Radiation impacts due to changes of the atmospheric composition, expressed as radiative forcings or effective radiative forcings (ERFs), are the basis for all calculations of CO₂ equivalent emission metrics (incl. GWA) and climate effects of aviation emissions examples can be found in. A recent compilation of (effective) radiative forcings for aviation has been presented by , who provides best estimates up to the year 2018 and 90 % confidence intervals for all aviation climate impact components.

Not all metrics can be used for trade-off considerations. Appropriate metrics must be additive and must be a monotonically increasing function of the corresponding emission. Thus, a metric like the absolute global warming potential (AGWP) is not usable, since it is “per kg of emission”, that is, a mass-specific quantity (in spite of its qualifier “absolute”). To be additive, a metric must be an extensive quantity. For instance, energies and temperature differences are additive, but temperature itself, being an intensive thermodynamic quantity, is not additive. PGP use GWA, which is quite similar to AGWP, but the final step in its calculation, the division by the mass of the emitted component is not taken, such that GWA is indeed an extensive quantity that increases monotonically with the emitted mass. This justification for the introduction of GWA is not explicitly mentioned in the paper by PGP.

The “activity” considered by PGP is the complete aviation in 2018. For this activity, the ERF values are at hand in the compilation by . One can imagine (a thought experiment) that in that year an operational or technical measure would have been permanently in force that would have led to changed ERF values overall. Mathematically, we can formulate a general metric as the result of application of an operator Φ on an ERF value. Φ stands for a mathematical operation (e.g. an integral) that maps the ERF into a metric. This mapping varies between species, e.g. if the integral involves different time-scales for examples, please consider the overview paper by. These differences are contained in the definition of Φ via a parameter (vector or scalar) Π. For instance, the time integral for the calculation of GWA needs one time-scale for contrails, two for nitrogen oxides, but a number of time-scales (typically three) for carbon dioxide. Such parameters e.g. perturbation life-times and weighting factors as listed byin their Table 2 are comprised in Π. Thus one can write generally: 1MC=Φ(ERFC;ΠC), where Φ is the operator (e.g. the integral that leads to AGWP or GWA), and the particular coefficients for component C are stored in the parameter (vector or scalar) ΠC.

If technical and/or operational measures had been in force in the considered year 2018, the ERF values would have changed and, in turn, the metric values as well, such that 2MC′=Φ(ERF′C;ΠC). If the operator Φ is linear, then 3MC′-MC=Φ(ERF′C;ΠC)-Φ(ERFC;ΠC)=dΦ(ERFC;ΠC)dERFC(ERFC′-ERFC)=:DΦ(C)ΔERFC, where the symbol DΦ has been introduced as a convenient abbreviation for the derivative. The postulate that the chosen metric must monotonically increase with the emissions implies that it also increases monotonically with ERF (for climate-warming agents). Thus, the derivative is positive. One can already note here that the ERF values per se do not appear in the balance, only their changes. Consequently, the final result does not depend on the ERF values. PGP's recommendation for an accurately as possible knowledge of ERFs is certainly desirable, but it does not follow as a consequence of the trade-off and risk analysis.

The total effect of two or more impacts is the sum of the metrics (this is the reason to postulate additivity), and the sum of the changes should be negative. With two opposing effects, say X and Y, this reads 4(MX′-MX)+(MY′-MY)≤0 and implies (see Fig.  for an illustration) 5DΦ(X)ΔERFX+DΦ(Y)ΔERFY≤0, which is equivalent to 6DΦ(X)DΦ(Y)≤-ΔERFYΔERFX. The expressions on both sides of this inequality are positive. Taking the inverse of both fractions and assuming that component Y is reduced while X increases, leads to 7DΦ(Y)DΦ(X)≥1|ΔERFY|ΔERFX. In this equation, the lhs (the ratio of the derivatives) is the uncertain part which needs a probabilistic treatment, while the rhs (changes of the ERFs) depends on the concrete mitigation measure which we must assume (at least as a working hypothesis, to be discussed later) to be represented by a certain number. Let us thus assume that in a certain mitigation measure one unit of increase of ERFX allows a reduction of ERFY by n units. Then the condition for a positive outcome is: 8DΦ(Y)DΦ(X)≥1n, an expression that resembles the one arrived at by PGP, but with different interpretation.

The lhs of this trade-off equation is not the ratio of any metrics as in PGP's derivation. It is rather the ratio of a pair of sensitivities, namely the sensitivity of the chosen metric to the change of the basic (effective) radiative forcing at the current values of these quantities (see again Fig. ). This form of the trade-off inequality might be called the “sensitivity form”, because it involves the ratio of a pair of sensitivities instead of a pair of metrics. The trade-off ratio itself, that is, the change in the two ERF values due to a particular technical or operational modification of the system, is hidden in the simple 1:n ratio.

