# Carbon Budgets and the TCRE

Posted August 6, 2019

on:**OTHER POSTS ON THE CARBON BUDGET**

**FIGURE 1: TCRE**

**FIGURE 2: EMISSIONS+FORCINGS = TEMPERATURE RESPONSE**

**FIGURE 3: TEMPERATURE TRAJECTORIES**

**FIGURE 4**

**Future cumulative budgets from January 2015 for percentiles
of the distribution of RCP8.5 (LEFT) AND RCP2.6 (RIGHT)**

**CITATION**

**Emission budgets and pathways consistent with limiting warming to 1.5 °C**

Richard J. **Millar**, Jan S. Fuglestvedt, Pierre Friedlingstein, Joeri Rogelj, Michael J. Grubb, H. Damon Matthews, Ragnhild B. Skeie, Piers M. Forster, David J. Frame & Myles R. Allen

Nature Geoscience volume 10, pages 741–747 (**2017**) **LINK TO FULL TEXT PDF: 2017CARBON-BUDGET-PAPER-PDF **

**ABSTRACT**

The Paris Agreement has opened debate on whether limiting warming to 1.5 °C is compatible with current emission pledges and warming of about 0.9 °C from the mid-nineteenth century to the present decade. We show that limiting cumulative post-2015 CO2 emissions to about 200 GtC would limit post-2015 warming to less than 0.6 °C in 66% of Earth system model members of the CMIP5 ensemble with no mitigation of other climate drivers. We combine a simple climate–carbon-cycle model with estimated ranges for key climate system properties from the IPCC Fifth Assessment Report. Assuming emissions peak and decline to below current levels by 2030, and continue thereafter on a much steeper decline, which would be historically unprecedented but consistent with a standard ambitious mitigation scenario (**RCP2.6**), results in a likely range of peak warming of 1.2–2.0 °C above the mid-nineteenth century. If CO2 emissions are continuously adjusted over time to limit 2100 warming to 1.5 °C, with ambitious non-CO2 mitigation, net future cumulative CO2 emissions are unlikely to prove less than 250 GtC and unlikely greater than 540 GtC. Hence, limiting warming to 1.5 °C is not yet a geophysical impossibility, but is likely to require delivery on strengthened pledges for 2030 followed by ** challengingly **deep and rapid mitigation. Strengthening near-term emissions reductions would hedge against a high climate response or subsequent reduction rates proving economically, technically or politically unfeasible.

**THE TCRE: TRANSIENT CLIMATE RESPONSE TO CUMULATIVE EMISSIONS **

**FIGURE 5: SPLIT HALF TEST: TCRE & ITS CORRELATION**

**FIGURE 6: UNCONSTRAINED RANDOM NUMBERS & THEIR CUMULATIVE VALUES**

**FIGURE 7: CONSTRAINED RANDOM NUMBERS AND THEIR CUMULATIVE VALUES**

**FIGURE 8: SUMMARY OF RESULTS FROM FIGURES 6&7**

**A CRITICAL EVALUATION OF THE NEAR PERFECT PROPORTIONALITY BETWEEN SURFACE TEMPERATURE AND CUMULATIVE EMISSIONS IN FIGURES 5, 6, &7 **

