Journal of Hydrology 399 (2011) 353–363

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Journal of Hydrology

journal homepage: www.elsevier .com/ locate / jhydrol

Configuration of the relationship of soil moistures for vertical soil profiles ona steep hillslope using a vector time series model

Sanghyun Kim a,⇑, Hanna Sun a, Sungwon Jung b

a Department of Environmental Engineering, Pusan National University, Jangjun-dong san 30 Kumjungku, Pusan 609-735, Republic of Koreab Hydrological Survey Center, Daewha-dong 2239, Koyang Kyongki Province 411-805, Republic of Korea

a r t i c l e i n f o s u m m a r y

Article history:Received 19 July 2010Received in revised form 11 November 2010Accepted 16 January 2011Available online 23 January 2011

This manuscript was handled by P. Baveye,Editor in Chief, with the assistance ofHongbin Zhan, Associate Editor

Keywords:Soil moistureTime series analysisGap fillingHillslope hydrology

0022-1694/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.jhydrol.2011.01.012

⇑ Corresponding author. Address: Dept. of EnviroNational University, P.B 609-735. Jangjun-dong sRepublic of Korea. Tel.: +82 051 510 2479; fax: +82 0

E-mail address: kimsangh@pusan.ac.kr (S. Kim).

Variation in soil moisture content throughout soil profiles during several sequential rainfalls representsthe internal hydrological response on a hillslope scale. A multiplex TDR system has been operating on amountainous hillslope to obtain the time series of soil moisture along two transects in the study area. Thesoil moisture modeling conducted in this study highlights our understanding of the inter-relationshipsbetween soil moistures at identical spatial locations, but at different depths. A sequential procedurewas used for the time series modeling to delineate an appropriate model for application to all monitoringpoints. The feedback relationship of soil wetness between two different depths was expressed with theproposed vector autoregressive model. Based on the successful modeling of 31 coupled soil water histo-ries, the vertical distributions of the stochastic model throughout the study area were obtained. The dis-tribution of the delineated models implied a spatial distribution of the hydrological processes, such asvertical infiltration for the upper soil layers and some of the lower soil layers (38 out of 62 models), lat-eral redistribution and subsurface flow over bedrock mostly for the lower soil layers (24 out of 62 mod-els) on the steep hillslope. With the use of the resultant models, applications were proposed to improvethe data acquisition system, i.e. gap filling for missing data and limited prediction for an ungaugedlocation.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

Soil moisture is a critical variable of the hydrological state indetermining hydrological processes such as runoff response, waterenergy interaction (i.e. evapotranspiration), transport of solute,groundwater and eco-hydrological dynamics (Ambroise, 2004;Brolsma and Bierkens, 2007; Chen and Hu, 2004; Hotta et al.,2010; Montaldo et al., 2001; Rodriguez-Iturbe and Porporato,2004; Tromp-van Meerveld and McDonnell, 2006). The spatialand temporal distributions of soil moisture have been studied overa wide range of scales such as 0.46, 0.47 and 0.57 ha (Penna et al.,2009), 10.5 ha (Western et al., 1999) and between 5 and 60 ha(Wilson et al., 2004) and those along hillslopes may indicate theinternal hydrological process for the generation of runoff (Hilbertset al., 2007; Hopp and McDonnell, 2009; Lin et al., 2006).

The movement of water in a soil layer is governed, not only bythe matrix flow, but is also controlled by the preferential flow(Ireson et al., 2006; Lepore et al., 2009; Mathias et al., 2006). How-ever, the difficulty in predicting the hydrological process is mainly

ll rights reserved.

nmental Engineering, Pusanan 30, Kumjung-ku, Pusan,51 514 9574.

associated with the heterogeneity of the soil (Allaire et al., 2009).The activity of earthworms, soil cracks, flow over distinct soil lay-ers and hydrophobicity are all responsible for the complexity offlow in the soil media of a natural system. Several mechanisms ex-ist for the generation of a flux faster than the matrix flow in a soillayer, such as crack flow, burrow flow, finger flow, soil interfacelateral flow and macropore flow (Blake et al., 1973; Gish et al.,2005; Greco, 2002; Rezanzhad et al., 2006; Weiler and McDonnell,2007; Zehe and Flüher, 2001; Zhu and Lin, 2009).

