2.2.3. Consideration of Terrace Units in the Time-Area Method

The revised time-area method simplifies a level terrace as a dynamic water tank, and its storage capacity is the product of the terrace area and the embankment height (Figure 5a,b). In this study, we made the assumption that the stream flow in a terrace should only be considered as the overflow, regardless of the drainage discharge. Namely, when the water trapped by the terrace exceeds the terrace storage capacity during heavy rainfall events, the surplus water will overflow (Figure 5c,d).

**Figure 5.** Schematic diagrams of the hydrological processes and flow distribution of the level terrace unit. (**a**) Section profile of a level terrace. (**b**) The generalization of the level terrace in (**a**). The flow distribution assumes that the soil profile of the terrace is deep enough for subsurface flow generation. The influence of the flow in different scenarios is listed from (**c**,**d**). *Aterrace* is the area of terrace; *d* is embankment height of terrace; *P* is rainfall; *f* is the infiltration; *Ql* is the interflow; *Qj*−<sup>1</sup> is the inflow; *Qd* is the overflow; *Tj* is the water stored in the terrace at the present time step *j*; *Tj*−<sup>1</sup> is the water stored in terrace at previous time *j* − 1; and *Qj* is the outflow.

Referring to Equation (3), when there are terrace units occurring in time zone *i*, the outflow *Qi,j* from time zone *i* at time step *j* is revised, as follows:

$$Q\_{i,j} = Q\_{i+1,j-1} + \Delta R\_j \Delta A\_i - \Delta T\_{i,j} \tag{5}$$

where *Qi*<sup>+</sup>1,*j*−<sup>1</sup> is the outflow from time zone *<sup>i</sup>* <sup>+</sup> 1 at time step *<sup>j</sup>* <sup>−</sup> 1 (m3); <sup>Δ</sup>*Ti,j* is the increased water ponded in terraces that are located in time zone *i* at time step *j* (m3). When Δ*Ti,j* equals 0, the storage capacity of the terraces is filling up and they can no longer have a retention function.

Before filling up, the current terrace storage volume is the sum of its previous storage volume and the current terrace watershed inflows. Due to the possible distribution of several terraces in a time zone and due to the fact that each terrace may correspond to multiple cells, the statistics of the water interception of each terrace requires a large amount of calculations. In order to simplify the complex problem, we estimated the terrace watershed inflows as a percentage of the runoff generation in the time zone, and the percentage equals the area ratio of all terraces control area to the time zone. Then Δ*Ti,j* is calculated as follows:

$$
\Delta T\_{i,j} = T\_{i,j} - T\_{i,j-1} \tag{6}
$$

$$T\_{i,j} = \begin{cases} \begin{array}{c} T c\_i \end{array} & \text{(Filling up)}\\ T\_{i,j-1} + \Delta R\_j \Delta A\_i \times T u\_i & \text{(Unfilling up)} \end{array} \tag{7}$$

$$T u\_i = \Delta A\_{T \text{control},i} / \Delta A\_i \tag{8}$$

$$T\mathbf{c}\_{i} = \Delta A\_{\text{termnc},i} \times d\_{i} \tag{9}$$

where *Ti,j* is the amount of stored water in all terraces in time zone *<sup>i</sup>* at present time step *<sup>j</sup>* (m3); *Ti*,*j*−<sup>1</sup> is the amount of stored water in all terraces in time zone *i* at previous time step *j* <sup>−</sup> 1 (m3); *Tci* is the maximum water storage of all terraces in the time zone *i* (m3); *Tui* is the ratio of all terrace control area in the time zone *i*; Δ*ATcontrol,i* is the area of all terrace control area in time zone *i* (m2); Δ*Aterrace,i* is the total area of terraces in time zone *i* (m2); and *di* is embankment height of terrace (m). As runoff is the main carrier of sediment, the increased sediment stored by terraces in time zone *i* at time step *j* equals the sediment yield in time zone *i* at time step *j* multiplied by the runoff trapped rate of terraces in time zone *i* at time step *j* (namely, the ratio of Δ*Ti,j* to Δ*Rj*Δ*Ai*).

Taking Equation (6) and Equation (7) into Equation (5), we can get the outflow *Qi,j* as follows:

$$Q\_{i,j} = \begin{cases} Q\_{i+1,j-1} + \Delta R\_j \Delta A\_i - T c\_i + T\_{i,j-1} & \text{(Filling up)}\\ Q\_{i+1,j-1} + \Delta R\_j \Delta A\_i \times (1 - T u\_i) & \text{(Unfilling up)} \end{cases} \tag{10}$$
