Assessment of Drainage Network Configuration Using Gibbs’ Model with Increased Number of Flow Directions

Abstract

Stochastic channel network modeling is an informative tool to replicate river networks for the purpose of understanding the variability of geometry and distinguishing observed patterns in networks. In contrast, the application of stochastic network models to artificial or urban drainage networks is not common despite their practical implications for engineering purposes. Gibbs’ model is a useful tool to investigate the network characteristics of drainage networks and also has an advantage to produce alternative networks with the same network characteristics due to its stochastic nature as a network model. This study utilized Gibbs’ model to estimate the network configuration of urban drainage networks in Seoul, South Korea, with an increased number of flow directions from four (N, E, S, W) to eight (N, NE, E, SE, S, SW, W, NW), which enables improved accuracy. Based on that, the network configuration affects the hydrological response directly, the results of this study imply a new design criterion which concerns the connection of upstream and downstream subcatchments with different network configurations to mitigate downstream flooding. Additionally, in order to evaluate the model’s usefulness to be employed to estimate the hydrologic responses of actual drainage networks, the width function-based IUH (WFIUH) was applied to a highly urbanized and pipe-networked catchment of the Shinweol Watershed in Seoul, South Korea.

1. Introduction

The river network configuration has a close relationship with the hydrologic response of catchments [1,2]. Likewise, urban drainage networks are essential for managing stormwater in cities, mitigating flood risks, and maintaining water quality. The importance of urban drainage networks has recently been widely recognized, and urban drainage network optimization is a critical area of research aimed at enhancing the efficiency and resilience of urban water management systems. Studies are being conducted on the optimal configuration of these networks to prepare for climate change and mitigate flood damage using structural optimization [3], topological analysis [4] or multi-objective network optimization [5].

Stochastic river network modeling has proven to be an invaluable tool in simulating river networks, thereby facilitating the exploration of their geometric variability and aiding in the characterization of observed network patterns [6,7,8]. These models have been extensively employed in the study of geomorphological evolution [9,10] and the topological properties of river networks [11]. Furthermore, stochastic network models have been applied for the analysis of metapopulation stability within branching river networks [12], as well as for the investigation of hydrological responses within randomly self-similar networks [13,14,15,16].

The initial approaches to modeling river networks involved the stochastic generation of networks on orthogonal matrices. Subsequently, the theoretical foundations of these models were articulated with greater precision and simplicity [17,18,19]. At the core of these models lies the assumption that all potential flow directions are equally probable, a principle hence referred to as the uniform model. Nonetheless, this assumption of uniform probability distributions resulted in an increased sinuosity in the simulated networks, rendering them less representative of actual natural riverine systems.

Troutman and Karlinger [20] proposed an enhanced model characterized by a singular parameter, β, which governs the overall sinuosity of a simulated network. This model is derived from Gibbs’ measure as outlined by Kinderman and Snell [21], in which the internal energy of a system is represented inversely by the power of an exponential function. Within this framework, an increase in internal energy corresponds to a decreased probability of state transition, and vice versa. In their innovative approach, they conceptualized the internal energy in terms of network sinuosity, incorporating the β parameter as a multiplicative factor [20]. Consequently, β modulates the transition probabilities, influencing whether the network evolves towards greater or lesser sinuosity. Specifically, as β decreases, the model predicts increased sinuosity, approaching the characteristics of a uniform model when β approaches zero. Conversely, as β increases, the network’s sinuosity diminishes, aligning with the Scheidegger model [22], which predicts no sinuosity when β theoretically approaches infinity.

Gibbs’ model serves as an effective framework for categorizing networks based on their overall sinuosity. In their 1992 study, Karlinger and Troutman [23] assessed the β value for river networks in Montana, observing a range between 0.2 and 2.21, with a mean of 0.92. This finding indicates that Montana’s river networks exhibit relatively low sinuosity and demonstrate comparatively efficient drainage. The sinuosity of a drainage network is critically linked to the geomorphologic attributes of a catchment and consequently influences its hydrologic response, as noted by previous studies [24,25]. Furthermore, the stochastic nature of such networks offers the potential benefit of proposing alternative network configurations, as discussed by Seo and Schmidt [26]. This aspect underscores the significance of Gibbs’ model in both theoretical and applied hydrological research.

