As the basis for the derivation of regression formulas for climate effect estimation, data from the project WeCare Utilizing WEather information for ClimAte efficient and ecoefficient futuRE aviation, was used, which was an internal project of the German Aerospace Center (Deutsches Zentrum für Luft- und Raumfahrt; DLR). The project addressed both an improvement of the understanding of aviation-influenced atmospheric processes and an assessment of different mitigation options. An essential output of the project was a new set of emission inventories for global aviation . The network of flight trajectories was developed following a four-layer approach implemented in the AIRCAST method . It is starting from an origin–destination passenger demand network that was built up from exogenous socio-economic scenarios, via the passenger routes network (sequence of flight segments, a passenger actually travelled from origin to destination) to an aircraft movements network, which assigns aircraft seat categories to the resulting flight routes and provides flight frequency information. The final step is a simulation of trajectories based on the aircraft movements obtained from the aircraft movements network layer using DLR's Global Air Traffic Emissions Distribution Laboratory GRIDLAB;. Each mission, defined by departure and arrival cities, aircraft type, and load factor, was simulated under typical operational conditions, resulting in a network of flight trajectories. For this purpose, DLR's Trajectory Calculation Module TCM; was used that applies simplified equations of motion known as the Total Energy Model.

Based on the aircraft's engine state determined by parameters such as thrust and fuel flow, the engine emission distribution of NO_x, CO, and HC species along the trajectory was determined by applying the Boeing Fuel Flow Method 2 . The amount of CO2 and H2O emissions was calculated assuming a linear relationship to the fuel burn. The mapping of emission distributions of all flights onto a geographical grid resulted in 3-D inventories. In WeCare, using the approach mentioned above, emission inventories and the corresponding climate effect were estimated for the years 2015 to 2050 in 5-year steps. The forecast was based on the dataset from the reference year 2012. Seven different aircraft seat categories (based on the number of seats) were considered in the inventories (20–50 seats; 51–100 seats; 101–151 seats; 152–201 seats; 202–251 seats; 252–301 seats; 302–600 seats). Each seat category was modeled using one representative jet powered aircraft type (plus one backup aircraft type). The representative aircraft type was selected such that it contributes to a significant share of the respective seat category. Respective engine emission characteristics were taken from the Aircraft Engine Emissions Databank of the International Civil Aviation Organization .

In order to obtain the climate effect for each flight corresponding to the flight plan, the climate effect for the emissions calculated along each single trajectory is estimated with the non-linear climate response model AirClim using gridded emission data for each species. Therefore, AirClim combines 3-D aircraft emission data with a set of pre-calculated non-linear emission–response relations for a set of atmospheric locations to estimate the temporal development of the global near-surface temperature change. AirClim includes the effects of the climate agents CO2, H2O, CH4, O3 and primary mode ozone (PMO) (the latter three result from NO_x emissions), as well as CiC. For deriving the atmospheric responses for H2O and NO_x-induced changes, 85 steady-state simulations for the year 2000 were performed with the chemistry climate model E39/CA , prescribing normalized emissions of NO_x and H2O at various atmospheric regions . For the effect of CiC, we use atmospheric and climate responses considering the local probability of fulfilling the Schmidt-Appleman criterion as well as ice-supersaturated regions, which were obtained from simulations with ECHAM4-CCMod . We follow a climatological approach in the estimation of the climate effect, meaning that the calculated values represent a mean over all weather situations averaging over individual spatially and temporally resolved responses.

