The spatial distribution of extreme storms varies in the four cities. The map of mean Rx3h during the 2000–2020 period in study areas is shown in
Figure 7. In SH and GZ, the largest rainfall is located around the city center and decreases toward the watershed boundaries. In BJ, the rainfall increases from the west to the east and clusters in the northeast corner which is a flat plain. In SZ, the mean of Rx3h clusters in the north-central and decreases toward the south. The results show that the spatial distribution of extreme rainfall is non-uniform in the four cities. In this section, hence, the rainfall spatial heterogeneity will be quantitatively identified and assessed.
4.2.1. Rainfall Clustering Analysis
The spatial variability of Rx3h is examined through the analysis of CV and Moran’s I (
Table 3). The Rx3h in the four cities show four different patterns of spatial clustering. The scatter diagrams of Moran’s I (
Figure 8) demonstrate the concrete features of clustering. The scatter falls in the first, second, third, and fourth quadrants, indicating high values aggregation, the low value itself surrounded by high values, the high value itself surrounded by low values, and low values aggregation, respectively. Z-score, which is the result of hypothesis testing, with values greater than 1.96, indicates the spatially clustered of rainfall is reliable statistically [
40].
In BJ, the high Moran’s I and low CV indicate that Rx3h is highly clustered in space. However, there is relatively little difference in rainfall magnitude between the clustered area and the rest part. Contrary to BJ, SZ has low Moran’s I (positive) and high CV, indicating that the clustering feature of Rx3h here is relatively insignificant, but the variability of rainfall magnitude is remarkable. In other words, the variability in space is more complex in SZ, which is also illustrated in the complex scatter distribution in
Figure 8d. In GZ, Moran’s I and CV are both high, showing both the noticeable clustering of spatial distribution and variability of rainfall magnitude. The scatters concentrated on the line “x = y” in
Figure 8c address the highly clustered pattern. Extreme storms concentrate in local areas in GZ with large rainfall magnitude, where likely has a higher risk of rainfall-induced flooding. Compared with GZ, Moran’s I and CV were both low (positive) in SH, showing both the moderate clustering of rainfall distribution and variability of rainfall magnitudes. The above results highlight that GZ has the largest rainfall spatial heterogeneity, while SZ has the smallest one.
The locations of the local spatial clusters of study cities are identified by LISA analysis [
41]. All the clusters show a monocentric pattern in study areas (
Figure 9). The comparison to the results in
Figure 7 proves the reliability of the results in LISA analysis. In SH, the Rx3h clusters in the central city area and southern suburbs with a trend of decreasing toward the watershed boundaries. It may be associated with the urban heat island effect and sea-land breeze circulation. In BJ, Rx3h clusters locate in the east of BJ and show a north-to-south spatial gradient across the plain. With the mountainous terrain to the west and the urban growth from the southeast, extreme rainfall appears to be positively influenced by mountain-plain circulations. In GZ, the Rx3h events cluster in the central area of the city with a decreasing spatial trend to other portions, which is closely linked to the topography of high in the north, low in the south, and tropical weather system. In SZ, the spatial clusters are located on the north side, especially in the northwest. Affected by both its elongated shape and significant southwest monsoon, the rain belt is generally across the whole city. The results imply that local spatial clusters can be influenced by climate, topography, and the pattern of urban growth.
The range of rainfall clusters is examined by semi-variance analysis. Generally, the semi-variogram is an increasing curve of the lag distance (h), and it may reach a sill or increase indefinitely as h increases. We use regular grids as the analysis unit in this study. For example, if SH includes 12 grid cells in longitude and 6 grid cells in latitude, a statistically reliable semi-variance analysis can be carried out within .
The semi-variance is first computed without considering the change of rainfall in different directions (
Figure 10). The spatial distribution of short-duration extreme rainfall in urban regions shows pronounced spatial aggregation. Variograms of the four cities can reach a stable value (sill) within the range of lag distance (
), which demonstrates that each range of the Rx3h clusters is smaller than the city scale. In SH, for instance, the distance where the variogram reaches the sill is 4.84 (
), indicating the Rx3h extends to a scale of 48.4 km × 48.4 km. The results of other cities are presented in
Table 4.
The other key parameters in
Table 4 also reflect different characteristics of spatial data variance. The small nugget variance (
) in SH and GZ indicates that it is less affected by other random factors (such as human factors, measurement errors, or others) [
42,
58,
61]. On the contrary,
values are larger in BJ and SZ, with the nugget-sill ratio (
, NSR) even reaching 12.3% and 26.6%, respectively. It implies that the uncertainty is significant and complex for the spatial variability in BJ and SZ. The results of
show the goodness of fit for the spherical model performed well in SH, BJ, and GZ. The low
in SZ is likely associated with fewer fit points and no-significant spatial autocorrelation.