Despite their significant potential for engineering applications, stochastic network models have seldom been applied to artificial drainage systems. Seo and Schmidt [27] investigated the relationship between network properties and the impacts of rainstorm movement on peak flows. They employed Gibbs’ model to categorize catchments based on network characteristics, which directly correlate with how rainstorm movement influences peak flows. When applying this model to twelve urban drainage networks in Chicago, they discovered that several networks exhibited a β value of 10⁻² or lower, indicating less efficiency compared to river networks. For instance, Karlinger and Troutman [23] reported that river networks typically demonstrate an average β value on the order of one. Further extending this line of inquiry, Seo et al. [28] conducted a study on network characteristics in Seoul, South Korea, and found cases where the β value reached as low as 10⁻⁴. The findings from previous studies suggest that artificial drainage networks, particularly in level regions such as Chicago, may exhibit high sinuosity and potentially lower efficiency compared to natural drainage systems. This observation is paradoxical, given the conventional perception of artificial systems as more efficient owing to their reduced surface roughness and accelerated flow velocities. However, the overall efficiency of these systems, when considered in the context of their entire network configuration, appears to be compromised. The recognition that artificial drainage systems can display a diverse range of network characteristics underscores the necessity for a meticulous and extensive examination of current drainage infrastructures. This is imperative given the implications for flood intensity and subsequent risk management strategies. Additionally, the characteristics of these networks are intrinsically linked to the resilience and sustainability of urban areas, which are increasingly vulnerable to anomalies anticipated under climate change scenarios, as highlighted in some studies [29,30,31].

This study utilized Gibbs’ model to estimate the network configuration of urban drainage networks in Seoul, South Korea, increasing the number of flow directions from four (N, E, S, W) to eight (N, NE, E, SE, S, SW, W, NW), which enables improved accuracy. Since the model is grid-based and a number of simulations for alternative networks are required, an actual drainage network should be reconstructed on a lattice. Previous studies on Gibbs’ model were based on four flow directions (4D), which interrupted the precise reconstruction of a drainage network close to the actual network. Furthermore, this study investigated the network configuration of the study area more in detail, focusing on the upstream and downstream connection of the subcatchment with relevance to actual flooding records in 2010 and 2011. In order to show the model’s usefulness to reproduce the hydrologic response of a real drainage network, width functions and hydrographs were compared to the measured flow data for a highly urbanized and pipe-networked catchment in Seoul, South Korea.

2. Application of Gibbs’ Model

2.1. Uniform, Scheidegger and Gibbs’ Model

Gibbs’ framework assigns probabilities based on Gibbs’ measure. Any positive measure exhibiting the characteristics of a Markov random field is considered Gibbs’ measure provided an appropriate energy function is identified, as established by [21]. In subsequent developments, Karlinger and Troutman [20] substituted the internal energy function with the sinuosity of a network, applying a multiplicative factor β. The state space for a defined Markov chain comprises spanning trees, denoted as S. A tree is defined as a connected graph with the unique property that any two vertices are linked by a single, simple path, while a spanning tree is a tree that connects all vertices in the network without forming any loops (cycles). Let s be an element of S, and consider two trees, s₁ and s₂, to be adjacent if one can be transformed into the other by altering the configuration at a randomly selected vertex in s₁ to define a new potential flow direction that avoids loops, thus forming a new spanning tree s₂. The transition probability from s₁ to s₂, denoted as R_s1s2, is defined accordingly [20]:

$$R_{{\mathrm{s}}_1{s}_2}\left\{ \begin{array}{c}{r}^{-1}\min \left\{1,\exp \left(-\beta \left[H\left(s_2\right)-H\left(s_1\right)\right]\right)\right\} s_2\in N\left(s_1\right)\\ 1-\sum _{s\in N\left(s_1\right)}{R}_{s_{1}s} s_2=s_1 \\ 0 otherwise\end{array}\right.$$

