Finite‐Element Regression to Estimate Production Profiles of Greenhouse Gases in Soils

The gradient method used to quantify gas fluxes in soils is based on Fick's second law. The production term is proportional to the second derivative of the concentration profile. Existing methods reveal important limitations, however. We suggest dividing the soil into homogeneous finite elements consisting of second‐order polynomials for the gas concentration. A new statistical procedure combining the mechanistic constraints with quadratic splines was used to estimate the derivatives of scattered gas data. The method was tested in a Picea abies (L.) H. Karst. stand for CO 2 and N 2 O depth profiles. The field results yielded a mean annual efflux of 5.9 Mg ha −1 yr −1 CO 2 −C, of which 86% originated from the mineral soil. The Ah horizon and the forest floor of both plots yielded around 0.8 kg ha −1 yr −1 N 2 O−N. In the subsoil, N 2 O−N production revealed a high spatial heterogeneity. To test for robustness, synthetic data sets were simulated for re‐estimation of belowground gas fluxes with the new procedure and with linear and exponential fits. When only the efflux at the surface was considered, both the finite element method and the local linear fit yielded results close to the expected value. For the deeper soil, the finite element fit yielded unbiased results as long as the preconditions were fulfilled. Otherwise, and this applied also to the alternative methods, results were rather arbitrary. Until now, no alternative to the gradient method for gas flux estimation in the deeper soil has been available. Therefore, deep soil gas flux estimations are questionable as long as the preconditions for the differentiating method are not thoroughly confirmed. As an alternative, we suggest an inverse parameter optimization based on a numerical solution of Fick's second law.