2.2.1. Time-Area Method for Overland Routing

Isochrones are defined as the contours of equal travel time to the catchment outlet, and the travel time zone is the area between two adjacent isochrones [67]. The isochrones of the whole basin are calculated, and then these are adjusted within each sub-catchment, as shown in Figure 3. It is assumed that the area of each time zone is ΔA1, ΔA2, ···, ΔAn, and their corresponding travel time is τ1, τ2, ···, τ*n*, respectively. Then, we set Δτ = τ*<sup>i</sup>* − τ*i*−*<sup>1</sup>* (i = 1, 2, ···, *n*). It is assumed that *m* is the time of runoff generation in hours, Δ*Rr* (*r* = 1, 2, ... , *m*) is the runoff depth in m, and the runoff discharge of the outlet at time step *j* is:

For *n* ≥ *m*

$$\Delta Q\_{j} = \frac{1}{\Delta \tau} \sum\_{r=1}^{k} \Delta R\_{r} \Delta A\_{j-\mathbf{r}+1\prime} \ (j = 1, 2, 3, \dots, n+m-1) \tag{1}$$

If *j* < *m*, *k* = *j*; if *j* ≥ *m*, *k* = *m*, and if *j* > *n*, Δ*A* = 0. For *n* < *m*:

$$Q\_{j} = \frac{1}{\Delta \pi} \sum\_{r=1}^{k} \Delta A\_{r} \Delta R\_{j-r+1 \prime} \ (j = 1, 2, 3, \dots, n+m-1) \tag{2}$$

If *j* < *n*, *k* = *j*; if *j* ≥ *n*, *k* = *n*, and if *j* > *m*, Δ*R* = 0.

For travel time zone *i*, its runoff discharge at time step *j* can be calculated as follows:

$$Q\_{i,j} = Q\_{i+1,j-1} + \Delta R\_j \Delta A\_i \tag{3}$$

where *Qi,j* is the outflow from travel time zone *<sup>i</sup>* at time step *<sup>j</sup>* (m3); and *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).

Unlike runoff routing, sediment transport simulation gave consideration to the sediment particle size and the hydraulic characteristic of the basin. It was calculated using Equation (4), as suggested by Williams [68]:

$$RY = \sum\_{i=1}^{n} Y\_i e^{-BT\_i \cdot \sqrt{D50\_i}} \tag{4}$$

where *RY* is the sediment yield for the entire basin (t); *Yi* is the sediment yield for the sub-catchment *i* before routing to channel (t); *B* is the routing coefficient; *Ti* is the travel time from the sub-catchment *i* to the basin outlet (h); and *D*50*<sup>i</sup>* is median particle diameter of the sediment for sub-catchment *i* (mm).

**Figure 3.** Isochrones of Pianguanhe's sub-catchments.
