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THIS POST IS A CRITICAL REVIEW OF A RESEARCH PAPER ON TCRE CARBON BUDGETS CITED BELOW WITH FULL TEXT PDF AVAILABLE FRIEDLINGSTEIN 

ABSTRACT: Climate science has misinterpreted anomalies created by statistical errors as a climate science issue that needs to be resolved with climate models of greater complexity. In this context we find that their struggle with the remaining carbon budget puzzle demonstrates a failure of climate science to address statistical issues of the TCRE in terms of statistics. This failure has led them down a complex and confusing path of trying to find a climate science explanation of the remaining carbon budget anomaly that was created by statistical errors. The research paper presented below serves as an example of this kind of climate research. The real solution to the remaining carbon budget puzzle is to understand the statistical flaws in the TCRE correlation and to stop using it. [LINK] [LINK] . In the second link we show that the TCRE procedure that shows that fossil fuel emissions cause warming also shows that UFOs cause warming [LINK] .  

Environmental Research Letters: Quantifying process-level uncertainty contributions to TCRE and Carbon Budgets for meeting Paris Agreement climate targets.
Chris D Jone1 and Pierre Friedlingstein: 1 April 2020: Abstract: To achieve the goals of the Paris Agreement requires deep and rapid reductions in anthropogenic CO2 emissions, but uncertainty surrounds the magnitude and depth of reductions. Earth system models provide a means to quantify the link from emissions to global climate change. Using the concept of TCRE – the transient climate response to cumulative carbon emissions – we can estimate the remaining carbon budget to achieve 1.5 or 2 oC. But the uncertainty is large, and this hinders the usefulness of the concept. Uncertainty in carbon budgets associated with a given global temperature rise is determined by the physical Earth system, and therefore Earth system modelling has a clear and high priority remit to address and reduce this uncertainty. Here we explore multi-model carbon cycle simulations across three generations of Earth system models to quantitatively assess the sources of uncertainty which propagate through to TCRE. Our analysis brings new insights which will allow us to determine how we can better direct our research priorities in order to reduce this uncertainty. We emphasize that uses of carbon budget estimates must bear in mind the uncertainty stemming from the biogeophysical earth system, and we recommend specific areas where the carbon cycle research community needs to re-focus activity in order to try to reduce this uncertainty. We conclude that we should revise focus from the climate feedback on the carbon cycle to place more emphasis on CO2 as the main driver of carbon sinks and their long-term behaviour. Our proposed framework will enable multiple constraints on components of the carbon cycle to propagate to constraints on remaining carbon budgets.

PART-1: WHAT THE PAPER SAYS

(1) ABOUT THE TCRE: A body of literature from 2009 found consistently that warming was much more closely related to the cumulative CO2 emissions than the time profile or particular pathway. This relationship between warming and cumulative emissions is found in the IPCC’s Fifth Assessment Report (AR5) as TCRE: the Transient Climate Response to cumulative carbon Emissions. The physical basis of TCRE is described by Caldeira & Kasting (1993) who noted that saturation of the radiative effect of CO2 in the atmosphere could be balanced by saturation of uptake by ocean carbon leading to insensitivity of the warming to the pathway of CO2 emissions. Literature since then has put this on a firm footing with numerous authors showing that trajectories of ocean heat and carbon uptake have similar effects on global temperature due to the diminishing radiative forcing from CO2 in the atmosphere and the diminishing efficiency of ocean heat uptake. Terrestrial carbon uptake is equally important for the magnitude of TCRE – in fact we will show here that land and ocean contribute equally to the magnitude of TCRE and that land dominates over the ocean in terms of model spread.

