Climate assessment of single flights: Deduction of route specific equivalent CO2 emissions

Abstract Climate impact of anthropogenic activities is more and more of public concern. But while CO2 emissions are accounted in emissions trading and mitigation plans, emissions of non-CO2 components contributing to climate change receive much less attention. One of the anthropogenic emission sectors, where non-CO2 effects play an important part, is aviation. Hence, for a quantitative estimate of total aviation climate impact, assessments need to comprise both CO2 and non-CO2 effects (e.g., water vapor, nitrogen dioxide, and contrails), instead of calculating and providing only CO2 impacts. However, while a calculation of CO2 effects relies directly on fuel consumption, for non-CO2 effects detailed information on aircraft trajectory, engine emissions, and ambient atmospheric conditions are required. As often such comprehensive information is not available for all aircraft movements, a simplified calculation method is required to calculate non-CO2 impacts. In our study, we introduce a simple calculation method which allows quantifying climate assessment relying on mission parameters, involving distance and geographic flight region. We present a systematic analysis of simulated climate impact from more than 1000 city pairs with an Airbus A330-200 aircraft depending on the flight distance and flight region to derive simplified but still realistic representation of the non-CO2 climate effects. These new formulas much better represent the climate impact of non-CO2 effects compared to a constant CO2 multiplier. The mean square error decrease from 1.18 for a constant factor down to 0.24 for distance dependent factors and can be reduced even further to 0.19 for a distance and latitude dependent factor.


Introduction
Aviation is an integral part of our globalized world and contributes with about 3-5% to the total anthropogenic climate impact in terms of radiative forcing (RF) (Lee et al., 2009). As aviation is one of the fastest growing sectors, the share in global CO 2 emission could rise from currently about 2 up to even 22% in 2050 (Cames et al., 2015). Although world passenger flights drastically decreased in April 2020 due to COVID19-Pandemic by more than 80% of seat capacity, it is assumed that it recovers to 60 to 80% of the previous year values already by end of 2021 (ICAO, 2021). Therefore, it will be important to reduce aviation's contribution to climate impact to achieve the 2 C goal.
As climate impact is more and more of public concern, people become aware that they have to contribute to reduce their climate impact. A first order estimate of the climate impact of an individual flight is interesting to know, e.g., for compensating climate impact or carbon footprint of a flight, or for companies which want to become climate neutral. But while the climate impact of CO 2 emissions can be directly calculated, impacts of non-CO 2 emissions are much more complex to determine.
Climate impact of aviation is caused beside CO 2 emissions also by non-CO 2 effects (e.g., Grewe et al., 2017;Lee et al., 2009) which contribute to the total climate impact of aviation. Among those non-CO 2 effects especially the climate impact caused by contrails, nitrogen oxides (NO x ), and particle emissions, which influence cloud formation, play an important role, as impacts are of the same order of magnitude as the CO 2 climate impact. Non-CO 2 emissions contribute to global warming by an increase of the greenhouse gases ozone (O 3 ) and water vapor (H 2 O), as well as particles, and the formation of contrails and contrail-cirrus (contrail induced cloudiness, CiC). While an increase in ozone and water vapor concentration always lead to a warming, contrails may lead to a cooling, depending on time of day, meteorology, and other factors (Grewe et al., 2014;Meerk€ otter et al., 1999;Schumann, 1996). Additionally, indirect NO x effects lead (beside ozone formation and warming) to destruction of methane (CH 4 ) and a subsequent reduction in the O 3 productivity, which reduce the ozone concentration (PMO, primary mode ozone), causing a reduced warming, i.e., net-cooling effect.
In contrast to the impact of CO 2 , the impacts of non-CO 2 emissions depend, apart from the emitted amounts, on the emission location, especially on the altitude and on the latitude where the emission takes place. Although many studies focus on impacts of non-CO 2 emissions (e.g., Fr€ omming et al., 2012;K€ ohler et al., 2008), there is no study which proposes a simple mathematical formula to calculate the climate impact of an individual flight.
To compensate the personal carbon footprint, a voluntary market exists, in which people or companies try to compensate for their contribution to climate change, e.g., they can pay for their emitted CO 2 emissions to support ecological projects to reduce the climate impact e.g., tree planting or provision of efficient stoves in poor regions to reduce fuel burn and deforestation (e.g., Atmosfair, 2018;Grenzen, 2018). In total, a compensation contribution of 84 Mt CO 2equivalent was paid in 2015 (Hamrick & Goldstein, 2016). These voluntary market providers often account only for CO 2 emissions or use simplified assumptions about the climate impact of non-CO 2 emissions. Atmosfair for example uses a constant factor of 3 for flight sectors in altitudes higher than 9 km to account for non-CO 2 effects (Atmosfair, 2016). As no factor is used for flight sectors lower than 9 km, a mean factor of 2.7 is used. Myclimate uses a factor of 2 (Myclimate, 2015). These factors are based on the Radiative Forcing Index (RFI) reported by IPCC (1999) and Sausen et al. (2005). This RFI represents the contribution of non-CO 2 effects to the total aviation Radiative Forcing (RF) in 2000. Although RFI is widely used for compensation market, several studies show that RFI is not well suited for this purpose, as it does not account for the different lifetime and climate sensitivity of the different species (e.g., Forster et al., 2006;IPCC, 2007). For that reason, this study analyses the climate impact in terms of average temperature response (ATR), which represents the near-surface average temperature change over a given time horizon, here 100 year. We use this climate metric of ATR as it accounts for the lifetime of the different species, the different climate sensitivities and the thermal inertia of the atmosphere-ocean system. However, other physical climate metrics, e.g., with a different time horizon, could be directly calculated from the results presented here following the same concept.
Hence, this study proposes a calculation method relying on a set of mathematical formulas which allow to establish a direct relationship between mission parameters and associated total climate impact. This method uses mission length together with departure and arrival airports to calculate associated climate impacts applying representative climate impact metrics. The objective of this paper is (1) to present dependence of CO 2 and non-CO 2 climate effects on specific mission parameters, comprising mission length and geographic flight regions of individual flights in a global representative route network and (2) to introduce calculation methods and corresponding formulas using mission parameters to calculate non-CO 2 effects, enabling to calculate total climate impact of aviation emission for individual flight missions.
We describe the global data set of aircraft trajectories used in this study (Section 2). Then we present in detail the overall climate impact of those aircraft trajectories (Section 3). Section 4 introduces the mathematical formulas allowing to estimate climate impact from mission parameters. In Section 5 we discuss shortcomings and strengths of the three mathematical relationships (simplified calculation methods) before we conclude the study in Section 6.
Note, that we analyzed in this study the impact of one typical long-haul aircraft type of A330-200 aircraft. The A330-200 was used as it is the most sold aircraft in the medium-and long-range category, besides the Boeing 777. Different types of aircraft or aviation fuels could lead to different results if, e.g., the emission composition, emission indices of NO x or the flight profiles significantly differ which would influence the implicitly assumed flight altitude.

