Introduction
Soil respiration, which includes both root and microbial respiration,
represents the largest outward flux of CO2 from terrestrial ecosystems,
with a magnitude far above that of anthropogenic emissions .
Small changes in the soil CO2 flux could therefore have a significant
impact on the carbon balance and global atmospheric CO2 concentrations.
In predictions of atmospheric CO2 over the 21st century,
uncertainties surrounding the response of land flux to climate change are
second only to uncertainties surrounding future anthropogenic emissions
. In order to accurately predict future atmospheric CO2
concentrations, it is crucial to gain a better understanding of how land
systems will respond to changing temperature and moisture regimes.
Soil CO2 production originates from plant root respiration and microbial
decomposition of organic matter. The temperature sensitivity of soil
respiration describes how the flux of CO2 from soils will respond to a
change in temperature. Normally soil microbial and plant root processes are
treated together because they are not readily distinguished from one another.
Temperature sensitivity is often quantified by the parameter Q10 (temperature sensitivity), which
describes the factor increase in soil respiration with a temperature increase
of 10 ∘C. This Q10 parameter is used in global climate models to
quantify soil feedbacks to climate change. It has been found that Q10
values are influenced by a range of environmental factors including soil
temperature , soil volumetric water content
, and soil organic matter content
. As these factors exhibit high spatial heterogeneity
across ecosystems as well as within a given ecosystem, it has long been
expected that Q10 will also exhibit high spatial variability. Despite
this, most existing models continue to use a globally constant Q10
value. This may reduce or enhance predicted release of CO2 from soils,
leading to large over- or under-estimates of the contribution of soil
respiration to terrestrial CO2 flux in the face of climate change. There
has been considerable debate over the usage and magnitude of Q10
, with different studies producing widely variant
values. While most studies agree that CO2-flux feedback will be
positive, there is no consensus on how best to estimate the magnitude of
Q10.
Historically, Q10 values have been determined through regression
analysis of soil temperature and CO2 surface-flux measurements. A known
source of error in this approach originates in the physics of soil heat and
gas transport, which might separate a change in surface soil temperature
(normally a 5 or 10 cm temperature is used for deriving Q10) from the
resultant change in CO2 flux measured at the surface. The lags depend
most heavily on soil heat transport , because changes in
surface temperature are shifted and dampened significantly as a function of
depth, with each successive soil layer experiencing a reduced temperature
change in amplitude. Gas diffusion also plays an important role, and even if
soil microbes and roots produced CO2 instantaneously upon receipt of
thermal energy at the characteristic production depths, gases still take time
to diffuse upward. Soil properties including heat and gas diffusion, and the
e-folding depth of CO2 production (Zp), all contribute to these lags
(Fig. 1). demonstrated that such lags can lead to severe
misinterpretation of data when attempting to extract true Q10 values
through regression of surface flux and a temperature measurement at a single
depth.
Thermal and gas diffusion lags through a soil profile.
These thermal and gas diffusion processes, and the resulting lags, can be
captured in a simple one-dimensional (1-D) physical heat and gas transport soil model
. Though not done to date for the soil respiration
system, it is possible to use such a model in inverse fashion for estimating
the value of parameters like Q10 and Zp by looping the forward model
iteratively through possible parameter combinations, with observed
measurements as a constraint. Normally, an objective function is used for
helping decide which parameter set best minimizes the difference between
modelled and measured data. This method has been identified as a promising
tool for determining unknown soil parameters , with an increasing
availability of high-frequency data sets allowing for rigorous constraints on
known model parameters.
This study seeks to develop a reliable inversion framework for determining
the Q10 and Zp of different sites given continuous soil measurements. It
also seeks to provide guidance for researchers who would like to build field
observational sites suited for inversion analysis. Working exclusively with
synthetic soil data that mimics the form of collected field data and of which
all parameters are known, we first undertake sensitivity tests to determine
optimal sensor placing in the field, and decide whether soil CO2 surface
flux, and/or profile measurements, are more suited for anchoring inversion
approaches with the necessary field data for parameter constraint. Using the
best sensor combination, we are able to evaluate the accuracy of the
inversion approach in returning the original Q10, and Zp, across many
realistic soil type scenarios.
