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 Journal of Zhejiang University SCIENCE A 2013 Vol.14 No.5 P.353-361 http://doi.org/10.1631/jzus.A1200306

Canonical correlation analysis of hydrological response and soil erosion under moving rainfall*

 Author(s):  Qi-hua Ran, Zhi-nan Shi, Yue-ping Xu Affiliation(s):  . Institute of Hydrology and Water Resources, Zhejiang University, Hangzhou 310058, China Corresponding email(s):   yuepingxu@zju.edu.cn Key Words:  Moving rainfall, Runoff, Sediment erosion, Canonical correlation analysis (CCA) Share this article to： More <<< Previous Article|Next Article >>>

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",
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pages="353-361",
year="2013",
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doi="10.1631/jzus.A1200306"
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%T Canonical correlation analysis of hydrological response and soil erosion under moving rainfall
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%P 353-361
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%I Zhejiang University Press & Springer
%DOI 10.1631/jzus.A1200306

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A1 - Qi-hua Ran
A1 - Zhi-nan Shi
A1 - Yue-ping Xu
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VL - 14
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SP - 353
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Y1 - 2013
PB - Zhejiang University Press & Springer
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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.

## 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, ; Singh, ; de Lima and Singh, ; Nunes et al., ; de Lima et al., ; Ran et al., ; ; Seo et al., ). Watershed characteristics usually include topography, shape, slope, drainage pattern, etc. (Montgomery and Dietrich, ; Assouline and Ben-Hur, ; Ran et al., ; Seo and Schmidt, ). 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., ; Seo et al., ) and the distributed response (e.g., the temporal and spatial variability of soil moisture) (Ran et al., ; ).

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, ). Singh () 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 () 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., ). Crusts always affect runoff generation by decreasing surface K s and subsequently also water infiltration (Carmi and Berliner, ). Previous studies have shown that ignoring storm movement can result in considerable overestimation and underestimation of runoff peaks (de Lima and Singh, ; Seo and Schmidt., ). de Lima et al. () 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 () 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., ; Ran et al., ; ; ; He et al., ), using field investigations (Ran et al., ), and through computational modeling (Singh, ; Nunes et al., ; Ran et al., ). 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., ; Wischmeier and Smith, ). Since then, quantitative studies of the effects of rainfall characteristics on soil erosion have been conducted (Nunes et al., ; Ran et al., ; ). Dessu and Melesse () 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, ; Arthur et al., ; Pappas et al., ), 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., ).

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., ).

### 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., ).

Two sets of vectors, X =[x 1, x 2, , xp ], X $${\mathbb{R}^{N \times p}}$$, and Y =[y 1, y 2,, yq ], Y $${\mathbb{R}^{N \times p}}$$, 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 ), $$\rho {\text{(}}U,V{\text{) = }}\frac{{Cov{\text{(}}U,V{\text{)}}}}{{\sqrt {Var{\text{(}}U{\text{)}}} \sqrt {Var{\text{(}}V{\text{)}}} }}{\text{ = }}\frac{{{a^{\text{T}}}{\sum _{12}}b}}{{\sqrt {{a^{\text{T}}}{\sum _{11}}a} \sqrt {{b^{\text{T}}}{\sum _{22}}b} }}$$, where 12 is a sample covariance matrix between X and Y , 21 between Y and X , and $${\Sigma _{12}} = \Sigma _{21}^T$$; 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: $$Var{\text{(}}U{\text{) = }}{a^T}{\Sigma _{11}}a{\text{ = }}1{\text{,}}\;\;Var{\text{(}}V{\text{) = }}{b^T}{\Sigma _{11}}b{\text{ = }}1$$. Setting $$A{\text{ = }}\Sigma _{11}^{ - 1}{\Sigma _{12}}\Sigma _{22}^{ - 1}{\Sigma _{21}}$$ and $$B{\text{ = }}\Sigma _{22}^{ - 1}{\Sigma _{21}}\Sigma _{11}^{ - 1}{\Sigma _{12}}$$ the solution of Eq. (1) can be obtained by solving either of the following two eigenvalue problems: $$Aa{\text{ = }}{\lambda ^2}a{\text{, }}\;Bb{\text{ = }}{\lambda ^2}b$$, 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: $${U_i}{\text{ = }}{a^{{{{\text{(}}i{\text{)}}}^{\text{T}}}}}X{\text{,}}\;{V_i}{\text{ = }}{b^{{{{\text{(}}i{\text{)}}}^{\text{T}}}}}Y,i = {\text{1}},{\text{2}}, \ldots ,p,$$ 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:

