## 3 Materials and Methods
### Empirical Models or Statistical Models  
There have been cases where conventional models are considered inefficient when the case is soils with cracks and high heterogeneity in terms of soil texture, particle shape and orientation, burrows, etc. Thus, the diffusion coefficient is difficult to adopt for extensive tortuosity. In such cases, many experimental data are needed or suitable empirical correlations must be employed [17]. Empirical models demonstrate simplistic concepts and are, in some cases, e.g., in regions that lack time series data, more attractive since they are easy to use and they do not necessitate profound experience in soil science and hydrological modeling. Empirical models might be simple equations with region-specific parameters [58]. They exploit collected datasets and allow prompt result acquisition and quicker final assessment and therefore prompt decision making. Calibration is less demanding, relying on observational data. On the other hand, empirical model prediction ability is fairly limited with narrow scenario flexibility and, in general, lacks accuracy and suffers quite often from oversimplification. It is apparent that in complex hydrological processes, significant soil-water interaction mechanisms are overlooked, a fact that is functioning as a deterrent to their adoption and usage. The HELP model is a hybrid model, both deterministic and statistical-empirical, that simulates water vertical flow through landfill layers. It proved to be quite useful as regards landfill capping water balance systems and leachate generation in northern Germany (see also Section 4.4). Nonetheless, empirical models are lacking satisfactory results when extrapolation is needed in the future. Despite this, it is still a useful tool in the decision making in farming or grazing [18]. The NLES5 model [98] is partially modified based on an earlier NLES4 model version [131] since it has almost the same structure and partially shares datasets for calibration. It is considered to be an empirical prediction model for quantifying nitrate leaching on an annual basis, considering the interest zone as the root zone, i.e., 1 m depth. The model accounts for incurred effects of added N under certain crop sequences, seasonal crop cover, topsoil types, and local weather conditions. Soil (N) leaching and mitigation prediction, caused by fertilizing over croplands, improved integrated water basin management, and promoted the good quality preservation of surface water and aquifers as the main goals of model development. The Quick Plant RZWQM2 sub-model is a simplified empirical plant module developed to simulate the water and nitrate fate for the crops and therefore to compute on a daily basis water and nitrogen uptake of the cultivated soil (see also Section 4.2 below). Meteorological data collection is essential to the final outcome. It incorporates the Green-Ampt infiltration equation, Richards' equation [78] for water redistribution, and a plethora of submodules such as Quick Plant, soil water and heat transfer, N balance, generic plant growth, evapotranspiration (PET), soil equilibrium chemistry, pesticides and finally the management module. Soil water retention curves are acquired by using the modified Brooks-Corey equation. Empirical prediction models demonstrate certain advantages since they do not require extensive calibration database information, which in many cases is unavailable [132]. Any future strategy must address the critical issue of relatively accurate nitrate leaching prediction across common cropping systems, soil types, and climate conditions in specific geographic areas, as well as the impact of seasonal vegetation characteristics on nitrate leaching, which is dependent on N fertilization practices. A carefully defined exponential function was fitted to the selected data. The function incorporated parameters that were estimated by non-linear regression analysis. The prompted leaching curve yields the marginal nitrate leaching rate, which corresponds to the recommended N application rate.