Methods
This study uses a 1-D CO2 and heat transport model described
by , originally developed by . This model,
with existing versions in Perl and R , was recoded in C to increase
computational efficiency for the parameter solving routine.
Model description
This model (Fig. 2) simulates the movement and production of CO2 through
the soil profile and into the free atmosphere. The model consists of one
atmospheric layer and a soil profile 1 m in length, divided into 100 layers
of uniform thickness. Each layer can exchange CO2 with its two nearest
neighbouring layers using the 1-D discrete form of Fick's first law:
Fij=-DijΔCijΔzij,
where Dij is the effective diffusion coefficient between two
soil layers, ΔCij is the CO2 concentration
difference (µmol m-3) and Δzij is the
difference in depth (m) between the two layers.
Conceptual representation of the 1-D layered soil model. Overall
profile length is denoted with L, and N represents the number of
individual layers in the model soil profile.
For every modelled time step, each soil layer has a defined temperature,
biological CO2 production, CO2 flux, thermal diffusivity, and gas
diffusivity. Temperature varies sinusoidally on daily and annual timescales.
Changes in surface temperature are shifted and dampened through the soil
profile using
T[i]=Tavg+ΔTDe-zidTDsin(ωDt-zidTD)+ΔTYe-zidTYsin(ωYt-zidTY),dTD=2DT[i]ωD,dTY=2DT[i]ωY,
which simulates the lags
related to the rates of thermal diffusion. In this equation, Tavg
is the average temperature in the air and soil profile for the duration of
the simulation, ΔTD is the amplitude of the daily
temperature fluctuation, ΔTY is the amplitude of the yearly
temperature fluctuation, ωD is the radial frequency for
daily oscillations (ωD=2π/86400 s), ωY
is the radial frequency for annual oscillations, zi is the layer depth
(m), and DT is the thermal diffusivity of the soil
(m2 s-1).
Biological CO2 production in each layer is calculated using an
exponentially decreasing function :
P[i]=Γ0∑i=1Ne-ziZpe-ziZpQ10T[i]-Tavg10,
where Γ0 is the total basal soil production (µmol m-3 s-1),
N is the number of soil layers, Q10 is the
temperature sensitivity of soil respiration, zi is the depth of the layer
(m), and Zp is the e-folding depth of CO2 production (m), defined as the
depth below which the total fraction of CO2 production remaining is
1/e (also called the characteristic production depth in some studies), from
which the production at any depth P(i) can be calculated based on Eq. (4).
Initially, the diffusivity of CO2 in the soil profile is calculated
using the Millington model , an empirically derived
approximation for calculating diffusivity in the field:
Dc=θw103DfwH+θg103DfgθT2,
where Dfw and Dfg are the diffusivity of CO2 in free water and free
air (m2 s-1), H is the dimensionless form of Henry's solubility
constant for CO2 in water, and θw, θg, and
θT are the water filled, air filled, and total soil porosities,
respectively.
At each time step, the diffusivity of each soil layer is calculated using a
temperature correction on this Millington diffusivity:
D[i]=Dc(T[i]Tavg)1.75.
As previously mentioned, the flux from each layer is determined by Fick's
first law, written explicitly as
F[i]=D[i](C[i]-C[i-1])dzdt,
where C[i] is the CO2 concentration of layer i (µmol m-3),
C[i-1] is the concentration of the layer above, and dt is the time step (s).
Finally, at each time step CO2 concentration in each layer i is
calculated using
C[i]=Ct-1[1]θgdz-F[i]+F[i+1]+P[i]θgdz,
where Ct-1[i] is the layer concentration at the previous time
step, F[i] is the flux of CO2 leaving layer i, F[i+1] is the flux of
CO2 entering the layer from the layer below, and P[i] is the CO2
production within layer i.
Model execution and validation
Before beginning the simulation, the system is initialized using input
parameters seen in Table 1. Atmospheric CO2 concentration remains
constant for the duration of the simulation; it is assumed that any flux from
the soil will quickly dissipate into the atmosphere. Flux from the bottom
soil boundary is set to zero, as production at this depth is negligible
according to the exponentially decreasing production function. These system
parameters were changed depending on the soil type being simulated.