Dataset of aircraft trajectories and climate impact
Climate impact of aviation operations depend on the emitted amount of CO 2 and non-CO 2 climate agents, flown distance and location and time of emission. For the analysis of the dependence of climate impact on mission length and geographic flight region, we use results from a comprehensive climate impact assessment which was performed on a set of individual missions in a representative global route network calculated by Dahlmann, Koch et al. (2016) and Koch (2013), which provide individual trajectory parameters and the associated climate impact for more than 1000 routes. They analyzed the climate impact of all the routes which were flown in 2006 (according to OAG, Official Airline Guide) with a typical long-haul aircraft (Airbus A330-200) by applying a workflow, which consists of coupled multidisciplinary models. In this model workflow, engines and aircraft are modeled to generate a performance map which includes information about necessary thrust settings and correspondent emissions of CO 2 , H 2 O, and NO x . For each individual route, the necessary fuel is estimated and it is checked whether the aircraft specific flight performance envelope is fitted to stall and buffet limits, and cruise altitude capability. The Trajectory Calculation Module (TCM, Linke 2016) uses the performance tables and calculates the exact 3D flight path as well as the emissions of CO 2 , H 2 O and NO x . Emission amounts and locations (altitude, latitude, longitude) are used in the climate response model AirClim Grewe & Stenke, 2008) to analyze the climate impact of each route. AirClim combines aircraft emission data (3D: longitude, latitude, and altitude) with a set of previously calculated atmospheric responses to calculate the temporal development of the global near-surface temperature change. The atmospheric responses for H 2 O and NO x -induced changes in O 3 and CH 4 are derived from 85 steady-state simulations for the year 2000 with the DLR climate-chemistry model E39/CA, prescribing normalized emissions of NO x and H 2 O at various atmospheric regions (Fichter, 2009). For the impact of CiC we use atmospheric and climate responses considering local probability of fulfilling the Schmidt-Appleman Criterion (SAC) as well as ice supersaturated regions, which were obtained from simulations with ECHAM4-CCMod from K€ archer (2009, 2011).
Note, that we follow a climatological approach in the calculation of the climate impact and the calculated values for the climate impact represent a mean over all weather situations averaging over individual spatially and temporally resolved responses. A detailed description of the workflow can be found in Dahlmann, Koch et al. (2016) and Koch (2013). For each of these routes, trajectories with different initial cruise altitude (ICA) and aircraft's flight speed (Mach number) were analyzed regarding the emissions, cash operating costs (COC) and climate impact of ATR100. The climate impact is calculated as average near surface temperature change over a time horizon of 100 years (Average Temperature Response, ATR 100 ) for a typical lifetime of aircraft of 32 years. For the systematic analysis presented in this study for each route flight trajectories with the lowest COC are analyzed, since they represent a reasonable representation of today's aircraft routings in a climatological sense.
In total 1178 different routes were analyzed, which can be seen in Figure 1. There are less flights between USA and Asia as Boeing aircraft are often used on these routes. For each of these routes the flight distance, fuel consumption, flight altitude, flight speed (Mach number), emissions of CO 2 , H 2 O, and NO x , as well as climate impact of CO 2 , H 2 O, O 3 , CH 4 , PMO, and CiC were used for the analyses.
To compare the climate impact of different climate species, we calculate the climate impact of all species as CO 2 equivalents, which is the climate impact of each climate species relative to the climate impact of one kg CO 2 . The CO 2 equivalent represents the amount of CO 2 which causes the same climate impact for the given climate metric (i.e., ATR 100 and constant emissions over 32 years). Here we calculate the impact per emitted kg CO 2 , as often only information about fuel consumption exists, but no detailed information about other emissions. Therefore, the results are only valid for similar specific fuel consumption and emission indices (see Section 5 for more details).

