Supplementary Material for: Scientific Uncertainty and Climate Change: Part I. Uncertainty and Unabated Emissions

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1 Noname manuscript No. (will be inserted by the editor) Supplementary Material for: Scientific Uncertainty and Climate Change: Part I. Uncertainty and Unabated Emissions Stephan Lewandowsky James S. Risbey Michael Smithson Ben R. Newell John Hunter November, 213 S1 Risk Rates for the Lognormal Distribution Consider a lognormal distribution, with probability density function (pdf) f = 1 xσ G 2π exp ( (log(x) µ G) 2 where µ G and σ G are mean and standard deviation of a Gaussian distribution. The partial differential of the lognormal complementary cumulative density function (1 cdf(x)) with respect to σ G is: h (x, µ G, σ G ) = [ / Erf 2σ 2 G ), ( )] (µ G log (x)) /2σ G 2 σ ( G = exp (µ G log (x)) 2/ ) 2σG 2 (µ G log (x)). σg 2 2π This function gives us the rate of change in 1 cdf(x) with respect to σ G. Clearly h is at the median of the lognormal distribution, x = exp(µ G ). It is positive for log(x) > µ G and negative for log(x) < µ G. So the threshold above which increasing σ G also increases 1 cdf(x) for any lognormal thereby fattening its tail is simply the median, exp(µ G ). Moreover, h(exp(µ G + s), µ G, σ G ) = h(exp(µ G s), µ G, σ G ). Thus, the rate of increase in 1 cdf(x) at exp(µ G + s) is equal to the rate of decrease in 1 cdf(x) at exp(µ G s). Note that cdf(exp(µ G +s)) = 1 cdf(exp(µ G s)), so for any probability, P, that x > exp(µ G + s), the rate of change in P with respect to σ G has equal magnitude to the rate of change in probability Q = 1 P that x > exp(µ G s). University of Western Australia and University of Bristol stephan.lewandowsky@bristol.ac.uk CSIRO Marine and Atmospheric Research, Hobart, Tasmania Australian National University University of New South Wales Antarctic Climate & Ecosystems Cooperative Research Centre, Hobart, Tasmania

2 2 Lewandowsky et al. Table S1 The effects of increasing climate uncertainty on expected damage costs (in arbitrary units) illustrated with a quadratic cost function d(t ) = T 2 and lognormally distributed climate sensitivities Climate Damages a Uncertainty ECS Mean sensitivity (µ L ) Mean damage b Uncertainty D a Climate uncertainty ECS refers to the standard deviation, σ L, of the simulated distribution of climate sensitivity. Sensitivity is expressed in C. Table entries are summaries of 1, samples drawn from a lognormal distribution of climate sensitivity. b Obtained standard deviation of the damage cost distribution. Now we incorporate a damage function, g(x), increasing in x. Obviously if g(exp(µ G +s) > g(exp(µ G s) then the magnitude of the rate of change in expected damage, g(x) h(x, µ G, σ G ), will be greater at exp(µ G +s) than at exp(µ G s). In fact, the ratio of those magnitudes is simply the ratio of g(exp(µ G +s)/g(exp(µ G s). The convexity assumption can be relaxed, although convexity further increases this asymmetry. It follows that increasing uncertainty ECS, which is proportional to σ G, increases the rate of acceleration of damages at any point above the median of the sensitivity distribution more rapidly than it decreases them at a comparable point below the median. This asymmetry increases with uncertainty ECS. S2 Sensitivity and Speed of Warming in a General Linear Model of the Climate System This section describes the response of the climate to radiative forcing, using a general linear model (i.e. one of any complexity). The speed of that response is related to the magnitude of equilibrium climate sensitivity.

3 Uncertainty and Unabated Emissions 3 S2.1 Theory A general linear system is governed by: T (t) = F (t t )C(t )dt, (1) where T is an output (in this case global temperature), F is the input (in this case, radiative forcing), C is a convolution kernel, t is time and t is the integration variable in time. C(t ) is only defined for < t <, because the system is unaware of future forcing. C(t ) contains everything we need to know about the response, T, of the system to forcing, F (the only assumption being that the system is linear). While C(t ) addresses all the time scales of the system (of which there may be many), there are two important ones: The effective time constant, τ, given by: τ = t C(t )dt C(t )dt (2) and an ultimate time scale, τ, beyond which the system has no memory, given by: Under steady forcing F = F s, Eq.1 becomes: C(t ) = for t τ. (3) T s = F s C(t )dt, (4) where T s is the temperature once a steady state is reached (if F s is the forcing due to CO 2 doubling, then T s is what is normally called the equilibrium climate sensitivity (ECS)). F t Fig. S1 Linear variation of F with t.