Default parameter values for simulations.
Parameter
Value/range
Soil porosity (θT)
0.40 (v/v)
Thermal diffusivity (DT)
5 × 10-7 (m2 s-1)
Average air and soil temperature (Tavg)
15 ∘C
Daily air temperature amplitude (ΔTD)
5 ∘C
Yearly air temperature amplitude (ΔTY)
12 ∘C
Atmospheric CO2
380 ppm
Total basal CO2 production (Γ0)
1–10 µmol m-2 s-1
Production exponential folding depth (Zp)
0.05–0.20 m
Q10
1.5–4.5
Volumetric water content (θw)
0.10–0.25 (v/v)
After initialization, the system undergoes spin-up, during which layer
temperatures are held constant at their initial values, and the model is run
until the CO2 concentration in each layer is constant. The duration of
the spin-up period is dependent on soil diffusivity (and therefore
θw), and is determined by plotting concentration vs. time through
the soil profile. This period ranges from 5 to 23 model days within the range
of θw (0.1 to 0.25). The CO2 concentration in each layer after
spin-up is the initial layer concentration at the beginning of the actual
simulation.
For each modelled time step (dt=1.0 s), temperature, CO2 diffusivity,
CO2 production, and CO2 flux are calculated in each soil layer.
Every soil layer is then revisited, and the new CO2 layer concentrations
are calculated. The progress of the simulation is monitored by outputting the
CO2 concentration and temperature of specified layers.
Validation
To ensure the model was performing correctly, steady-state concentrations
through depth (following spin-up) were compared to the steady-state solution
proposed by . Daily and yearly temperature fluctuations were
removed from the model, and the model was run until CO2 concentrations
in each layer were constant. Deviations of modelled from analytic
concentrations were found to be far less than 1 %.
Incorporating constraining data
In this study, we incorporated external data in the same way we would with
field studies. We started with real measurements of temperature through
depth, and soil volumetric water content, from a local field site. One of the
largest challenges in preparing data for inversion is to accurately model
soil temperature through depth and time, as temperature is the known
determinant of soil lags . For each set of temperature
measurements through depth, a linear regression (in R) was performed,
resulting in a fifth-order polynomial for temperature through depth every 1800 s.
A linear interpolation through time was performed to obtain temperature
values in each layer for every modelled time step. The resultant temperature
values replaced our originally sinusoidally varying temperature function in
the model. The value of thermal diffusivity was therefore implicitly built
into these measurements and is no longer required as a direct model input.
These physical variables were used in a forward instance of the soil model to
create CO2 surface-flux and CO2 concentration time series. Data sets
were created using many Zp and Q10 values of interest, so that we had
many idealized data sets on hand in which concentration, fluxes, and
associated temperatures, Zp, and Q10 values were known. Soil volumetric
water content was not formally incorporated as a driver of respiration in
these synthetic data sets, so all simulations were performed over periods of
constant soil volumetric water content. During inversion we pretended not to
know Zp and Q10 values of these synthetic data sets, and hoped the
inversion process would return the known values. Since the same forward soil
model that generated the synthetic data sets was also embedded within the
inversion scheme, errors in Zp, or Q10, would be due entirely to the
inversion process itself.
Inversion process
The soil profile CO2 concentrations and soil CO2 surface flux are
outputs of the simulation. Their values are dependent on all of the system
input parameters. A method called inverse parameter estimation is employed to
determine the values of Q10 and Zp that would have given rise to the
observed concentrations and fluxes. Through this process, model outputs are
compared to measured field data or synthetic data over a range of model input
parameters. The field measurements used in this process will be referred to
as the model constraints; these constraints consist of CO2 concentration
measurements at various depths in the soil profile, as well as CO2
surface-flux measurements.