For analyzing the climate effect, a physical climate metric is selected which assumes growing emissions for characterizing radiative effect of the single flight emission in a future atmosphere, instead of applying a pulse metric. The reason behind this choice is that there exists a mismatch between the summed effect of pulse emissions and the respective scenario analysis (equal to the effect of aggregated pulse emissions). Therefore, we calculate CO₂-equivalents based on a scenario with ATR100 as a metric and apply these equivalence factors to the pulse emissions in FlightClim. As a consequence, when selecting ATR100 as a physical climate metric, the climate effect of these single flight emissions is evaluated assuming a future increase of emissions and concentrations from future flights and future contributions from non-aviation sectors over the next 100 years. This is relevant for the radiative effects estimated as changing concentrations in the future are taken into account. Hence, we assume emissions starting in 2012 and a future increase in emissions according to the scenario Fa1 of the Intergovernmental Panel on Climate Change , which is a reference scenario developed by the International Civil Aviation Organization Forecasting and Economic Support Group (ICAO FESG) with mid-range economic growth and technology for both improved fuel efficiency and NO_x reduction . Historical emissions are neglected. For background concentrations of CO2 and CH4, which influence the climate effect of CO2 and CH4 emissions, we assume IPCC scenario RCP4.5 . A number of different climate metrics can be applied to account for the different components of the aviation climate effect. However, selecting a suitable metric is challenging due to the uncertainties and varying lifetimes of non-CO2 effects. recommend using the average temperature response (ATR) or the efficacy-weighted global warming potential (EGWP) with a time horizon of more than 70 years. As a time horizon of 100 years was used for the Kyoto Protocol and other political applications, we quantify the climate effect using ATR100, which is the mean near-surface temperature change over 100 years. For any climate metric, non-CO2 effects can be expressed as an equivalent amount of CO2 emissions, so called CO2-equivalents (CO2,e), that would produce the same effect over a defined time horizon and a given emission scenario. Hence these CO2-equivalents are derived from a scenario analysis and applied in FlightClim to a pulse emission to obtain the best possible consistency between the effects of pulses and the scenario.

AirClim does not account for the influence of different aircraft sizes on contrail climate effect. To account for that we use a parametrization derived from (see Sect. S1). While this parametrization is already included in the ATR100-values used for the Symbolic Regression (see Sect. ), the MR-formulas for CiC have to be scaled afterwards (see Sect. ).

In the data structure for each of the about 57 thousand simulated flight trajectories, characterized by origin and destination airport as well as aircraft size, the resulting amounts of engine emissions were stored together with the ATR100 climate effect per species. This database was then used to derive the climate effect regression functions as well as regression formulas for fuel use and NO_x emissions necessary for the MR-approach.

Due to the large variety of importance of the different climate effect components among different flights, it is challenging to find a single set of equations that would reasonably estimate the climate effect under most circumstances. Therefore, in the first step, we apply a K-Means clustering algorithm to separate the flights into several clusters. This clustering is based solely on the share of the six aforementioned components of the climate effect in the total climate effect: ATR100CO2ATR100tot,ATR100H2OATR100tot,ATR100CiCATR100tot,ATR100O3ATR100tot,ATR100PMOATR100tot,andATR100CH4ATR100tot.

This ensures that flights in a given cluster have similar climate effect characteristics. The clustering is not directly dependent on proxy quantities to the climate effect, such as the amount and location of the emissions. We use an implementation by the free software machine learning library for the Python programming language scikit-learn and scale the input quantities to the standard normal distribution before clustering. We find a partition into three clusters to be most useful, as larger numbers of clusters lead to some cluster distinctions lacking a clear physical interpretation. The resulting three clusters occupy distinct areas in the latitude-distance space (Fig.  left). We therefore name them the short-flight cluster (green), the tropical cluster (orange), and the mid-latitude cluster (blue).

In the second step, simple thresholds are derived which separate the flights into three categories that approximate the found clusters. This is necessary to enable easy categorization of additional flights not contained in this data set. One threshold is a maximum distance for the short-flight cluster, and another threshold is the absolute mean latitude of great circle trajectories confining the tropical cluster. We choose the values for these thresholds in such a way that the amount of wrongly categorized flights is minimized. This leads to a threshold distance of 462.5 km below which flights are categorized as belonging to the short-flight cluster, and a threshold mean latitude of ±29.7° within which flights are categorized as belonging to the tropical cluster. All other flights are categorized into the mid-latitude cluster. This approximation wrongly categorizes 16.8 % of all flights used for clustering. The resulting simplified clustering is shown in Fig.  in the right plot.