Aircraft trajectory characteristics and climate impact
The climate impact of an aircraft mission depends, in addition to the amount of emissions, on the emission location.
From a comprehensive analysis of climate impact of individual missions in the global route network, we present in a first step how specific fuel consumption and emission altitude depend on mission length (Section 3.1). In a second step we present how climate impact of individual missions depends on geographic region where the aircraft is operated (Section 3.2).

Dependence of specific emissions and flight altitude on mission length
The specific fuel consumption (fuel consumption per flown kilometer) on short-haul missions (up to 1000 km) is larger than that on long-haul missions (black dots, Figure 2a). This is on the one hand due to the comparably high amount of fuel needed for take-off and on the other hand due to lower flight altitudes of short haul missions and the increased aerodynamic drag at these lower altitudes. Hence, specific fuel consumption decreases with increasing mission length, and only for missions longer than 5000 km the specific fuel consumption slightly increases due to the increasing weight of the necessary fuel, which increases aircraft take-off weight (TOW). The specific fuel consumption decreases by about 30% when mission length increases from 500 km to 2000 km. The specific emissions of NO x (measured as kilogram NO x per flown kilometer, read dots, Figure 2a) show a similar decreasing trend for missions up to 4000 km, while reduction is more substantial dropping by 45% when mission length increases from 500 km to 2000 km, and even by about 60% when mission length increases from 500 km to 4000 km. For missions longer than 4000 km specific  emissions of NO x increases from 180g/km for 5000 km mission length by about 10% for 10,000 km mission length. Absolute values of the NO x emission index (EINO x , relation between NO x emission and fuel consumption) decreases from about 38 g/kg for very short distances down to about 18 g/kg for long-haul missions.
Trajectories evaluated in our study have an average ICA of 36 kft (about 11,400 m), but the ICA as well as the mean and maximum flight altitude depends on the flown distances ( Figure 2b). As continuous climb is assumed for the trajectories, increasing distances lead to increasing flight altitude, at least for short distances lower than 2000 km. For larger distances, the ICA needs to be reduced to avoid flight envelope violations, like altitude limitations. Therefore, mean and maximum flight altitude show steps at around 11,000 m.