Inversion steps
The model is run for two unknowns, including Q10 values ranging from 1
to 5.5 in steps of 0.1, and Zp from 0.02 to 0.3 m in steps of 0.01 m. This
results in a total of 1260 parameter combinations. Inversion seeks to
identify the parameter set that minimizes the objective function
(S1-M1)2+(S2-M2)2+(S3-M3)2+…,
where Si and Mi correspond to modelled and measured CO2
concentrations at various profile depths. For each parameter set, this
objective function is calculated every 1800 time steps and averaged at the end
of the simulation. The pair that minimizes Eq. (9) is output as the inversion
result.
Validation of the inverse method
Before applying the inversion method to real field data, tests must be done
to ensure method accuracy, and this manuscript focuses on such tests. We
created synthetic time series using the original soil model, that mimic the
form of real data sets. The values of Q10 and Zp were known for each
synthetic time series, as these parameters are required to run the model. This
synthetic data included temperature measurements at six depths in the
profile, volumetric water content, CO2 surface-flux, and CO2
concentration measurements at various depths in the soil profile.
The inverse method was applied to these synthetic data sets, and the output
value of Q10 and Zp could then be compared to the actual values of these
parameters used to create the time series.
Constraint, sensitivity, and random error testing
To determine which model constraints resulted in the highest accuracy of the
inversion method, the error (Eq. 9) was calculated using a large range of
constraining parameters and combinations thereof. A total of 35 different
constraint combinations were tested, representing various combinations of
surface CO2 flux, and subsurface CO2 concentration measurements up
to 0.6 m depth. These combinations are illustrated in Table 2. Testing which
constraints consistently returned the most accurate values of Q10 and Zp
aids in determining optimal sensor placing in the field.
Measurement combinations used for the simulations. The combination
number is listed at the beginning of each row. The columns represent the type
of measurement (e.g. CO2 surface flux), or the depth of concentration
measurement in centimetres. The “×” values denote whether the type or depth
of measurement was included in the combination.
Combination
Flux
5
10
15
20
25
30
35
40
45
50
55
60
1
×
2
×
3
×
4
×
5
×
6
×
7
×
8
×
9
×
10
×
11
×
12
×
13
×
14
×
×
15
×
×
16
×
×
17
×
×
18
×
×
19
×
×
20
×
×
21
×
×
22
×
×
23
×
×
24
×
×
×
25
×
×
×
26
×
×
×
27
×
×
×
28
×
×
×
29
×
×
×
30
×
×
×
×
31
×
×
×
×
32
×
×
×
×
33
×
×
×
34
×
×
×
35
×
×
×
To ensure model validity across all possible parameter values that may be
encountered in the field, extensive sensitivity testing was done using these
synthetic time series. These time series were created across a range of
combinations of Q10, Zp, volumetric water content (diffusivity), and
total soil production. Table 3 illustrates the ranges tested for each
parameter.
Field-deployable CO2 sensors typically have 1–5 % error. To see how the
model and inversion would perform under these conditions, errors of 1, 5, and
10 % were added into all components of the synthetic data. The effect of
these errors on the inverse method were observed.
Default parameter values for sensitivity testing.
Parameter
Abbr.
Minimum
Maximum
Increment
Total basal CO2 production (µmol m-2 s-1)
Γ0
1
10
10
Production exponential folding depth (m)
Zp
0.05
0.2
0.05
Q10
1.5
4.5
1
Volumetric water content (v/v)
θw
0.1
0.25
0.05
Results and discussion
Inversions on synthetic time series were successful across all tested soil
parameters, though some CO2 concentration measurement depth combinations
(surface flux, single, or multiple profile measurements) helped to minimize
the overall error, as well as the error in Q10 and Zp individually.
Errors discussed in this section represent an average from 64 inversions
across values of Q10, Zp, and soil diffusivity as presented in Table 3.
In this section, we use either fractional error
(|actual-result|actual), or absolute deviation from the actual
value (|actual-result|).
Best measurement configurations to obtain Q10 and Zp via inversion
In Fig. 3 we show the average fractional error in the returned Q10 value
for every combination of subsurface CO2 sensor measurements.
