Qi-hua Ran, Zhi-nan Shi, Yue-ping Xu. Canonical correlation analysis of hydrological response and soil erosion under moving rainfall[J]. Journal of Zhejiang University Science A, 2013, 14(5): 353-361.
@article{title="Canonical correlation analysis of hydrological response and soil erosion under moving rainfall", author="Qi-hua Ran, Zhi-nan Shi, Yue-ping Xu", journal="Journal of Zhejiang University Science A", volume="14", number="5", pages="353-361", year="2013", publisher="Zhejiang University Press & Springer", doi="10.1631/jzus.A1200306" }
%0 Journal Article %T Canonical correlation analysis of hydrological response and soil erosion under moving rainfall %A Qi-hua Ran %A Zhi-nan Shi %A Yue-ping Xu %J Journal of Zhejiang University SCIENCE A %V 14 %N 5 %P 353-361 %@ 1673-565X %D 2013 %I Zhejiang University Press & Springer %DOI 10.1631/jzus.A1200306
TY - JOUR T1 - Canonical correlation analysis of hydrological response and soil erosion under moving rainfall A1 - Qi-hua Ran A1 - Zhi-nan Shi A1 - Yue-ping Xu J0 - Journal of Zhejiang University Science A VL - 14 IS - 5 SP - 353 EP - 361 %@ 1673-565X Y1 - 2013 PB - Zhejiang University Press & Springer ER - DOI - 10.1631/jzus.A1200306
Abstract: The impacts of rainfall direction on the degree of hydrological response to rainfall properties were investigated using comparative rainfall-runoff experiments on a small-scale slope (4 m×1 m), as well as canonical correlation analysis (CCA). The results of the CCA, based on the observed data showed that, under conditions of both upstream and downstream rainfall movements, the hydrological process can be divided into instantaneous and cumulative responses, for which the driving forces are rainfall intensity and total rainfall, and coupling with splash erosion and wash erosion, respectively. The response of peak runoff (P_{r}) to intensity-dominated rainfall action appeared to be the most significant, and also runoff (R) to rainfall-dominated action, both for upstream- and downstream-moving conditions. Furthermore, the responses of sediment erosion in downstream-moving condition were more significant than those in upstream-moving condition. This study indicated that a CCA between rainfall and hydrological characteristics is effective for further exploring the rainfall-runoff-erosion mechanism under conditions of moving rainfall, especially for the downstream movement condition.
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Article Content
1. Introduction
Rainfall-runoff processes, together with the related upland erosion and sediment transport, are highly complex, and are impacted by two main aspects: rainfall and watershed characteristics (Yen and Chow, 1969; Singh, 2002; de Lima and Singh, 2003; Nunes et al., 2006; de Lima et al., 2009; Ran et al., 2012a; 2012b; Seo et al., 2012). Watershed characteristics usually include topography, shape, slope, drainage pattern, etc. (Montgomery and Dietrich, 2002; Assouline and Ben-Hur, 2006; Ran et al., 2009; Seo and Schmidt, 2012). Rainfall characteristics, including rainfall intensity, duration, direction, and velocity of movement, are more variable, in spatial and temporal contexts, and often impact both the integrated response (e.g., runoff hydrograph) (de Lima et al., 2009; Seo et al., 2012) and the distributed response (e.g., the temporal and spatial variability of soil moisture) (Ran et al., 2009; 2012a).
As characteristics of natural rainfall, directions of rainfall movement and velocity have an important influence on the runoff response and soil loss (Seo and Schmidt, 2012). Singh (2002) pointed out that the rainfall movement velocity has a significant influence on the surface/near-surface hydrologic response and soil erosion, especially for extreme storms. Storms that move rapidly have much less impact on peak discharge than those moving at an equal speed. Yen and Chow (1969) showed that a lower velocity causes a larger peak discharge, and less time reaches peak. While given the impacts by soil properties, runoff is also generated as a result of crust development on the soil surface during or after precipitation (Ran et al., 2012b). Crusts always affect runoff generation by decreasing surface K_{s} and subsequently also water infiltration (Carmi and Berliner, 2008). Previous studies have shown that ignoring storm movement can result in considerable overestimation and underestimation of runoff peaks (de Lima and Singh, 2003; Seo and Schmidt., 2012). de Lima et al. (2009) simulated storms crossing in different directions, and showed that soil loss resulting from rainstorms moving in different directions were clearly linked to the characteristics of the corresponding overland flow hydrographs and peak discharge. Seo and Schmidt (2012) also studied the relations among the direction of rainfall movement, the maximum peak discharge, and the network configuration. However, still few studies have revealed the impacts of rainfall movement directions on the characteristics of different runoff and erosion development stages, coupled with the crust development properties during those periods.
