Taking into account the nonlinearity of a hydrological system, a rainfall–runoff process can be expressed by a second-order Volterra functional series (Xia 1991; Xia et al. 2005; Carassale & Kareem 2009) as: where y is the system output (e.g., runoff), x is the system input (e.g., rainfall), h is a linear response function, g is a nonlinear response function, L is the system's memory length, t, τ, σ are time variants. Equation (1) describes the nonlinear responses of a hydrological system, but is, in general, not analytically tractable.

Xia et al. (1997) proposed a relatively simplified nonlinear systematic rainfall–runoff model (i.e., the TVGM), equivalent to a special form of the complex second-order nonlinear Volterra model. The hydrological processes of the TVGM are given by: where R is the rainfall excess, G is a time variant gain coefficient related to soil moisture conditions, X is the hydrological system input (i.e., rainfall), U is the response function, and Y is the hydrological system output (i.e., runoff). Here G can be written as: where g₁ and g₂ are two parameters related to watershed properties that are not time variant, API is the time variant antecedent precipitation index, which can be simulated as a response of a simple linear reservoir to the rainfall X (Ahsan & O'Connor 1994; Xia et al. 1997, 2005) by: Here, U₀ is a response function given by: where K_e is a parameter indicating the rate of soil moisture recession.

Substituting Equation (5) into Equation (4), and then combining with Equation (2), we have: Substituting Equation (7) into Equation (3), we obtain an isomorphic representation of the second-order Volterra model as: As a simple nonlinear system approach, the TVGM can be used for hydrological forecasting with reasonable accuracy even if catchment evapotranspiration or soil moisture data are unavailable. The TVGM was evaluated at the outlet of Bailianhe Reservoir, which controls a relatively small area of 1,797 km² with negligible human interference and close to Huai River Basin with climatic similarity (marked by a star in Figure 1). The model was applied in this area to simulate a unimodal flood event on July 18, 1982 and a bimodal flood event on August 19, 1982. By comparing the results of simulation with observations as shown in Figure 3, there is a good agreement for the high discharge range, while larger discrepancy is found in low streamflow periods. The reason is that the TVGM does not generate runoff without rainfall, as Y vanishes if X is zero according to Equation (8). This is not physically reasonable because baseflow still exists even if there is no precipitation. As indicated in Napiórkowski (1992), negative outflows can also be obtained through this model. Therefore, a modification is necessary in order to improve the predictability of the TVGM in low streamflow periods.