**Figure 5**is a split-half reliability test of the near perfect proportionality between surface temperature and cumulative emissions from which the TCRE and the carbon budget are derived. In this context, it is best to understand surface temperature as cumulative warming. Therefore the correlations we see in these charts are correlations between cumulative values – cumulative warming as a function of cumulative emissions.- Three different datasets of mean global temperatures are used – two temperature reconstructions (HADCRUT & BERKELEY) and the RCP8.5 business as usual projection of CMIP5 forcings. The full span is restricted to 156 years as 1861-2016 constrained by the the RCP8.5 series. The split halves are therefore 78 years. What we see in
**Figure 5**is that both the correlation and the regression coefficient (TCRE) between cumulative warming and cumulative emissions show large differences among full span, first half, and second half values not only for the temperature reconstructions but also for the theoretical projections from climate models in the RCP8.5 values. We conclude from the analysis in**Figure 5**that the TCRE is an unreliable statistic because it fails the spit half test and is therefore likely to be spurious and illusory. A further conclusion is that since these differences are also seen in the theoretical RCP series, the problem with the TCRE proportionality is likely to be a structural issue and not unique to these data. - The structural nature of the spuriousness of correlations between cumulative values of time series data is examined in
**Figure 6**and**Figure 7**by studying the behavior of random numbers. It is noted that**emissions data are always positive**and the temperature data in a period of**warming has a bias for positive differences**from one year to the next and that therefore there is some bias for the sum of temperature changes from year to year to be positive. These sums for the three data sets used, RCP8.5, HadCRU, and Berkeley are 14.6, 14.5, and 18.6 respectively. The random numbers used in Figure 6 and Figure 7 are therefore studied with positive random values for emissions against random temperature values with and without a bias for positive changes. - In the analysis of random numbers,
**Figure 6**shows the behavior of the data when no bias exists in year to year changes in temperature but with emissions restricted to positive numbers. The two GIF images display an animation of the data under this condition. The first video displays the randomness of the relationship between the simulated positive annual emissions and simulated annual warming data without a sign constraint. No relationship is evident in the video. The second video shows the relationship between the cumulative values of the data presented in the first video. Although some random spurious correlations are seen both positive and negative, on the whole we see no evidence of a proportionality between cumulative warming and cumulative emissions. - The corresponding videos with a positive bias in temperature changes appear in
**Figure 7**. Here, though no relationship is seen in the source data, a strong proportionality is found in the cumulative values of random numbers, just as climate science had found in the actual data for emissions and temperature. It is on this basis that we propose that the TCRE proportionality in climate science (Matthews 2009) is indistinguishable from the same proportionality in random numbers. The data presented in the GIF animations of Figure 6 and Figure 7 are summarized in**Figure 8.**These charts make it clear that the strong proportionality between cumulative emissions and cumulative warming found by climate science is illusory and not real because it is a creation of the bias for positive temperature changes that can be recreated in random numbers. Therefore, though carbon budgets may be constructed on the basis of the Matthews 2009 proportionality, no conclusions can be drawn from such budgets because the correlation is spurious and illusory and has no interpretation in the real world. - It is shown in a related post
**[LINK]**that in statistical procedures that use source data repeatedly, a loss in effective sample size (EFFN) is incurred due to multiplicity in the use of the data and that this loss in EFFN translates into a loss in degrees of freedom. An extreme case of such multiplicity in the use of source data is the construction of a time series of the cumulative values of another time series. It is shown in an online paper that the in all such cases the effective sample size of cumulative values is EFFN=2 and that therefore the degrees of freedom is DF=0. It should also be noted that the time series of cumulative values has no time scale since the there is no moving window of fixed size that moves through the time series but the size of the window changes from TS=1 to TS=N-1. Thus the time series of the cumulative values of another time series contains neither degrees of freedom nor time scale. - We conclude from the analysis presented above, that the TCRE is a spurious and illusory statistic that has no interpretation and that therefore, carbon budgets constructed from he TCRE are mathematical illusions.
**The Millar 2017 paper cited above shows that despite its statistical flaws, climate science makes use of the TCRE in its construction of carbon budgets**. The authors write “the relationship between CO2-induced future warming compatible with cumulative emissions is broadly consistent with that expected from the IPCC-AR5 likely range of TCRE”. In a related post**[LINK]**, it is shown that the complexity of the Remaining Carbon Budget issue in climate science derives from the statistical flaw of the TCRE described in this work, **CONCLUSION: The near perfect proportionality between cumulative warming and cumulative emissions described by Matthews and others in 2009****[LINK]****is a creation of the transformation to cumulative values. That proportionality is also found in the cumulative values of random numbers. This correlation derives from a sign pattern wherein emissions are always positive, and in a time of global warming, changes in temperature have a positive bias. It is shown here that under the same conditions the same correlation is found in random numbers. Therefore although strong correlation and regression coefficients can be computed from the time series of cumulative values, these statistics have no interpretation because they are illusory. The presentation of climate action mathematics by climate science in the form of carbon budgets derived from the TCRE has no interpretation in the real world because the TCRE is a creation of a spurious correlation. The instability and unreliability of the TCRE demonstrated in this work, has been noted in climate science research****[LINK], [LINK], and in other posts on this site [LINK] . This work provides further evidence of instability along with a statistical basis for instability in the TCRE.**

**OTHER POSTS ON THE CARBON BUDGET**

### 9 Responses to "Carbon Budgets and the TCRE"

[…] CARBON BUDGETS AND THE TCRE […]

[…] CARBON BUDGETS AND THE TCRE […]

[…] And yet, a statistical detail overlooked in the rush to TCRE as a replacement for the failed ECS parameter is that correlations between cumulative values of time series data are spurious. Since temperature is cumulative warming, the TCRE derives from a spurious correlation between cumulative values. A regression coefficient that depends on a spurious correlation does not have an interpretation because a time series of cumulative values of another time series contains neither time scale not degrees of freedom as explained in this related post [LINK] . […]

[…] BUDGET CONUNDRUM IN CLIMATE SCIENCE DESCRIBED IN THREE RELATED POSTS CARBON BUDGET POSTS: [LINK] [LINK] [LINK] . TOGETHER, THEY EXPOSE FUNDAMENTAL FLAWS AND INCONSISTENCIES IN THE THEORY OF AGW AND […]

[…] Yet the move to the TCRE and the use of the TCRE to construct cumulative emissions carbon budgets for climate action reveals even greater weaknesses in climate science as it has been shown in a related posts at this site that the TCRE is illusory because it is based on a spurious correlation [LINK] [LINK] [LINK] . […]

[…] Carbon Budgets and the TCRE […]

[…] It is noted here that emissions are always positive and the way degree days are defined in terms of degrees Kelvin, that series also consists of positive numbers. It is shown in related posts that a time series of the cumulative values of another time series has neither time scale nor degrees of freedom and that therefore their correlation has no interpretation; and that experiments with random numbers show that as long as the two time series being compared have a similar sign bias, i.e., mostly positive or mostly negative, their cumulative values will show a correlation by virtue only of the sign convention. Therefore such correlation cannot be interpreted in terms of the responsiveness of the object time series to changes in the explanatory time series. The statistical details of this argument may be found in these posts on this site: [LINK] [LINK] . […]

1 | Alan Tomalty (@ATomalty)

August 7, 2019 at 3:17 pm

Brilliant