The three clusters have distinct characteristics (Fig. ). The short-flight cluster has a negligible contribution of contrails to the climate effect at an average of 3.5 %, and a strong contribution of CO2 at an average of 57.4 % of the total climate effect. Flights in this cluster are often too short to reach the required altitude of at least about 8km e.g.; for contrail formation. The climate effect of the tropical cluster is dominated by contrails (average contribution of 56.6 %) because strong contrail formation occurs at tropical latitudes. The mid-latitude cluster contains the remaining flights and has large climate effect contributions from NO_x and H2O (average contributions of 49.1 % and 6.8 %, respectively; see below for further discussion).

For the cluster analyses only flights with seat category 3 to 7 are used. The remaining seat categories 1 and 2 (less than 100 seats) were only added to the dataset later in development. They contribute only 4.2 % to global ASK and therefore a minor share of total aviation emissions. Nevertheless the number of flights with seat category 1 and 2 is high. Therefore, an additional cluster for regional jets was used for MR. For SR all flights were clustered in one of the three clusters.

Multiple Regression is a common method in environmental science, with primary advantages being its simple application and interpretability. The structure of MR functions is predefined as a sum of predictor dependent terms each multiplied by a coefficient. Each coefficient indicates the impact of a specific predictor, allowing to understand the effects of each variable on the climate. It also enables the inclusion of numerous variables and can incorporate interaction terms of multiple predictors. However, MR assumes a predefined mathematical form making knowledge about the interactions necessary, which can be a limitation when relationships are non-linear. The necessary assumption may lead to misspecification of the model if the actual relationships are not well-captured by these forms. Additionally, MR can be vulnerable to multicollinearity (when predictors are highly correlated), which can distort coefficient estimates.

The MR-approach extends the idea of and leads to the following structure for the derived formulas for all clusters: 2ATR100tot=cCO2⋅f+cNOx(d,ϕ‾)⋅e+cH2O(d,ϕ‾)⋅f+cCiC(d,ϕ‾)⋅d⋅fACsize(b), where fACsize is the adaptation factor for the contrail climate effect due to the wing span b (see Eq. S2), and cCO2, cNOx, cH2O, cCiC are the cluster-dependent climate effect regression functions. Therefore, the climate effect of a species is estimated as a product of the respective climate effect regression function and the relevant reference quantity (f, e, d⋅fACsize). These MR-formulas are composed of polynomial and arctan functions and are designed to fit the respective partial climate effects cCO2=ATR100CO2/f, cNOx=ATR100NOx/e, cH2O=ATR100H2O/f, and cCiC=ATR100CiC/d. The climate effect function for CO2 is fixed at cCO2=8.145×10-11mKkg(fuel)-1, because the climate effect of CO2 is independent of the emission location in AirClim, so that no fit is required. Details on the derivation of the MR-formulas are given in Sect. S2.2.

Note that for the derivation of the climate effect regression functions apart from the predictors d and ϕ‾ we use the WeCare estimates for the burnt fuel f and emitted NO_x e, implying that those are also required for the application of these formulas. If those are not available we provide additional Fuel and NO_x MR functions, that only use the flight distance d and seat category (see Sect. S2.1). For the comparison with the SR-approach in Sect.  the derived Fuel, NO_x and climate effect regression functions are considered, combined determining the quality of the climate effect estimation.

The Symbolic Regression method used here, avoids a pre-defined structure of the formula. Instead an evolutionary algorithm provides a best fit and thereby defines the structure of the formulas. This structure can be represented by an expression tree, called gene. Each node in the gene represents a variable, a mathematical operation or a constant. The nodes are merged to a formula by the tree structure . The tool we apply, GPTIPS 2 , specifically uses Multi-Gene Symbolic Regression, which combines multiple genes with an additional scaling factor per gene (b1, b2) and a bias term (b0) to assemble the whole formula (Fig. ).

The optimization process to find an optimal formula uses an evolutionary algorithm based on a fitness function, in this case the root mean square error for the given dataset. Beneficiary solutions based on a random start population of multiple formulas are evolved over several generations. The evolution-inspired mechanisms forming the final formulas are fitness-based selection, as well as mutation and crossover .

