Deriving analytical expressions of the spatial information entropy index on riverine water quality dynamics

Information entropy theory has been largely applied in hydrology and water resource engineering. Recently, entropy theory has received significant attention for describing reactive solute mixing and transport processes in ground water and surface water and for its applications in water quality management. However, analytical expressions of the entropy index of riverine water quality dynamics are often not included in the literature. In this study, the analytical expressions of the spatial information entropy index (G_x) in the 1D steady-state transport process under full-extent observation and tracking observations are derived after the definition of the system boundary and probability space. The expressions of G_x were successfully validated by field data from two tracer experiments, and the different characteristics of G_x under different riverine hydraulic situations were uncovered. The formula of G_x shows a reasonable linkage with the Peclet number (Pe) in hydrodynamics. This indicates that G_x is a physical quantity describing or linking to the real world more than a statistical index, and in practice, similar decision or management measurements based on entropy theory can be made for rivers with similar Pe conditions. Hydrologists will benefit in many ways from analytical expressions, such as theoretical analysis, quick calculation of the entropy index, preliminary engineering design, etc. A clearly defined example of maximum entropy time (t_Gmax) and its application in water quality evaluation, monitoring optimization and pollution source identification are demonstrated. These new findings will provide a convenient tool for catchment water quality management.