Based on the original TVGM, here we propose an advanced TVGM by introducing two separate time variant gain coefficients for two runoff types, i.e., quick flow and baseflow, respectively. For quick flow: where  is the effective rainfall for generating quick flow, G_s is a time variant gain coefficient for quick flow and can be expressed as: Similarly, baseflow is calculated by: where R_g is the effective rainfall for generating baseflow, G_g is a time variant gain coefficient for baseflow and can be expressed as: In Equations (10) and (12), coefficients g₁, g₂, g₃, and g₄ are constants for a specific watershed.

In addition, unlike the flow routing process by using a single response function for all effective rainfall in the original TVGM as indicated in Equation (3), here we use two separate response functions for routing of quick flow Y_s and baseflow Y_g, respectively, in correspondence with two time variant gain flow generation mechanisms: where u_s and u_g are the response functions for quick flow and baseflow, respectively. These functions are given by: Here, n is a numerical parameter indicating the capacity of watershed storage, which is equivalent to the number of linear reservoirs; K is a storage-discharge parameter with the dimension of time (Nash 1957; Young & Beven 1991); and K_g is the storage coefficient of ground water with the dimension of time.

Since rainfall data are usually recorded in discrete forms, the analytical forms of instantaneous response functions, i.e., u_s, u_g, need to be practically discretized for a given duration. The quick flow response function u_s, viz. the instantaneous unit hydrograph can be discretized using S-curve method (Sherman 1932; Cleveland et al. 2008), while the baseflow response u_g can be converted into a discrete form through the water balance equation by assuming a linear relationship between ground water storage and discharge (Pedersen et al. 1980; Purcell 2006). With commonly adopted unit hydrograph, S-curve and underground linear reservoir, the quick flow Y_s and baseflow Y_g can be calculated by discrete convolution integrals as: where t is the current time step, T_u is the duration of the u_s, c = A/(3.6 × Δt) is a coefficient to transform the unit of runoff volume from m³ s⁻¹ to mm, A is the catchment area in km², Δt is the time interval in hour, R_s and R_g are effective rainfall in mm, KKG is a coefficient given by KKG = (K_g − 0.5Δt)/(K_g + 0.5Δt).