The influences of rainfall characteristics on hydrologic responses have been investigated experimentally (de Lima et al., 2009; Ran et al., 2011; 2012b; 2012c; He et al., 2013), using field investigations (Ran et al., 2011), and through computational modeling (Singh, 2005; Nunes et al., 2006; Ran et al., 2009). Olson and Wischmeier (1963) measured the soil loss per unit of rainfall erosivity based on simulated rainfall and plot experiments, and then scientists led by Wischmeier developed the famous universal soil loss equation (USLE), in which rainfall was considered a major influencing factor impacting overflow (Foster et al., 1977; Wischmeier and Smith, 1978). Since then, quantitative studies of the effects of rainfall characteristics on soil erosion have been conducted (Nunes et al., 2006; Ran et al., 2012a; 2012b). Dessu and Melesse (2012) recently found that the soil and water assessment tool (SWAT) has the potential to simulate the long-term rainfall runoff process. Meanwhile, some scholars have begun to further discuss the mechanisms of hydrological response using statistical methods (e.g., SPSS), again in order to systematically study the relationships between the response factors (Rice, 1972; Arthur et al., 2011; Pappas et al., 2011), while still remaining at the macro level, and cannot reveal the detail internal correlations between rainfall and slope response properties. Furthermore, directions of rainfall movement are always ignored in those statistical methods.
The objective of this study is to discuss the canonical correlation between sets of rainfall characteristics and hydrological response characteristics by systematic statistical methods, and to compare the response degree under different directions of rainfall movement along a small-scale slope.
2. Materials and methods
2.1. Laboratory experiments
The experiments carried out in this study involved the use of a rainfall simulator, a tilted soil flume, a runoff recording system, and a set of soil water content monitoring devices, set up as shown in Fig. 1. Multiple scenarios relating to various rainfall movement directions, intensities (I_{r}) and event durations (D_{r}) were considered. Details of the experimental facilities (e.g., rainfall simulator, soil flume, and gathering devices) as well as the initial treatment of the soil can be found in descriptions in our previous study (Ran et al., 2012b).
Fig.1 Schematic representation of the laboratory experimental set-up used in this study (a) Section drawing of the experimental set-up; (b) Section drawing of the soil flume at the outlet; (c) Elevation drawing of the experimental set-up (unit: m)
The rainfall scenarios used in this study were constructed by varying four parameters: direction of rainfall movement, rainfall intensity (I_{r}), duration (D_{r}), and the interval between rainfall events. To simplify the laboratory experiments, only two directions were considered for rainfall movement: upstream and downstream, with rainfall moving in only one direction during each event. Three combinations of I_{r} and D_{r} were considered: low I_{r} (1×10^{−5} m/s) with long D_{r} (120–240 min), moderate I_{r} (2.5×10^{−5} m/s) with medium D_{r} (60–120 min), and extreme I_{r} (4×10^{−5} m/s) with short D_{r} (15–60 min). The raindrop generator was moved upstream or downstream by 0.2 m at regular time intervals for each moving rainfall scenario. The equivalent velocity for this movement ranged from 0.2×10^{−3} m/s to 4.3×10^{−3} m/s. Details for setting, as well as the data used in the experiments were described in our previous study (Ran et al., 2012b).
2.2. Canonical correlation analysis (CCA)
Canonical correlation analysis (CCA), a method for studying the correlativity between two different sets of variables, is aimed to identify and quantify their internal relationship. The brief mathematical principles of CCA are presented as follows (Wang et al., 2012).
Two sets of vectors, X=[x_{1}, x_{2}, …, x_{p}], X∈, and Y=[y_{1}, y_{2},…, y_{q}], Y∈, and their linear combinations U=a^{T}X and V=b^{T}Y, were used to study the correlativity between the primitive variables X and Y.
CCA seeks a pair of vectors, a and b, which maximize the correlation ρ(U, V),
,
where ∑_{12} is a sample covariance matrix between X and Y, ∑_{21} between Y and X, and ; ∑_{11} and ∑_{22} are the covariance matrices of X and Y, respectively.