Dependence of specific CO 2 and non-CO 2 climate effects on altitude and geographic region
Beyond specific emission also their associated climate forcing varies with emission altitude. Background concentrations, photochemical, and physical processes varying with altitude are at the origin of strong altitude dependence of non-CO 2 climate effects, comprising nitrogen oxides, water vapor or contrail-cirrus. As a result, climate impact given per kg fuel consumption shows a clear dependence on the altitude of emissions. Figure 3  In Figure 4 the specific climate impact of each route, shown as climate impact in terms of ATR 100 per flown distance, is presented. We normalize to flown distances in order to identify efficiencies of short-, middle-, and longhaul flights. Values of specific climate impact vary between 0.1 pK/km and 1 pK/km. It can be seen that the specific climate impact of routes at around 60 N is higher than that of tropical routes. In regions with dense air traffic like the North-Atlantic flight corridor (NAFC) smaller impacts can be found. Low specific climate impact appears on some of the short-haul routes.
In order to better explain these large variations in specific climate impact, we show impact per flown distance of individual species separately. As mentioned before specific climate impact of CO 2 does not depend on the flight altitude and region but only on specific fuel consumption ( Figure  5a). As the fuel consumption and therewith the CO 2 emission is smallest for medium range distances ( Figure 2a) the climate impact due to CO 2 is also smallest for medium range distances, while the impact of CO 2 is larger for short haul flights. Figure 5b shows the lower specific climate impact of CiC in regions with dense air traffic like the NAFC due to contrail saturation effects (e.g., Dahlmann, 2012;Marquart et al., 2003). This is due to the fact that contrail formation reduces the ambient water vapor and hence lowers the possibility to form additional contrails in the same region. Furthermore, additional contrails above or below preexisting contrails have a lower radiative impact. Lower contrail climate impacts also occur for small flight distances as in low altitudes the temperature is too high for contrail formation. NO x emission leads to an enhanced ozone formation contributing to global warming, and to an enhanced  methane loss, causing a cooling effect. The specific ozone climate impact for individual trajectories decreases with increasing latitudes (Figure 5c), due to the decreasing photochemical activity of the atmosphere. The methane destruction is larger in the tropics and lower at higher latitudes ( Figure 5d). Both effects partly compensate and their sum leads to smaller variations in the total specific climate impact of NO x , which is shown in Figure 5e. The specific NO x climate impact is higher at high latitudes and smaller in the tropics, as the variation in O 3 specific climate impact is larger than the variation of the CH 4 specific climate impact. The specific NO x climate impact for short routes is small positive or even negative as the ozone impact increases faster with altitude than that of methane (Figure 3, Dahlmann, Koch et al., 2016).
Specific climate impact due to H 2 O emissions shows higher values in Northern high-latitudes, while lowest values are found in the tropics. Impact of H 2 O strongly increases with altitude when approaching the tropopause as the atmospheric lifetime in the troposphere is considerable shorter than in the stratosphere. As the altitude of the tropopause decreases with increasing latitude, the impact of H 2 O also increases with latitude (Figure 5f). Hence larger specific climate impact at higher latitudes is a result from flights occurring more often in the stratosphere.
Overall, the large total climate impact at higher latitudes is a result of higher specific values of CiC, H 2 O and only small compensation effects from CH 4 . Lower values in the NAFC results from contrail saturation effects.

Mathematical formulas to calculate total climate impact of aviation
In this section we present three types of mathematical formulas for aviation's CO 2 and non-CO 2 climate impact. The climate impact is given as CO 2 equivalent factor (climate impact of an emission of 1 kg of a species relative to that of 1 kg CO 2 emission) using ATR100 as climate metric. The values represent annual and global mean values. The first is a very simple constant factor, the second includes a distance dependency and the third additionally includes a latitude dependency. Hence, instead of using global mean values we aim at stepwise capturing some main sensitivities of the non-CO 2 impacts, as shown in the previous section, by using easily available information like flight distance and geographical latitude.

