A nonlinear atmospheric response function
approach to the attribution of CO2
concentrations to emissions
Brian C. O’Neill
International Institute for Applied Systems Analysis &
Watson Institute for International Studies, Brown University
3rd Expert Meeting on the Brazilian Proposal, 8-9 September 2003, Berlin.
Motivation
Attribution calculations are credible if
method is transparent and theoretically
grounded (what was done, what does it
mean)
Common theoretical framework helps
understand differences across studies
– Many approaches being used, differences among
them unclear (common global turnover time;
parallel models; scaled impulse response
functions; time slice; etc.)
Conclusions
Response functions (and reservoir theory) can
provide a theoretical framework for attribution
calculations, even for nonlinear systems
No unique solution to nonlinear attribution problem,
no single “right answer”.
– Subjective choices involved, making transparency
especially important
Application to a nonlinear carbon cycle model:
– Define and calculate alternative response functions for use
in attribution
– Shows quantitative differences between principal methods
likely to be small (for historical period) and why
– Can use framework to define other existing approaches
Rs(t) = Ms(t)/Ma(t)
Rs
Ms
Ma
t
τ
Ψ
Attribution to source s
Reservoir mass attributed to
source s
Total anthropogenic reservoir
mass
calendar time
age (time since emission)
reservoir mass age distribution
Reservoir mass, Ma(t)
Reservoir theory
Ψ(τ, t)
s3
s2
s1
Age, τ (time since emission)
M s (t ) =
t −t0
Ψs (t '
, t )dt '
0
Nonlinear response functions
t
τ
Ψ
Is
calendar time
age (time since emission)
reservoir mass age distribution
Emissions rate from source s
H
Nonlinear response function
Ψs (τ , t ) = I s (t − τ ) H (τ , t − τ )
For a single well-mixed reservoir (O’Neill et al., 1994):
∂H (τ ,t )
∂τ
= −kˆ(t ) H (τ , t )
H (0, t ) = 1
For multiple well-mixed reservoirs (e.g., the carbon cycle):
a system of differential equations for Hi(τ,t), i.e. the fraction of an emission
at time t remaining in reservoir i, τ years later
Solving for H(τ,t) for the carbon cycle
All fluxes written in form of k(t) M(t)
Alternative attribution methods are defined by alternative
ways of deriving k(t)
Provides an intuitive way to understand attribution
methods, because k(t) describes the relationship between
the flux (F(t)) and the reservoir mass (M(t))
Expressions for k(t) are substituted for kˆ(t ) and the system
of equations is solved for H
tra
c
Tracer method:
er
Alternative attribution methods
• flux proportional to contribution
to total reservoir mass
• gives fluxes of tracer that would
result if each emission were of
an ideal tracer
• does not have adding up
property for excess reservoir
mass – some excess flux is
unattributed
kt
keq
• flux proportional to contribution
to excess reservoir mass
• has adding up property for excess
reservoir mass – all excess flux is
attributed
pro
po
rtio
n
er
tra
c
Proportional method:
al
Alternative attribution methods
ket
kt
keq
Alternative attribution methods
al
tra
cer
ma
rgi
nal
pro
po
rtio
n
Marginal method:
ket
kt
• applies marginal removal rate to
all partial masses
• does not have adding up property
keq
km
Alternative attribution methods
tra
cer
ma
rg
dif inal
fer
pro entia
l
po
rtio
nal
Time-ordered differential
method:
ket
kt
• each partial mass removed at
different rate according to
flux-mass relationship
• e.g., emission at time=1 removed
as if no further emissions occur;
each subsequent response function
calculated in turn
• has adding up property
keq
km
Relations among methods
All methods identical in linear system with
proportional fluxes
Tracer and proportional methods are
identical when equilibrium reservoir
content is zero
All methods that use a single removal rate
(tracer, proportional, marginal) can be
made equivalent by scaling
Principal carbon cycle nonlinearities
Surface ocean carbon chemistry: based
on contribution to surface ocean carbon
content
CO2 fertilization flux: based on
contribution to atmospheric CO2,
photosynthetic biomass, or both?
Temperature feedback (to oceanatmosphere flux and to terrestrial
biosphere fluxes): based on contribution
to temperature change, to reservoir
carbon content, or both?
Response functions, 1765 and 1900,
calculated using ISAM (Jain et al., 1995)
1
Fraction
0.8
0.6
0.4
IRF
0.2
tracer
1800
marginal;
proportional
1850
1900
1950
Year
• Marginal and proportional methods nearly identical
• Impulse response functions very different from all other methods
2000
Further work
Express approaches taken by other
studies in response function form (i.e.,
what is the implicit k(t)?)
Carry out attribution calculation based on
response functions
Run alternative future scenarios
(nonlinearities likely to become more
important in the future)
Sensitivity to choices about CO2
fertilization and temperature feedback
related fluxes
Impulse response functions (IRFs)
IRFs do not have
adding up property
Source: O’Neill, 1996.