The correlation coefficients of these random variables do not change if they are multiplied by a constant; some constraints (Eq. (2)) are included in Eq. (1) in order to prevent unnecessary repetition:
.
Setting and the solution of Eq. (1) can be obtained by solving either of the following two eigenvalue problems:
,
where the square roots of the eigenvalues λ^{2}, obtained from Eq. (3), are called canonical correlations, and the vectors a and b are the eigenvectors corresponding to A and B, respectively. Consequently, we acquire the ith set of canonical variables:
as well as the ith canonical correlation coefficient λ.
3. Results
In this study, a total of 65 1-d laboratory experiments, comprising 33 upstream-moving and 32 downstream-moving rainfall events, were carried out in 23 d. Table 1 summarizes the results obtained from the physical experiments. Table 1 includes two sets of information from the experiment data: rainfall and slope response (runoff, erosion). Therefore, two sets of primitive variables for the CCA are presented as follows:
Primitive variables of rainfall characteristics:
, where Q is the total rainfall for each experiment.
Primitive variables of hydrological response characteristics:
, where P_{r} is the peak runoff, R the total runoff, C_{s} the sediment concentration, P_{s} the peak sediment discharge, and S the total sediment discharge at the outlet of the soil slope for each rainfall experiment.
Table 1
Experimental observed results in the study for upstream- and downstream-moving conditions, respectively
Serial number
Test date
Test number
No-rain interval (s)
P_{r} (×10^{−6} m^{3}/s)
R (×10^{−6} m^{3})
C_{s} (g/L)
P_{s} (g/s)
S (g)
Upstream-moving
1
08-13
US001
30.2
22 088
57.1392
2.1800
1262.0902
2
US002
850
38.8
31 002
30.5435
1.5950
946.9093
3
US003
1750
40.0
31 898
20.0078
1.0200
638.2087
4
US004
3550
38.9
31 953
14.6201
0.7330
467.1559
5
US005
12 250
40.8
31 635
10.8623
0.6140
343.6231
6
06-27
US006
24.3
36 147
57.4379
1.4795
2076.1852
7
US007
1700
28.8
45 929
31.7919
1.0000
1460.1692
8
US008
3500
27.5
44 425
19.4827
0.6505
865.5200
9
US009
10 700
23.8
38 085
12.4460
0.4505
474.0071
10
08-21
US010
44.1
137 000
19.9227
0.9440
2729.4038
11
US011
3400
49.6
159 500
8.8707
0.5475
1414.8714
12
US012
11 500
48.5
159 300
6.2134
0.3585
989.7924
13
07-27
US013
43.3
274 680
30.2731
1.2753
8315.4166
14
US014
14 000
45.0
283 553
15.8695
0.7437
4499.8510
15
UM001
3380
13.6
16 780
69.2465
0.9965
1161.9565
16
UM002
1700
22.0
34 124
36.8433
0.9275
1257.2402
17
07-25
UM003
22.4
35 071
24.2902
0.7085
851.8823
18
UM004
12 500
21.5
34 635
16.9290
0.5005
586.3376
19
06-25
UM005
14.3
40 698
29.0862
0.5013
1183.7500
20
UM006
22 200
15.3
46 202
18.1655
0.3777
839.2833
21
07-03
UM008
22.0
139 270
36.9969
0.8758
5152.5629
22
UM009
11 300
21.3
137 850
14.9373
0.3293
2059.1092
23
08-04
UM010
26.0
239 279
32.6173
0.9820
7804.6448
24
UM011
9000
27.3
268 464
22.7816
0.7195
6116.0404
25
06-21
UW001
0.0
0
0.0000
0.0000
0.0000
26
UW002
3400
0.2
154
4.6753
0.0013
0.7200
27
UW003
14 200
7.5
18 446
6.7196
0.0686
123.9500
28
06-15
UW004
0.0
0
0.0000
0.0000
0.0000
29
UW005
8 700
4.2
12 571
10.6913
0.0513
134.4000
30
08-08
UW006
9.5
87 500
39.3379
0.4008
3442.0670
31
UW007
9900
10.0
95 500
24.9994
0.3022
2387.4386
32
08-17
UW008
10.8
130 644
41.0484
0.4652
5362.7320
33
UW009
7700
10.8
138 506
23.6539
0.2995
3276.2119
Downstream-moving
34
08-15
DS001
29.0
13 643
79.5117
2.1290
1084.7785
35
DS002
850
38.9
27 896
34.5689
1.4430
964.3332
36
DS003
1750
47.3
29 921