(2) ABOUT TCRE CARBON BUDGETS: The IPCC AR5 assessed a total carbon budget of 790 PgC to stay below 2C above pre-industrial, of which about 630 PgC has been emitted over the 1870-2018 period. However, the uncertainty in the remaining carbon budget to achieve 1.5C or 2C is very large – in fact possibly larger than the remaining budget itself. This large uncertainty hinders the potential usefulness of this simplifying concept to policy makers. All studies and reports which present estimates of the remaining carbon budget (e.g. The IPCC’s Fifth Assessment Report, its Special Report on Global Warming of 1.5oC, or the UNEP Gap Report) have to make an assumption on how to deal with and present this uncertainty. Some explicitly describe the chosen assumptions (such as 50% or 66% probability of meeting targets) or tabulate multiple options, but all are hindered by the uncertainty. The AR5 Synthesis Report quoted a value of 400 GtCO2 (110 GtC) remaining budget from 2011 for a 66% chance to keep warming below 1.5C. It is now clear that this was an underestimate as this would mean a remaining budget of about 20 GtC from 2020. Since AR5 there has been extensive literature on the application of the TCRE concept and its limitations including the choice of temperature metric and baseline period and issues of biases in Earth system models (ESMs). Some studies accounted for climate model biases by relating warming from present day onwards to the remaining carbon budget . Other studies have used the historical record to constrain TCRE and the remaining budget using simple models or attribution techniques. Both these approaches find a substantial increase in the remaining carbon budget for 1.5C compared to the IPCC AR5 SPM approach. Studies that have tried to additionally account for non-CO2 warming. show that CO2-only TCRE budgets are a robust upper limit but taking account of non-CO2 forcing results in lower allowable emissions. Some have proposed techniques for combining emissions rates of short-lived climate pollutants with long-term CO2 cumulative emission budgets. In light of these advances, the IPCC Special Report on Global Warming of 1.5C (SR15) quotes a value of 420 GtCO2 remaining carbon budget for a 66% chance to keep warming below 1.5C – a value very similar to the AR5 value from 5 years earlier.
There is also a lot of focus on how to achieve such carbon budgets and the increasing realization of the need for carbon dioxide removal and research into the feasibility and implications of negative emissions technology.

(3) ABOUT CARBON CAPTURE AND SEQUESTRATION:  The discussion around carbon dioxide removal (CDR) requires more detailed assessment of the magnitude and timing of any requirement for negative emissions technology and hence more precise estimates of remaining carbon budgets. Glen Peters  argues that large uncertainty in budget estimates may be used to justify further political inaction and Sutton (2018) argues for consideration of plausible high impact outcomes in the tails of the likelihood distribution. The same argument applies to TCRE and carbon budgets: we need information on best estimates but also possible extremes however unlikely. The feasibility of achieving 1.5C without net negative emissions depends on the remaining budget being at the high end of current estimates. Knowing the likelihood of the range as well as central estimate is required to inform the debate on requirements for negative emissions. We should break down individual the individual contributions to uncertainty in carbon budgets in terms of historical human induced warming to date, likely range of TCRE, potential additional warming after emissions reached zero, warming from non-CO2 forcing, and carbon emissions from Earth system feedbacks not yet in Earth System Models as in thawing permafrost. Our ability to model the climate-carbon cycle system is imperfect with uncertainties but it plays a dominant role in the remaining carbon budget issue.
The SR15 assumptions of no further warming after CO2 emissions cease is consistent with the multi-model mean. Similarly, CMIP6 and sophisticated ESMs begin to include additional Earth system feedbacks – but the elephant in the room is that past generations of models have not seen a decreased spread in TCRE remaining carbon budget and adding complexity doesn’t help. In terms of climate sensitivity, GCMs continue with the large range of 3°C (from about 1.5 to about 4.5 °C) since the Charney report of 1979. We need to figure out where the large uncertainty in the TCRE remaining carbon budget comes from so that we can control it with observational constraints.

(4) WHAT’S NEW IN THIS PAPER:  Here we perform a new analysis of three generations of Earth System Model results, spanning over a decade, to examine whether or not existing simulations and analyses are well placed to answer the increasing requirements of policy makers on the carbon cycle research community. We present a new analytical framework which allows us to quantify sources of uncertainty in carbon budgets to land or ocean response to CO2 or climate. It is the carbon cycle response to CO2, rather than its response to climate, which dominates the uncertainty in TCRE and hence carbon budgets.

EARTH SYSTEM MODELS  [SOURCE]

Conventional climate models separate the carbon cycle from the model by including only the net carbon feed into the system from the carbon cycle. Earth System Models (ESM) include the carbon cycle in the climate model. The ESM is thought to be a more accurate representation of climate dynamics and is therefore relied upon by climate scientists to unlock the mystery of the remaining carbon budget puzzle.

CRITICAL COMMENTARY

(1)  A TIME SERIES OF VALUES COMPUTED FROM ANOTHER TIME SERIES CAN LOSE DEGREES OF FREEDOM AND THEREBY LOSE STATISTICAL POWER.  Here we demonstrate this principle with a time series of the number of foreign golfers at a golf club in Huahin, Thailand per day for a period of 30 days in the month of October. October is the month when European tourists begin to arrive in large numbers in Thailand. The golf course manager would like to know if the rise in the number of golfers  in October is due to tourist arrivals in Huahin in terms of a correlation between the two time series.