4 4 Lewandowsky et al. Under linear forcing starting at t = and prescribed by: (see Fig. S1), Eq. 1 becomes: T = β F = for t < and F = βt for t (5) t t t (t t )C(t )dt = βt C(t )dt β t C(t )dt, (6) which, for long times, t, defined by t τ, becomes: T = βt C(t )dt β t C(t )dt = βts (t τ), (7) F s which is just a scaled version of the forcing, lagged by the time constant, τ. This is illustrated schematically in Fig. S2. F, (F s /T s )T F τ (F s /T s)t t Fig. S2 Schematic of F and scaled response (F s/t s)t. Under linear forcing, F becomes equal to F s (the forcing used to define the steady-state solution), when t = F s/β (from Eq. 5). At this time, the temperature, T s is (from Eq. 6): Fs/β Fs/β Ts = F s C(t )dt β t C(t )dt. (8) If F s is the forcing related to the ECS, Ts is the transient climate response (TCR). If the time, F s/β, to reach forcing F s is long (i.e. F s/β τ ), then Eq. 8 becomes: ( Ts = F s C(t )dt β t C(t )dt = T s 1 βτ ). (9) F s

5 Uncertainty and Unabated Emissions 5 S2.2 Discussion The purpose of the above analysis is to investigate whether an increase in the ECS necessarily implies an increase in the TCR (or in some other measure of the rate of warming). Two cases are discussed in the following, in which it is assumed that F s is equivalent to CO 2 doubling, so that T s is the ECS and Ts is the TCR. S2.2.1 The time to CO 2 doubling is short A short time to CO 2 doubling implies F s/β < τ. In this case T s (ECS) is given by Eq. 4 and Ts (TCR) is given by Eq. 8. If some change is made to the climate system (e.g., a climate model) then this implies a change to C(t ). A major difference between T s (Eq. 4) and Ts (Eq. 8) is that the integral in Eq. 4 spans, while the integrals in Eq. 8 only span F s/β. There is therefore a span of C(t ) (F s/β ; the long time-scale part) that is not included in the derivation of Ts. If this part of C(t ) is changed, then only T s is changed and not Ts. In this case, it is possible to change T s (the ECS) without changing Ts (the TCR). S2.2.2 The time to CO 2 doubling is long A long time to CO 2 doubling implies F s/β τ. In this case T s (ECS) is given by Eq. 4 and Ts (TCR) is given by Eq. 9. Under these conditions, it seems reasonable to assume that, if T s (ECS) is changed, then Ts (TCR) changes in the same direction (assuming that τ doesn t change significantly in order to confound this relationship). S2.3 Summary The time to CO 2 doubling is of the order of a century, which is similar to the time scale for fast climate feedbacks (e.g. changes in clouds and sea ice). However the time scale for slow climate feedbacks (e.g. changes in land ice; Hansen, 27; Hansen, Sato, Russell, & Kharecha, 213; PALAEOSENS, 212) is centuries to millennia. Conventional atmosphere-ocean general circulation models (AOGCMs) only include fast climate feedbacks, and it is over these time scales that the ECS generally applies. This ECS (often called the Charney sensitivity) however completely ignores the longer term changes which depend on the slow feedbacks (which are, in this case, treated as forcings rather than feedbacks). The question of whether it is possible for the ECS to change independently of the TCR depends strongly on how the ECS is defined that is, whether it includes the slow climate feedbacks. If the ECS is defined in the conventional way (excluding slow climate feedbacks) then it is quite likely that any change in ECS is reflected in a change in TCR in the same direction. This is because the time to CO 2 doubling is of the same order as the longest time scales associated with the fast feedbacks, and Eq. 9 approximately applies. If, however, the ECS is defined including the slow climate feedbacks, then the time to CO 2 doubling is much less than the longest time scales associated with

6 6 Lewandowsky et al. Table S2 The effects of increasing climate uncertainty on expected damage costs illustrated with a quadratic cost function d(t ) = T 2 and normally distributed climate sensitivities. Climate Damages a Uncertainty ECS Mean sensitivity (µ G ) Mean damage b Uncertainty D a Climate uncertainty ECS refers to the standard deviation, σ G, of the simulated distribution of climate sensitivity. Sensitivity is expressed in C. Table entries are summaries of 1, samples drawn from a normal distribution of climate sensitivity. Mean sensitivity µ G was set to a high value to minimize the number of negative samples (which were discarded without replacement). b Obtained standard deviation of the damage cost distribution. the slow climate feedbacks, and Eq. 8 applies. Under these conditions, it is quite possible for the ECS to change (through changes in the long time-scale part of C(t )) without any change in the TCR. S3 Symmetrical climate sensitivity and convex damage functions Although the probability density function for climate sensitivity is assumed to have a fat upper tail, we relax this assumption here to underscore the generality of our analysis. Table S2 shows the effects of increasing uncertainty on expected damage costs when the distribution of climate sensitivities is normal, rather than lognormal. S4 Uncertainty in expert elicitation Morgan and Keith (1995) elicited subjective judgments about climate sensitivity from 16 leading U.S. climate scientists. Their measures included subjective probability distributions for various policy-relevant climate parameters. Here we focus on the mean climate sensitivity and its standard deviation elicited from the 16 experts, which are shown in Figure S3 (Panel A). Panel B of Figure S3 shows the corresponding logit-transformed probabilities with which a threshold temperature increase of 4 C (arbitrarily chosen) would be exceeded upon doubling of CO 2 based on the subjective means and standard deviations in Panel A (assuming normality). The panel highlights an outlying

