Estimating the nitrous oxide emission rate from the soil surface by means of a diffusion model (Q2742753)

A stochastic model for soil cover concentrations of nitrous oxide is proposed. Let \(C_t\) denote the nitrous oxide concentration beneath a soil cover at time \(t\) since placement of the cover and let \(C_0\) denote the ``background'' concentration at the time of placement of the soil cover. The author presents a model which is given by a diffusion process defined by the stochastic differential equation \(dC_t = k(\phi -C_t) dt + \sqrt{\omega^2 C_t} dW_t\), where \(W\) is a one-dimensional Wiener process and \(\omega^2\) is a positive unknown parameter, \(\phi\) denotes the nitrous oxide concentration at depth \(d\) below the surface, and \(k\) is a positive parameter. Statistical estimators of \(k\), \(\phi\), \(\omega^2\) and \(r_0 =dC_t/dt|_{t=0}\) are presented. Assume that the nitrous oxide concentrations are measured simultaneously in \(d\) soil covers at time-points \(0 = t_0 <t_1 <\ldots < t_n\), \(t_i = i\Delta\), and let \(C_{t_i}^j\) be the results of the measurements. The author proposes, for example, an estimator NEWLINE\[NEWLINE\hat k = \Delta^{-1} \log [(S_2 S_4 -(nd)^2)/(S_1 S-4 - nd S_3)],NEWLINE\]NEWLINE where, for example, NEWLINE\[NEWLINES_1 = \sum_{j=1}^d \sum_{i=1}^n C_{i\Delta}^{(j)},\quad S_3 = \sum_{j=1}^d \sum_{i=1}^n C_{i\Delta}^{(j)}/ C_{(i-1)\Delta}^{(j)}.NEWLINE\]NEWLINE The method is applied to some nitrous oxide data from an experiment conducted by \textit{S. Petersen} [J. Environ. Qual. 28, 1610-1618 (1998)]. The author presents a simulation study of some properties of the proposed methods for practicable sampling schemes.