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One of the most challenging tasks in studying streamflow is quantifying how the complexities of environmental and dynamic parameters contribute to the overall system complexity. To address this, we employed Kolmogorov complexity (KC) metrics, specifically the Kolmogorov complexity spectrum (KC spectrum) and the Kolmogorov complexity plane (KC plane). These measures were applied to monthly streamflow time series averaged across 1879 gauge stations on U.S. rivers over the period 1950–2015. The variables analyzed included streamflow as a complex physical system, along with its key components: temperature, precipitation, and the Lyapunov exponent (LEX), which represents river dynamics. Using these metrics, we calculated normalized KC spectra for each position within the KC plane, visualizing interactive master amplitudes alongside individual amplitudes on overlapping two-dimensional planes. We further computed the relative change in complexities (RCC) of the normalized master and individual components within the KC plane, ranging from 0 to 1 in defined intervals. Based on these results, we analyzed and discussed the complexity patterns of U.S. rivers corresponding to each interval of normalized amplitudes.

One of the most challenging tasks in studying precipitation is quantifying how the complexities of individual components contribute to the overall system complexity. To address this, we employed information measures based on Kolmogorov complexity (KC), specifically the Kolmogorov complexity spectrum (KC spectrum) and the Kolmogorov complexity plane (KC plane). We applied these measures to monthly time series data, both measured and simulated by the EBU POM regional climate model, spanning the period from 1982 to 2005 for Sombor (45.78° N, 19.12° E) in Serbia. The variables analyzed included precipitation—a complex physical system—and its individual components: mean temperature, minimum and maximum temperatures, humidity, wind speed, and global radiation. By applying the listed measures to all time series, we calculated normalized KC spectra for each position in the KC plane, displaying interactive master amplitudes against individual amplitudes. We proposed a simplified four-step method to compute the relative change in complexities within the overlapping area beneath the KC spectra. Our results facilitated a discussion on the relationship between the complexity of precipitation and that of its individual components.