Time domain reflectometry (TDR) has been the most reliablemethod for in situ soil moisture measurement (Cosh et al., 2005;Zehe et al., 2010). Further developments in vertical TDR probing(Greco and Guida, 2008), wireless sensor networking (Bogenaet al., 2007) or ecological measurements such as stem water con-tent (Hernandez-Santana et al., 2008) and litter moisture (Canoneet al., 2009) also indicate extensive applications of TDR. Soil mois-ture measurements were used for the characterization of spatialvariability (Western et al., 1999; Petrone et al., 2004; Zehe et al.,2010). The stochastic features of the soil moisture dynamics werestudied both in a probabilistic approach (Ridolfi et al., 2003) andtime series modeling of measurements at the hillslope scale (Kimand Kim, 2007; Kim, 2009a). However, the stochastic structure be-tween measured soil moistures had not been studied and coupledsoil moistures can be understood through systematically analysis

354 S. Kim et al. / Journal of Hydrology 399 (2011) 353–363

of the intensive time series measurements. The interaction be-tween vertically adjacent soil moistures may be mutual, ratherthan being a uni-directional function, not only because of the het-erogeneous flux pattern, but also due to hystereisis in the soil ten-sion and water content, and the pressure wave transmission(Abdul and Gillham, 1984; Weiler and McDonnell, 2007).

This paper addresses two questions via soil moisture monitor-ing and modeling. First, what is the hydrological process alongthe two transects and on the downslope part of hillslope in thecontext of the delineated model structure? More specifically, whatadvantage can be gained using a vector model over other simplerapproaches, such as univariate or multivariate models? Secondly,how can the results of modeling be used to support better dataacquisition for missing data or to improve the redundancy of soilmoisture monitoring?

2. Study area

The study area was a hillslope on the Gamak Mountain, locatedin the north-western part of South Korea. The hillslope is the head-water of the tributary stream, Sulmachun (Fig. 1), which is con-nected to the Imjin River. The average annual rainfall in this areahas been approximately 1500 mm/year over the last 10 years, andthe minimum and maximum temperatures are �10 and 35 �C,respectively. The vegetation in this study area is primarily com-posed of a mixture of Polemoniales and shrubby Quercus. The slopeof study area (Fig. 1) varies between 20� and 35�. A previous soilmoisture study has been conducted on the hillside adjacent to thestudy area (Kim and Kim, 2007; Kim, 2009b). In fact, extensive sur-vey work and redesign of the monitoring program were performedto address the monitoring network along multiple transects.

Gneiss composites, underlined by granite bedrock, are the pri-mary geological features and the depth of the soil layer varies from25 to 95 cm. The porosity at a depth of 10 cm was about 50% andthis decreased in deeper layers. Considering the relatively smallsize of the study area (11,000 m2), the spatial variability in the veg-etation can be neglected (Kim, 2009b). A particle analysis of soilsamples indicated that the soil composition percentages of sand,silt and clay ranged between 55% and 75%, 25% and 45% and 2%and 5%, respectively, and there was no systematic difference inthe soil texture distribution along the hillslope (Kim and Kim,

Fig. 1. The Sulmachun catchm

2007; Kim, 2009b). Refined Digital Elevation Models (DEM) forthe surface and bedrock, with a resolution 0.5 m, were obtainedafter an intensive topographical survey (540 points) of the studyarea using a theodolite (DT-208P, TOPCON) and via direct and mul-tiple measurements of the soil depth using iron poles.

3. Methods and materials

3.1. Field data acquisition of soil moisture

A soil monitoring system was installed and operated over a fewsequential rainfall events during mid-September to the end ofOctober 2007. Based on digital elevation models, the spatial distri-bution of the flow was evaluated assuming the soil wetness wasprimarily governed by the terrain on the humid hillslope (Wilsonet al., 2004; Quinn et al., 1991; Anderson and Kneale, 1980). Mon-itoring locations were determined for the two transects, transectsA and B, from the hilltop to a natural and variable channel initia-tion point and the hydrologically active area, region C, locateddownslope of the study area. Fig. 2 shows the locations of the mon-itoring points, as well as the topographic wetness index, ln (a/tan b), where ‘a’ is the upslope area and ‘tan b’ is the local slopeestimated, using the MD8 algorithm (Quinn et al., 1991).