For the derivation of regression functions the flight database is split into 80 % training and 20 % test data. Four different formulas are computed for the climate effect of the climate agents CO2, H2O, NO_x, and CiC (see Eq. ). The two main predictors, d and ϕ‾ from the first approach are used in the second one as well complemented by m, that replaces the segmentation into seat categories. The flight distance d is meant to cover effects based on the flight length like fuel use, ϕ‾ geographically differing climate effects of emissions and m different aircraft sizes. The dependent variable of the SR-formulas is the ATR100 for CO2, H2O, CiC and NO_x.

To check the effectiveness of the clustering derived in Sect. , regression formulas with and without use of the three clusters are computed. In the clustered version, separate formulas are derived for each cluster. This leads to in total twelve formulas for the clustered version and four for the unclustered one. For each resulting formula of the clustered and unclustered version a multiple runs of GPTIPS 2 (1296 for unclustered and 648 for clustered) are executed as part of a gridsearch for the regression hyperparameters. The main settings of the GPTIPS-software are used as hyperparameters. The reason for the selected gridsearch-approach with many runs is the high variability in the resulting estimation quality of regression formulas.

From all derived formulas the Pareto-optimal individuals according to the coefficient of determination R2 (Eq. ) and the number of nodes are considered as candidates for the final formula of the species and cluster (see Sect. S3.1). To obtain one formula for the total climate effect the formulas for CO2, H2O, CiC and NO_x have to be combined according to Eq. (). In this step the numbers of nodes for the ATR100_tot add, but the quality of estimation measured as R2 has to be newly computed. It is not apparent, which Pareto-formula to choose for each species to achieve an optimum in estimation quality and number of nodes for ATR100_tot. However, by trying all combinations it is possible to identify Pareto-optimal combinations that represent a optimal trade-off between a high value of R2 and a low number of nodes. Figure  shows these ATR100_tot-Pareto-fronts for the unclustered (blue) and the aggregated clustered version (purple; for the individual clusters please see Fig. S10). The final choices made are indicated by red and green dots. The selected formulas are given in Sect. S3.1. 3R2=1-∑i=1N(ATR100totpred-ATR100totact‾)2∑i=1N(ATR100totact-ATR100totact‾)2

For ATR100_tot in the short-flight cluster the clustered approach shows a significantly better estimation quality than the unclustered one (see Fig. S11). For the two other clusters the quality is comparable. Therefore as a combination of low complexity and a high quality of estimation the clustered formulas are applied for flights in the short-flight cluster in the further analysis and the unclustered formulas are used for mid-latitude and tropical cluster flights.

The use of different regression formulas for the derived clusters leads to discontinuities at the cluster boundaries that do not reflect real behavior and might result in inconsistencies. The significant difference in estimated climate effect over the cluster boundaries (e.g. see Fig. ) makes it necessary to smooth this effect. The smoothing is implemented by using a weighted sum of the cluster-specifically computed climate effects. The weighting factor evolves linearly from a starting point inside the particular cluster until the cluster boundary. At the cluster boundary the weighting of both cluster formulas is equal. Figure  sketches this general scheme of the smoothing. The climate effect of a flight within the smoothing area is accordingly estimated by 4ATR100=ATR100C1⋅(0.5+|dC1,2|2⋅bC1)+ATR100C2⋅(0.5-|dC1,2|2⋅bC1),ifdC1,2∈-bC1,0ATR100C1⋅(0.5-|dC1,2|2⋅bC2)+ATR100C2⋅(0.5+|dC1,2|2⋅bC2),ifdC1,2∈0,bC2, with ATR100C1/ATR100C2 as the cluster 1/2 estimates, dC1,2 the distance from the cluster boundary and bC1/bC2 the smoothing boundary 1/2 values. The smoothing boundaries mark the starting points of the smoothing area and are derived as the R2-optimal values within preset boundaries.

For both approaches smoothing is applied to the existing cluster boundaries of the recommended versions. The details on the individual smoothing are outlined in Sects. S2.3 and S3.2.

