This section presents the results for the developed rainfall and flood scenarios for TS Erika over a small southern catchment in Dominica island state. The rainfall and flood scenarios were selected after performing two iterations of the proposed workflow in Fig. 1. For each experiment, we present the rainfall and flood analysis outputs as detailed in Sections 3.1–3.3. The output temporal clusters were labeled with the initial T; for instance, T1 represented temporal cluster 1 (this applied to all clustering results). Also, the naming format of the cluster representative precipitation signals was, for instance, T1-Q2 represented RPS Q2 of temporal cluster T1. When determining the optimal number of clusters, the total within-cluster sum of squares (see Section 2.2) was plotted against fifteen K values, as shown in Fig. 3. The output elbow graph was smooth; therefore, we read the optimal value at K = 5, where the graph flattened, as indicated by the red arrow. During the two iterations, the value of K was reduced to K = 4 and K = 3, also shown on the elbow graph using the blue and orange arrows, respectively. The clustering goodness measures were 83.9% for K = 5, 80.1% for K = 4, and 75.1% for K = 3.
3.1 Results for using K = 5
The first experiment was based on K = 5, read directly from the elbow graph in Fig. 3. The K-means clustering results for using the five optimal clusters are presented in Table 1. From the rainfall statistics, we observed the cluster extremes decreased in the order of T1, T3, T5, T4, and T2. For TS Erika, the rainfall was mainly concentrated south of the storm track, and the heaviest precipitation poured in two spatially distinct regions, both over the ocean. We observed this pattern in the most extreme cluster, T1, comprising pixels to the east and west of the study area, see Fig. 2. There were no T1 pixels over Dominica as the island only had pixels in clusters T3, T4, and T5. The characteristic of spatial dissimilarity was also observed in other clusters, such as T2, comprising pixels far from TS Erika’s track. T2 was also the largest cluster, including 41.8% of the rain pixels, with very low rainfall intensity and volume (see Table 1). As explained in Section 2.3, we assumed that rainfall over T2 pixels was likely not associated with TS Erika considering the very low rainfall intensity and volume; hence pixels of this cluster were discarded from the analysis.
Table 1
Summary statistics for the temporal clusters resulting from K = 5
Cluster
|
|
T1
|
T2
|
T3
|
T4
|
T5
|
Cluster size
|
|
132
|
798
|
433
|
399
|
149
|
%
|
|
6.9
|
41.8
|
22.7
|
20.9
|
7.8
|
Maximum Intensity (mm/hr)
|
min
|
55
|
0.4
|
36.4
|
18.2
|
32
|
max
|
120
|
50.2
|
120
|
111
|
120
|
Total Rainfall
(mm)
|
min
|
451.5
|
1.4
|
201.1
|
47.8
|
121
|
max
|
767.9
|
112.4
|
571.5
|
275.9
|
603.2
|
mean
|
583.9
|
29.4
|
344.2
|
167.5
|
331.4
|
After disregarding T2, we performed the temporal alignment and quantile analysis for the remaining clusters, i.e., T1, T3, T4, and T5, to derive representative precipitation signals (RPSs) for running the openLISEM flood model. The RPSs needed to summarize the rainfall characteristics for the individual clusters in terms of the total volume, maximum intensity, and duration. Before deriving the RPSs, we varied the beginning of the storm’s precipitation for four intensity values, i.e., 2, 5, 10, and 20 mm/hr. When we examined the result of the temporal alignment, we found the starting threshold of 10 mm/hr as most appropriate to align the time series for the onset of TS Erika’s rainfall. Table 2 presents the calculated rainfall characteristics for signals derived for each cluster at quantile positions, i.e., Q2 (0.5), Q3 (0.75), and Q4 (0.9). From the quantile analysis, we observed that signals derived at Q4 were not realistic; their quantified total rainfall was way higher than the totals for the original time series, refer to Table 1. Therefore, Q2 and Q3 were the selected RPSs for the individual clusters. As shown in Fig. 4, RPSs for clusters T1, T3, and T4 had similar shapes; however, they differed significantly in duration, cumulative rainfall, and maximum intensity. On the other hand, the T5 RPSs had a different structure comprised of multiple peaks with relatively low rainfall intensity. We selected the best cluster RPS(s) after examining the modeled flood characteristics when Q2 and Q3 signals were used separately as precipitation inputs for openLISEM.
