4  Process rate estimation

4.1 State function set

The process rate estimator includes three-state functions, describing the change in N₂O concentrations over time. The change in N₂O concentrations in each depth increment over time \(\frac{\Delta}{\Delta t}[\ce{N2O}]\) depends on the flux of N₂O entering the depth increment from the top or the bottom through diffusion (\(\text F_\text{t,i}\) and \(\text F_\text {b,i}\), respectively), the flux of N₂O leaving the depth increment through diffusion (\(\text F_\text {out}\)), the rate of N₂O produced through nitrification (N2Onit), the rate of N₂O produced through denitrification (N2Oden), and the rate of N₂O reduced to N₂ (N2Ored).

\[\frac{\Delta[\ce{N2O}]}{\Delta t} = \text F_{\text{t,i}} + \text F_{\text{b,i}} + \ce{N2O}_{\text{nit.}} + \ce{N2O}_{\text{den.}} + \ce{N2O}_{\text{red.}} \tag{4.1}\]

For the equation of the change of site preference over time, we’ll use the substitutions \(A_1\), \(A_2\) and \(A_3\), as the expression is rather long.

\[\begin{gathered} \frac{\Delta \text{SP}}{\Delta t} = \frac{A_1 + A_2 + A_3}{[\ce{N2O}]} \\ \\ A_1 = \text F_{\text{t,i}}(\text{SP}_{\text{t,i}} - \eta\: \text{SP}_\text{diff.} - \text{SP}) + \text F_{\text{b,i}}(\text{SP}_{\text{b,i}} - \eta\: \text{SP}_\text{diff.} - \text{SP}) \\ A_2 = \ce{N2O}_{\text{nit.}}(\text{SP}_\text{nit.} - \text{SP}) + \ce{N2O}_{\text{den.}}(\text{SP}_\text{den.} - \text{SP}) \\ A_3 = - \eta\: \text{SP}_\text{diff.} \text F_{\text{out}} - \eta\text{SP}_\text{red.} \ce{N2O}_{\text{red.}} \end{gathered} \tag{4.2}\] Similar to \(\text{SP}\), the change in \(\delta^{18}\text O\) over time in each time point and depth increment can be described as:

\[\begin{gathered} \frac{\Delta\:\delta^{18}\text{O}}{\Delta t} = \frac{B_1 + B_2 + B_3}{[\ce{N2O}]} \\ \\ B_1 = \text F_{\text{t,i}}(\delta^{18}\text{O}_{\text{t,i}} - \eta\: ^{18}\text{O}_\text{diff.} - \delta^{18}\text{O}) + \text F_{\text{b,i}}(\delta^{18}\text{O}_{\text{b,i}} - \eta\: ^{18}\text{O}_\text{diff.} - \delta^{18}\text{O}) \\ B_2 = \ce{N2O}_{\text{nit.}}(\delta^{18}\text{O}_\text{nit.} - \delta^{18}\text{O}) + \ce{N2O}_{\text{den.}}(\delta^{18}\text{O}_\text{den.} - \delta^{18}\text{O}) \\ B_3 = - \eta\: ^{18}\text{O}_\text{diff.} \text F_{\text{out}} - \eta\ce{^{18}O_\text{red.}} \ce{N2O}_{\text{red.}} \end{gathered} \tag{4.3}\]

4.2 How to run the process rate estimator in R

To run the process rate estimator in R, you’ll first have to install the package PRE as described in the introduction. Once installed, add the package to the search path.

library(PRE)

Prepare the data

In a first step, we can calculate the N₂O-N from the original measurements.

data <- getN2ON(data = PRE::measurements, parameters = PRE::getParameters())

Next, the data are interpolated. This process uses kernel regression to interpolate between the measured values of N₂O-N, SP and d18O.

data <- getMissing(data = data, hyperparameters = PRE::hyperparameters)

With all of these steps done, we can calculate the Fluxes according to equations 1, 2, 3.

data <- calculateFluxes(data = data, parameters = PRE::getParameters())

Now, everything is ready to run the PRE.

Running the process rate estimator

# run the solver once
x <- PRE(data = data, column = 1, depth = 7.5, date = "2016-01-01", n = 1000)

# print out information
print(x)

# plot
plot(x)

# pairs panel
pairs(x)

Run the process rate estimator over the entire time span.

x = longPRE(data, column = 1, depth = 7.5, n = 15)

Run the estimation of process rates for the entire project, i.e. applying the longPRE function to all combinations of depths and columns. For very robust results, n = 1000 is used in the corresponding R script.

Rscript scripts/run-process-rate-estimator/run-PRE.R

The results will be written as estimated-process-rates.csv into the output directory.