**4.1 Remaining CO2 budget**

*y*¼ *A x*ð Þ � *u*

The coefficient *A* is determined by equating the slope *m* with *dy*

Then *B* is deduced by plugging in *x*0*, y*<sup>0</sup> and *A* in Eq. (2).

**3. Time model for global warming**

*Global Warming and Climate Change*

applicable from the year 2000 (i.e., *t* = 0)

**Figure 1.**

**66**

*A*¼ *m=*2ð Þ *x*<sup>0</sup> � *u*

*B*¼ *y*<sup>0</sup> � *m x*ð Þ <sup>0</sup> � *u =*2

Indeed, the form of the equation is due to the symmetry about the line *x* ¼ *u:*

As it can be seen from (1), an estimation of the atmospheric CO2 concentration ratio through time will give a time model for the global warming. This estimation can be done by linear regression of the annual average ratio for CO2, using the NASA dataset [20]. This leads to the following no-climate-mitigation model *r*0*,*

*r*0ð Þ*t* ≈*a*0*t* þ *b*<sup>0</sup>

model *w*<sup>0</sup> for global warming, applicable from the year 2000 (i.e., *t* = 0)

*Estimated global warming since 2000 and expected trend in case of no global climate mitigation.*

*<sup>w</sup>***0**ð Þ¼ *<sup>t</sup>* **<sup>Δ</sup>***T r*ð Þ **<sup>0</sup>**ð Þ*<sup>t</sup>* <sup>≈</sup> *<sup>β</sup> <sup>t</sup>* <sup>þ</sup> *<sup>λ</sup>***<sup>0</sup>**

Such a linear regression was found to be statistically highly significant (*p*< 10�21) and extremely strong (*r*<sup>2</sup> ≈0*:*99). By composing the energy-balancebased model Δ*T* (given in (1)) with *r*0, one gets the following climate-policy-free

*t* þ *μ***<sup>0</sup>**

The estimations based on this model, for the period 2005–2015, appear to be very close to the annual averages calculated using the NASA data [3] for the same period. According to (4), the rise in global average temperature will be estimated at

<sup>2</sup> <sup>þ</sup> *<sup>B</sup>*

(2)

(4)

¼ 2*A x*ð Þ <sup>0</sup> � *u :*

*dx x*¼*x*<sup>0</sup>

*<sup>a</sup>*<sup>0</sup> <sup>≈</sup>0*:*0076*, b*<sup>0</sup> <sup>≈</sup>1*:*<sup>3176</sup> (3)

*λ***<sup>0</sup>** ¼ ð Þ *b***<sup>0</sup>** � 1 *=a***0***, μ***<sup>0</sup>** ¼ ð Þ *b***<sup>0</sup>** � *k =a***<sup>0</sup>** *a***0***, b***<sup>0</sup>** as in 3ð Þ*, β* as in 1ð Þ

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

$$E\_0(t) \approx a\_0 t + \beta\_0. \quad a\_0 \approx 0.91, \beta\_0 \approx 24.47\tag{6}$$

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 estimated as follows:

$$\mathbf{R}(\mathbf{t}) \approx (\mathbf{h} - \mathbf{t})(\mathbf{a}\_0(\mathbf{h} + \mathbf{t}) + \mathbf{2}\boldsymbol{\theta}\_0)/2, \quad \mathbf{h} \text{ as in (5), } \mathbf{a}\_0, \boldsymbol{\theta}\_0 \text{ as in (6)}\tag{7}$$

Indeed, with no climate mitigation, the CO2 emissions between times *t* and *h* would reach a total amount of *R t*ðÞ¼ <sup>Ð</sup> *<sup>h</sup> <sup>t</sup> E*0ð Þ *x dx,* which is nothing else but the area of a trapeze with bases *E*0ð Þ*t* and *E*0ð Þ *h* and height *h* � *t,* and this gives (7).

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