# gcb13120-sup-0001-Supinfo

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Doc S1. Detailed mathematical explanations to our decomposition analysis
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This Supplementary Information provides the concrete mathematical method developed for
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this study to decompose aggregate emissions. We exemplify the method using the KPI-
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identity on emissions from crop production (KPI-C).
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KPI-C:
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 GHGLUC GHGsoil  GHGc;in Ec;in   Ec;out
DM c;out
 
GHGcrop  

 


 areacrop; food
 Ec;out
Ec;out
Ec;out   DM c;out areacrop;all
 Ec;in

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For each identity emissions are expressed as a function, f(x1, …, xm), of several variables, x1,
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…, xm, say, each of which may vary with time (t). The total change in emission over a certain
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period of time may then be written as a sum of m parts, where each part is the contribution to
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the change from one of the variables. The change per time unit is the derivative of the
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function f with respect to t (Wachsmann, et al., 2009). According to Leibnitz' rule this change
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may be decomposed into
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df
f
f

x1 ' t   ... 
xm ' t  ,
dt x1
xm
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where f’(t) is the derivative of f(t) and likewise for x1’(t), …, xm’(t).
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Thus, the contribution from the first factor, x1, is the change in emission per unit change in x1
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multiplied by the change in x1 per time unit. Summing up over a time period from time t0 to
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t1, say, the total contribution from x1 to the change in emission over this period may be
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written as the integral:
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
t1
t0
f dx1
dt
x1 dt
1
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In the KPI-C the emissions from crop production are a sum of three terms, each of which is a
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product of a number of factors. The second term, for example, may be written y=x1&middot;x2&middot;x3&middot;x4,
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where x1=GHGsoil/Ec;out, x2= Ec;out/DMc;out, x3=DMc;out/areacrop;all and x4= areacrop;food. The
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contribution from x1 to the change from t0 to t1 may be written as (Wachsmann, et al., 2009):
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y
t1
     ,
ln y  ln y 
 yt0  ln x1t1  ln x1t0
t1
t0
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which is based on an interpolation of x1t by an exponential function from time t0 to time t1.
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Exponential interpolation is possible only for variables with positive values, but the first term
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in KPI-C has a factor, x1=GHGLUC/Ec;out, which may take negative values. For such a variable
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we use linear interpolation and obtain its contribution, Cx1, to the change as
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C x1 
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when the other three variables are positive, and we have used the notation r2=ln(x2t1)-ln(x2t0)
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for the logarithmic change in x2 and similarly for x3 and x4. In this case the contribution from
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x2 is given by:
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Cx2 
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with similar expressions for Cx3 and Cx4.
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Finally, all contributions from each specific variable are added to obtain the contribution from
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that variable to the total change.
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References
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Wachsmann, U., Wood, R., Lenzen, M. &amp; Schaeffer, R. (2009): Structural decomposition of
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energy use in Brazil from 1970 to 1996. Applied Energy. Vol. 86:4, pp. 578-587.
x1
t1
y
t1

 x0 t0  x 2 t1  x3t1  x 4 t1  x 2 t  x3t0  x 4 t0
r 2  r 3  r 4

 yt0  C x1  r 2
(r 2  r 3  r 4)
0
,
,
2
```