The hydrological process involved in the vertical soil profile canbe configured by investigating the interactions between the soilmoisture responses at different depths. In this study, the soil mois-ture along two transects of a mountainous hillside was intensivelymonitored. Reliable in situ measurements, using a multiplexTRASE-TDR system (Soil Moisture Equipment Corp., 2005) wereemployed to obtain the soil moisture as a multiple time series.Depending on the soil layer depth, 2–3 sensors were installed atthe monitoring points to configure the vertical profile and subse-quent redistribution of the soil moisture. Wave guides were in-serted horizontally, heading in an upslope direction at depths of10, 30 and 60 cm, without disturbing the soil layer.

Temporal variations in the soil moisture content were recordedbi-hourly for the modeling period between September 18 andOctober 8 2007, and for an extended period between October 9and 25 2007. An Automatic Rain Gauge System (ARGS), located50 m from the study area, was used to measure the rainfall, asshown in Fig. 1. During the modeling period, rainfalls of 38.6,

ent and the study area.

00.61.21.82.433.64.24.85.466.67.27.88.499.610.210.811.41212.6

0m 20m 40m

Fig. 2. The locations of the soil moisture monitors and wetness index area using theMD8 algorithm (Quinn et al., 1991).

S. Kim et al. / Journal of Hydrology 399 (2011) 353–363 355

31.1, 57.9, 9.2 and 5.2 mm were recorded on September 19, 21 and27, and October 3 and 7, respectively. Rainfall on October 19 and25 were 12.7 and 4.9 mm, respectively.

3.2. The soil moisture modeling procedure

This study performed a modeling approach to explore the feed-back process between soil moistures at two different depths. Delin-eation of the vector autoregressive soil moisture model should beperformed through a systematic time series analysis procedure,as follows: examination of raw data and pre-treatment, investiga-tion of any correlation relationship to identify the causality struc-ture, the parameter estimation and model selection consideringboth statistical and physical contexts, followed by the refinementof the model structure and diagnostically checking the final model(Salas et al., 1988).

During the preliminary data investigation step, the normality ofthe time series is estimated. The raw data frequently require pre-treatment due to the presence of trends, which may violate the ba-sic requirement for the time series analysis processes, such asmodel identification and parameter evaluation. Data differencingcan be used to remove mean variations or seasonality and Box-Cox transformation is employed to improve the normality of thedata.

The structure of time series correlation can provide useful infor-mation for identifying the rational vector model. The matrix struc-ture of parameters in the vector model significantly increases thenumber of parameters required to build the model (i.e. six param-eters for two series). The number of parameters needs to be re-duced, with the strongest consideration given to models withlow orders (Tiao and Box, 1981). A potential model structure forall pairs of a series can be effectively identified using a cross corre-lation matrix and partial autocorrelation matrix for the vector ser-ies. The partial autocorrelation is generally useful in determiningthe order of a model for an autoregressive approach. If the seriesvector, Zt, follows a vector autoregressive model, with an order p,

then the coefficients of the partial autocorrelation matrix associ-ated with and an order higher than p can be ignored. Therefore,all elements of Uij for k > p, where Uij is the partial autocorrelationmatrix with lag k, would be expected to be insignificant. The vari-ances of the elements of Uij can be estimated using the least squaretheory, and the comparison with the partial autocorrelation matri-ces may then provide a useful determination criterion of thesignificance.

