###### Abstract
Firstly, it is to increase the TOPMODEL model structure, such as the excess infiltration runoff module [52; 53], snowmelt runoff module [54], solute transport module [55], canopy interception [56], and groundwater evaporation module [57]. Secondly, it is to replace the assumptions of the original model, for example, introducing the Green-Ampt infiltration model to replace the original model [58], introducing the power function model to replace the original exponential function model [58; 59; 60; 61; 62], introducing the motion wave model [63]. Thirdly, it is to simplify the model [64]. From this, we know that although there is much research on the TOPMODEL model, most still use the isochrones method [65] or the instantaneous unit hydrograph [66] to calculate the confluence module, which needs to be improved. Watershed confluence is a very complex physical process. The Saint-Venant equations derived from the law of conservation of mass and energy are the theoretical basis of confluence calculation. On the basis of the Saint-Venant equations and the Nash linear reservoir, the \(n\)-order linear differential equation with constant coefficients and the famous Nash instantaneous unit hydrograph of the confluence system can be obtained by continuous calculation [67]. The Nash instantaneous unit hydrograph is a generalization model with physical concepts, rigorous mathematical derivation, and deep theory. Recently, a relatively complete confluence theory has been formed with the rapid development and gradual improvement of the Nash instantaneous unit hydrograph [68; 69; 70; 71; 72; 73]. The Nash instantaneous unit hydrograph is based on integer order differential equations, which mathematically reflect the local properties of a function at a certain point. They reflect the short-range memory characteristics of the catchment in the confluence system. However, the hydrological response of many natural systems involves multiple scales. The movement process of water flow has historical dependence and path dependence, characterized by trailing characteristics, that is, the long-term memory effect. Fractional calculus is a non-local operator expressed in differential-integral form, which is very suitable for representing processes with memory and hereditary properties. Recently, it has been widely concerned, developed, and successfully applied in many fields [74; 75; 76; 77; 78; 79]. Fractional calculus has become a key research field due to its unique advantages and irreplaceability. The application of fractional calculus theory in the hydrology field is mainly reflected in the solute transport in underground aquifers [80; 81; 82], unsaturated soil water transport [83; 84; 85], solute transport in soil [86; 87], and so on. Given the long memory of the confluence system, the theory has been applied to the confluence calculation process. On the basis of fractional calculus theory and linear reservoir theory, fractional instantaneous unit lines are derived and applied to the simulation of the rainfall-runoff process [88]. For basins with multiscale characteristics, such as karst and hilly areas, [PERSON] et al. [89] proposed to establish the response model of rainfall-runoff by using the infinite characteristic time transfer function with a long memory, whose governing equation is a fractional differential equation. This model incorporates the long memory characteristics of runoff, and its accuracy is significantly improved compared with the commonly used linear or nonlinear reservoir models. Because of the heterogeneity of the bus system, the movement of surface runoff shows abnormal behavior scale dependency, and [PERSON] et al. [90], on the basis of the wave equation of river flood routing, used the operator decomposition method to establish the river confluence of fractional differential equations and numerical simulation. The results showed that the model can accurately reflect nonlocality and tail confluence characteristics.