2.1. Storm Water Management Model
The open source hydrologic model (the SWMM), which was developed by the US Environmental Protection Agency [
29], was selected as the modeling platform for flood simulation in this study. The SWMM is a 1D dynamic rainfall-runoff simulation model for urban watersheds and combined sewer overflow phenomena [
30]. The main implementation principles and ideas fully consider the features of the urban watershed environment [
29]. The SWMM includes four calculation modules: flow generation, transport, extended delivery, and storage/processing. These are used to simulate the surface water production and sinking of urban watersheds, storm sewer network convergence, and water storage and sewage treatment processes [
29]. The rainfall–runoff produced in subcatchments flows into the storm sewer network through manholes and combines with dry-weather flow and ground-water infiltration [
30]. Flow routing from upstream and downstream boundaries of the storm sewer system is governed by the conservation of mass and momentum equations for gradually varied, turbulent, and unsteady flow (Saint Venant) equations [
30,
31]. For more details, see [
30]. For overland flooding simulations with SWMM, the excess volume of overflow junctions flows evenly into a ponded area that can be set to the area within the subcatchment that can be flooded (subtracting the building footprint) [
29].
In this study, flow routing was computed according to dynamic wave theory and infiltration based on the Green–Ampt method [
27]. For the main estimation procedure, the measured parameters [
32,
33] (e.g., area, width, and slope) are based on measurement and spatial analysis technologies with a geographic information system (GIS) [
34,
35], and the inferred parameters [
32,
33] (e.g., roughness coefficients) are based on empirical estimation [
35]. Parameters are calibrated and verified by trial and error with an automatic optimization procedure, where the objective is to maximize the Nash–Sutcliffe (NS) coefficient of efficiency likelihood values. Details are presented in [
35].
The catchment discretization procedure [
36] is mainly based on [
34] and [
35] and uses GIS technology to fully consider various factors that affect the flow process of a city’s water flow, including rivers, roads, storm sewer networks, and buildings. As shown in
Figure 1, DEM data are first used to analyze the flow paths of urban surface water and obtain the natural basins with GIS hydrologic tools [
34,
37]. Second, given that rivers and main roads obstruct rainfall across themselves, the centerlines of rivers and main roads are applied to split natural basins into catchments with the GIS clip tools. Finally, the catchments are divided into subcatchments based on the Thiessen polygon method to help assign rainfall data from each rain gage to the nearest subcatchment [
38,
39]. The shapes of subcatchments are then revised by considering the effect of buildings on the direction of storm runoff.
2.2. The Stroke Scaling Method
The main approach to expressing a storm sewer network is based on the node–arc structure. This structure separates conduits and has poor visual coherence, which makes it difficult to match a human’s overall perception. Moreover, the large number of junctions and conduits in the storm sewer network result in inefficient or low accuracy of the generalization or classification method, even within a small area. Therefore, we used the stroke technique [
40,
41,
42] to design a stroke scaling method for simplifying and classifying a storm sewer network in a highly efficient and accurate manner.
The stroke technique concatenates separate line segments (e.g., conduits) into longer lines to detect and resolve spatial inconsistencies [
40,
41,
42]; this provides a more integrated structure to further improve the efficiency of subsequent processing. During the generalization procedure for a storm sewer network, we usually keep or remove conduits and connected junctions with similar shapes or attributes. For example, we may remove conduits with a diameter of less than 400 cm or that are not laid on the main roads. Therefore, we can consolidate the connectivity of conduits with similar shapes or attributes into one pipe stroke to simplify the structure of the storm sewer network and improve the later processing speed.
The main idea of pipe stroke construction is to convert
L conduits into
S pipe strokes (
) based on certain constraints. This can be expressed as follows:
where
is all conduits of the storm sewer network,
is the number of conduits,
represents all pipe strokes of the storm sewer network,
is the number of all pipe strokes, and
is one of
.
We select the angle (
) between conduits and the horizontal direction, the road level (
) that the conduits are laid on, and the diameter (
) of the conduit as constraints for combining two conduits (
) that are directly connected to each other into one pipe stroke. The following conditions need to be met:
where
is the difference threshold between the
values of
and
, and
is the difference threshold between the
values of
and
. The values of
and
depend on the shape and attribute characteristics of the sewer network in the study area.
For convenient generalization or classification of the sewer network, in addition to the conduits, we also add the junction information, start the search junction, and the level of the pipe stroke. Thus, the pipe stroke comprises all of the information of the original components but with more integrity. The data structure of the pipe stroke can be described as follows:
where
PS is the pipe stroke,
L is the set of conduits of the pipe stroke,
P is the set of junctions of the pipe stroke,
StartP is the start search junction of the pipe stroke, and
Le is the level of the pipe stroke. The level is assigned according to the distance from the pipe stroke to the outfall. The level of the pipe stroke (
) that is directly connected to the outfall is set to 1, the level of the pipe stroke that is directly connected to
is set to 2, and so on. We also set
Le of the pipe stroke at the end of the sewer network and the total length to be less than a given threshold, which depends on the actual situation at the last level. This ensures that the unimportant segments are deleted during the generalization procedure.
The complete construction algorithm of pipe strokes is displayed in
Figure 2 and
Figure 3.
PSs are the set of all pipe strokes of the storm sewer network.
AL and
AP represent the shape and attribute information of all conduits and junctions of the original storm sewer network.
Lset1 represents the conduits that connect to
StartP directly from the set of all conduits (
AL).
Lset2 represents the conduits that connect to
l1 directly from the set of all conduits (
AL).
Pset is
P of the
PS for which construction was just finished. FindSS() is the function for searching the pipe strokes set of the original sewer network. FindNextL() is the function of finding the next conduit and junction of a pipe stroke. The main idea of this algorithm is to proceed from the outfall to the upstream pipes to find each pipe stroke and assign
Le to strokes based on the upstream and downstream relationships of the storm sewer network (i.e., function FindSS()), as shown in
Figure 2. The main idea of the construction algorithm for each pipe stroke is to start from the given junction, which is set as
StartP, and then select one conduit that connects with
StartP directly as the seed conduit (
l1). Then, a traversal search is performed for all conduits (
L) and other junctions (
P) that belong to this pipe stroke based on the connectivity between conduits and the constraints (Equations (5)–(7)) to determine whether two conduits belong to one pipe stroke (i.e., function FindNextL()), as shown in
Figure 3.
Figure 4 shows a simple example to explain the pipe stroke construction method and the corresponding classification results. The sewer network in this figure with conduits
l1,
l2, …,
l17 and junctions
p1,
p2, …,
p19 is spread on five roads (Roads 1–5). The pipe diameters of
l1,
l2,
l3,
l4,
l5,
l6, and
l7 are 0.8 m; the pipe diameters of
l8,
l9, and
l10 are 0.6 m; and the diameters of other pipes are 0.4 m. We set
= 50° and
= 0.3 m based on the sewer network shape shown on the left side of
Figure 4 and the pipe diameter of each pipeline. We can get seven pipe strokes (
PS1–
PS7) based on the algorithm of the pipe stroke construction (
Figure 2). The right side of
Figure 4 displays the
P,
L,
StartP, and
Le of each stroke. As shown in
Figure 5, we can divide the structure of the pipe system into three levels from simple to complex. Leve1 1 contains
PS1; Level 2 contains
PS1,
PS2, and
PS3; Level 3 contains
PS1,
PS2,
PS3,
PS4,
PS5,
PS6, and
PS7.