Observations of CO2 concentration shallow in the soil were found to be
necessary for highly accurate Q10 estimates. The lowest inversion error
for Q10 was 1.85 %, in a scenario where subsurface measurements were
made at 5, 10, and 15 cm. Single concentration measurements at or above 10 cm
also proved successful, with errors < 2.3 %. The least accurate inversions
for Q10 occurred when the constraint consisted of (single or multiple)
CO2 concentration measurements deep in the soil profile. We propose that
the poor performance of inversion when using deep profile constraints could
be related to the low magnitude of thermal and concentration variability at
these depths. Deep soil layers are subject to much smaller thermal
fluctuations than layers close to the surface. In this less variable
environment, CO2 concentrations are less variable and provide less of a
signal upon which to anchor inversion. In contrast, CO2 concentrations
shallow in the soil exhibited larger variations in temperature and
concentration, which presumably allowed Q10 to be extracted more easily.
If the primary interest is to obtain Q10 from inversion, multiple
CO2 concentration measurements in the soil were found to be important.
It should be noted that, while differences in error rate were noted, errors
for all scenarios could be considered tolerably low relative to the normal
variance expected from regression-based Q10, considering the gas
transport lags inherent in those data .
Fractional error in Q10 and Zp individually for different
sensor combination scenarios, plus cumulative fractional error in Q10
and Zp for the same scenarios.
The average fractional error in Zp for different model sensor combination
constraints is also shown in Fig. 3. Out of the 35 combinations tested, only
5 resulted in an average Zp error greater than 2 %. Single concentration
measurements shallow or deep in the soil profile caused this larger error,
but on average, single concentration measurements at any depth in the soil
were less accurate. Inversions constrained by at least one measurement
shallow (< 15 cm) and one deep (≥30 cm) in the profile returned Zp
with 100 % accuracy across all sensitivity tests. We did expect that
single measurements deep in the profile would perform poorly relative to
others, because with the exponentially decreasing production defined in the
model, CO2 production approaches zero at significant depths regardless
of the value of Zp and thus cannot perform well as an inversion anchor. The
large Zp error of almost 25 % associated with soil surface CO2-flux
measurements, was also not surprising. In this situation the inversion scheme
must reconstruct Zp mainly via the temporal delay, and damping, between
sinusoids of temperature through depth, and soil surface CO2 flux.
Without a concentration measurement in the soil, the gas transport regime is
black boxed from the perspective of the inversion scheme, resulting in the
large error. Overall, surface CO2-flux measurements alone are less
suited for elucidating information on e-folding depth of production, whereas
a combination of shallow and deep measurements is best for reconstructing the
distribution of CO2 production in the soil profile.
In examining inversion accuracy for both parameters Q10 and Zp
simultaneously (Fig. 3), we found that multiple concentration measurements
shallow in the soil (≤15 cm), or combinations shallow in the soil with
one deep concentration measurement (≥30 cm) were the best constraints.
Deep soil measurements and surface-flux constraints should therefore be
avoided if the aim is to minimize overall error. This overall result is a
combination of what was found for Q10 and Zp individually, where shallow
measurements were best for Q10 and a combination of shallow and deep
measurements resulted in most accurate Zp.
Depending on error tolerance for the final parameter estimates, it is
conceivable that the accuracy of all inversions performed here might be
sufficient for the community of soil scientists. Out of the 35 combinations
tested, 19 resulted in an overall average error less than 5 %. The top
constraint (measurements at 5, 10, and 15 cm) had an average error of 2.01 %,
and the top six combinations all had error less than 3 %. These errors are
small compared to the degree of random error in CO2-flux studies
. These results are summarized in Table 4, where the top and
bottom five combinations are listed individually and overall.
Best and worst sensor combinations for determining Q10, Zp, and
overall through inversion.
