Page 53 - 2417

Page 6 of 22 François et al. Carbon Footprints 2026, 5, 22

The climate warming impulse response function R is expressed as follows:
θ

@ B − C @ 5 − C Õ @ : − C (6)
' \ (C) = 4 3 B + 4 3 5 = 4 3 :
3 B 3 5 3 :
:
with q , q, (K.m .W ) representing respectively a slow and a fast contribution to the temperature increase in
-1
2
s
f
respectively d and d (year) the surface and deep layers of the atmosphere .
[28]
s
f
The GTP indicator, in Equation 7, is a relative metric based on the AGTP expressed relatively to the CO ,
2
chosen as a reference substance.
)% 9 ())
)% 9 ()) = (7)
())
)% $ 2
with GTP(T) the Global Temperature Potential indicator of GHG j over a time T (kg CO -eq) .
[24]
j
2
In addition to the degradation or absorption rates of GHGs and their climate impulse responses, some
feedback mechanisms contribute to climate change metrics. Firstly, through oxidation reactions, a part of
methane emissions degrades into carbon dioxide. These indirect emissions contribute to climate change and
are estimated through the cumulative product of methane degradation and AGWP of CO (see Equations in
2
Supplementary Part 2). Secondly, any GHG emissions that contribute to global warming can modify the
balance of carbon flows between Earth’s reservoirs. Then for each GHG other than CO , metrics should
2
integrate a carbon cycle response.

Dynamic metrics for climate change
To distinguish dynamic factors from static ones, static quantities are indicated in upper case in the previous
section, and dynamic quantities in lower case in the following section.

Unified mathematical framework between static and dynamic approaches
In this section, several time entities are defined. The times corresponding to the first and last emission of the
life cycle of the considered product are defined as t 0 and t respectively. Then, the duration of the LCD (life
max
cycle duration) can be defined as:

(8)
! = C <0G − C 0

with LCD the Life Cycle Duration (yrs), t 0 (yrs) the time of the first emission of the life cycle, and t (yrs) the
max
last emission of the life cycle.

Let’s also define t as the time at which the integration calculation stops. Thus, independently from the static
x
or dynamic method, the total integration time can be expressed as:

(9)
) = C G − C 0

For static impact assessment, as represented in Figure 2A, the integration time is equal to the Time Horizon
of the Impact:

(10)
) BC0C82 = )