Climate impact as a direct proportional function of CO 2 emissions
The easiest way to estimate the climate impact of non-CO 2 emissions, though not recommended (see below), is to use constant factors for CO 2 equivalent factors. They can be calculated by averaging the non-CO 2 impacts for all analyzed flight trajectories. The CO 2 equivalents per emitted kg CO 2 are: The obtained factors indicate that the impact of all emissions is 3.4 times the impact of CO 2 over the considered time horizon. The impact of NO x and CiC are in the same order of magnitude as the climate impact of CO 2 .
This calculation method is easy to use, as the total CO 2 emissions only have to be multiplied with those constant factors. Figure 6 presents the CO 2 equivalent factors for the climate agents in dependency of the flight distance together with the simplified CO 2 equivalent estimations (constant factor is shown as grey line). For NO x emissions ( Figure 6a) the constant factor overestimates the impact for distances shorter than 2000 km and even provides the opposite sign for distances less than 500 km. For distances larger than 2000 km the factor underestimates the impact. For CiC (Figure 6b) and H 2 O ( Figure 6c) the climate impacts show a very wide spread and the constant factors do not well represent the calculated impacts. For the total non-CO 2 effect ( Figure 6d) the constant factors overestimate the impact for flown distances lower than 2000 km and underestimates the impact for longer than 2000 km flown distances. This can also be seen in Figure 7a, where the correlation between CO 2 equivalent factors calculated with AirClim and the constant CO 2 equivalent factor is presented. The gray line represents perfect agreement. It can be seen that only very few missions are correctly represented. Applying a constant factor underestimates the specific climate impact by up to about 40% or shows even a wrong sign compared to the AirClim results. Applying a constant factor shows that 80% of the estimates lie within a ±20% range. The mean square error is about 1.18.

Climate impact as a function of flight distance
In Figure 6d, the CO 2 equivalent factors per emitted CO 2 for non-CO 2 effects are shown as a function of the flight distances. For very short distances, lower than 500 km, the CO 2 equivalent factor is small positive or even negative, as those flights are operated in low altitudes (Figure 2). At these altitudes it is too warm for contrail formation and the impact of H 2 O is very low (see Figure 3). The impact of NO x emissions is even negative as the low altitude of these short distances leads to a low positive contribution of the ozone production and larger negative contribution by methane destruction, especially since many short flights occur in tropical regions in the route network used here.
For flown distances between 500 and 4000 km the CO 2 equivalent factors of non-CO 2 species increase with the flown distance, as the mean flight altitude increases leading to the larger climate impact. For flown distances larger than 4000 km the impact of non-CO 2 effects shows only a small increase between 2.8 and 3.4 times the impact of CO 2 . The mean flight altitude hardly changes for those long-haul flights and hence the CO 2 equivalent factors are almost unaffected. Note, that the impact of larger fuel consumption for longer flights as well as large specific fuel consumption for very short flights is contained in the CO 2 emission and cannot be seen in this kind of figure. Hence, very short distances as well as very long distances are less climate friendly than a comparison of the CO 2 equivalent factors would suggest.
As the dependency of CO 2 -equivalents on the distance shows a large change for low distances but a smaller one for large distances, we fit arc tangent functions to the results ( Figure 6, black lines). The resulting CO 2 equivalents for the different climate species are: eq CO 2 CO 2 ¼ 1:0 (6) eq CO 2 NO x ¼ 2:3arctan 3:1D ð ÞÀ 2:0 (7) eq CO 2 CiC ¼ 1:1 arctanð0:5DÞ (8) where D is the flown distance in 1000 km. While the constant factors show large deviations from the calculated CO 2 equivalents, the distance dependent calculation factors show, especially for the NO x emissions and the total non-CO 2 climate impact, quite good agreement (see Figure 6, black lines). This can also be seen in Figure 7b, where the correlation between the CO 2 equivalent factors calculated with AirClim and the distance dependent CO 2 equivalent factor is presented. Applying a flight distance dependent factor shows that 95% of the estimates lie within a ±20% range. The mean square error is reduced to about 0.24.