21.8798
1.1060
654.6669
37
DS004
3550
41.5
30 100
15.9385
0.8010
479.7477
38
DS005
11 650
43.0
30 726
10.5380
0.5440
323.7915
39
06-29
DS006
24.1
21 109
40.9603
1.1595
864.6310
40
DS007
1900
27.3
41 261
19.5182
0.6170
805.3391
41
DS008
4100
27.5
41576
12.2782
0.4150
510.4771
42
DS009
3500
27.8
40 082
8.3786
0.2995
335.8327
43
08-02
DS010
39.9
92 813
49.4416
1.9830
4588.8208
44
DS011
3400
43.6
133 924
21.3428
1.1515
2858.3078
45
DS012
10 600
42.3
131 230
17.1447
1.0670
2249.9005
46
07-29
DS013
31.9
157 332
43.0180
1.3260
6768.1027
47
DS014
13 880
33.2
203 942
20.9780
0.9073
4278.2919
48
07-01
DM001
21.8
21 757
59.2334
1.3275
1288.7402
49
DM002
1700
24.7
37 425
25.4327
0.7715
951.8205
50
DM003
3500
24.5
37 426
15.6115
0.4910
584.2754
51
DM004
11 900
23.3
34 848
10.2470
0.3325
357.0881
52
07-21
DM005
20.1
34 180
64.7911
1.4627
2214.5606
53
DM006
3600
20.2
62 027
31.8704
0.8443
1976.8254
54
DM007
11 800
19.7
60 074
20.9033
0.5263
1255.7441
55
07-05
DM008
21.2
77 472
29.2863
0.7062
2268.8703
56
DM009
11 900
21.8
132 089
14.2250
0.4022
1878.9665
57
08-06
DM010
22.3
152 310
27.7122
0.6617
4220.8460
58
DM011
9000
23.6
172 201
21.0547
0.5575
3625.6409
59
06-23
DW001
1.1
730
14.0137
0.0176
10.2300
60
DW002
16 000
5.5
11 589
21.0640
0.1808
244.1000
61
DW003
3390
6.1
14 552
16.2507
0.1475
236.4800
62
08-10
DW006
10.9
55 166
40.9443
0.4577
2258.7356
63
DW007
10 020
11.2
103 494
24.0305
0.3160
2487.0143
64
08-19
DW008
9.9
51 286
0.3720
1710.2001
65
DW009
9060
10.4
100 958
0.2872
2475.9238
The CCAs were conducted using SPSS based on the observed data (Table 1); the results showed that two sets of canonical relationships (I, II) between rainfall characteristics and response characteristics were obtained, for both conditions (Tables 2–6), these being: set I (CVU I) and set II (CVU II), canonical variables of the upstream movement condition; and set I (CVD I) and set II (CVD II), canonical variables of the downstream movement condition. Eqs. (5)–(8) represent the canonical conversion relations between the primitive and canonical variables. Figs. 2a–2d present the canonical loading relationships corresponding to Eqs. (5)–(8), respectively.
Generally, the canonical correlation coefficient is close to 1 (not less than 0.96) for all pairs of canonical variables (Tables 2–5), clearly indicating a strong correlation between rainfall characteristics and hydrologic response characteristics.
3.1. Analysis of CVU I, CVU II, CVD I and CVD II
For CVU I (Table 2, Eq. (5), Fig. 2a (p.359) ), the conversion coefficient of Q (1.021) in U_{u1} is much larger than that of D_{r} (−0.038) and I_{r} (−0.631), which means U_{u1} mainly represents the properties of total rainfall. Here, D_{r} is perceived as a rectified variable, the reason is that its canonical conversion coefficient (−0.038) and canonical correlation coefficient (0.881) are opposite in sign, as is C_{s} in V_{u1}. Similarly, D_{r} and S in Eq. (6), D_{r} and C_{s} in Eq. (7), as well as D_{r} and P_{s} in Eq. (8) are all rectified variables for the same reason. The opposite canonical variable V_{u1} mainly represents the property of R, for its largest canonical conversion coefficient (0.937), and the absolute of the coefficient for P_{r} is a little smaller than that of R, while they are opposite in sign. Generally, the degree of response of sediment erosion is weaker than runoff processes for the relatively small canonical conversion coefficients of P_{s} and S, so it is included with the canonical loading diagram (Fig. 2a).