Figure 1: Daily Data: Figure 1 below depicts the data for the number of golfers at the course and the number of net arrivals (net of departures) in Huahin, the city nearby where there are a large number of hotels and ladies of the night. What we see in Figure 1 is that both time series show a rising trend but the correlation analysis in the third frame does not show that the two time series are related such that the rise in golfer numbers can be explained by tourist arrivals in Huahin. The correlation is found to be r=0.265 with a sample size of n=30, and degrees of freedom of df=30-1=29. The standard deviation of the correlation coefficient can be computed as (1-r*r)/sqrt(n) = 0.1697. These figures imply a t-statistic of t=0.265/0.1697 = 1.5633. At degrees of freedom = 30-1=29, that yields a p-value of pv=00644 >0.05 and so we fail to reject the null hypothesis. No evidence of correlation is found in the data.

Figure 1: Daily golfer counts and net arrivals

Figure 2:  5-day averages: In Figure 2 below, time series of 5-day averages is tried because the daily figures may contain too much of a variance to detect the suspected correlation. Here the correlation is much higher at r=0.55 however the sample size has shrunk to n=6 with degrees of freedom df=5. The t-test shows a p-value of p=0.055 and so, at alpha=0.05, we once again fail to reject H0 the null hypothesis that the two time series are not related. The next option we can try is a moving 5-year average that moves through the time series one day at a time. {Note: The charts for 5-day averages are incorrectly labeled as “5yr averages”}

FIGURE 3: 5-DAY MOVING AVERAGES:  Next we try a 5-year moving average that moves through the time series one year at a time. The results appear in the chart below. Here we find a higher correlation of r= 0.615 and with apparently a longer time series of n=26 that will yield higher degrees of freedom and greater statistical power. At n=26, we find that the standard deviation of the correlation coefficient is sd=0.2725 and that yields a t-statistic of t=2.258. if we use the sample size of n=26, we get degrees of freedom df=25 and that yields a p-value of p=0.016, less than alpha=0.05 meaning that the observed correlation is statistically significant.

FIGURE 4: MULTIPLICITY: However, there is a problem with the results shown in Figure 3 above and it has to do with multiplicity. Multiplicity means that some of the data values were used more than once and that created the illusion of a longer time series and more degrees of freedom than we actually have. As shown in the multiplicity chart below, the first five numbers are used once, twice, thrice, four times, and five times respectively; and the last five numbers are used 5 times, 4 times, 3 times, 2 times, and once respectively. All the other numbers are used five times. The average multiplicity of use is 4.333. Therefore, the effective sample size is n/multiplicity = 30/4.333 = 6.923 or approximately 7 with degrees of freedom=6. Using the lower degrees of freedom in the p-value computation of Figure 3 above yields a higher p-value of p=0.03 that is less than our critical value of alpha=0.05 and so in this case the moving average series provided more statistical power and was able to detect a correlation between net arrivals of tourists and the number of golfers on the golf course. However, this is no always the case and it is necessary to check the effective sample size and effective degrees of freedom for statistical tests of significance in time series analysis.

FIGURE 5: CUMULATIVE VALUES: AN EXTREME CASE OF MULTIPLICITY.  In the construction of a time series of the cumulative values of  another time series, say of length N, the multiplicity of use is significantly greater. Here, the Nth number is used once, the (N-1)the number is used twice, the (N-2)th number is used three times and so on until we get to the first number which is used N times. The total number of numbers used in this sequence is the sum of consecutive numbers from 1 to N. For example, in a time series of 30 numbers, the total number of numbers used in the construction of the cumulative value series is the sum of the integers from 1 to 30 computed as M=(30/2)*(1+30) = 15*31=465. On average the numbers in the time series are used 465/30 times or M=15.5 times each and thus the effective sample size is 30/15.5=1.935. As seen in the chart below, in the case of the cumulative values of the golf club dataset, we compute a near perfect correlation between tourist arrivals and golfer count of Corr=0.976. If multiplicity adjustment is not used to correct for effective sample size, we compute the standard deviation of the correlation coefficient as sigma= 0.008658 that yields t-statistic=112, and a p-value close to zero indicating a statistically significant correlation exists between arrivals and golfer counts. However, when the sample size is corrected for multiplicity in the use of the data, the effective sample size is reduced to n=1.9735 leaving degrees of freedom for the determination of correlation as less than unity.