The climate effect functions were developed to represent a fitting of more detailed results from the non-linear response-model AirClim with algebraic relationships. Hence, the reliability of representing the estimated total climate effect is influenced by the applied fitting procedure. To evaluate the quality and reliability of the estimates, the derived, smoothed formulas from MR and SR are compared in this section. For SR, the formulas estimate the ATR100 directly leading to one formula per cluster and species. For MR the climate effect functions have to be computed for each cluster, which are four formulas per species apart from CO2, on the one hand, as well as the regression formulas for the used reference quantity on the other. Those are seven formulas for the fuel regressions, one per seat category, and 14 formulas for the NO_x regressions, two per seat category. The last formula of the MR-approach is the subsequent contrail wing span adaption. In total, this leads to 35 formulas for MR compared to 8 formulas for SR without smoothing (see Table ).

One advantage of the MR-approach are the separate fuel and NO_x functions, which the SR-approach does not include directly, hence fuel can still be derived from the CO2 climate effect. Furthermore all MR-formulas have the same predefined structure, while each SR-formula is different in shape and operators. Also, even though the SR-approach is optimized towards minimum formula complexity, it generally tends to include irrelevant, over-fitting terms and does not include certain input parameters into formulas for species, where correlations are present (e.g. ϕ‾ into ATR100NOx). As advantages the SR-approach evolves according the optimum predictive accuracy and yields a better ratio of complexity in terms of the number of formulas to quality. In addition it enables a continuous estimation over the aircraft size by using the MTOW instead of categorical seat categories. The number of formulas and the input parameters are a result of the different study designs for the two approaches and not inherent to the methods, even though the SR approach supports a lower number of formulas by being able to adapt to complex dynamics independently, whereas the MR-approach depends on a predefined structure. Therefore the MR-approach features separate seat categories and a separate estimation of fuel consumption and NO_x-emissions to distinguish their dynamics.

The estimation quality of both approaches is similar (Table ). Overall, the SR-formulas show a slightly higher coefficient of determination R2 (Eq. ). This might result from the optimization towards R2 in the SR-approach, even though for CO2 and CiC the MR-formulas show a slightly greater R2. In terms of the mean absolute relative error (MARE; Eq. ) the MR-formulas surpass the SR ones. This mainly results from the better relative estimation for short and medium range flights compared to the long range flights (see Fig. ). The better estimation of longer flights of the SR-formulas is a result of the absolute error-based evolutionary optimization process, which gives a greater weight to longer flights with higher climate effect. The MARE for H2O and CiC cannot be calculated due to flights with almost or exactly zero ATR100 in the dataset distorting the relative metric. 5MARE=1N∑i=1NATR100totact-ATR100totpredATR100totact

The MARE of the climate effect estimation is generally higher for short flights, as for these the variety in flight trajectories and non-CO2-effects generation increases (Fig. ). The correlation of both approaches with the AirClim estimated values is shown in Fig.  (for MR) and Fig.  (for SR). For the estimation of ATR100CO2 the MR-formulas show a better performance than those of the SR-approach, especially for long distance flights, because the points in Fig. a are located closer to or almost on the diagonal compared to Fig. a. The SR-formulas generally tend to underestimate those flights with two noticeable groups of flights being overestimated. These two groups are also distinguishable in the MR-Fig. a. One of them is estimated better and the smaller one is instead underestimated.

ATR100H2O-estimation (see Fig. b and b) shows relevant differences between both approaches with the MR-formulas generally overestimating the climate effect especially for long flights. The SR-formulas show a better accuracy for those flights and do in general neither tend to over- nor underestimate.

The quality of estimation for the climate effect of contrails is similar for both approaches (see Figs. d and d). As the occurrence of contrails is hard to predict and model, the accuracy of the CiC formula is low. The calculation of a meaningful MARE for contrails is not possible for short flights due to some flights with zero or close to zero ATR100 values, but for longer flight distances the MARE is by 2 to 4 times higher than that of ATR100tot.

For ATR100NOx the SR-approach leads to a better quality of estimation, with fewer points far away from the diagonal in Fig. e than for MR in Fig. e, indicating fewer large estimation errors. In contrast to the SR-formulas the MR-formulas have a tendency to under- or overestimate some distinguishable groups of flights.

The ATR100tot correlations in Figs. c and c show only minor differences between the two approaches. Hence we can conclude that the total quality of both approaches is similar, only with certain advantages for single species.