The tentative order of a vector autoregressive can be deter-mined by employing the statistics used by Bartlett (1938):

MðlÞ ¼ �ðN � 1=2� l � kÞ � ln U ð1Þ

where U is the ratios of the determinants, U = |S(l)|/|S(l � 1)|, and S(l)is the matrix of the residual sum of squares and cross products afterfitting to an AR(l) model, and N = n � p � 1 is the effective numberof observations, assuming a constant term is included in the model.M(l) is, via the null hypothesis, asymptotically distributed as a v2

distribution with k2 degrees of freedom.Multiple candidate models are obtained during the preliminary

identification procedure. The structure of the autoregressive vectormodel can be expressed as:

In �U1n �U2

n � � �Ukn

� �� Zt ¼ C þ noise ð2Þ

where I is the n by n identity matrix, Ukn is the autoregressive coef-

ficient matrix with lag k, and Zt is the vector series and C is a vectorconstant.

A parameter estimation of the Ukn matrix and a comparison with

the corresponding standard errors indentifies the number of insig-nificant estimates. A heuristic approach was performed through are-estimation with zero for the insignificant parameters. Both theconditional and exact likelihood functions are used to evaluatethe parameters for the correlation matrix (Phadke and Kedem,1978; Hillmer and Tiao, 1979). The other criterion used in modelselection is the parsimony of the delineated models. Akaike’s infor-mation criterion (AIC) was used to find a balance between the var-iance of the residuals and the number of autoregressive parameters(Akaike, 1974):

AICðp; qÞ ¼ N � ln r2e

� �þ 2 � ðpÞ ð3Þ

where N is the sample size, r2e is the maximum likelihood estimate

of the residuals variance and p is the order of the autoregressiveparameters.

Diagnostic checks of the model are performed to reveal possiblemodel inadequacies, including visual checking of the residual ser-ies, investigation of the residual autocorrelation function and com-putation of the cross correlation matrix for the residuals of models.The procedures, from model identification to diagnostic checking,need to be repeated until an appropriate model is obtained.

4. Results

Fig. 3 shows the soil moisture time series for transects A, B andfor region C, respectively. The soil moisture was more pronouncedat points located downslope and within deeper layers, such aspoint A7 at a depth of 30 cm (A7-30) and B7 and B8 both at depthsof 60 cm (B7-60; B8-60). The soil moisture trend did not vary overthe monitoring period, and the patterns of response to stormevents were a temporary rise and subsequent recession, withoutsignificant variation in the mean saturation, i.e. the soil moistureprior to the rainfall events as shown in Fig. 3.

Preliminary statistical evaluation of the soil moisture data indi-cated that an appropriate pre-treatment was required. Both Box-Cox transformation and centralization were used to satisfy theassumption for the statistical distribution of the modeling data(Salas et al., 1988). Highly skewed coefficients for most dataset

Rai

nfal

l(m

m/2

hr)

0

10

20

Soil

moi

stur

e (%

) 010203040

A1A2A3A4A5A6A7

010203040

010203040

Soil

moi

stur

e (%

) 010203040 B1

B2B3B4B5B6B7B8

010203040

010203040

Soil

moi

stur

e (%

) 010203040 C1

C2C5C6

010203040

Time (bi-hours)0 40 80 120 160 200 240 280

010203040

10cm

30cm

60cm

10cm

30cm

60cm

10cm

30cm

60cm

Fig. 3. Time series of the soil moisture for transects A and B, and Region C during the modeling period.

356 S. Kim et al. / Journal of Hydrology 399 (2011) 353–363

were substantially reduced for 43 of 52 points. Any combination ofdata transformations could not reduce the skewed coefficientsbelow two for points A3-30, A6-10, A6-20, A7-10, B6-10 B8-30,B8-60, C2-60 and C5-60. Skewed coefficients were more prevalentin the deeper layers at downslope points, which may be associatedwith the lateral flow over the bedrock. However, centralizationmade the means and standard deviations of all data 0 and 1,respectively.

The stochastic structure can be identified via the computationof the cross correlation matrices for a vector series. Table 1a andb show cross correlation matrices(CCMs) of the soil moistures atdepths between 10 and 30 cm for points A1 and A5, denoted asA1-10, A1-30, A5-10, and A5-30, respectively. Table 1a and b indi-cate that a low order of the moving average process is appropriatefor explaining a soil moisture vector series composed of two differ-ent depths. The partial autocorrelation matrices (PAM) between

Table 1Cross correlation matrix between A1-10 and A1-30 (a) with 2�SE = 0.128; and A5-10; and A5-30 with 2�SE = 0.129 (b); 2�SE: standard error; SS: significance signal; +: significantpositive correlation; -: significant negative correlation; .: insignificant correlation.