Table 2
Canonical correlation analysis result for CVU I
CVU I
Variable
Standard canonical coefficient
Canonical loadings
Cross loadings
U_{u1}
D_{r}
−0.038
0.881
0.878
I_{r}
−0.631
−0.320
−0.319
Q
1.021
0.815
0.812
V_{u1}
P_{r}
−0.613
−0.219
−0.218
R
0.937
0.767
0.765
C_{s}
0.005
−0.018
−0.018
P_{s}
−0.116
−0.291
−0.290
S
0.144
0.789
0.786
Canonical correlation between
U
_{u1} and
V
_{u1}: 0.997
Fig.2 Canonical loadings between the observed and canonical variables CVU I (a), CVU II (b), CVD I (c) and CVD II (d)
For CVU II (Table 3, Eq. (6), Fig. 2b), it is apparent that the canonical conversion coefficient of I_{r} is the largest in the conversion relationship with U_{u2}, which means that the whole rainfall properties appear to be more significant as I_{r} increases, and the coefficient of Q is much smaller than that of I_{r}. The opposite V_{u2}, is mainly embodied by the characteristics of P_{r}, of which the canonical conversion coefficient is the largest (0.774), and that of R the second largest (0.326). In general, the responses of C_{s} and P_{s} are much weaker than both P_{r} and R in terms of their relatively small canonical coefficients. Thus, it can be seen that CVU II represents the main response characteristics of P_{r} under a rainfall force dominated by I_{r}, and also some of the weaker responses of soil erosion, so it is included with the canonical loading diagram (Fig. 2b).
Table 3
Canonical correlation analysis result for CVU II
CVU II
Variable
Standard canonical coefficient
Canonical loadings
Cross loadings
U_{u2}
D_{r}
0.045
−0.145
−0.143
I_{r}
0.881
0.947
0.936
Q
0.297
0.580
0.573
V_{u2}
P_{r}
0.774
0.971
0.960
R
0.326
0.641
0.633
C_{s}
0.009
0.095
0.094
P_{s}
0.112
0.568
0.561
S
−0.061
0.420
0.415
Canonical correlation between
U
_{u2} and
V
_{u2}: 0.988
As for CVD I (Table 4, Eq. (7), Fig. 2c), similar to CVU II, the absolute canonical conversion coefficient for the U_{d1} of I_{r} (1.055) is much larger than that for Q (0.363). P_{r} also has the largest effect on V_{d1} as it has the largest canonical coefficient (0.788), and P_{s} is the second largest (0.369), while those of R and S are clearly much weaker. Thus, CVD I mainly shows the P_{r}-dominated response under I_{r}-based rainfall action, coupled with the relatively weak response of P_{s}, so it is included with the canonical loading diagram (Fig. 2c).
Table 4
Canonical correlation analysis result for CVD I
CVU I
Variable
Standard canonical coefficient
Canonical loadings
Cross loadings
U_{d1}
D_{r}
0.149
−0.652
−0.644
I_{r}
1.055
0.967
0.956
Q
−0.363
−0.211
−0.209
V_{d1}
P_{r}
0.788
0.954
0.943
R
−0.080
−0.043
−0.043
C_{s}
−0.095
0.128
0.127
P_{s}
0.369
0.679
0.671
S
−0.203
−0.030
−0.030
Canonical correlation between
U
_{d1} and
V
_{d1}: 0.988
Moreover, like CVU I, CVD II (Table 5, Eq. (8), Fig. 2d) mainly shows an R-dominated response under the main force of Q, while the responses of P_{r}, C_{s} and S respond to a similar degree.