Combination
Rank
Q10
Zp
Overall
1
5 + 10 + 15 cm
n/a
5 + 10 + 15 cm
2
5 + 15 cm
n/a
5 + 15 cm
3
5 cm
n/a
5 + 15 + 30 cm
4
10 cm
n/a
5 + 15 + 30 + 60 cm
5
5 + 15 + 30 cm
n/a
5 + 30 cm
31
45 cm
55 cm
50 cm
32
50 cm
50 cm
50 + 60/50 + 55 + 60 cm
33
50 + 60/50 + 55 + 60 cm
60 cm
55 cm
34
55 cm
5 cm
60 cm
35
60 cm
Surface Flux
Surface flux
This assessment was performed using synthetic data, and even the most ideal
field settings will depart from these modelled profiles. For example, we
represented CO2 production through depth using an exponential production
function, but a field site may show a linear decrease in production at
increasing depths. Clearly users of the inversion process will want to
characterize as many site-specific parameters as possible so as to provide
proper guideposts and constraints for the inversion; otherwise, additional
error will be introduced. The sensitivity of the inversion to error is an
important question, and will be addressed in a later section.
Effect of soil-specific parameters on inversion success
Having determined the best CO2 sensor concentration measurement depth to
constrain inversions, we can examine how site-specific parameters, such as
soil diffusivity, Zp, and Q10, affect inversion results. For this
assessment, we will use the best-performing measurement configurations
established. Even when not a top choice, we will always include CO2
surface-flux measurements in this section, because of the likelihood that
scientists will want to use inversion to analyse these data, which are
increasing in number rapidly.
Figure 4a and d illustrate how deviation in Q10 and Zp were affected
by the diffusivity of soils. When subsurface sensor combinations were used as
a constraint, there was an overall downward trend in Q10 and Zp error
with increasing diffusivity. As diffusivity increases (drier soils), CO2
travels through the soil layers to the surface more quickly, which results in
decreased lag times, more rapid concentration changes, and more distinct soil
responses. Under these conditions of rapid diffusion, inversions were most
successful. Sites that are frequently waterlogged with limited air-filled
pore space tended to be less ideal for inversion, but the optimal instrument
configuration still helps ensure reasonably small error throughout the entire
range of diffusivities, so there is no strict limitation on the use of the
inversion approach in low diffusivity soils.
Error in Q10 and Zp as a function of Q10 (c, f),
Zp (b, e) and Dc (a, d), for a grouping of the best sensor measurement
depth combinations. Individual 5 and 10 cm observational scenarios are shown
in light blue and dark blue, respectively. The 5 + 15 cm measurement
scenario is shown in green. Orange and red illustrate sensitivity of the
5 + 10 + 15 and 5 + 10 + 30 cm scenarios, respectively.
Finally, the 4-point 5 + 15 + 30 + 60 cm measurement sensitivity
is represented in grey while the surface-flux scenario is shown in black. For
these sensitivity tests, the known Q10 was 2.0, and a Zp of 0.2 m was
used.
Sample of random error effects on inversion, constrained by one
concentration measurement at 5 cm. For this sensitivity test, the known
Q10 was 2.0.
Figure 4b and e demonstrate the impact of the Zp parameter value on inversion
success in terms of deviation in returned Q10 and Zp values. For small
Zp values, shallow CO2 concentration measurements (≤ 15 cm) were the
best constraints, presumably because the soil is most active in these top
layers. As Zp increases, the production of CO2 is no longer limited to the
shallow soil, the exponential production function decreases more slowly. With
increasing Zp, CO2 production in deeper soil layers is higher, and more
useful as an inversion constraint. Some matching of deployment depth was also
found, where for example shallow concentration measurements were more
accurate for returning the correct value of shallow CO2 production.
Error in Q10 and Zp as a function of Q10, Zp, and
diffusivity for the constraint 5 + 10 + 15 cm. For these sensitivity
tests, the known Q10 was 2.0, and a Zp of 0.2 m was used.
Sensitivity tests indicate that increasing the temperature sensitivity of
respiration had opposite effects on Q10 and Zp error. Deviation in
returned Q10 values increased rather uniformly across the best
subsurface measurements, while for most subsurface combinations the Zp error
decreased. With increasing Q10, respiration becomes more sensitive to
temperature changes, leading to larger variations in production in the event
of a temperature fluctuation. Figure 4c and f illustrate the impact of this
parameter on Q10 and Zp error.
