The CO_{2} concentration is approximated by a sum of exponentially decaying functions, one for each fraction of the additional concentrations, which should reflect the time scales of different sinks. The coefficients are based on the pulse response of the additional concentration of CO_{2} taken from the Bern model (Siegenthaler and Joos, 1992).
f

r 
= 
concentration 
C_{CO2} 
= 
constant (approximately 0.47 ppmv/GtC, but use this parameter to fine tune your results) 
E_{CO2} 
= 
emissions of CO_{2} 
t_{CO2,S} 
= 
atmospheric exponential decay time of the s^{th} fraction
of the additional concentration (171.0, 18.0 and 2.57 years) 
f_{CO2,0} 
= 
first fraction (0.152) 
f_{CO2,S} 
= 
respective fractions (0.253, 0.279 and 0.316) 
Check how closely CO_{2} concentrations modelled using the CDIAC emissions approach observed CO_{2} concentrations. If the agreement is poor, use C_{CO2} to tune the results.
The values for f_{CO2,S} and t_{ CO2,S} given above are the Bern TAR standard and should be used as the default. Alternative values for the CO2 impulse response functions, for exploring the uncertainties, are given in the table below.
Reference:
The carbon cycle can be approximated as given in Den Elzen, M. et al., The Brazilian proposal and other options for international burden sharing: an evaluation of methodological and policy aspects using the FAIR model, RIVM Report No. 728001011, June 1999. 
Impulse response functions for CO2 have been estimated by forcing the Bern Carbon Cycle Models as used in the IPCCSAR or as used in IPCCTAR by releasing instantantaneously 40 GtC into the preindustrial atmosphere.
The impulse response function (IRF) is then fitted by a series of exponentials:
IRF= a(0) + Sum a(i) * exp(t/tau(i))
The following coefficients were obtained:
c 
Bern SAR 
Bern TAR 
standard 
low 
high 
standard 





a(0) 
0.1369 
0.1253 
0.1504 
0.152 
a(1) 
0.1298 
0.0989 
0.1787 
0.253 
a(2) 
0.1938 
0.1839 
0.1798 
0.279 
a(3) 
0.2502 
0.2674 
0.2201 
0.316 
a(4) 
0.2086 
0.2380 
0.1725 
 
a(5) 
0.0807 
0.0865 
0.0975 
 





tau(1) 
371.6 
407.2 
330.8 
171.0 
tau(2) 
55.70 
50.86 
67.03 
18.0 
tau(3) 
17.01 
15.19 
21.72 
2.57 
tau(4) 
4.16 
3.73 
5.61 
 
tau(5) 
1.33 
1.42 
1.51 
 
This figure is also available as a PDF.
 All IRFs are obtained by running the Bern model (HILDA and 4box biosphere) as used in SAR or the Bern CC model (HILDA and LPJDGVM) as used in the TAR.
 A preindustrial background (CO_{2} around 280 ppm, zero emissions) was used and a pulse of 40 GtC was released instantanteously into the model atmosphere. Control simulations were also performed.
 For the Bern CC/TAR version, the IRF was calculated for the standard model version.
 Three IRFs (standard, low, high) were calculated with the Bern SAR model.
 The CO_{2}fertilization parameter beta was varied within plausible ranges to obtain a "standard," "low", and "high case". This approach builds on SAR simulation where beta was varied in the same manner.
standard case: 
beta=0.287 
low case: 
beta=0.465 
high case: 
beta=0.110 

High value corrected May 16th
 The IRFs were fitted by sum of exponentials. The coefficients of these fits and the IRFs obtained by using the exponential approximation are provided.