(1)

where N(s₁) is the set of trees adjacent to s₁ and β is a parameter that controls how much of the current that the sinuosity of the network affects in the generation of the new spanning tree, s₂. The degree r means the maximum number of directions that can be selected. The parameter H is a measure of the sinuosity of a given spanning tree, s [20]:

$$H\left(s\right)={\displaystyle \sum _{d\in D\left(B\right)}{\xi }_s\left(d\right)}-{\displaystyle \sum _{d\in D\left(B\right)}{\xi }_B\left(d\right)}$$

(2)

where s is the spanning tree, d is the point of the finite and connected graph B and D(B) is the set of the total points of B. ξ_s is the distance to the outlet along s from v, while ξ_B is the shortest distance to the outlet from d.

The methodology utilized in this study to construct a network for a given value of β follows the process outlined by Barndorff-Nielsen [3]. Initially, a uniform network is created using the uniform distribution model, denoted as s₁. A node v is randomly selected within this network, and a new directional flow from v is established to form an adjacent network, termed s₂. It is important to note that the uniform model postulates that all directions are equally probable during network formation, as supported by the findings of [14] and further discussed in [16]. Second, the newly formed network s₂ is evaluated for the presence of any loops. If no loops are detected, a random variable x is drawn from a uniform distribution over the interval [0, 1]. The condition to accept s₂ as the new network is that x must be less than e^−β[ΔH], where ΔH is equal to H(s₂) − H(s₁). Third, let s₂ be equal to s₁ and repeat the above steps enough times until the resulting tree has a distribution close to the stationary Gibbs’ distribution (Figure 1).

This study utilized the width function, which is defined as the catchment area at a distance from the outlet [32]. The width function can be regarded as the direct interpretation of a network response [33]. The width function can be converted to a hydrograph easily, provided with channel flow characteristics such as a celerity and a flow diffusion coefficient, and both are functions of channel geometry [34]. The width function and the area function can be differently defined based on channelization [35], but the width function basically represents the distance-area function [36]. For a given flow distance from the outlet, ξ, the width function can be defined as follows:

$$L\left(\xi \right)=N\left\{d\left(\xi \right)\right\}, \xi ={\xi }_1,{\xi }_2,\dots , {\xi }_n$$

(3)

where L(ξ) is the width function for a drainage network. The width function can be obtained by counting the number of grid points with distance ξ from the outlet. The parameter d(ξ) is the grid point for which the distance from the outlet is equal to ξ. ξ_n is the longest distance from the outlet. A width function combined with a rainfall excess at time t is defined as follows [27].

$$L_{\beta }^I\left(\xi ,t\right)={\displaystyle \sum _{i,j}{r}_{ij}\left(\xi ,t\right)}, \xi ={\xi }_1,{\xi }_2, \dots , {\xi }_n$$

(4)

where r_ij(ξ, t) is the rainfall intensity at a grid point (i, j) located at a distance from the outlet ξ at time t.

The geomorphological features of the river network, such as bifurcation ratio and self-similarity, are well known and have led to the development of GIUH (Geomorphologic Instantaneous Unit Hydrograph) [24]. Differences in drainage network configuration can result in varied hydrologic responses to the same rainfall event. An effective drainage network can be defined as one in which the average flow distance from the outlet is shorter, allowing it to convey flows to the outlet in a shorter time period, resulting in greater and sharper peak flows compared to a less effective one [37].

In this regard, it is essential to identify the effectiveness of a drainage network in terms of flow distance. Gibbs’ model can serve as a valuable tool for this classification. As shown in the previous section, Gibbs’ model can encompass both the uniform and the Scheidegger model, depending on the parameter value of β. When the value of β approaches zero, it represents the uniform model, which is highly sinuous, whereas as the value of β tends toward infinity, it represents the Scheidegger model with no sinuosity, as illustrated in Figure 1. Previous studies revealed that artificial drainage networks are not necessarily effective and can exhibit high sinuosity [27,28].

