Statistical models and analysis of carbon dioxide in the atmosphere

Yong Xu, Chris P.Tsokos

Department of Mathematics and Statistics, University of South Florida

Tampa, FL 33620 USA.

This research is connected with important ecological problems. Global warming is a function of two main contributable entities, atmospheric temperature and carbon dioxide, CO₂. The object of the present study is to develop a statistical model taking into consideration all the attributable variables that have been identified and their corresponding response of the amount of carbon dioxide (CO₂) in the atmosphere in the continental United States. The development of the statistical model that includes interactions and higher order entities, in addition to individual contributions to CO₂ in the atmosphere, are included in the present study. The proposed model has been statistically evaluated and produces accurate predictions for a given set of the attributable variables. Furthermore, we rank the attributable variables with respect to their significant contribution to CO₂ in the atmosphere.

For introduction we note:

Wikipedia defines Global Warming as the increase in the average temperature of the Earth's near-surface air and oceans since the mid-20^th century. The projected statistical model is to predict CO₂ in the atmosphere taking into consideration eight attributable variables to the subject matter. The eight attributable variables are namely, CO₂ emission (E), deforestation and destruction of biomass and soil carbon (D), terrestrial plant respiration (R), respiration from soils and decomposers (S), the flux from oceans to atmosphere (O), terrestrial photosynthesis (P), the flux from atmosphere to oceans (I), the burial of organic carbon and limestone carbon in sediments and soils (B).

We need to mention here that some of the attributable variables are the function of several other variables within themselves. For example, CO₂ emission, E, is a function of six attributable variables namely, Gas fuels (Ga), Solid fuels (So), Liquid fuels (Li), Gas Flares (Fl), Cement (Ce) and Bunker (Bu).

The proposed model that we are developing takes into consideration individual contributions and interactions along with higher order contributions if applicable. In developing the statistical model, the response variable is CO₂ in the atmosphere and is given in unit parts per million (PPM). In the present analysis, we used real yearly data that have been collected from 1959 to 2004 for the continental United States. The air samples were collected at Mauna Loa Observatory, Hawaii. The CO₂ emission data were obtained from Carbon Dioxide Information Analysis Center (CDIAC). The CDIAC is the primary climate-change data and information analysis center in the U.S. Department of Energy (DOE), located at Oak Ridge National Laboratory (ORNL) and includes the World Data Center for Atmospheric Trace Gases. All emission estimates are expressed in thousand metric tons of carbon (MT). Carbon emissions are calculated by the fuels consumed times the heat coefficient times the carbon coefficient times the combustion efficiency. The product of the fuels consumed times heat coefficient is in the unit of trillion Btu. The carbon coefficients are given by the Environmental Protection Agency (EPA) reports.  It is the amount of carbon that is emitted per unit of heat realized from combustion. Petroleum data were obtained from DOE report, and are published in the Monthly Energy Review.

The proposed statistical model is useful in predicting the CO₂ in the atmosphere given the information of attributable variables. It has been statistically evaluated using R square, R square adjusted, PRESS statistic and residual analysis. Finally, its usefulness has been illustrated by utilizing different combinations of various attributable variables. To our knowledge, no such model has been developed under the proposed analytical structure. In addition we rank the attributable variables according to their CO₂ contributions in the atmosphere.

Some historical survey.

Thomas J.Goreau stated the eight attributable variables for CO₂ in the atmosphere. The parametric analysis for CO₂ has been studied extensively by R.Wooten and C.P.Tsokos. They have found that the CO₂ data follow the three parameter Weibull probability distribution contrary to the fact that some scientists believed that CO₂ in the atmosphere follows Gaussian probability distribution. C.P.Tsokos and Y.Xu have developed statistical analysis in CO2 emission modelling by using differential equations that characterize the rate of their behavior as a function of time. Additional research publications of interest of GLOBAL WARMING was published early; also as some classical historical research papers. Some other important and recent references for the readers who will have an interest in GLOBAL WARMING are in early works. In our research we proceed to develop a statistical model taking into consideration the eight attributable variables as presented previously.

In regard to usefulness of the proposed model:

we can conclude from our extensive statistical analysis that there are only three significant attributable variables to CO₂ in the atmosphere namely, Gas fuels, Cement and Deforestation. Furthermore, we also tested 36 possible interactions of the attributable variables and we found only one interaction to significantly contribute to CO₂ in the atmosphere, namely, Gas fuels and Cement. Thus one may obtain a good estimate of CO₂ in the atmosphere by knowing the measurement of Gas fuels, Cement, Deforestation and Interaction of Gas and Cement.

One can utilize the above model equation 3.3 to perform surface response analysis to identify the values of the contributable variables that will minimize CO₂ in the atmosphere.

As conclusions for discussion we note:

In the present study, we have performed parametric analysis for CO₂ in the atmosphere. The initial measurement of CO₂ in the atmosphere was collected at Mauna Loa Observatory, Hawaii (C.D.Keeling, T.P.Whorf, 2005). Those data do not follow normal probability distribution. Thus, we transform the response of the data by using Box-Cox transformation that resulted in CO₂ being normal. We proceed to develop a "nonlinear" statistical model (nonlinear in terms of the higher power of the response variable). Through the process of developing the statistical model, we have found that only three variables, namely, Gas fuels, Deforestation and Cement and one interaction Gas and Cement significantly contribute to CO₂ in the atmosphere. The proposed statistical model was evaluated using the R-square, R-square adjusted and PRESS statistics. All of them support the high quality of the developed statistical model.

This model can be used to obtain a good estimate of CO₂ in the atmosphere knowing only the three significantly attributable variables mentioned above.

 

Acknowledgements.The authors wish to acknowledge the assistance and suggestions of T.J.Blasing from Oak Ridge National Laboratory during the progress of the present study.