With large amounts of existing surface-flux data, it is also worth examining
the effectiveness of the soil CO2 surface flux as a constraint, even
when it is not the preferred constraint. It is immediately evident from Fig. 4
that inversions constrained by the surface flux resulted in Q10 and Zp
deviations that responded much differently to changes in soil diffusivity,
Zp,
and Q10. These deviations were often significantly larger than when
subsurface constraints were used. Deviations in Q10 and Zp generally
increased as all three parameters increased. This suggests that for low
diffusivity, Zp, and Q10, surface flux was a reasonable model constraint,
producing errors comparable to the subsurface measurements. This constraint
was much less effective for determining depth of CO2 production. However,
Zp was always returned within at least 3.5 cm of its actual value, which for
some uses may be an acceptable level of uncertainty. Inversions constrained
by surface flux were quite effective in returning Q10. Returning to Fig. 3,
the overall average Q10 error associated with surface flux was less
than 5 %, which is significantly better than results using deep
subsurface measurements. Figure 4f suggests that inversions using large
Q10 values were responsible for the majority of this error. For Q10
of 1.5, these inversions returned Q10 with 100 % accuracy. For the
largest Q10, deviation from the true value climbed as high as 0.6–0.7,
which is non-negligible. A shorter model time step could potentially reduce
this error, as it may be able to better capture the larger and faster
responses associated with high Q10 and diffusivity. As we cannot
estimate the Q10 of a site prior to inversion, however, this insight may
not be overly useful in site selection. Overall, inversions constrained by
the CO2 surface flux are possible but should be performed with caution,
and with reasonable expectations as to the resultant error level.
It is also of interest to examine how the amount of CO2 production in
the soil profile affects inversion. The bulk of our sensitivity tests were
performed using a basal CO2 production of 10 µmol m-3 s-1,
which is a fairly high. In order to test the other extreme, several
inversions were performed using a production level of 1 µmol m-3 s-1.
These inversions performed with exactly the same accuracy as those
with a production level of 10. From this, we can conclude that the magnitude
of production has no effect on inversion success.
Random error and inversion
The measurements performed by sensors in the field will always be uncertain
to some degree. It is therefore important to examine how these uncertainties
in recorded temperature, CO2 and soil volumetric water content
measurements will impact the accuracy of the inversion method. Inversions
performed on synthetic data to which random errors of 1, 5, and 10 % had been
added were indeed less accurate than those performed on idealized data.
However, the resulting errors in returned Q10 and Zp were not
proportional to the amount of error added to the input data, but actually
much lower. That is, errors of 5 % in the input data did not result in an
additional 5 % error in output values. An example of this is illustrated in
Fig. 5. This plot demonstrates that with random measurement errors in the
ranges of 1–5 %, Q10 values were still determined with reasonable
accuracy. Prior to error addition, deviation in Q10 was around 0.12.
This deviation increased to 0.14 for 1 % error and 0.17 for 5 % error. As
sensors in the field are typically uncertain by 1 to 5 %, the inversion
method remains feasible. We can thus conclude that the inversion process is
rather tolerant of error in measurement.
Multi-parameter error landscape
It is worth investigating in detail the error landscape of the inversion
process using a multi-parameter sensitivity tests. For this test, we chose
the combination of measurements at 5, 10, and 15 cm, which had resulted in the
most accurate inversions on average.
The results from the sensitivity tests are shown in Fig. 6, panels a to f. In
all combinations, the error in Zp was very small, with the maximum error for
any single inversion being just over 2 %. Despite this small error, it
remains evident which soil conditions should be avoided for most accuracy.
Sites with low diffusivity, production deep in the soil and low Q10 are
the most problematic. This is consistent with the results from Fig. 4a, b, and
c. Trends were not as evident for error in Q10. In panels a, c, and e the
most notable error was found in panel a for high Zp, low Q10. There is
an error in Q10 here of almost 15 %, which equates to a deviation in
Q10 of about 0.225 from its actual value. This result is not
unreasonable, but it is significantly higher than results from the other
inversions. Plot e demonstrates an interesting result, where there seems to
be a valley in the Q10 error, illustrating a tradeoff between Zp and
diffusivity. This is not evident in the other plots, and does not have an
intuitive physical explanation. The effect of Q10 on inversion varies,
but success hinges quite clearly on soil diffusivity and depth of CO2
production. Choosing a site in the appropriate ranges of these two parameters
will maximize chances of success.