Apart from the results for the used dataset, the performance of the estimated ATR100tot for a validation dataset is analyzed. The validation dataset includes 439 flights of the German cargo airline EAT. The mainly short and medium haul flights took place with Airbus A300, A330 and Boeing 757 aircraft in 2021 and 2022. The AirClim climate effect estimates based on the real trajectories of these flights serve as the validation reference. The formulas of both approaches show reasonable correlations for the validation dataset, indicating a valid estimation. Longer flights are rather under- than overestimated (see Figs. f and f). This trend is stronger for the MR-formulas, which do also have a lower R2 value for the validation dataset.

Assessing the sum of all ATR100 estimates for the regression dataset shows the SR-approach to be closer to the actual AirClim values than the MR-estimates. The sum of MR-approach estimates is, apart from ATR100_H2O, in average smaller than the actual values, which results in the total sum of ATR100 estimates being 5.4 % smaller than the sum of all actual AirClim estimates. For SR the estimated sum is 0.3 % greater than the actual sum. The median signed relative error shows a different trend for both approaches, as it indicates a median relative overestimation of 0.6 % for MR, while SR-formulas in median overestimate by 9.8 %, mainly caused by many overestimated short flights with larger relative errors. This indicates different characteristics for different group of flights of different total amounts of climate effect. But in total about half of the flights are over- and half of them are underestimated for the MR-approach, while for the SR-approach about two third of the flights are overestimated.

The goal of this study is to develop an easy-to-use calculation method for estimating the total climate effect of individual flights, including CO2 and non-CO2 effects. Two approaches with smoothed formulas from MR and from SR have been compared. Due to the similar estimation quality of both approaches their greatest differences are the number of formulas and the input parameters, which can hence serve as crucial points for making a choice. Therefore, the SR-formulas can be recommended for application, if the complexity of the calculation in terms of the number of formulas is an important factor or if the aircraft size should be modeled continuously. If the estimation quality of short- and medium-haul flights is of greater importance, like for the FlightClim implementation, the MR-formulas are the better choice. In general, the specific requirements of an application towards the complexity, interpretability or the quality of estimation should serve as decisive points, which approach to use.

For both approaches applied in this study the ratio between non-CO2 and CO2 effects is approximately 4 for the used global aviation emission dataset. This number is higher than in other alternative publicly available methods for simplified climate footprint assessment of single flights. They use a constant factor of 2 to 3, which is based on assessments of total historical aviation emissions e.g., from 1940 to 2018 for, which is in principle consistent with AirClim results (see Sect. ). It has to be noted that the relation between non-CO2 and CO2 strongly depends on the level of the CO2 reference and the climate metric. Since the regression functions are designed to estimate the climate effect of present and future flights, we do not consider any emissions of historic aviation. Given the long lifetime of CO2, historical assessments such as , who analyzed aviation emissions from 1940 to 2018 in terms of ERF, report a stronger dominance of CO2 (31 %) than in the present study (19 %). A direct comparison is, however, limited because different metrics (ATR vs. ERF) and emission patterns (historical vs. present and future) are considered. Nevertheless, the relative importance of non-CO2 species is broadly similar, with shares of 4 % for H2O, 33 % for NO_x and 44 % for CiC in our dataset versus 2 % for H2O, 16 % for NO_x and 52 % for CiC in , when excluding the studied aerosol effects.

Clearly, aviation non-CO2-effects are subject to large uncertainties. The uncertainties regarding the climate effect of individual routes as described here, are likely to be even larger than those for the global average. Here we conceptually (and mathematically) distinguish between uncertainties in the importance of individual aviation effects that are globally applicable (global uncertainty) and the sensitivity of these effects to certain influences like location or aircraft technology (differential uncertainty). This discrimination is shown in Fig.  that describes a full approach in assessing uncertainties in the climate effect of individual flights estimated with FlightClim. Figure  is based on the high-level modeling workflow of FlightClim as illustrated in Fig.  (grey box).