In a highly developed urban catchment, the drainage network is an important factor that directly affects the hydrologic response of the catchment, similar to the role played by the river network in a natural watershed. The configuration of a network directly affects the shape of hydrographs from the network. Figure 1 shows that the uniform model is the most sinuous and has the longest flow distance from the outlet. In contrast, the Scheidegger model has the shortest distance from the outlet. Therefore, the response from the uniform model is more dispersed compared to that from the Scheidegger model and the peak flow is maximized for the Scheidegger model. Figure 1 shows that Gibbs’ model becomes closer to the uniform or Scheidegger model depending on the value of β. Hence, the sinuosity of Gibbs’ model depends on the value of β as the peak response of Gibbs’ model is also dependent on β. Figure 2 shows the width function from Gibbs’ model, as shown in Figure 1, where the peak response is highest with β equal to 10² and lowest with β equal to 10⁻⁴.

2.2. Study Area for Application of Gibbs’ Model

In this study, the network characteristics of drainage networks from the nine subcatchments (6.50 km² in total) in the Shinweol Basin in Seoul were investigated, as shown in Figure 3. The Shinweol Basin is a highly developed area with a mixture of residential and commercial areas. Drainage loads in the basin are mostly conveyed through underground networks of drainage structures, such as pipes and culverts that replaced open channels and streams throughout the history of the city’s development. The Shinweol Basin, including the subcatchments for this study, suffered from severe flood events in 2010 to 2011. The Seoul Metropolitan Government implemented several structural measures including deep tunnels with diameters up to 10 m and depths of 50 m to store flows. In addition, design criteria and design standards for urban drainage structures were re-evaluated and re-established. The study used a GIS database of the drainage network from the city government that includes pipes with diameters greater than or equal to 300 mm and it also includes multiple box structures with a single width up to 4.5 m.

3. Results and Discussion

3.1. Assessment of Drainage Network Configuration Using Gibbs’ Model

To classify the actual drainage network, it was reconstructed on a lattice, and the width function obtained from this reconstruction was used to estimate the value of β by comparing it with the width function from the synthetic network generated by Gibbs’ model. One of the advantages of Gibbs’ model is that it can generate as many realizations as needed given a specific parameter value of β. This advantage can provide multiple choices for designers and planners to select the most suitable network for their purpose. The width function obtained from the actual drainage network can be converted to a hydrograph using a solution of the advection–diffusion equation of flow perturbation as a routing function combined with the width function [34]. Additionally, the hydrographs from Gibbs’ model with four directions and eight directions were compared to demonstrate the improvement resulting from the increased number of flow directions in Gibbs’ Model.

The value of a width function is equal to the contributing area of the entire catchment given a distance from the outlet of the catchment. Figure 4 shows the Scheidegger and uniform models generated with 8D and 4D, respectively, and the width functions for those corresponding Gibbs’ models. The simulation shows the basic differences between two models. First, the maximum flow distance of the 4D Scheidegger models is 15, which is greater than the maximum distance, which is 11 ($\approx 7\times \sqrt{2}+1)$, of the 8D Scheidegger model. Second, the distance that contributes to the peak width function values is eight, which is the same for both 4D and 8D Scheidegger models. However, the number of grids that contribute to the peak width function is 11 for the 8D model, whereas it is 8 for the 4D model. This is important in that the increase in the peak width function implies the increase in peak hydrologic response for the corresponding network. We can expect a larger difference in peak response for a small-scale reconstruction of a diagonal direction-dominated drainage network. It is quite obvious that 8D models capture the network characteristic of a specific drainage network to be modeled better than 4D models because there are many river networks or urban drainage networks in the diagonal direction in reality.

