**2.3 Global CO2 emissions data (2000–2013)**

It is necessary to determine the trend of the global CO2 emissions over the past two decades prior to modelling with time the desired effect of any projected mitigation in line with the UN climate goal. This trend will be estimated by linear regression of the annual gas emissions recorded by Carbon Dioxide Information Analysis Center (CDIAC) up to 2013 [16].

### **2.4 Quadratic interpolation**

Classically, a quadratic interpolation consists of determining a quadratic function using the values that it takes on at exactly three particular values of its variable. The following result provides an original quadratic interpolation using also three given data on the parabola representing the function: its symmetry axis, one of its points (other than the vertex), and the slope of the tangent line at that point. This technique will be used to add a smooth transition to an exponential model for CO2 emissions.

*If a parabola is symmetric about the line: x* ¼ *u, passes through a point x*0*, y*<sup>0</sup> *, with x*<sup>0</sup> 6¼ *u, and is tangent at this point to the line of slope m, then an equation of this parabola is*:

$$\begin{aligned} y &= A(\mathbf{x} - \boldsymbol{u})^2 + B \\ A &= m/2(\mathbf{x}\_0 - \boldsymbol{u}) \\ B &= y\_0 - m(\mathbf{x}\_0 - \boldsymbol{u})/2 \end{aligned} \tag{2}$$

1.19°C by 2020. Then, under the assumption of no climate mitigation, it will reach 1.5°C by 2041 and 2°C by 2084. More generally, any climate target L Lð Þ <*β* will

*Mathematical Model for CO2 Emissions Reduction to Slow and Reverse Global Warming*

which makes the year *d* ¼ 1999 þ ½ � *h* the deadline for the implementation of the

For the rest of the paper, L denotes any UN climate target 1ð Þ *:*5≤L< 2 *,* and *h* is

The consistency of future CO2 emissions with a prescribed UN target and their

By definition, the CO2 budget is the total amount of cumulative anthropogenic CO2 emitted in the atmosphere since the industrial revolution up to the time *h* when the UN climate target will be hit. To estimate the remaining budget at any time, future emissions need to be modelled explicitly with time in the scenario of no climate policy, which can be done by linear regression of the annual gas emissions since 2000 using CDIAC database [16]. This leads to the following no-climate-

*E*0ð Þ*t* ≈*α*0*t* þ *β*0*, α*<sup>0</sup> ≈0*:*91*, β*<sup>0</sup> ≈24*:*47 (6)

*<sup>t</sup> E*0ð Þ *x dx,* which is nothing else but the area

policy model *E*<sup>0</sup> (in GtCO2), applicable from the year 2000 (i.e., *t* = 0):

Such a linear regression was found to be statistically highly significant (*p*<sup>&</sup>lt; <sup>10</sup>�<sup>11</sup><sup>Þ</sup> and extremely strong (*r*<sup>2</sup> <sup>≈</sup> <sup>0</sup>*:*98). As a consequence of (6), the

remaining CO2 budget *R*(*t*), from time *t* ð Þ 0 ≤*t*<*h ,* consistent with the L-target, is

Indeed, with no climate mitigation, the CO2 emissions between times *t* and *h*

In particular, the remaining CO2 budgets from 2020, to meet the targets 1.5 and 1.8°C, will be estimated at 1155 and 2929 (GtCO2) respectively, and these represent

One obvious way to regularly reduce the CO2 emissions would suggest a constant rate of reduction, which will definitely put an end to them at time *t* ¼ *z*

of a trapeze with bases *E*0ð Þ*t* and *E*0ð Þ *h* and height *h* � *t,* and this gives (7).

about 63 and 81% of the corresponding remaining budgets from 2000.

**4.2 CO2 emissions pathways consistent with the UN climate targets**

*R t*ð Þ≈ ð Þ *h* � *t α***0**ð Þþ *h* þ *t* **2***β***<sup>0</sup>** ð Þ*=***2***, h* as in 5ð Þ*, α***0***, β***<sup>0</sup>** as in 6ð Þ (7)

**4. Smooth pathways for CO2 emissions to achieve the UN goal on**

rapid reduction, as urged by the UN, are crucial in the elaboration of suitable pathways for the emissions. Prior to the modelling, however, two parameters need to be determined; an estimation of their level in the beginning of the mitigation and their expected cumulative amount during the mitigation (remaining CO2 budget).

*<sup>β</sup>* � L ð Þ *<sup>λ</sup>*0*, <sup>μ</sup>*<sup>0</sup> as in 4ð Þ (5)

be hit by the year 2000 þ ½ � *h* where *h* is the hitting time:

its hitting time as formulated in (5).

*DOI: http://dx.doi.org/10.5772/intechopen.88961*

**climate change**

**4.1 Remaining CO2 budget**

estimated as follows:

**67**

would reach a total amount of *R t*ðÞ¼ <sup>Ð</sup> *<sup>h</sup>*

*<sup>h</sup>*<sup>≈</sup> *<sup>μ</sup>*0L � *<sup>λ</sup>*0*<sup>β</sup>*

*2015 Paris Agreement*. See **Figure 1** for graphical estimation of *h*.

Indeed, the form of the equation is due to the symmetry about the line *x* ¼ *u:* The coefficient *A* is determined by equating the slope *m* with *dy dx x*¼*x*<sup>0</sup> ¼ 2*A x*ð Þ <sup>0</sup> � *u :* Then *B* is deduced by plugging in *x*0*, y*<sup>0</sup> and *A* in Eq. (2).
