to systematically deconstruct these narrative descriptions into standardized components. Below are several typical methodological descriptions provided by modelers, followed by their classification according to our classification scheme. •Original description: "We apply kriging with external drift for the reference year, using modeling data as the drift variable. For each grid cell, we compute the ratio between kriged data and raw modeling data, and multiply this ratio by the scenario modeling data." Classification code: SCA.Mult.Cell.ked Analysis: The description reveals a Spatialize-Calibrate-Apply sequence: first spatializing observations via KED (S), then calibrating a multiplicative factor by cell (C), and finally applying it to the scenario (A). •Original description: "We compute the mean difference between observed and modeled data for the reference year. This offset is then applied globally to the scenario modeling data to correct for systematic errors. Given the limited domain size, this correction is considered sufficient." Classification code: CA.Add.All Analysis: This method follows a simple Calibrate-Apply sequence using an additive adjustment calibrated across all stations (C), with no explicit spatialization step as the same global offset is applied everywhere (A). •Original description: "We calculate the difference between scenario and base case model outputs. This delta is then applied to the ordinary-kriged observed data to produce biascorrected scenario results." Classification code: SCA.Add.Cell.ok Analysis: This represents a Spatialize-Calibrate-Apply approach where observations are first spatialized using ordinary kriging (S). Mathematically, this approach can be expressed in two equivalent forms: Ccorr =Okriged + (Mscenario −Mbase)(as described) = (Okriged −Mbase) + Mscenario (alternative form) The second form reveals that the method is equivalent to first calculating the bias between kriged observations and the base case model (Okriged −Mbase), then applying this additive correction to the scenario. This demonstrates how different procedural descriptions can represent the same underlying mathematical operation, classified as an additive correction calibrated cell-by-cell (C, A). •Original description: "At the station level, we compute the difference between observed and modeled concentrations. These residuals are interpolated over the grid using the Empirical Bayesian Kriging approach and then added to the scenario." Classification code: CSA.Add.Each.ebk Analysis: This clearly follows a Calibrate-Spatialize-Apply sequence: calibration of additive residuals at each station (C), spatialization of these residuals via EBK (S), and final application to the scenario (A). 8 •Original description: "We fit a linear model to observed versus modeled data in the reference year. The fitted model is then applied to scenario data." Classification code: CA.Lin.All Analysis: This is a direct Calibrate-Apply sequence where a linear relationship is calibrated using all available data (C) and then applied to correct the scenario (A). This exercise demonstrates how the classification serves as a powerful normalization tool, transforming varied narrative descriptions into consistent, comparable classifications. By identifying the fundamental components - sequence, algorithm, calibration method, and spatialization technique - we can systematically organize and compare methodologies that might initially appear quite different in their narrative descriptions. To illustrate how different approaches work in practice, we present three detailed examples. Each diagram in figures 1, 2 and 3 shows the step-by-step application of a specific unbiasing method, without specifying the exact spazialization algorithms used. 5 Discussion and conclusions The classification scheme presented in this report—structured around the unbiasing sequence, the adjustment method, the calibration method, and the spatialisation algorithm—provides a coherent and flexible framework for deconstructing and describing bias correction methods for air quality scenarios. This scheme offers several benefits. First, it promotes standardisation through a common vocabulary that reduces communication ambiguity. Modelers can succinctly describe complex methodologies using a compact code, thereby facilitating clearer communication. Second, it enhances transparency and reproducibility by requiring an explicit declaration of each component, which is a cornerstone of scientific reproducibility. Third, it enables a structured intercomparison of methodologies. Activities such as the FAIRMODE WG5 Unbias Exercise can benefit from its use to group and analyse results according to their constituent elements, helping to isolate the impact of specific choices, such as the spatialisation algorithm. Finally, for operational modelling centres, the scheme can serve as operational guidance, providing a checklist for developing or selecting an unbiasing procedure. The primary application of this classification scheme is within the ongoing FAIRMODE and Italian unbias exercises. Beyond these, it can readily be applied to other air quality benchmarking studies. In conclusion, this classification scheme supports a more systematic understanding of bias correction techniques. By providing a common framework, it fosters clearer communication, more consistent methodological evaluation, and more reliable bias-corrected air quality scenario applications for both policy and research. References [1] G. Bonafè and L. Colombo. jobonaf/scenario-unbias: v1.0.1, April 2025. Available at: https://doi.org/10.5281/zenodo.15188017. [2] C Borrego, Alexandra Monteiro, MT Pay, I Ribeiro, AI Miranda, S Basart, and JM Baldasano. How bias-correction can improve air quality forecasts over Portugal. Atmospheric Environment, 45(37):6629–6641, 2011. [3] N Cressie. Statistics for Spatial Data. Wiley, 1993. [4] DW Wong, L Yuan, and SA Perlin. Comparison of spatial interpolation methods for the estimation of air quality data. Journal of Exposure Science & Environmental Epidemiology, 14(5):404–415, 2004. 9