Figure 4 shows that the uniform models have delayed peak responses compared to the Scheidegger model. However, the uniform models show different results compared with the Scheidegger models. The width functions of the Scheidegger model were the same for 8D and 4D because the flow path in the Scheidegger model always takes the shortest distance to the outlet and the flow directions are downstream directions only. However, the realization of the uniform model is not the same because the model is based on a uniform probability for each flow direction. Figure 5a,b depict the 1000 realizations of the uniform model for 8D and 4D, respectively. The results for the averaged width function show that the averaged peak is higher for 4D (4.356) instead of 8D (3.799), whereas it was not for the Scheidegger model. Figure 5a,b show that the uniform model with 8D has longer flow distances compared with 4D, which, in turn, results in a lower peak with 8D. The results imply that increased flow directions lead to an increased number of realizations for both the Scheidegger and uniform model. For the Scheidegger model, the allowance of flow direction in the diagonal direction makes the model take shorter paths, which contributes to the increased peak response of the network. In contrast, the increased number of flow directions offers increased chances to flow upstream for the uniform model, which ends up with increased flow lengths and a decreased peak, as shown in Figure 5.

3.2. Procedures to Estimate β of the Shinewoel Watershed Using Gibbs’ Model

Network configuration is one of the most important geomorphological characteristics of a catchment and determines the shape of hydrographs [34,38]. In a general sense, a less sinuous network gives rise to higher peak flows and less drainage time and vice versa. Moreover, Da Ros and Borga [37] showed that the network configuration determines the impact of rainstorm movement on peak flows. If a drainage network is more sinuous, the peak flow of the network is less affected by moving storms and vice versa. Therefore, it is important to identify a network configuration of a drainage network more accurately. Gibbs’ model can be utilized to categorize drainage networks [20,27,37] depending on the value of β, as was shown in previous sections.

Figure 6 shows the realization of Gibbs’ model for a given value of β for a subcatchment (Mokdong 2) in the study area. The left-most networks are actual networks reconstructed on grids with eight (8D) and four flow directions (4D). Given the same boundary and outlet, Gibbs’ model was generated for the catchment. As the value of β decreases, the realization of Gibbs’ model is more sinuous, whereas the value of β increases, the realization of the model becomes less sinuous. Due to the stochastic nature of Gibbs’ model, every realization is different from others. The realization using an increased number of flow directions better mimics the original networks, as shown in Figure 6. We generated 100 networks and averaged the width functions from those. Then, the averaged width function was compared with the width function from the actual network. The value of β that yields the closest width function to the actual network is determined to be a representative value of β for the target catchment. Figure 7 depicts the procedure to estimate the value of β for the study subcatchments. Solid line shows the actual width function which is compared with that from the simulated networks (dashed).

The results showed that Gibbs’ model with an increased number of flow directions from four (4D) to eight (8D) can contribute to categorizing and identifying the network configuration of drainage networks more accurately.

Table 1. Estimated value of β for the subcatchments.

Subcatchment	β		Nash–Sutcliffe Efficiency (E)
	4D	8D	4D	8D
Shinweol1	10−1	101	0.852	0.934
Shinweol2	103	103	0.546	0.785
Shinweol3-1	10−3	101	0.799	0.804
Shinweol3-2	10−1	10−2	0.931	0.935
Mokdong2	10−2	10−3	0.682	0.674
Hwagok2	10−3	10−3	0.889	0.932
Mokdong3	103	103	0.836	0.933
Shinjeong_highland	10−4	10−3	0.544	0.503
Gochuk2	101	102	0.735	0.879

lists the estimated value of β for the target subcatchments. The Nash–Sutcliffe efficiency shows that the networks simulated by Gibbs’ model are closer to the actual drainage network when the number of flow directions is increased (8D) for most subcatchments. For example, the efficiency of Mokdong3 (Figure 7) increased from 0.836 with 4D to 0.933 with 8D. This is due to the fact that the formation of the drainage network of Mokdong3 is primarily made up of diagonal directions. However, two subcatchments have better efficiency values with 4D, which is due to the formation of the drainage networks, which are closer to the perpendicular directions.

In addition, the results show that the estimated value of β can be different for the same catchment when four and eight flow directions are allowed. For example, the estimate of β for the subcatchment Shinweol1 is equal to 10⁻¹ when four flow directions are used. However, the estimation changes to 10¹ when eight flow directions are used. For the subcatchment Shinweol1, Figure 8 shows that the Nash–Sutcliffe efficiency increases as β increases. For 4D, the Nash–Sutcliffe efficiency is highest when β is equal to 10⁻¹ but slightly decreases as β increases further. The behavior of the Nash–Sutcliffe efficiency depending on β is similar for 8D compared to 4D. However, the efficiency has its maximum value with β equal to 10¹. These differences originate from the difference in flow distance from the outlet, as described in Section 3.1.