Table 5
Canonical correlation analysis result for CVD II
CVD II
Variable
Standard canonical coefficient
Canonical loadings
Cross loadings
U_{u2}
D_{r}
−0.171
0.600
0.582
I_{r}
0.125
0.250
0.242
Q
1.100
0.974
0.944
V_{u2}
P_{r}
0.249
0.248
0.240
R
0.691
0.973
0.943
C_{s}
0.306
0.063
0.061
P_{s}
−0.348
0.208
0.207
S
0.345
0.925
0.897
Canonical correlation between
U
_{d2} and
V
_{d2}: 0.969
3.2. Redundancy analysis
Table 6 shows the redundancy analysis results, presenting the reference value of the CCA. When rainfall moves upstream, the degree of explanation of X—the primitive variable for rainfall properties—given by the canonical variable U_{u1}, is 51.4% (CVX1-1), and for U_{u2} 41.8% (CVX1-2), V_{u1} (CVX2-1) 51.1%, and V_{u2} 40.8% (CVX2-2). However, the effect of rainfall on slope response is single-directional and their opposite canonical variables U_{u1} 26.7% (CVY2-1), and U_{u2} 36.4% (CVY2-2). When rainfall moves downstream, the degree of explanation of X given by U_{d1} and U_{d2} is 46.9% (CVX1-1) and 45.7% (CVX1-2), respectively, also that of Y given by V_{d1} and V_{d2} is 27.8% (CVY1-1) and 38.2% (CVY1-2), respectively, and U_{d1} 27.2% (CVY2-1), U_{d2} 35.9% (CVY2-2).
Table 6
Redundancy analysis of CCA results in this study
Rainfall movement
Proportion of variance of X explained by U_{u1}, U_{u2}, U_{d1}, U_{d2} (%)
Proportion of variance of X explained by V_{u1}, V_{u2}, V_{d1}, V_{d2} (%)
CVX1-1
CVX1-2
CVX2-1
CVX2-2
Upstream-moving
51.4
41.8
51.1
40.8
Downstream-moving
46.9
45.7
45.7
43.0
Rainfall movement
Proportion of variance of Y explained by V_{u1}, V_{u2}, V_{d1}, V_{d2} (%)
Proportion of variance of Y explained by U_{u1}, U_{u2}, U_{d1}, U_{d2} (%)
CVY1-1
CVY1-2
CVY2-1
CVY2-2
Upstream-moving
26.9
37.2
26.7
36.4
Downstream-moving
27.8
38.2
27.2
35.9
Thus, it is acceptable for the explanation degree of primitive rainfall property variables given by U_{u1} and U_{u2} for upstream-movement conditions; although it is a little poorer for that of Y, because the ideal state of the degree of explanation should not be less than 30%, in theory (Zhang, 2004), it might be perceived as more or less reluctantly accepted. When rainfall moves downstream, apart from the weaker degree of explanation of Y given by its set I of canonical variables U_{d1} and V_{d1}, other degrees of explanation for canonical variables are generally acceptable.
4. Discussion
Based on the CCA between the variables of rainfall and hydrologic response, it can be seen that, for both rainfall directions, the rainfall-runoff-erosion process in this study can be divided into an instantaneous and a cumulative response, of which the driving force is rainfall intensity and total rainfall, respectively, accompanied by splash erosion and wash erosion for each response.
When rainfall moves upstream, the canonical conversion coefficients of R and P_{r} have opposite signs for the cumulative response (Eq. (5)), which means the slope response appears stronger with increasing R, while the decrease in P_{r}, occurs mainly because D_{r} in the experiments with weak I_{r} is much longer than those with strong I_{r}, which leads to the much larger Q of the former. Moreover, P_{r} is mostly determined by rainfall intensity because the slope responses are already in steady state when P_{r} emerges (Ran et al., 2012b); hence, the larger the Q, the smaller the I_{r}, leading to the weaker response of P_{r}, just the opposite trend to the runoff response. When rainfall moves downstream, the canonical conversion coefficients of R and P_{r} have the same sign, which means the slope hydrologic response appears stronger with increases of both R and P_{r}. Though under nearly the same initial conditions, the directions of rainfall and the overflow are the same along the slope while rainfall moves downstream. Consequently, a thin layer of sedimentary crust would form in the initial rainfall stages, due to initial overflow from the relatively high positions on the slope, resulting in evidently smaller permeability coefficient (K_{s}) and larger compaction of the surface soil than for the condition of upstream movement (Robinson and Woo-dun, 2008) together with the cumulative washing by overflow before peak runoff emerges. Thus, peak run-off appears larger than that for upstream movement experiments under similar conditions (Ran et al., 2012b).