Table 2
Rainfall statistics for RPSs Q2, Q3, and Q4 for the temporal clusters resulting from K = 5. Only Q2 and Q3 were used to run to flood model
|
T1
|
T3
|
T4
|
T5
|
|
Tr*
(mm)
|
Imax* (mm/hr)
|
Dr* (hr)
|
Tr
(mm)
|
Imax (mm/hr)
|
Dr (hr)
|
Tr
(mm)
|
Imax (mm/hr)
|
Dr (hr)
|
Tr
(mm)
|
Imax (mm/hr)
|
Dr (hr)
|
Q2
|
518.9
|
73.1
|
17.5
|
307.1
|
50.2
|
15.5
|
107.6
|
20.2
|
14.5
|
159.7
|
24.6
|
18.0
|
Q3
|
729.9
|
94.9
|
25.0
|
440.0
|
63.8
|
17.5
|
209.9
|
38.9
|
20.5
|
427.3
|
61.6
|
22.5
|
Q4
|
1029.1
|
112.4
|
31.5
|
587.0
|
75.8
|
23.5
|
345.3
|
55.2
|
23.5
|
834.9
|
94.5
|
25.0
|
*Tr:
|
Total Rainfall (mm)
|
*Imax:
|
Maximum Intensity (mm/hr)
|
*Dr:
|
Period (hr)
|
|
The flood statistics in Table 3 were obtained from eight separate runs of the openLISEM model, equivalent to the total number of selected precipitation curves (Q2 and Q3, see Fig. 4) for the clusters T1, T3, T4, and T5. For each flood characteristic, we present eight quantities (two for each cluster), e.g., the flood extents due to RPSs T1-Q2 and T1-Q3 are 3.98 km2 and 4.84 km2, respectively. The eight RPSs caused flood extents ranging from 1.02 to 4.84 km2. On average, T1 and T3 signals caused larger flooded areas with water at greater depths ranging from 2.79 to 4.21 m, attributed to the high rainfall intensity of these RPSs, as seen in Table 2. We observed that RPSs that caused large flood volumes ( i.e., in the range of 1.10 to 2.28 million m3) consequently generated heavy runoff represented by a high runoff ratio (above 0.7). Infiltration was generally low for all RPSs hence the longer flood durations ranging from 15.71 to 27.24 hours.
Table 3
Calculated flood characteristics for RPSs Q2 and Q3 resulting from using K = 5
|
T1
|
T2
|
T4
|
T5
|
FLOOD EXTENT (km2)
|
Q2
|
3.98
|
3.12
|
1.02
|
1.40
|
Q3
|
4.84
|
3.76
|
2.19
|
3.12
|
MAXIMUM FLOOD DEPTH (m)
|
Q2
|
3.88
|
2.79
|
1.26
|
1.32
|
Q3
|
4.21
|
3.70
|
1.69
|
2.85
|
FLOOD VOLUME (million m3)
|
Q2
|
1.67
|
1.10
|
0.42
|
0.56
|
Q3
|
2.28
|
1.50
|
0.77
|
1.12
|
RUNOFF RATIO
|
Q2
|
0.80
|
0.71
|
0.33
|
0.43
|
Q3
|
0.85
|
0.77
|
0.59
|
0.74
|
INFILTRATION (mm)
|
Q2
|
92.14
|
75.31
|
60.83
|
77.82
|
Q3
|
97.42
|
86.61
|
73.03
|
98.50
|
FLOOD DURATION (hr)
|
Q2
|
18.43
|
15.71
|
16.68
|
23.64
|
Q3
|
27.24
|
18.85
|
22.25
|
18.60
|
We looked at differences in the quantities calculated for the flood characteristics, i.e., we took 0.1 as a maximum to still consider possible similarity. For instance, from Table 3, we observed that T3-Q2 and T5-Q3 caused similar flood extents even though these RPSs belonged to separate clusters. Likewise, the calculated flood depth, volume, and runoff ratio associated with the RPSs T3-Q2 and T5-Q3 exhibited a similar response. We also observed that the quantified flood characteristics for T1-Q2 and T3-Q3 differed by minimal margins. For example, the flood depth for T1-Q2 and T3-Q3 only differed by 0.17m which we may regard as minimal in flood hazard assessment. Only infiltration and flood duration exhibited significant differences across all RPSs and clusters. We observed that RPS Q3 generated the highest quantities of the calculated flood characteristics for all clusters compared to RPS Q2. Precipitation curves for clusters T1 and T4 had the most significant and most minor quantities of the flood characteristics, respectively. The flood extent, duration, volume, runoff ratio, and infiltration showed higher correlation coefficients (0.85 < r < 0.98) with the rainfall characteristics, i.e., cumulative total and maximum intensity. We observed a strong linear correlation between the flood and rainfall characteristics; however, the maximum rainfall intensity was a more dominant driver of the flood responses. On the other hand, the flood duration showed very low correlation coefficients with the cumulative rainfall (r = 0.39) and maximum rainfall intensity (r = 0.34).
