Physical soil properties create lags between temperature change and
corresponding soil responses, which obscure true Q

Soil respiration, which includes both root and microbial respiration,
represents the largest outward flux of CO

Soil CO

Historically, Q

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

This study seeks to develop a reliable inversion framework for determining
the Q

This study uses a 1-D CO

This model (Fig. 2) simulates the movement and production of CO

Conceptual representation of the 1-D layered soil model. Overall
profile length is denoted with

For every modelled time step, each soil layer has a defined temperature,
biological CO

Biological CO

Initially, the diffusivity of CO

At each time step, the diffusivity of each soil layer is calculated using a
temperature correction on this Millington diffusivity:

As previously mentioned, the flux from each layer is determined by Fick's
first law, written explicitly as

Finally, at each time step CO

Before beginning the simulation, the system is initialized using input
parameters seen in Table 1. Atmospheric CO

Default parameter values for simulations.

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 CO

For each modelled time step (d

To ensure the model was performing correctly, steady-state concentrations
through depth (following spin-up) were compared to the steady-state solution
proposed by

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

These physical variables were used in a forward instance of the soil model to
create CO

The soil profile CO

The model is run for two unknowns, including Q

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 Q

The inverse method was applied to these synthetic data sets, and the output
value of Q

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 CO

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. CO

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 Q

Field-deployable CO

Default parameter values for sensitivity testing.

Inversions on synthetic time series were successful across all tested soil
parameters, though some CO

In Fig. 3 we show the average fractional error in the returned Q

Fractional error in Q

The average fractional error in

In examining inversion accuracy for both parameters Q

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 CO

Best and worst sensor combinations for determining Q

This assessment was performed using synthetic data, and even the most ideal
field settings will depart from these modelled profiles. For example, we
represented CO

Having determined the best CO

Figure 4a and d illustrate how deviation in Q

Error in Q

Sample of random error effects on inversion, constrained by one
concentration measurement at 5 cm. For this sensitivity test, the known
Q

Figure 4b and e demonstrate the impact of the

Error in Q

Sensitivity tests indicate that increasing the temperature sensitivity of
respiration had opposite effects on Q

With large amounts of existing surface-flux data, it is also worth examining
the effectiveness of the soil CO

It is also of interest to examine how the amount of CO

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, CO

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

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 CO

A future opportunity for inversion studies is to determine depth-specific
Q

Despite the relatively nascent stage of our soil CO

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
Q

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 Q

The next step for this work would be to perform inversions on real time series
with appropriate measurement constraints, to obtain temperature sensitivity
and CO

The authors wish to thank the Natural Sciences and Engineering Research Council (NSERC) and the Atlantic Computational Excellence Network (ACENet) for project resources. We also wish to thank Nick Nickerson and Chance Creelman for valuable comments early on in this research. Edited by: A. Ibrom