7 Uncertainty and Unabated Emissions 7 A B Climate Sensitivity ( C) logit(exceedance Probabilities) Fig. S3 Panel A: Subjective mean (data points) and standard deviation (error bars) of 16 expert judgments of equilibrium climate sensitivity. Data from Figure 1 of Morgan and Keith (1995). Duplicate data points provided by 2 experts for state changes or surprises are omitted. Panel B: Corresponding threshold exceedance probabilities, transformed into logodds, based on a threshold T c=4 C. Each data point represents a π i, as defined in main text. observation (Expert 5) that introduces second-order uncertainty into the otherwise rather homogeneous subjective estimates. We next examine the consequences (a) of a hypothetical reduction of secondorder uncertainty and (b) of a hypothetical increase. To model the reduction of uncertainty, we replaced the observation for Expert 5 with the mean of the remaining 15 observations. We call this the homogenized data from here on. To model an increase of uncertainty we replicated Expert 5 by replacing the data from 4 randomly chosen experts with Expert 5 s subjective values ( heterogeneous data from here on). S4.1 Decreasing second-order uncertainty To examine the effects of a decrease in second-order uncertainty, we first fit Beta distributions to the (untransformed) exceedance probabilities (Figure S3B) for values of T c that ranged from 4 C to 6 C in steps of.2. For each value of T c, in turn, we examined the probability of exceeding a tolerable risk, π t, for values of π t ranging from.3 to.5 in steps of.2. Each π t represents a cutoff on the Beta distribution that characterizes the overall set of 16 exceedance probabilities, and each P (π > π t ) describes the likelihood that the actual exceedance probability is greater than that tolerable risk. We next repeated this process for the homogenized data. Each panel in Figure S4 plots the differences between the set of P (π > π t ) for the original data and

8 8 Lewandowsky et al T c Excess Risk π t Excess Risk Fig. S4 Each panel plots the excess risk arising from greater second-order uncertainty, defined as the difference between P (π > π t) when all data (including outlier) are retained and when the data are homogenized by eliminating leverage of the outlier. The top panel plots excess risk as a function of global temperature change threshold, T c, and the bottom panel plots excess risk as a function of the tolerable risk, π t. In each panel, darkness of plotting symbols represents the value of the other parameter; viz. π t for the top, and T c for the bottom panel, respectively. the homogenized data. Positive numbers arise if P (π > π t ) (i.e., the probability of the actual exceedance probability being greater than a tolerable risk) is greater when all data are retained than when second-order uncertainty is reduced by eliminating the leverage of Expert 5. The top panel shows this excess risk as a function of temperature thresholds (T c), and the bottom panel shows the same data as a function of different tolerable risks (π t ). The figure shows that for most combinations of tolerable risks (π t ) and temperature thresholds (T c), greater second-order uncertainty renders it more likely

9 Uncertainty and Unabated Emissions 9 that a tolerable risk of threshold-exceedance is in turn exceeded. The excess risk is positive for 116 out of 121 combinations examined (i.e., 96%). S4.2 Increasing second-order uncertainty To examine the effects of a further increase in second-order uncertainty, we repeated the preceding comparison for the two hypothetical data sets; the homogenized set (minimal uncertainty) and the heterogeneous set (large second-order uncertainty). Figure S5 shows the results, again plotting differences between P (π > π t ) for the two hypothetical data sets in each panel. Although greater second-order uncertainty did not uniformly increase the likelihood of a risk becoming intolerable, it did so in the majority of cases (95 out of 121 or 79%). This is notable because unlike for the theoretical exploration in the main text, the mean exceedance probability here was not constrained to remain equal when second-order uncertainty was varied. On the contrary, the mean estimate of the mean, µ B, of the fitted Beta distributions across all values of T c was.54 for the original data (i.e., in Figure S3B), compared to.91 for the homogenized set (Expert 5 removed) and.29 for the heterogeneous data (Expert 5 replicated). The positive values in Figures S4 and S5 were thus obtained despite opposing differences between means (.38 and.63, respectively). References Hansen, J. E. (27). Scientific reticence and sea level rise. Environmental Research Letters, 2, 242. doi: 1.188/ /2/2/242 Hansen, J. E., Sato, M., Russell, G., & Kharecha, P. (213). Climate sensitivity, sea level and atmospheric carbon dioxide. Philosophical Transaction of the Royal Society A, 371, Morgan, M. G., & Keith, D. W. (1995). Subjective judgments by climate experts. Environmental Science & Technology, 29, 468A 476A. PALAEOSENS. (212). Making sense of palaeoclimate sensitivity. Nature, 491, doi: 1.138/nature11574

10 1 Lewandowsky et al. Excess Risk T c Excess Risk π t Fig. S5 Each panel plots the excess risk arising from greater second-order uncertainty, defined as the difference between P (π > π t) when an outlying expert was replicated a further 4 times, and when the data are homogenized by eliminating leverage of the single outlier. The top panel plots excess risk as a function of global temperature change threshold, T c, and the bottom panel plots excess risk as a function of the tolerable risk, π c. In each panel, darkness of plotting symbols represents the value of the other parameter; viz. π c for the top, and T c for the bottom panel, respectively.