To further examine the change of rainfall variability in different directions, the north–south and east–west directions are selected to calculate the semi-variance (
Figure 11). The spatial distributions of annual max 3-h rainfall in the four cities are all anisotropic. In SH, the spatial variability of Rx3h in the north-south direction of the city is greater than that in the east-west direction, and the range of the rainfall clusters shows a balanced pattern between transverse and longitudinal directions (52 km × 58 km). Within 50 km, the spatial variability in GZ was more influenced by rainfall in the east–west than north–south. The semi-variance in the east-west direction reaches the sill at about 58 km (
), while there is still a growing trend in north-south. It indicates that the observations still have spatial autocorrelation within the predetermined distance. The boundary of the rainfall clusters may go beyond the predefined spatial extent. Similarly, it can be proved that the Rx3h in GZ is more manifested as a longitudinal (north–south) clustering pattern. Rx3h clusters in BJ and SZ both show a transverse (east–west) clustering pattern.
A radio is used to assess the size of the aggregation area and it is denoted by the following: . In which , are the range () obtained from the two directions; , are the number of the grid cells in latitude and longitude. It can be found that the area of rainfall clusters in GZ is the largest with a ratio of 37.97%. The area of rainfall clusters in SZ is the smallest with a ratio of 12.88%. SZ and BJ have moderate ratios of 28.41%, 28.40%.
4.2.2. Distribution of Rainfall Centroids
Rainfall-movement strongly impacts surface flow and peak discharges [
62,
63], making it an important issue in the extreme rainfall study. Spatial heterogeneity is manifested not only in the magnitudes and distribution of areal mean rainfall but also in the moving of the rainfall centroids [
23]. Rainfall centroids indicate the locations of the maximum rainfall in a certain duration, which should be alarmed if there is excessive or prolonged rainfall at the centers.
The location of Rx3h centroids is demonstrated in
Figure 12. The dot colors indicate the magnitude of rainfall. If the rainfall centroids occur at the same location more than once, the rainfall will be accumulated and shown in
Figure 12. In SH, the rainfall centroids cluster in the central portions and tend to distribute along the Huangpu River. The rainfall centroids in BJ distribute in the northeast plain and central urban area (southeast area). It spreads out to the east, with only two rainfall events occurring in the same location. In GZ, most of the rainfall centroids cluster in the center of the city. In SZ, centroids distribute on the north side with repetitions in the same location. It should be noted that there is a large rainfall event of about 202 mm occurs in the northeast, even more than the accumulation of four rainfall events at the center.
The wind rose is used to illustrate the relative position of rainfall centroids and geographic location centers (
Figure 13). In SH, compared to the geographic location centers, the occurrence probability and magnitudes of rainfall on the east side were both larger than that on the west side. In BJ, 80% of rainfall centroids with a magnitude ranging from 20 to 52 mm show a pattern of clustering in the east. As mentioned above, it is likely linked to the mountain-plain topography. In GZ, the maximum magnitude of the rainfall centroids reaches 68–84 mm, which is distributed in the southwest, northwest, and northeast of the geographical center. In SZ, centroids with the highest probability of occurrence are located in the north while the maximum magnitudes of rainfall occurred close to the northwest. Generally, the distribution of rainfall centroids and rainfall clusters is consistent in space.
The spatio-temporal information of rainfall centroids is extracted per time step to examine the path of rainfall events. For a 3-h time scale, 6 centroids can be obtained from the 3-h rainfall process (temporal resolution = 30 min). The movement of centroids can be simply described by comparing the locations of the first and last centroid. CV is computed by the rainfall magnitude at 6 centroids (
Table 5).
The results of centroid movements are shown in
Table 4. During 2000–2020, 42.9% of rainfall centroids in BJ move toward the southwest of the city, which are paired with the growing trend of urbanization reported by Wang [
11]. Greater urban flood risk may be increased under the similar direction of urban expansion to the location of clustered rainfall. In SH, 28% of rainfall centroids move toward the southeast coast. Centroids in GZ clusters in the central portion of the city and move toward the southeast in a relatively small range. The rainfall centroids in SZ mostly cluster in the north portion. The two most common directions of movement in SZ are along the northeast–southwest and northwest–southeast which is closely related to the South China Sea monsoon.