Scaling fluctuation analysis and statistical hypothesis testing of anthropogenic warming

Abstract

Although current global warming may have a large anthropogenic component, its quantification relies primarily on complex General Circulation Models (GCM’s) assumptions and codes; it is desirable to complement this with empirically based methodologies. Previous attempts to use the recent climate record have concentrated on “fingerprinting” or otherwise comparing the record with GCM outputs. By using CO₂ radiative forcings as a linear surrogate for all anthropogenic effects we estimate the total anthropogenic warming and (effective) climate sensitivity finding: ΔT _anth  = 0.87 ± 0.11 K, $\uplambda_{{2{\text{x}}{\text{CO}}_{2} ,{\text{eff}}}} = 3.08 \pm 0.58\,{\text{K}}$. These are close the IPPC AR5 values ΔT _anth  = 0.85 ± 0.20 K and $\uplambda_{{2{\text{x}}{\text{CO}}_{2} }} = 1.5\!-\!4.5\,{\text{K}}$ (equilibrium) climate sensitivity and are independent of GCM models, radiative transfer calculations and emission histories. We statistically formulate the hypothesis of warming through natural variability by using centennial scale probabilities of natural fluctuations estimated using scaling, fluctuation analysis on multiproxy data. We take into account two nonclassical statistical features—long range statistical dependencies and “fat tailed” probability distributions (both of which greatly amplify the probability of extremes). Even in the most unfavourable cases, we may reject the natural variability hypothesis at confidence levels >99 %.

Appendix: Scaling modified Gaussians with fat tails

In Fig. 9 we showed the empirical probability distributions (Pr(ΔT > s), for the probability of a random (absolute) temperature difference ΔT exceeding a threshold s for time lags Δt increasing by factors of 2. Note that we loosely use the expression “distribution function” to mean Pr(ΔT > s). This is related to the more usual “cumulative distribution function” (CDF) by: CDF = Pr(ΔT < s) so that Pr(ΔT > s) = 1 − CDF. Two aspects of Fig. 9 are significant; the first is their near scaling with lag Δt: the shapes change little, this is the type of scaling expected for a monofractal “simple scaling” process, i.e. one with weak multifractality (as discussed in Lovejoy and Schertzer (2013), over these time scales, the parameter characterizing the intermittency near the mean, C ₁ ≈ 0.02 so that this is a reasonable approximation).

This implies that there is a nondimensional distribution function P(s):

$$P(s) = \Pr \left( {\frac{{\Delta T(\Delta t)}}{{\upsigma_{{\Delta T}} }} > s} \right);\quad\upsigma_{{\Delta T}} = \left\langle {\Delta T(\Delta t)^{2} } \right\rangle^{1/2}$$

σ_Δt is the standard deviation. Due to the temporal scaling, we have $\upsigma_{{{\lambda \varDelta }t}} =\uplambda^{H}\upsigma_{{\Delta t}}$ where H is the fluctuation exponent and P(s) is independent of time lag Δt. From Fig. 9 it may be seen that as predicted by the RMS fluctuations (σ_Δt, Fig. 7), H ≈ 0. This is a consequence of the slight decrease in the RMS Haar fluctuation (with exponent H _Haar  ≈−0.1; Fig. 8). Unlike the Haar fluctuation, the ensemble mean RMS differences cannot decrease but simply remains constant until the Haar fluctuations begin to increase again in the climate regime (compare Figs. 7, 8, beyond Δt ≈ 125 years).

The second point to note is that the lag invariant distribution function P(s) has roughly a Gaussian shape for small s, whereas for large enough s, it is nearly algebraic. This can be simply modelled as:

$$P_{qD} (s) = \begin{array}{*{20}l} {P_{G} (s);} \hfill & {s < s_{qD} } \hfill \\ {P_{G} (s_{qD} )\left( {\frac{s}{{s_{qD} }}} \right)^{ - qD} ;} \hfill & {s \ge s_{qD} } \hfill \\ \end{array}$$

where P _G (s) is the cumulative distribution function for the absolute value of a unit Gaussian random variable. The simple way of determining s _qD used here is to define s _qD as the point at which the logarithmic derivative of P _G is equal to −q _D so that the plot of log P _qD versus log s is continuous:

$$\left. {\frac{{d\log P_{G} (s)}}{d\log s}} \right|_{{s = s_{qD} }} = - q_{D}$$

this is an implicit equation for the transition point s _qD .

In actual fact the only part of the model that is used for the statistical tests is the extreme large s “tail” which Fig. 9 empirically shows could be bracketed between:

$$P_{qD1} (s) < P(s) < P_{qD2} (s);\quad q_{D1} > q_{D2} ;\quad s > s_{qD1} > s_{qD2}$$

(with q_D1 = 6, q_D2 = 4) hence the Gaussian part of the model is not very important, it only serves to determine the transition point s _qD . In any case, for the extremes we can see from the figure that this bracketing is apparently quite well respected by the empirical distributions.