Chaos Analysis and Prediction of Monthly Runoff Using a Two-Stage Variational Mode Decomposition Framework

Variational modal decomposition (VMD) has proven to be an effective technique for improving the accuracy of runoff prediction and has been widely used in runoff analysis and prediction. However, the presence of noise and outliers can complicate the runoff decomposition, so obtaining purely stochastic and deterministic components is difficult. Therefore, exploring the chaotic properties of the components obtained from decomposition can provide a new perspective to reveal intrinsic variability patterns and enhance our understanding of the unity of determinism and stochasticity. In this study, a two-stage VMD framework is proposed to decompose the monthly runoff, with the chaotic characteristics analyzed by employing an unthreshold recurrence plot and the largest Lyapunov exponent. Additionally, a chaotic prediction model is developed using phase space reconstruction, Volterra filter, and wavelet neural network methodologies. The model is validated using monthly runoff data from four stations in the Yellow River Basin, China. The results demonstrate that, although the VMD method effectively isolates trend components in monthly runoff, it still exhibits challenges in separating periodic and random elements, leading to the identification of chaotic components characterized by the amalgamation of periodicity and randomness. Notably, the two-stage VMD-phase space reconstruction-Volterra-wavelet neural networks model outperforms the VMD-phase space reconstruction-Volterra-wavelet neural networks model, with a substantial increase in Nash-Sutcliffe efficiency during validation, rising from an average of 0.9271-0.9508 and reaching a maximum of 0.96 across the four stations. Overall, this study demonstrates the potential of VMD for improving the monthly runoff prediction accuracy and elucidates the interplay between determinism and stochasticity in runoff analysis, offering valuable insights for further research in this area.