3.3. Importance of Network Configuration in Floods

This study investigated network configurations, and Figure 8 shows the resulting values of β for the subcatchments of the study area. The configuration of the network shows that upstream subcatchments have efficient drainage networks, whereas the downstream subcatchments have less efficient ones, in general, regardless of the number of directions applied. The flow direction depicted in Figure 8 reveals the connection of upstream and downstream subcatchments. Focusing on the flooded area in 2010 and 2011, the flow starts from the uppermost catchment No. 1 (Shinweol1) and runs down to catchment No. 2 (Shinweol2), No. 3 (Shinweol3) and eventually to No. 4 (Hwagok2). One of the findings here is that an efficient network, which can drain water faster than less efficient networks, does not guarantee a no-flooding condition because urban catchments are connected to each other from upstream to downstream. An efficient drainage network configuration can convey stormwater faster from a site to the outside of it but can also be a disaster for downstream areas. The flooding records indicate that the flooded areas in 2010 and 2011 are concentrated in those connecting areas, as shown in Figure 8. In particular, flooded areas are concentrated on the connection area of subcatchment 2 (more efficient, β 10³) to subcatchment 3 (less efficient, β 10⁻¹). Subcatchment 4 (β 10⁻³) also suffered from flooding, and upstream subcatchment 2 has a more efficient network configuration.

These findings suggest a new concept for alternative measures to reduce peak flows by combining different network configurations for the upstream and downstream subcatchments, as shown as a conceptual diagram in Figure 9. Figure 9a shows a drainage system which has an efficient and fast-conveying drainage network at upstream areas but less efficient network at downstream areas. The fast stormwater conveyance is accumulated with the delayed response downstream to produce higher peaks. The hydrograph at the outlet combining those two hydrographs results in higher peaks compared to Figure 9b. This case corresponds to the study area of Shinweol2 (No.2)-Shinweol3 (No.3) and Shinweol3 (No.3)-Hwagok2 (No.4). In contrast, Figure 9b depicts an opposite case, where a drainage system has a less efficient network and is sinuous at upstream areas and has a more efficient network downstream. This combination of the upstream and downstream networks results in a reduced peak compared to the previous case. The downstream network drains stormwater faster than the peak response from the upstream, which results in dispersed response of the whole catchment and lower flow peaks.

3.4. Application to a Study Area for Flow Estimation Using the Width Function-Based IUH

In this study, we utilized a solution of the advection–diffusion equation of flow perturbation as a routing function combined with the width function [34]. For a given time step i, the width function associated with the spatial rainfall distribution is applied to obtain the hydrologic response at the outlet of a study subcatchment as follows:

$${\tilde{h}}_i\left(t\right)={\displaystyle \sum _{j=1}^n\frac{\Delta x}{\sqrt{4\pi D{t}^3}}{L}_i^I\left(j\Delta x\right)\exp \left[-\frac{{\left(j\Delta x-ct\right)}^2}{4Dt}\right]} \Delta x$$

(5)

where ${\tilde{h}}_i$ is a discrete response function and L^I_j is the width function associated with rainfall at time step i [24,30]. Δx is the grid size, c is the celerity, and D is the diffusion coefficient. The width function based-IUH has been applied for rivers and urban drainage networks and has proved its applicability and simplicity [27,31,33,39,40,41]. The celerity and the diffusion coefficients are physically determined from the geometry of the conduit cross section [31]. In this study, the celerity values are determined to be 0.34 m/s and 0.41 m/s and the diffusion coefficients are 1600 m²/s and 1000 m²/s for observation station 1 (Shinweol 3) and 2 (Hwagok 2), respectively.