We start by considering a one-year 3-D emission inventory, as usually done in climate effect assessments, and derive the climate response to individual flights by using AirClim that is based on athmospheric response surfaces derived from chemistry climate model simulations. Overall assessments of aviation climate effect, such as that from may serve as an uncertainty estimate for the importance of individual effects that basically represents the uncertainty in the ratio of CO2 to individual non-CO2 effects. Note that in the calculation of the uncertainty ranges in such assessments, different models are applied that are characterised by different spatial sensitivities. The main point here, however, is to distinguish between globally applicable uncertainty estimates and the effects of certain influences (differential uncertainty) like spatial variations. The approach applying globally applicable uncertainty estimates for individual flights has been used recently by and is applied to our results to derive global uncertainties in the applied aviation climate effect modeling.

In Fig. , the global part in the uncertainty description is complemented by an assessment of flight-specific, here called differential, uncertainties like the spatial sensitivity of the responses or the influence of differences in aircraft technology, which requires a much more detailed analysis than merely estimating globally applicable values. In addition, only limited data of the uncertainties in those sensitivities is available. A detailed analysis is clearly beyond the scope of this work. Therefore, in the following, we concentrate on the first part only, bearing in mind that the differential uncertainty is not represented.

As we break down the overall mean effect into contributions from individual routes also opposite should apply for the uncertainties: The sum of the uncertainties of the climate impact from individual routes should result in the overall uncertainty of the best estimate e.g. as given in . The regression formulas always try to model the median estimate. However, this might differ from median estimates in assessments (Sect. ). Hence, for the estimate of the 90 % confidence interval, we apply a mapping to the  data, before applying the confidence interval globally to the FlightClim estimates. For the mapping from AirClim to the median we use reported data for estimates of the 2005 radiative forcing (RF) and for the confidence interval ERF 2018 data (Eq.  and Sect. S5 for more details). Hence, this approach guarantees a consistency of the confidence interval with values from , while the actual calculated value represents the AirClim's best estimate including its spatial sensitivity. The CO₂ as well as the total RF AirClim estimate fits well with that of Lee et al. and the 90 %-confidence interval is roughly ±50 %. However, AirClim's contrail and water vapor RF is lower and the ozone and PMO effect are greater than those from Lee et al. (2021) 6confidence interval (low/high)=ATRx×RFxL21RFAirClimG21×ERFxL21(low/high)ERFxL21=ATRx×rxref×rxL21(low/high)=ATRx×rxtot(low/high)

The second part of global uncertainties of the FlightClim estimates is the regression error, which adds on the before described uncertainty of the regression database. In this section we expand the analysis on the regression error from Sect. , where the MARE and R2 of the two applied regression approaches, MR and SR was discussed. To analyse the uncertainty of the regression the 90 % confidence interval of the relative error of the total ATR100 FlightClim result is calculated based on all flights in the regression dataset. Figure  shows that the 95 %-percentile varies over the length of flights for both regression approaches in a similar manner. The percentiles vary for short- (< 1000 km) medium- (1000–4500 km) and long-range (> 4500 km) flights (see Table ). While short-range flights have a larger confidence interval of the relative error of roughly -40 % to 80 %, the others are mostly in the range of ±1/3.

The derived MR-formulas are integrated into the easy-to-use Excel-tool FlightClim v1.0. When applying the estimator it is of key importance to consider its limitations. FlightClim is based on regression formulas, that fit the results of the climate response model AirClim, which is itself derived from chemistry-climate model simulations. This means that the estimation quality and precision is not comparable to complex climate-chemistry models. For example, the developed tool is not suited to compare the climate effect of flights with similar aircraft of different generations or different travel times in the year, meaning that for an individual flight the real climate effect can strongly deviate from the estimated average. It is also not suited to study certain atmospheric characteristics and their impact on the climate effect or the influence of trajectory optimization. To answer those questions more complex models are needed. The main advantage of FlightClim is that it produces reasonable estimates including CO2 and non-CO2-effects while being easy to use and requiring very few input data per flight, in fact only origin and destination airport as well as aircraft size.

FlightClim is suited for climate footprint assessments by passengers to compare individual flights and even different travel modes (e.g. aircraft, train, car). Organizations can use it for climate effect monitoring as well as for travel planning. Moreover, the tool can be used for scientific research for quick and coarse quantification of the climate effect of even large flight plans, wherever the restrictions of the developed method are acceptable.