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with the presence of newly incorporated organic matter (e.g., decomposing plant litter,
crop residue, and livestock feces and urine). Furthermore, I do not attempt to review the
many techniques for characterizing heterogeneous systems (e.g., geospatial statistics) as
there are several excellent summaries already available (e.g., Robertson, 1987; Goovaerts,
1998; Ettema and Wardle, 2002; Webster and Oliver, 2007).
3.2 Theoretical underpinnings
A simple mathematical property of nonlinear functions known as Jensen's inequality pro-
vides the theoretical underpinnings for how heterogeneity affects soil microbial process
rates and biological systems in general (Jensen, 1906; Ruel and Ayres, 1999). This function
was defined over a century ago by Danish mathematician Johan Jensen. Jensen's inequal-
ity describes the effect of averaging independent variables (i.e., predictor variables) prior
to applying nonlinear functions to explain patterns and contrasts that to applying the
function to each individual datum and then summarizing the resulting functions. Jensen's
inequality is important to soil process rates for two reasons: (1) Most biochemical pro-
cesses are nonlinear functions of fundamental abiotic controls on soil biology, such as tem-
perature, moisture availability, substrate abundance, and pH (e.g., Webster et al., 2009). (2)
When spatial or temporal heterogeneity of these controls is measureable (the norm), then
aggregating across spatial or temporal scales can introduce bias in process rate estimations
(Robertson, 1987; Ruel and Ayres, 1999). Because this inequality is inconsistently applied
when interpolating process rates from point measurements to greater scales in time or
space, these biases can systematically or randomly alter ecosystem-level rate estimation.
As a mathematical expression, Jensen's inequality state
s
that for any nonlinear func-
tion,
f
(
x
) with variation around the mean predictor variable
x
that
f
( )
≠
f
( )
(3.1)
This gives rise to the memorable phasing of the i
nequ
ality that the function of the mean
(
f
( )
) does not equal the mean of the functions (
f
( )
). To illustrate Jensen's inequality, I
use the nonlinear influence of soil temperature on relative metabolic activity of soil organ-
isms (
Figure 3.1
). In particular, I apply the commonly used van't Hoff function (Webster
et al., 2009):
Relative Activity =
Q
10
*
e
T
/10
(3.2)
where
Q
10
is the fitted constant by which respiration increases when the temperature
rises by 10°C, and
T
is the increase in temperature above 0°C. A
Q
10
of 2.0 is illustrated in
Figure 3.1.
In this simple example, the heterogeneous system is represented by a soil sys-
tem with two discrete temperatures (dashed lines at 20°C and 30°C). The activity levels at
the two temperatures are then estimated at 15% and 40% for the 20°C and 30°C systems,
respectively. The mean activity level of these two systems (i.e., the heterogeneous system)
is about 27.5% (represented by the short horizontal dashed line). In contrast, when the
function is applied to the mean temperature of these two components (25°C), the estimated
activity level is only 24.4%. The approximately 13% difference between the function of the
mean and the mean of the functions constitutes an underestimation bias.
My own first encounter with Jensen's inequality was not in a statistics course or reading
of it in the primary literature, but an accidental one that is likely common to many scientists.
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