1. Primitive variables of rainfall characteristics: $$X = {\left( {{D_r},{I_r},Q} \right)^T}$$, where Q is the total rainfall for each experiment.

2. Primitive variables of hydrological response characteristics: $$Y = {({P_r},R,{C_s},{P_s},S)^T}$$, 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 m3/s) R (×10−6 m3) 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 26), 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. ${\kern 0pt} \left\{ \begin{matrix} {U_{u1}}{\text{ = }} - {\text{0}}{\text{.038}}{D_r} - {\text{0}}{\text{.631}}{I_r}{\text{ + 1}}{\text{.021}}Q, \hfill \\ {V_{u1}}{\text{ = }} - {\text{0}}{\text{.613}}{P_r}{\text{ + 0}}{\text{.937}}R{\text{ + 0}}{\text{.005}}{C_s} - {\text{0}}{\text{.116}}{P_s}{\text{ + 0}}{\text{.144}}S, \hfill \\ \end{matrix} \right.$ ${\kern 0pt} \left\{ \begin{matrix} {U_{u{\text{2}}}}{\text{ = 0}}{\text{.045}}{D_\{r}}{\text{ + 0}}{\text{.881}}{I_r}{\text{ + 0}}{\text{.297}}Q, \hfill \\ {V_{u{\text{2}}}}{\text{ = 0}}{\text{.774}}{P_r}{\text{ + 0}}{\text{.326}}R{\text{ + 0}}{\text{.009}}{C_s}{\text{ + 0}}{\text{.112}}{P_s} - {\text{0}}{\text{.061}}S, \hfill \\ \end{matrix} \right.$ ${\kern 0pt} \left\{ \begin{matrix} {U_{{\text{d}}1}}{\text{ = 0}}{\text{.149}}{D_r}{\text{ + 1}}{\text{.055}}{I_r} - {\text{0}}{\text{.363}}Q, \hfill \\ {V_{d1}}{\text{ = 0}}{\text{.788}}{P_r} - {\text{0}}{\text{.080}}R - 0.095{C_s}{\text{ + 0}}{\text{.369}}{P_s} - {\text{0}}{\text{.203}}S, \hfill \\ \end{matrix} \right.$ ${\kern 0pt} \left\{ \begin{matrix} {U_{d2}}{\text{ = }} - {\text{0}}{\text{.171}}{D_r}{\text{ + 0}}{\text{.125}}{I_r}{\text{ + 1}}{\text{.100}}Q, \hfill \\ {V_{d2}}{\text{ = 0}}{\text{.249}}{P_r}{\text{ + 0}}{\text{.691}}R{\text{ + 0}}{\text{.306}}{C_s} - {\text{0}}{\text{.348}}{P_s}{\text{ + 0}}{\text{.345}}S. \hfill \\ \end{matrix} \right.$

Generally, the canonical correlation coefficient is close to 1 (not less than 0.96) for all pairs of canonical variables (Tables 25), 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 Pr 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, ), 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., ); 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, ) 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., ).

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., ). 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., ). 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., ) 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.

1. 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.

2. 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.

3. 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.

4. The analytics work by CCA is more effective for the condition of downstream movement than upstream movement.

* Project supported by the National Basic Research Program (973) of China (No. 2011CB409901-01), and the National Natural Science Foundation of China (No. 4081011)

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