Climate impact as a function of flight distance and latitudinal dependency
A more comprehensive formulation relies on an algorithm which identifies climate impact of an individual flight using flight distance and latitude of flight. As shown in Section 3 Figure 7. Correlation between total CO 2 equivalent factors calculated with AirClim and total CO 2 equivalent factors calculated with constant factor (a), distance dependent (b), and latitudinal dependent (c), respectively. Note that the constant factor for non-CO 2 effects exclude CO 2 and therefore is 2.4.
the climate impact of individual flights depends, beside the fuel consumption, also on the flown distance and on the flight region. In Figure 8 the climate impact of NO x emissions is exemplarily shown as a function of latitude of origin and destination airports for routes with flight distances longer than 2500 km. The impact of NO x emissions increases with increasing mean latitude (average of origin airport latitude and destination latitude, blue lines) and increasing negative mean latitudes, which means that the impact of NO x emissions is lowest near the tropics (see Section 3). As seen in Figure 8 the mean latitude is a good proxy for the latitude dependency. Therefore, we further use the mean latitude in this study. The CO 2 equivalent factors in dependency of the flown distances and mean latitude are show in Figure 9. Especially for flight distances longer than 1500 km, the CO 2 equivalent factor of NO x depends on the mean latitude (Figure 9a). The impact increases from the tropics to mid and high latitudes. The CO 2 equivalent factor of CiC ( Figure 9b) provides a clear latitude dependency especially between 1000 and 7000 km. For longer than 7000 km flight distances the climate impact depends on whether the flights really occur in the tropical region or a large share of the flight sectors are in midlatitudes. The H 2 O shows a clear dependency from the mean latitude for distances larger than 1000 km (Figure 9c). The impact increases from tropics to higher latitudes, as the share of flight sectors in the stratosphere increases and the impact becomes larger due to the longer lifetime in the stratosphere (see Section 3). The total non-CO 2 climate impact ( Figure 9d) shows a lower dependency of the latitude, as the impact of NO x and H 2 O provides counteracting latitudinal dependencies as CiC. While the impact of NO x and H 2 O increases from the tropics to higher latitudes, the impact of CiC decreases.
To account for this additional latitudinal dependency of the CO 2 equivalent factors the difference between the AirClim results and the distance dependent CO 2 equivalent factors from Section 4.2 are calculated and the common 4th order polynomial equation was defined for fitting on NO x , CiC and H 2 O (called user-defined function here): where L is the mean-latitude in deg North and D is the flown distance in 1000 km. We fitted this user-defined function to the AirClim results by using the non-linear leastsquares Marquardt-Levenberg algorithm implemented in gnuplot, where the weighting was set to 1.0 for both meanlatitude and flight distance. The final set of coefficients (a to e) were found (see Table 1) and we obtained Equations (11)- (13): Including the latitudinal dependency in addition to the distance dependency in the calculation formulas further increases the accuracy of the results. This can be seen in Figure 7c, where the correlation between CO 2 equivalent factors calculated with AirClim and the distance and latitudinal dependent CO 2 equivalent factors are presented. Applying a distance and latitudinal factor shows that 92% of the estimates lie within a ±20% range. The mean square error is reduced to about 0.19.

Discussion and terms of use of calculation methods
This study introduces three calculation methods, how to calculate climate impact of a flight mission, using mission parameters. These simplified calculation methods provide approximations of total climate impact, while due to individual definitions each method has its own specific strengths and limitations. However, simplifications proposed, exhibit as well short comings, which are described in detail within this section. Here we discuss general principles of each climate impact calculation method, assess their limitations and provide a recommendation which method to adopt preferably for which purpose, in order to allow estimating total climate impacts of aircraft missions, comprising non-CO 2 effects.

Constant factor method using fuel consumption
The first, very simple option introduced to calculate the climate impact of aviation's non-CO 2 impacts from available information on CO 2 emissions (or fuel consumption) is to use constant factors, which represent a typical relationship between CO 2 emissions and associated non-CO 2 impacts occurring. These constant CO 2 equivalent factors are then simply multiplied with the CO 2 emissions, e.g., known from the fuel consumption (consumption of 1 kg jet fuel emits 3.15 kg CO 2 ). As this option is very simple to use, such constant factors are often asked for e.g., by politicians or companies. Such simple constant factors can only provide a first order approximation, and can only be used for an initial estimate of the magnitude of non-CO 2 effects from aviation, in particular on an average basis. However, for single aircraft missions, such a constant factor approach represents an inadequate simplification, as it does not include any influences of different flight altitudes, emitted non-CO 2 amounts, and flown regions. Therefore, such constant factors cannot be used for providing quantitative estimates as required when analyzing individual trajectories, e.g., in a technology assessment or as a possible metric for emission trading. Additionally, such constant factors would lead to an artifact as the only option to reduce overall climate impact is a reduction of fuel consumption (hence CO 2 emission), which neglects any mitigation potential resulting from non-CO 2 effects. This would introduce a falsification when providing strategic guidance and analysis of technology options toward development of sustainable aviation.

