The Dynamics of Shannon Entropy in Analyzing Climate Variability for Modeling Temperature and Precipitation Uncertainty in Poland

The aim of this study is to quantitatively analyze the long-term climate variability in Poland during the period 1901-2010, using Shannon entropy as a measure of uncertainty and complexity within the atmospheric system. The analysis is based on the premise that variations in temperature and precipitation reflect the dynamic nature of the climate, understood as a nonlinear system sensitive to fluctuations. This study focuses on monthly distributions of temperature and precipitation, modeled using the bivariate Clayton copula function. A normal marginal distribution was adopted for temperature and a gamma distribution for precipitation, both validated using the Anderson-Darling test. To improve estimation accuracy, a bootstrap resampling technique and numerical integration were applied to calculate Shannon entropy at each of the 396 grid points, with a spatial resolution of 0.25° × 0.25°. The results indicate a significant increase in Shannon entropy during the summer months, particularly in July (+0.203 bits) and January (+0.221 bits), compared to the baseline period (1901-1971), suggesting a growing unpredictability of the climate. The most pronounced trend changes were identified in the years 1985-1996 (as indicated by the Pettitt test), while seasonal trends were confirmed using the Mann-Kendall test. A spatial analysis of entropy at the levels of administrative regions and catchments revealed notable regional disparities-entropy peaked in January in the West Pomeranian Voivodeship (4.919 bits) and reached its minimum in April in Greater Poland (3.753 bits). Additionally, this study examined the relationship between Shannon entropy and global climatic indicators, including the Land-Ocean Temperature Index (NASA GISTEMP) and the ENSO index (NINO3.4). Statistically significant positive correlations were observed between entropy and global temperature anomalies during both winter (ρ = 0.826) and summer (ρ = 0.650), indicating potential linkages between local climate variability and global warming trends. To explore the direction of this relationship, a Granger causality test was conducted, which did not reveal statistically significant causality between NINO3.4 and Shannon entropy (p > 0.05 for all lags tested), suggesting that the observed relationships are likely co-varying rather than causal in the Granger sense. Further phase-space analysis (with a delay of τ = 3 months) allowed for the identification of attractors characteristic of chaotic systems. The entropy trajectories revealed transitions from equilibrium states (average entropy: 4.124-4.138 bits) to highly unstable states (up to 4.768 bits), confirming an increase in the complexity of the climate system. Shannon entropy thus proves to be a valuable tool for monitoring local climatic instability and may contribute to improved risk modeling of droughts and floods in the context of climate change in Poland.