## 2 Materials and Methods
### Empirical Soil Respiration Models  
The current research of the SR estimations is focused on two groups of empirical models connecting SR with the temperature and amount of precipitation, serving as a proxy for the soil moisture [20; 24; 39; 40; 41; 54] and by this, estimating SR affected by the climatic conditions. The first group includes the models dependent on \(T_{soil}\)[26; 31; 32; 33], and the second group are the models dependent on \(T_{air}\)[20; 24; 25; 42]. The comparison between the groups shows that both groups are generally based on the same formulations but with using different sources of temperature: \(T_{soil}\) or \(T_{air}\).  
Temperature and Temperature--Precipitation models [20; 24; 26; 31; 32; 33]:  
\[SR_{T}=R_{0}e^{\mathcal{Q}T} \tag{1}\]  
\[SR_{TP}=R_{0}e^{\mathcal{Q}T}\bigg{(}\frac{P}{K+P}\bigg{)} \tag{2}\]  
Figure 2: The 25-year-long measurements of soil respiration (SR, **top**), temperatures of air and soil (\(T_{air}\)—red and \(T_{soil}\)—brown, **middle**), and precipitation (Prec, **bottom**); on the SR graph: normal conditions—green, wet conditions—blue, dry conditions—brown.  
Temperature--Precipitation model dependent on the amounts of precipitation for the current (\(P\)) and previous (\(P_{m-1}\)) months [25; 42]:  
\[SR_{TPP}=R_{0}e^{\ QT}\bigg{(}\frac{\alpha P+(1-\alpha)P_{m-1}}{K+\alpha P+(1- \alpha)P_{m-1}}\bigg{)} \tag{3a}\]  
\[SR_{TPPh}=R_{0}e^{\ (QT-Q_{2}T^{2})}\bigg{(}\frac{\alpha P+(1-\alpha)P_{m-1}}{K+ \alpha P+(1-\alpha)P_{m-1}}\bigg{)} \tag{3b}\]  
Temperature--Precipitation--SOC model [48]:  
\[SR_{TPC}=R_{0}e^{\ QT}\bigg{(}\frac{P}{K+P}\bigg{)}\bigg{(}\frac{SOC}{\psi+SOC} \bigg{)} \tag{4}\]  
As an extension from the previous models (Equations (3a) and (4)), we suggest to look at the combined Temperature--Precipitation--\(SOC\) model dependent on the amounts of precipitation of two months:  
\[SR_{TPPC}=R_{0}e^{QT}\bigg{(}\frac{\alpha P+(1-\alpha)P_{m-1}}{K+\alpha P+(1- \alpha)P_{m-1}}\bigg{)}\bigg{(}\frac{SOC}{\psi+SOC}\bigg{)} \tag{5}\]  
In all models, \(T\) is the average monthly temperature of the soil surface layer [26; 31; 32; 33] or the average monthly temperature of air [20; 24; 25; 42]; \(P\) is the average monthly amount of precipitation; \(SOC\) is the organic-carbon storage in the top 20-cm of the soil.  
The \(R_{0}\) (g C/m\({}^{2}\)day) is SR at 0 \({}^{\circ}\)C in normal soil-humidity conditions--it is usually estimated from the measurements as an average SR for the not-frozen top-soil level. After \(R_{0}\) is identified, the non-linear regression is used to estimate other parameters of the models: \(Q\) and \(Q_{2}\) are the exponential-relationship temperature coefficients; \(K\) (cm) is the half-saturation constant of the hyperbolic relationship between SR and the amount of precipitation; \(\alpha\) is the redistribution coefficient between the amounts of precipitation for the current (\(P\)) and previous (\(P_{m-1}\)) months; and \(\psi\) (kg C/m\({}^{2}\)) is the half-saturation constant of the hyperbolic relationship between SR and \(SOC\).  
Practically all the empirical models described above, T (Equation (1))-TP (Equation (2))-TPP (Equation (3a))-TPC (Equation (5)), use the linear temperature dependency in the exponential term and because of it, can be put into the same class of the temperature relationship with SR. On the other hand, the model TPPrh (Equation (3b)) uses the quadratic temperature dependency in the exponential term, which obtains good results when the model was parameterized with the air temperature [25; 42], and it was taken to compare with other models.  
The quality control of modeling was based on the comparison of the following statistics: the mean bias error (MBE), the root mean square error (RMSE), the slope and intercept of the linear regression (lm) between the measured and modeled data, and the lm determination coefficient (\(R^{2}\)).