Distance dependent factors using fuel consumption and mission length
In order to avoid such falsification introduced by using a constant factor, we suggest as an initial improvement of the calculation method, to use the distance dependent factors for estimating non-CO 2 effects, which results in a better representation of total climate impact of aviation ( Figure 7). As the average flight altitude calculated over the full mission increases with distance, due to higher fraction of cruise phase, these factors implicitly integrate altitude dependency of aviation emission impact. Therefore, the climate impact is represented more realistically by second method proposed.
To apply this calculation, only the flight distance has to be known as additional information, which makes this improved representation of non-CO 2 effects still easy to use. Such simple factors could be used even for the general public to show more realistic estimates of the total climate impact of aviation, e.g., for a compensation market or for considering personal CO 2 footprint. Using such distance dependent factors leads to more realistic estimates; however, it generates insufficient guidance (or incentives) for airlines to reduce climate impact of non-CO 2 effects by considering alternative altitudes or flight regions during trajectory optimization. This calculation method with the factors is therefore unsuitable for emission trading or mitigation strategy development, as changes in flight regions or altitude and reduction in NO x emissions would erroneously not reduce calculated climate impact, e.g., CO 2 equivalents.

Latitudinal dependent factors using fuel consumption, mission length, and mission latitude
Using additionally latitudinal dependent factors, beside mission length and fuel consumption, increases the accuracy of climate impact calculation further on. In this third formula individual effects are represented on a climatological basis, representing typical averaged synoptic conditions. However, this method using those factors still does not provide incentives to reduce the climate impact of aviation's non-CO 2 emissions for specific routes, as would be needed for concepts on climate-optimization of individual trajectories (e.g., Matthes et al., 2020). Nevertheless, it enables a more realistic calculation of non-CO 2 impacts, leading to more realistic estimates of total climate impact of aircraft operations, e.g., for compensation market.

Further developments of proposed calculation method
Based on the proposed calculation methods a set of further developments are discussed here. First, the applied method can be expanded by analyzing different aircraft engine/type configurations. The results, presented here, were obtained from analyzing the impact of one typical long-haul aircraft type of A330-200 aircraft. The A330-200 was used as it is the most sold aircraft in the medium-and long-range category, besides the Boeing 777. In 2015 still about 5% of flown ASK (available seat kilometers) was served by A330-200. The A330 is a similar aircraft type as B777, with similar flight profiles. Both aircrafts together serve still (in 2015) about 25% of global ASK. Different types of aircraft (e.g., turboprop instead of turbofan) could lead to different results if, e.g., the emission composition (emission indices of NO x ) or the flight profiles significantly differ (which would influence the implicitly assumed flight altitude). The presented calculation method could also be expanded to consider alternative aircraft types or emission compositions. Although this would further increase the accuracy, it also increases the complexity of the calculations and the need for further information, which often are not available. The analyzed aircraft is a long-haul aircraft, which is used nevertheless for short-and middle-haul flights, resulting in inefficiencies in the overall analysis. Therefore, the results for short and middle haul flights might overestimate total climate impact, hence should be used with limitations.
The climate impact here is analyzed on a climatological base. It is assumed that the aircraft are operated whole year over in a lot of different weather situations. The obtained results are hence valid on an annual mean basis, while individual flights under specific weather conditions can have completely different climate impact, compared to the impact of the annual mean basis. The presented calculation method could in principle be used to derive mathematical formulas for different weather situations, but this would largely extend the complexity of the method and the need for detailed weather information. Therefore, the presented formulas are intended to capture the overall climate impact, neglecting specific weather situations, as such detailed information is often not available for individual aircraft missions.