With the exception of the rectified variables of V_{u1} and V_{u2}, it is shown both in the conversion formulas (Eqs. (5) and (8)) and canonical loading diagrams (Figs. 2a and 2d) that the degrees of erosion under conditions of the two rainfall directions are significantly different. In conditions of upstream movement, both P_{s} and S have little impact on V_{u1} for their significantly smaller values of absolute conversion coefficients. However, when rainfall moves downstream, both C_{s} and S (the erosion property variables) have a relatively large impact on V_{d2}, because gaps between their canonical conversion coefficients, as well as canonical loadings and that of R, are a lot smaller than those in conditions of upstream movement. It was also found for V_{d2} that the degrees of influence of P_{r}, C_{s}, and S are very close, the reason is that the erosive sediment on the slope surface could be relatively completely washed by overflow because the rainfall and overflow move in the same direction along the slope. When rainfall moves upstream, the change of surface permeability was less than that in downstream conditions (Ran et al., 2012b). In contrast, the erosive sediment could not be washed completely towards the outlet unless the washing duration was long enough, because the rainfall and overflow move in opposite directions along the slope (Ran et al., 2012a). On the whole, in response to the Q-dominated rainfall action, the erosion response is more strongly indicative of the whole hydrologic response characteristics for conditions of downstream rather than upstream movement.
In response to the force which is dominated by I_{r} (rainfall action), P_{r} appears to be the best determining factor of all the response characteristics, whether for conditions of upstream or downstream movement (Eqs. (6) and (7); Figs. 2b and 2c). Compared with the canonical loadings for P_{r} and R, both C_{s} and P_{s} have little impact on V_{u2} when rainfall moves upstream (Eq. (6), Fig. 2b). This was particularly so for C_{s}, a cumulative variable, the response of which could be ignored, so are the variables R and S in V_{d1}. Furthermore, both the canonical conversion coefficient and canonical loading of P_{r} are much larger than that for P_{s}, because as each response is made, the surface hydrologic responses are mostly in a steady state, resulting in much smaller K_{s} (Ran et al., 2012b) and greater compaction of surface soil. Compared to the response at the beginning of each day, the majority of experiments appear to obviously increase in runoff levels, and have a decreased sediment discharge in this steady state (Table 1). As mentioned above, the erosive sediment may be left on the slope unless the washing time continues for a sufficient period, owing to the opposite directions of movement between rainfall and overflow. Thus, P_{r} has the strongest influence on the canonical variables of response properties, and P_{s} has a slightly weaker influence. For conditions of downstream movement, both the canonical loadings and conversion coefficient of P_{s} in V_{d1} are larger than those of P_{r} (Eq. (6)). Generally, the peak sediment discharge in downstream movement experiments has a larger impact on the canonical variable of response properties than conditions of upstream movement, of which reason is identical as the cumulative response mentioned above (Eqs. (5) and (8)).
Generally, for the dominant variables U_{d1} and U_{d2}, absolutes for both canonical loadings and canonical conversion coefficients are much larger than those for other variables, showing the most distinctive divide between instantaneous and cumulative responses. The divide is, however, not so obvious in upstream movement experiments. This means that the analytics work of CCA for the conditions of downstream movement is more effective.
5. Conclusions
Based on the comparative rainfall-runoff experiments on a small-scale slope (4 m×1 m) under conditions of different rainfall movement directions along the slope surface, as well as the CCA between the rainfall and hydrologic response characteristics, it was found that the CCA method is valid for research into hydrologic responses to rainfall movement, and several conclusions were obtained via CCA.
Under both rainfall directions, the rainfall -runoff-erosion process in this study can be divided into instantaneous and cumulative responses, of which the driving force is rainfall intensity and total rainfall, respectively, accompanied by splash erosion and wash erosion for each response. Rainfall duration in this study did not have a dominant effect on the hydrologic response.
The response of P_{r} to the I_{r}-dominated rainfall action appeared to be the most significant, both for conditions of upstream and downstream movement. The response of sediment erosion to rainfall appeared obviously weaker than that of runoff characteristics. The instantaneous responses of erosion in downstream conditions appear more strongly than those for upstream conditions. However, in reality, those responses may be more complex for various impact factors in nature, future studies will be carried out to further explore this subject.
The response of R to the Q-dominated rainfall action appeared to be the most significant for conditions of both upstream and downstream movements. The cumulative responses of sediment erosion in downstream movement were also more significant than those for upstream movement.
The analytics work by CCA is more effective for the condition of downstream movement than upstream movement.
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