The observed similarities in the quantified flood characteristics for the RPSs belonging to supposedly different clusters signified the likelihood of redundancy during the clustering phase of the rainfall time series. The value K = 5 was an arbitrary choice since the elbow graph did not give a defined inflection point, see Fig. 3. However, the choice K = 5 helped define a starting K value for the first experiment. Based on these observations, we made iterations of the procedure in Fig. 1, which involved reducing the value of K before conducting the temporal clustering again. We still used the threshold value of 10mm/hr to define the onset of TS Erika’s rainfall during the iterations. We only derived and used RPS Q3 to run the openLISEM model during the iterations since we observed that when using K = 5, enormous quantities for the investigated flood characteristics were mainly associated with the Q3 signal across all the clusters. The iterations were meant to optimize the result to make a rational decision on the final TC-associate rainfall scenarios. In this case, we performed the iterations twice, i.e., when K = 4 and K = 3, as detailed in Sections 3.2 and 3.3.
3.2 Results for the first iteration: using K = 4
In the first iteration, when we reduced the optimal clusters to K = 4, we observed that the pixel time series were redistributed to form only four temporal clusters, whose statistics are in Table 4. The output clusters had a larger cluster size than when using K = 5, see Table 1. The cluster extremes (in terms of the total rainfall and intensity) decreased in the order of T1, T3, T4, and T2. The most extreme cluster was labeled T1, and the least intense cluster was labeled T2. We observed a similar range for maximum rainfall intensity for clusters T1 and T2 in both cases of using K = 5 and K = 4. However, there were slight changes in their total rainfall. Rainfall statistics for clusters T3 and T4 were comparable in both cases of K = 5 and K = 4. The pixels that comprised T5 when using K = 5 likely redistributed into other clusters hence the slight changes in the cluster statistics when using optimal value K = 4. The spatial locations of the clusters did not change; only the cluster boundaries changed.
Table 4
Summary statistics for the temporal clusters resulting from K = 4
Cluster
|
|
T1
|
T2
|
T3
|
T4
|
Cluster size
|
|
146
|
818
|
513
|
434
|
%
|
|
7.6
|
42.8
|
26.8
|
22.7
|
Maximum Intensity (mm/hr)
|
min
|
55
|
0.4
|
42
|
18.8
|
max
|
120
|
50.2
|
120
|
111
|
Total Rainfall
(mm)
|
min
|
433.7
|
1.4
|
202.2
|
47.8
|
max
|
767.9
|
132.1
|
571.5
|
275.9
|
mean
|
574.3
|
31.1
|
347.6
|
180.1
|
We derived Q3 representative precipitation signals for T1, T3, and T4, see Table 5, after excluding T2 from the analysis. The computed flood characteristics resulting from using the three RPSs, i.e., T1-Q3, T3-Q3, and T4-Q3, separately as the rainfall inputs in the openLISEM model are presented in Table 6. We quantified the differences in flood characteristics of the RPSs; for example, T1-Q3 caused more 1.09km2 of the flooded area than T3-Q3. Also, the flood duration for T3-Q3 was shorter than that for T4-Q3 by 10.17 hours. When compared to results from using K = 5, we observed substantial differences in the flood characteristics of the RPSs for clusters from K = 4. Also, we detected three levels of magnitude in the order T1, T3, and T4, with significant differences in rainfall and flood characteristics.