The width function combined with the temporal rainfall distribution is transformed to a hydrograph using a discretized convolution integration (Equation (5)). Before that, we need to reconstruct the original drainage networks on a lattice in order to obtain the width function [27,31,33]. Figure 10 shows the reconstructed drainage networks with 4D and 8D, respectively, to estimate the flow hydrographs at station 1 (for the subcatchment Shinweol1 and Shinweol3) and station 2 (for the subcatchment Shinweol1, Shinweol3 and Hwagok2).

The results showed that the flow estimation based on the reconstructed drainage network with 8D was improved compared with the flow estimation with 4D. Figure 11 shows the resulting hydrographs for station 1 and station 2 from the reconstructed drainage networks compared with the observed flows and also the hydrographs from the detained SWMM. First, the results show the estimated hydrographs with the WFIUH close to the observed flows and the hydrographs from the detailed SWMM. The WFIUH is a semi-distributed and efficient model with a small number of parameters. However, the performance of the model is as good as the heavy and detailed SWMM, which has 7120 links (pipes) and 6821 junctions (nodes). Furthermore, the flow estimation with 8D showed better results compared with 4D, as shown in

Table 2. Nash–Sutcliffe efficiency of simulated hydrographs compared with observations from reconstructed networks with 4D and 8D.

Flow Directions	Gage Station 1		Gage Station 2
	2 July 2013	12 July 2013	2 July 2013	12 July 2013
4D	0.83	0.84	0.84	0.68
8D	0.83	0.84	0.89	0.75
SWMM	0.81	0.83	0.87	0.78

and

Table 3. Comparison of peak flows (m³/s) from reconstructed networks with 4D and 8D.

Flow Directions	Gage Station 1		Gage Station 2
	2 July 2013	12 July 2013	2 July 2013	12 July 2013
4D	9.38	14.85	14.75	22.06
8D	9.83	15.19	16.41	24.29
SWMM	11.19	17.38	16.43	24.66
Observed	8.13	16.28	17.51	29.68

. Table 2 lists the Nash–Sutcliffe efficiency of estimated hydrographs from 4D and 8D reconstructed drainage networks compared with the observed flow. The results show that the Nash–Sutcliffe efficiencies with 8D drainage networks are enhanced compared to those with 4D drainage networks, especially for Gage station 2. For example, the efficiency improved from 0.84 (4D) to 0.89 (8D) for the storm event on 2 July 2013 and also improved from 0.68 (4D) to 0.75 for the storm event on 12 July 2013.

Furthermore, the results showed that the estimated peak flows from the reconstructed networks with 8D (−2.56% error on average compared to observation) are closer to the observed peak flows compared with those with 4D (−8.71% error on average) for the test event cases. Table 3 lists the estimated peak flows from WFIUH and also from the detailed SWMM (+5.31% error on average). The peak flow estimation from 8D is close to the observed flows as well as the results from the detailed SWMM model. The hydrographs show the sharp peak flows obtained from the 8D network, whereas the peak flows obtained from the 4D network are lower. The sharp peak flows from the 8D network are close to the observation and the results from the detailed SWMM. These results indicate that an increased number of flow directions for reconstruction of drainage networks on grids can contribute to the improvement of flow estimation.

4. Conclusions

Stochastic channel network models have been a useful tool to understand the variability of geometry and to distinguish observed patterns in networks. Despite its benefits and engineering implications, the application of stochastic network models to artificial or urban drainage networks is limited. This study investigated the network characteristics of subcatchments in the Shinweol Watershed, Seoul, South Korea, using Gibbs’ model based on four directions and eight directions, respectively. Due to the increased diagonal flow directions, the 8D model improved the assessment of drainage networks in terms of network characteristics. It also improved the estimation of flow hydrographs based on the width function. It is also interesting that the flow difference between the 8D and 4D network model is not as impressive, which means that 4D models are still meaningful considering the increased efforts and computational power required for the construction of 8D models. The investigation of the network configuration of the study area also highlighted the importance of upstream and downstream connections with different network characteristics. These results imply a new concept of a drainage network design criterion as a sustainable flood mitigation measure in the face of increasing hydrological extremes due to climate change.