Application of calculation methods
For all above introduced mathematical formulas, an important limitation applies in application of these calculation methods for identification of mitigation strategies, as it does not create any incentives to reduce the impact of non-CO 2 effects, but only incentives for reducing CO 2 emissions, as the calculated eq CO 2 -factors are multiplied with CO 2 emissions to gain the total eq CO 2 of a flight. Hence, for technology assessments of mitigation options or emission trading it is recommended to use a model which at least includes explicitly an altitude dependency and the real emitted amounts of individual species. In order to optimize an individual mission on a specific day, a calculation model would need to be used which considers real emissions and associated impacts. With such comprehensive climate impact assessment, one could work on inclusion of such external effects and associated costs with the help of market-based measures (Scheelhaase, 2019). For using such comprehensive climate impact assessments, a large amount of additional data has to be provided (e.g., time of flight, emissions along flight trajectory). The eq CO 2 factors, which are assessed in this study, could be used as a kind of fallback option if airlines are not able or willing to provide detailed flight information. The airlines have to pay for the eq CO 2 calculated with such an approach or provide sufficient data about the real flights, which can reduce the costs if they fly more climate friendly.
In this study we use the ATR with a time horizon of 100 years as climate metric to compare CO 2 and non-CO 2 effects. Benefit of using ATR is that it accounts for the lifetime of the different climate agents as well as for the climate sensitivity parameter. However, compensation on a voluntary basis currently relies on RFI as calculation method, which neglects lifetime and climate sensitivity of different effects. With the calculation formulas proposed here, this shortcoming is overcome providing more realistic estimates. Applying our simple calculation method leads to a factor of 3.4 (total climate impact is 3.4 times the CO 2 impact) and is comparably higher than the RFI value of about 1.9 from Sausen et al. (2005) or 2.7 from IPCC (1999). But both approximations only include line-shaped contrails but no contrail cirrus, as we include.
Note, that on the one hand side specific CO 2 equivalent factors for short flight distances are lower than those for long flight distances. However, their specific fuel consumption is considerably larger for the short flight distances (see Section 3). Therefore, short flight distances are less climate friendly than a comparison of specific CO 2 equivalent factors would suggest. Nevertheless, the total climate impact of short haul flights is, at least for this aircraft type, smaller than for long-haul flights.

Conclusions
In this study we present three mathematical formulas for assessing non-CO 2 effects from aviation of individual aircraft missions of different complexity. In order to identify these mathematical formulas, we analyze the dependence of climate impacts on different routes, regions and altitudes to estimate the total climate impact derived from a comprehensive performance assessment of a global route network with the climate response model AirClim. The overall idea is to have as few additional parameters as possible, while considering as many as needed. The first mathematical formulas are composed of simple constant factors, which can be multiplied with the CO 2 emissions to get a first order estimate of aviation's non-CO 2 effects. The second formulas use the flown distance as additional information to increases the accuracy of the climate impact calculation as this implicitly includes the influence of flight altitude. In the third set of mathematical formulas, we introduce latitudinal information which increases the accuracy of the CO 2 equivalent calculation further on. As the additional information (region factor) only the mean latitude is needed, which can be directly calculated from geographical positions of origin and destination airports. This calculation method is able to represent geographic, in particular, latitudinal dependence of non-CO 2 impacts, resulting in a more accurate estimate of aviation's total climate impact than the two other calculation formulas.
Including distance and latitudinal dependent CO 2 equivalent factors increases the accuracy of the results significantly. However, it generates insufficient incentives for airlines to reduce climate impact of non-CO 2 effects by considering alternative altitudes or flight regions during trajectory optimization. Nevertheless, formulas introduced allow to provide a more realistic quantitative estimate of total climate impact of aircraft missions.
Using additionally latitudinal dependent factors, beside mission length and fuel consumption, increases the accuracy of climate impact calculation further on. Now, individual effects are represented on a climatological basis, representing typical averaged synoptical conditions. Nevertheless, it enables a more realistic calculation of non-CO 2 impacts, leading to more realistic estimates of total climate impact of aircraft operations, e.g., for compensation market.
This calculation method with the factors is therefore unsuitable for emission trading or mitigation strategy development, as changes in flight regions or altitude and reduction in NO x emissions would erroneously not reduce calculated climate impact, e.g., CO 2 equivalents. The formulas introduced have limitations as they are derived from one single aircraft type of A330-200 aircraft. For different aircraft types the NO x emission per fuel consumption as well as the probability to produce contrails or the flight profiles may change. Using different aircraft types and different emission compositions would further increase the accuracy, but also the complexity of the calculations and the need for further information, which often are not available. The climate impact here is analyzed on a climatological base. It is assumed that the aircraft flies the whole year in a lot of different weather situations. The obtained results are valid on an annual mean basis, while one single flight under specific weather conditions can have completely different climate impact, compared to the impact of the annual mean basis. Similar to the different aircraft types, the inclusion of such information would increase on the one hand side the accuracy, but on the other hand increase the amount of information and increase the complexity of the calculation.

Competing interest
No potential competing interest was reported by the authors.

Data availability
The data that support the findings of this study are available from the corresponding author, K.D., upon reasonable request.

Funding
This work was supported by the DLR project Eco2Fly.