Table 5
Rainfall statistics for RPS Q3 for the temporal clusters resulting from K = 4
|
TOTAL RAINFALL (mm)
|
MAXIMUM INTENSITY (mm/hr)
|
PERIOD (hr)
|
T1
|
727.3
|
91.4
|
25.0
|
T3
|
446.2
|
65.0
|
18.0
|
T4
|
230.4
|
42.2
|
18.0
|
Table 6
Comparison of calculated flood characteristics resulting from RPS Q3 when using K = 4
|
FLOOD EXTENT (km2)
|
MAXIMUM FLOOD DEPTH (m)
|
FLOOD VOLUME (million m3)
|
RUNOFF RATIO
|
INFILTRATION (mm)
|
FLOOD DURATION (hr)
|
T1
|
4.80
|
4.22
|
2.22
|
0.85
|
98.13
|
23.73
|
T3
|
3.70
|
3.65
|
1.42
|
0.77
|
88.95
|
19.16
|
Diff*
|
1.09
|
0.57
|
0.79
|
0.08
|
9.18
|
4.57
|
T1
|
4.80
|
4.22
|
2.22
|
0.85
|
98.13
|
23.73
|
T4
|
2.47
|
1.84
|
0.84
|
0.62
|
74.08
|
29.33
|
Diff
|
2.32
|
2.39
|
1.38
|
0.23
|
24.06
|
-5.61
|
T3
|
3.70
|
3.65
|
1.42
|
0.77
|
88.95
|
19.16
|
T4
|
2.47
|
1.84
|
0.84
|
0.62
|
74.08
|
29.33
|
Diff
|
1.23
|
1.82
|
0.59
|
0.15
|
14.87
|
-10.17
|
*Diff: Difference
|
3.3 Results for Second Iteration: Using K = 3
The second iteration of the workflow was based on the optimal value K = 3 when running the K-means algorithm over the data. Statistics for the three output temporal clusters with quantified rainfall extremes reducing in the order T1, T3, and T2, are presented in Table 7. Compared to using K = 5 and K = 4, in this case, we observed that pixels merged further to form the three clusters hence the larger cluster size, especially for T2 and T3. The range of the maximum rainfall intensity did not change for the most extreme cluster, T1; see Tables 1 and 4. Compared to results from using K = 5, there were observable changes in the total rainfall associated with the output clusters. Further merging of the pixels likely influenced the average total rainfall of the individual clusters. There was a significant change in the cluster boundaries, especially for T3.
Table 7
Summary statistics for the temporal clusters resulting from K = 3
Cluster
|
|
T1
|
T2
|
T3
|
Cluster size
|
|
212
|
937
|
762
|
%
|
|
11.1
|
49.0
|
39.9
|
Maximum Intensity (mm/hr)
|
min
|
55
|
0.4
|
26
|
max
|
120
|
68.6
|
120
|
Total Rainfall
(mm)
|
min
|
352.5
|
1.4
|
83.6
|
max
|
767.9
|
171.9
|
489.2
|
mean
|
530.8
|
42.7
|
279.8
|
We excluded T2 from further analysis considering the cluster statistics showed shallow ranges of the calculated rainfall characteristics. For K = 3, we calculated rainfall characteristics for representative precipitation signals T1-Q3 and T3-Q3, as shown in Table 8. We observed that the derived RPSs were associated with lower values for the rainfall characteristics than when using K = 5 and K = 4. The optimal value K = 3 divided TS Erika’s rainfall into two levels of magnitude with differences in quantified flood responses shown by the statistics in Table 9. We observed that the calculated quantities for the flood characteristics were slightly lower than those for RPSs from using K = 4, see Table 6. For instance flood extent of the most extreme cluster T1 when using K = 4 is 4.80 km2 and when using K = 3 flood extent is 4.62 km2.
Table 8
Rainfall statistics for RPS Q3 for temporal clusters resulting from K = 3
|
TOTAL RAINFALL (mm)
|
MAXIMUM INTENSITY (mm/hr)
|
PERIOD (hr)
|
T1
|
676.6
|
84.9
|
24.5
|
T3
|
381.3
|
59.0
|
17.5
|
Table 9
Comparison of calculated flood characteristics resulting from rainfall signal Q3 for K = 3
|
FLOOD EXTENT (km2)
|
MAXIMUM FLOOD DEPTH (m)
|
FLOOD VOLUME (million m3)
|
RUNOFF RATIO
|
INFILTRATION (mm)
|
FLOOD DURATION (hr)
|
T1
|
4.62
|
4.18
|
2.05
|
0.84
|
97.69
|
22.76
|
T3
|
3.48
|
3.35
|
1.28
|
0.74
|
84.40
|
20.83
|
Diff*
|
1.14
|
0.83
|
0.77
|
0.10
|
13.29
|
1.93
|
*Diff: Difference
|