If the operators ΦC involve uncertain coefficients ΠC, the derivatives are uncertain and can be described by PDFs. Analogously, the ratio of the derivatives is then uncertain and can be described by a PDF as well, and the CCDF of that ratio describes the probability that it exceeds 1:n, that is, that the outcome is positive.

The original ERF values do not appear in the final result, only their respective changes due to the chosen mitigation measure, and the risk of the trade-off comes in via the uncertain parameters in the operators that translate ERF-values into climate metrics. In the light of this result, PGP's postulate that “clearly the most pressing need is to firmly establish the ERF calculations” appears as an exaggeration.

This general theoretical program will be applied to GWA in the next section. Another application is given in the Appendix.

In this section we choose for Φ(ERFC;ΠC) the integral operator that leads from ERF to GWA. GWA is obtained like AGWP from the initial radiative forcing as an integral over a time horizon, but without the final step of division by the emitted mass of the substance in question. The definition of AGWP is see, e.g., 9AGWPC(H)=∫0HACIC(t)dt=AC∫0HIC(t)dt where AC is the effective radiative forcing per kg increase in the component C in the atmosphere, that is, ERFC=AC⋅mC, where mC is the emitted mass of C due to the considered activity that leads from AGWP_C to GWA_C. That is, we use GWAC=MC=Φ(ERFC;ΠC)=ERFC∫IC(t;ΠC)dt. IC(t;ΠC) is the impulse response function of component C. IC(t) has a simple exponential form for, e.g., contrail cirrus, ozone and methane, that is, exp⁡(-t/αC) (that is, the parameter ΠC=αC is the atmospheric lifetime of the perturbation of C). For CO₂ the expression must include more than one atmospheric decay time-scale (usually three), that is, ΠCO2 is a parameter vector. Once the horizon H is fixed, the integral over IC(t;ΠC) has a certain value KC(H), that depends on the chosen horizon and which is specific to the component in question. GWAC is thus the product ERFC⋅KC(H). Thus, the sensitivity factor introduced above (Eq. ) is simply DΦ(C)=KC(H). The postulate for a climate benefit then reads: 10KX(H)ΔERFX+KY(H)ΔERFY≤0. Since KC(H)=GWAC/ERFC this can be rewritten as 11GWAXΔERFXERFX+GWAYΔERFYERFY≤0. Sorting the GWA and ERF terms on the two sides of the inequality, and assuming that component Y is reduced while X is increased, gives 12GWAYGWAX≥-ΔERFX/ERFXΔERFY/ERFY=ΔERFX/ERFX|ΔERFY|/ERFY=1:|ΔERFY|/ERFYΔERFX/ERFX. The final version of the equation is again in the format suggested by PGP.

Now the meaning of the curve and the % : % unit in PGP's risk calculation becomes clear. It is not just “an N percentage reduction in the non-CO₂ impact” vs. a “1 % increase in CO₂ emissions”, it is precisely the ratio of a relative reduction of a non-CO₂ (effective) radiative forcing (Y) to the relative increase of the corresponding CO₂ (effective) radiative forcing (X).

The problem with the latter equation is that uncertain quantities appear on both sides of the equation: On the lhs both GWA values, on the rhs the ERF values are uncertain. It is thus inadequate to write the lhs in the form 1:n. The latter would imply that the relative ERF changes would rather exactly be known, but this is not the case if the reference state ERFs are uncertain. Thus, it is not allowed for a certain mitigation measure to pick a certain 1:n on the x axis of PGPs Fig. 3; it would rather be necessary to consider an interval or a worst case value for which the CCDF of the GWA ratio would need to be determined. As an illustration of this, let us assume that contrail avoidance would lead to an absolute reduction of contrail ERF that is ten times larger than the corresponding increase of the CO₂ ERF, that is, let |ΔERFCiC|=10×ΔERFCO2. To mark a trade-off ratio in PGPs Fig. 3 it is necessary then to multiply this ratio of absolute changes with the ratio of the uncertain ERFs, for which it makes sense to assume a lognormal ratio consistent with PGPs assumptions. From this ratio distribution 1000 samples are drawn and used to estimate the percentiles of this distribution. These are then multiplied by 1:10, that is, by the ratio of the absolute ERF changes. The result is the distribution of the unknown rhs in Eq. (), presented as percentiles on the x axis (blue lines in Fig. ). The rightmost blue line marks the 99th percentile of this distribution, which could be considered in a risk-adverse decision or following the precautionary principle. Note that the x value 1:10 (x=0.1) lies approximately at the 36th percentile, that is, most of the distribution of the ratio of the relative ERF changes lies to the right of 1:10 where the risk is higher than at 1:10. It means that although the ratio of absolute ERF changes may be 1:10, the corresponding ratio of relative ERF changes may be much larger, implying a much higher risk than if the risk would be naively read-off at 1:10 on the x axis. In order to avoid this undesired consequence, the absolute ΔERF change for contrails must exceed that of CO₂ by a much larger factor than 10. The 99th percentile of the ratio of relative ERF changes would be shifted to approximately 1:5 with a success probability of 0.9 if the ratio of absolute ERF changes was lowered to about 1:22, such that most of the distribution of the relative ERF changes lies outside the figure, where the risk curve is flat. This undesired property renders the method of PGP ill-defined and thus inadequate for its designed purpose.