Limitations and opportunities
There are both limitations and future opportunities for the inversion
approach. In general, the better an inversion is constrained with data, the
better it will do in returning the true value for parameters of interest.
Some of the soil parameters are distributed in ways that must be assumed. For
example, the distribution of CO2 production may be unknown. While most
studies in the history of soil modelling have assumed an exponential
distribution (and it has been seen in many studies using the gradient
technique), some other considerations might help determine whether additional
parameterization measurements are needed. For example, knowledge of rooting
depth could be one aid. On average, root respiration accounts for 50 % of
soil respiration and provides root
distributions for different terrestrial biomes. found that
tundra, boreal forest, and temperate grasslands had upwards of 80–90 % of
roots within the top 30 cm of soil, whereas deserts and temperate coniferous
forests had much deeper rooting profiles, with only 50 % of roots within
the top 30 cm. These and other methods may be helpful in providing
constraint data when running inversions on real time series.
A future opportunity for inversion studies is to determine depth-specific
Q10 values, which would be of interest to many researchers. Currently
there are few field examples where researchers have determined in situ
Q10 as a function of depth, but two examples of such
gradient studies include and . Incubations might seem
useful in this regard, but are disputed as a representation of in situ
conditions, especially at depth . Ideally, an inversion
approach could determine depth-specific Q10, but the reality is
challenging. If an additional 100 Q10 values were included as unknowns
(one for each layer), it would increase computational demand by 100 times.
The inversion results would also be confusing, and extreme values at one
depth could potentially cause a spurious match to the measured data. The
number of non-unique and implausible solutions would rise significantly as a
result. A more reasonable approach might be to define Q10 as we do Zp,
which is as a function of soil depth. This would be computationally
compact, but it is not well known whether a function would be realistic because there are
so few examples of Q10-depth profiles against which we could evaluate
this approach. The best approach would be to use many profile concentration
sensors for constraining data, so that the e-folding depth of production
could be known, and the inversion could focus instead solely on determining
layer-specific Q10 values.
Despite the relatively nascent stage of our soil CO2 inversion approach,
indications are that it has better theoretical validity than traditional
regression approaches, which do not take thermal- and gas-transport lags into
account. Our error results here compare very favourably against error
analyses generated from detailed examination of regression approaches across
thermal- and gas-transport parameter space .
Conclusion
Overall, this inversion method proved successful in testing on synthetic
data. Depending on the tolerable level of error for a given application,
almost every tested combination resulted in reasonably accurate returned
Q10 and Zp values. The subsurface concentration measurements that
yielded the highest error were typically those that would be of least
convenience to install and maintain deep in the soil profile. The other
constraint associated with high overall error was CO2 surface flux, which
would likely be the data with the highest availability. Most of the error from
this constraint arises in estimating the Zp parameter. The CO2 surface
flux is still a reasonable means of estimating Q10 values via inversion.
While in most cases the error was lower for high diffusivity, shallow
production soils, the application of this method is not limited to such
regions.
This method is computationally intensive as it performs a sweep through all
possible combinations in parameter space. This study used roughly 2.5
core-years of time despite the fact that synthetic time series were short.
This full sweep ensures that the global minimum in the objective function is
located every time, and when solving inversely for two unknown parameters (as
we are), this is not an unreasonable approach. However, if it was of interest
in the future to examine longer time series, or additional parameters such as
the depth dependence of Q10, resulting in additional unknown parameters,
it may be beneficial to explore other search algorithms to increase
efficiency, such as simulated annealing.
The next step for this work would be to perform inversions on real time series
with appropriate measurement constraints, to obtain temperature sensitivity
and CO2 production distribution estimates for various sites. With the
increasing availability of high-frequency soil data, there would be no
shortage in data to analyse. Applying this method for periods of varying
constant moisture levels could also help build an understanding of moisture
effects on temperature sensitivity of respiration.