## 2 Materials and Methods  
### Study Site, Physical Framework and Data  
Our research utilized average monthly discharge measurements from four monitoring stations located on the Romanian section of the Mures River (Figure 1), spanning a comprehensive period from 1977 to 2014. In time series datasets, such as river flow records, missing values emerge from both random occurrences and systematic variations. Addressing these data gaps necessitates sophisticated imputation techniques, as conventional methods like mean substitution or linear interpolation frequently prove inadequate in capturing complex temporal interdependencies [31; 32]. Advancing imputation accuracy demands methodological approaches that can effectively leverage the intrinsic temporal and stochastic characteristics of environmental data. Interpolation under conditions of limited monthly observations requires a robust physical framework that facilitates generating contextually appropriate values for data-sparse periods. This framework essentially involves identifying and exploiting natural relationships among sparsely observed environmental measurements. Hydroological and climatological datasets exhibit distinctive periodicities and auto-correlational patterns, fundamentally driven by Earth's heliogeophysical characteristics--including its spherical morphology, orbital dynamics, and axial inclination. These fundamental planetary attributes generate annual solar radiation and surface heating cycles, manifesting as a predictable 12-month periodicity in hydroclimatic elements. Such cyclical patterns reveal intrinsic similarities between corresponding monthly observations within and across years (Figure 2), and their structured arrangement (Figure 3) further highlights the serial correlation and autocorrelation inherent in such time series. Figure 1: Course of Mures River in Romania and the considered four hydrological stations. The 12-month periodicity in hydroclimatic datasets provides insights into seasonal behaviors but simultaneously introduces non-stationarity, demanding sophisticated preprocessing for effective modeling. Historically, both deterministic seasonal models [33] and stochastic approaches [34; 35; 36] have been employed by researchers to address such challenges, recognizing the complex interplay between periodicity and autocorrelation in environmental data. Contemporary imputation methodologies continue to evolve, increasingly refining techniques to address non-stationary series characteristics. The data preprocessing approach demonstrated methodological flexibility, with two data arrangement strategies: (a) traditional time series approach by maintaining chronological sequence; (b) tabular configuration by organizing monthly data vertically by year. By strategically leveraging the inherent 12-month periodicity characteristic of hydroclimatic systems, ML models can be systematically trained using innovative tabular data configurations, organizing each year as a discrete row and individual months as columnar variables. This sophisticated architectural approach enables advanced ML algorithms to comprehensively explore and exploit periodic interdependencies, facilitating robust predictive capabilities for reconstructing missing data points. To rigorously validate our imputation methodologies, we implemented a systematic approach to creating artificial data gaps with 4 data removal scenarios: 5%, 10%, 15%, and 20% data point elimination using uniform distribution as a randomization method. In this study, we introduced artificial data gaps using a uniform distribution to provide an unbiased assessment of the method's foundational capabilities, while acknowledging that in real-world applications, certain data points may carry more importance and thus warrant a more tailored gap-generation approach. To ensure the robustness of validation we have generated 1000 distinct random gap masks for each scenario.