The CCDF of the GWA ratio in this figure (red curve) has been computed analytically, which is possible once the single GWA distributions are taken as log-normals. The ratio distribution is then log-normal as well and its CCDF, FX‾(x) is: 13FX‾(x)=12-12erfln⁡(x/μ∗)2ln⁡σ∗, which involves an error function erf(⋅) that is defined by geometric mean μC∗ and standard deviation σC∗, determined from the involved GWA_C uncertainty distributions. For the case of contrails (CiC) and a time horizon of 100 years, μ∗=μCiC∗/μCO2∗ and σ∗=σCiC∗σCO2∗ with the geometric means and standard deviations from PGP's table: (μCiC∗=33, μCO2∗=81, σCiC∗=1.55, σCO2∗=1.16). With the analytical expression, the CCDF of PGP can be obtained (apart from minor differences due to rounding), as shown in Fig. .

The correct form of the risk analysis, that is, the sensitivity form, has the uncertain components on one side of the equation and on the other side there is the expression that characterizes the change in the system. In the present case, this risk-equation is 14KY(H)KX(H)≥1|ΔERFY|/ΔERFX=:1:n, for which the 1:n representation of the x axis is justified if and only if the individual changes of ERF can be determined with a better precision than the uncertainty of the sensitivity factors. This might be possible, see the discussion in Sect. 3.1, but if not, then there is no possibility at all to conduct a climatological risk assessment that is based on the postulate that the eventual sum of the chosen metric should get reduced. In this case, completely new strategies are needed. Recall that the quantities KC(H) are the integrals of the impulse response functions over the chosen time horizon. Uncertain parameters are thus the parameters of the response functions. These are lifetimes in the exponential expressions and, in case of CO₂, the weighting factors of the different lifetimes see e.g.their Table A1.

It has been shown that the method works only appropriately if the central trade-off inequality (Eq. ) has the uncertain quantities collected on one side, and relatively certain ones on the other. For the derivation it was necessary to assume that ΔERF should be known to a degree much better than the original ERF values. But the question arises whether such an assumption is justified, considering that ΔERF is the difference of ERF′ and ERF, which both are uncertain. The answer is that indeed this assumption can be justified. ERF′ and ERF must be calculated using a global model. Certainly such a model has, as every model, flaws and inadequacies in its representation of relevant physical processes. This can introduce a systematic error in the calculation of ERF. However, the systematic error is produced both in the calculation of ERF and in that of ERF′ in a very similar way, such that it cancels once the difference is taken. This fact is always exploited when a pair of simulations is performed, one with and one without aviation, to isolate the aviation effect. The uncertainty bars shown in the diagram of reflects the differences between several global models to calculate ERFs. This uncertainty does not apply since ERF′ and ERF need to be computed with one and the same model and ideally the same weather pattern (over the course of a year or longer) in both runs. This can in principle be assured using nudging. If the weather patterns could not be guaranteed to be equal in both model runs, then ERF′ and ERF would not be comparable note that there is a strong interannual variation due to the weather, see and the resulting difference would be meaningless and not usable for a risk assessment. What results when ΔERF is determined with various models, would need to be tested. It is not clear whether the error cancellation works in the same way for different models.

Anyway, should for any reason ΔERF be as uncertain as ERF and in particular less certain than the involved pair of sensitivities, then there is no way to conduct a climatological risk assessment that is based on the postulate M′-M≤0. In this case the method is not applicable, neither in the sensitivity form nor in PGPs form and for no kind of metrics.