FIGURE 6: STATISTICAL ISSUES CANNOT BE RESOLVED WITH EARTH SYSTEM MODELS: For the regression of temperature against cumulative emissions that yields the TCRE regression coefficient, the degrees of freedom is 1.9735-2, a negative number. The TCRE “near perfect proportionality” between cumulative warming and cumulative emissions is therefore illusory and has no interpretation in terms of phenomena it apparently represents because a statistical test of significance for the TCRE is not possible. Yet another statistical issue in the TCRE is that a time series consisting of cumulative values of another time series does not have a time scale. Statistical flaws in the TCRE create confusing situations in climate science procedures that rely on the TCRE. Climate scientists interpret these anomalies created by statistical flaws as climate science issues further confusing the creation of a statistical flaw in the TCRE mathematics. Climate science procedures for the resolution of statistical defects with more sophisticated climate models and Earth System Models do not lead to resolution of the statistical issues but to further and deeper confusion about the TCRE.

We conclude from the analysis presented above that the near perfect proportionality between temperature and cumulative emissions cited by climate science is a correlation between cumulative values and that therefore this correlation has no interpretation in the real world because it has neither time scale not degrees of freedom. 

FIGURE 7: THE REMAINING CARBON BUDGET PUZZLE: The Jone- Friedlingstein paper presented here is an example the intense research agenda in climate science having to do with the “remaining carbon budget puzzle” that has so engaged climate science research ever since the TCRE “near perfect proportionality” between cumulative emissions and cumulative warming became the primary theoretical link between emissions and warming in that discipline. The remaining carbon budget puzzle is that midway into a TCRE carbon budget time span, the remaining carbon budget computed by subtraction does not equal the remaining carbon budget computed by the TCRE procedure that was used to construct the carbon budget for the full span of the TCRE carbon budget. This apparent anomaly is a statistics issue and not a climate science issue.

The TCRE correlation does not derive from the responsiveness of warming to emissions but from a fortuitous sign pattern in which annual emissions are always positive and, during a warming trend, annual warming is mostly positive. Since emissions are always positive, the TCRE regression coefficient in this proportionality is determined by the fraction of annual warming values that are positive. Larger fractions of positive warming values yield higher values of the TCRE regression coefficient and lower fractions of positive warming values yield lower lower values of the TCRE. It is the regression coefficient that determines the value of the carbon budget. Because of the random nature of the annual warming values, it is highly unlikely that the fraction of annual warming values that are positive in the full span of the carbon budget period will be the same as the fraction of annual warming values that are positive in the two halves of the full span. This is the source of the remaining carbon budget problem because this is where the remaining carbon budget enigma comes from. In general the TCRE regression coefficient for the full span of the carbon budget period, that for the first half of the carbon budget period, and that for the second half of the carbon budget period will be different and that is why the remaining carbon budget computed by subtraction cannot be expected to equal the remaining carbon budget computed with a new TCRE procedure for that period.

Since emissions are always positive, the critical factor is the fraction of annual warming values that are positive. In the chart above, the upper left frame shows random annual warming values with no bias for positive values. The right frame visually displays the corresponding correlation of cumulative warming with cumulative emissions. It is evident in this graphic that without a bias for positive values in annual warming no TCRE correlation can be found. In the lower frame, a slight bias is inserted for positive values of annual warming and the corresponding right frame shows the strong TCRE correlation that was created by the bias for positive values of annual warming. The essence of the remaining budget puzzle can be understood in terms of this demonstration because there is no guarantee that the fraction of annual warming values that are positive in the full span of the TCRE carbon budget time period, the fraction that is positive in the first half of the time period, and the fraction that is positive in the second half of the time period cannot be expected to be the same.

CONCLUSION: Climate science has misinterpreted these statistical anomalies as a climate science issue that needs to be resolved with climate models of greater complexity. Their struggle with the remaining carbon budget puzzle seen in the Jone-Friedlingstein paper presented above, demonstrates a failure of climate science to address statistical issues of the TCRE in terms of statistics. This failure has led them down a complex and confusing path of trying to find a climate science explanation of the remaining carbon budget anomaly. The research paper presented above serves as an example of this kind of climate research. The real solution to the remaining carbon budget puzzle is to understand the statistical flaws in the TCRE correlation and to stop using it.  [LINK] [LINK] . In the second link we show that the TCRE procedure that shows that fossil fuel emissions cause warming also shows that UFOs cause warming [LINK] .