GWAs are derived from values of effective radiative forcings (ERFs) via integration over a time horizon, a linear operation. provide estimates of ERFs of aviation emissions. These are uncertain and thus presented as best estimates and symmetric error bars which represent 90 % confidence intervals (1.645σ). These error bars comprise various sources of uncertainty, but not all. write in their Appendix E that the number of available studies is too small for a robust uncertainty estimate of the contrail forcing. They considered in their study two sources of uncertainty, which both originate from the individual method how radiative effects and contrail and cirrus microphysics are treated in the respective model. Other uncertainties, in particular that due to a wide annual variability from changing weather are not regarded. It is evident, that the actual global contrail ERF uncertainty is larger than what is estimated by . While the CO₂ uncertainty distribution is relatively narrow, in particular that of contrails is quite wide. If the uncertainty distribution would be modelled as a normal PDF, a small probability for negative ERF of contrails would result. In contrast the choice of a log-normal model to describe the uncertainty distribution does not allow values of both signs to appear; the values are strictly positive or strictly negative. For contrails with a log-normal uncertainty PDF, the possible ERF values are strictly positive. This is presumably the idea which PGP had in mind when they chose log-normal (that is asymmetric) uncertainty PDFs, although it is not stated in that paper.

The ERF distributions (probability density functions, PDFs) for CO₂ and contrail cirrus (CiC), interpreted both as normal and log-normal distributions are shown in Fig. .

The risk function for an underlying normal distribution of the GWA uncertainties has been computed using the geometric means and standard deviations from PGPs table, translated into normal distributions, keeping the 68 % (±σ) confidence levels. In this case, the ratio distribution is not simply given as an analytic expression and must be obtained using a Monte Carlo experiment. The result is presented in Fig. .

The black curve shows the PDF of the GWA ratio, which extends (with small probability) to negative numbers. Consequently, the CCDF extends into the negative range as well and crosses the trade-off ratio 1:∞. This signifies that no trade-off occurs when contrails were cooling in the global and multi-annual average. However, as the red curve shows, the probability that the GWA ratio is positive (that is, that contrails warm the climate) is practically unity. This example demonstrates clearly that the risk curve is different from the original one (red curve in Fig. ).

In fact, there is no particular reason to assume a normal uncertainty distribution at all; one could select as well a triangular or a uniform distribution, each time with a different resulting risk curve F‾(x). Ideally, the choice should be guided by physical arguments similarly to the choice of a prior distribution in Bayes-statistics, see. However, as there are few global models that can provide GWAs or other global metrics for aviation emissions, the statistical basis for a well-conceived choice is thin. As long as the choice of the distribution model for the GWA uncertainty distribution is a mere subjective choice (perhaps for the reason of more convenient mathematics), the resulting risk function F‾(x) cannot be regarded as an objective measure of trade-off risk.

Rerouting for the purpose of contrail avoidance implies deviations from the cost-optimal path, which in most cases requires the burning of additional fuel. This leads to increased gaseous emissions, not only of CO₂ but of all other combustion products as well, in particular NOx. The proper trade-off consideration of rerouting for contrail avoidance needs thus to account for all additional emissions, not only CO₂. Following PGP by computing the CCDF of the ratio of GWA(CiC) to GWA(other emissions) for that purpose implies the need to compute the PDF of the sum of the GWAs of CO₂ and the other gaseous emissions first, before the ratio is taken. It follows that first one has to compute the probability density function of the sum of the GWAs of CO₂ plus the other gaseous emissions, for instance via a Monte Carlo method. The desired CCDF of the GWA ratio is then achieved from a Monte Carlo calculation as well. For the calculation, the mean values and standard deviations from the table of PGP are used. The result for taking short- (ozone) and long- (mainly methane) term NOx effects into account is shown in Fig. . The black curve is just the copy from Fig. ; it takes only CO₂ into account. The red curve shows the risk curve, if additionally the ozone effect from NOx is considered. The addition of ozone into the trade-off ratio leads to a quite different risk curve than the original one. The curve becomes much flatter which means that the probability of success for a certain trade-off ratio gets considerably lower. That would imply in turn that much more contrail effect would need to be reduced for one unit of a combined CO₂ and ozone effect to get a high probability of success. The blue curve represents the case where both, the short and long-term NOx effects are accounted for. A problem for the proper treatment is the correlation between these two effects, which implies that they cannot be added as independent stochastic quantities. A simple solution for this case is to treat them as one effect with a GWA distribution, where the mean and variance are, respectively, the sum of the two means and the two variances. All these numbers have been computed using the entries in PGP's table. The blue curve is close to the original black one because of the partial cancellation of the short- and long-term NOx effects. The treatment of the correlation is certainly too simple for real applications, but it is sufficient for the present goal, that is, to demonstrate that when determining the risk of contrail avoidance it is not sufficient to just consider the trade-off with CO₂ alone.