Lag 1 2 3 4 5

(a)q11 q12q21 q22

0:93 0:780:76 0:96

0:87 0:760:71 0:92

0:82 0:750:66 0:89

0:76 0:720:60 0:85

0:69 0:690:54 0:81

SS þ þþ þ

þ þþ þ

þ þþ þ

þ þþ þ

þ þþ þ

(b)q11 q12q21 q22

0:94 0:830:93 0:97

0:88 0:770:93 0:93

0:82 0:710:92 0:88

0:75 0:650:88 0:82

0:68 0:590:83 0:76

SS þ þþ þ

þ þþ þ

þ þþ þ

þ þþ þ

þ þþ þ

⁄ The cross correlation function when A1-30 leads A1-10 is q12 and vice versa is q21. If all the coefficients of a matrix are lower than twice the standard errors of thecorresponding components, 0.128 and 0.129 for (a) and (b), respectively, then the matrix for lag k is considered insignificant. In order to determine an effective summary ofthe pattern for the correlation structure, symbols such as +, �, and ., are introduced for the identification of candidate model.

Table 3Autoregression summary for the stepwise lags between A1-10 and A1-30 (a); and A5-10; and A5-30 (b); RV: residual variance; SSP: significance signal; +: significantpositive partial autocorrelation; -: significant negative partial autocorrelation; .:insignificant partial autocorrelation.

Lag(l) RV M(l) AIC SSP

(a)1 .112 948.48 �5.366 þ þ

: þ.0632 .112 33.39 �5.479 : :

� þ.0573 .111 16.97 �5.520 : :

� :.0564 .108 8.33 �5.524 : :

: :.0545 .104 21.07 �5.585 : �

: :.054

(b)1 .092 953.92 �5.956 þ �

þ þ.0312 .092 53.32 �6.158 : :

� þ.0253 .090 32.96 �6.270 : :

: �.0224 .088 14.39 �6.300 � :

� :.0205 .087 1.63 �6.280 : :

: :.020

S. Kim et al. / Journal of Hydrology 399 (2011) 353–363 357

the soil moistures at depths of 10 and 30 cm for points A1 and A5are presented in Table 2a and b. The appropriate order of lags be-tween A1-10 and A1-30 and between A5-10 and A5-30 were foundto be two, as illustrated in Table 2a and b, respectively. The CCMsand PAMs were computed for all adjacent soil moisture pairs alongthe vertical profiles and the preliminary structures of the vectorautoregressive models were also determined.

The residual variances, M(l) statistics, AIC, and the significanceof the partial autocorrelation coefficients between A1-10 and A1-30 and between A5-10 and A5-30 are presented in Table 3a andb, respectively. Table 3a indicates that v2 is significant (9.5 at the5% level) up to the 3rd and for the 5th lags, and variances of theresidual series decrease from the 4th lags. Table 3b also showsthe significance of v2 up to the 4th lags, and the residual variancedecreases from the 3rd lags. AICs for points A1 and A5 were alsominima for lags 5 and 4, respectively. The statistics in Table 3 wereestimated for all monitoring points and used in the model identifi-cation. Multiple candidate vector models were completed for allmonitoring points based on similar preliminary estimations asshown in Tables 2 and 3. In most models, the constant vector esti-mates were close to zero or insignificant due to the successful cen-tralization of the vector series.

Table 4 shows the final vector models along transect A. The 1stand 2nd columns are dependent variables, and the 5th to 12th col-umns represent the autoregressive coefficients with various lags.Points A1 and A2 have significant one directional correlation struc-tures between two different depths, but point A3 did not show anyinfluence between different depths at the soil moistures. The rela-

Table 2Partial autocorrelation matrix between A1-10 and A1-30 (a); and A5-10; apositive correlation; -: significant negative correlation; .: insignificant corre

Lag 1 2

(a)q11 q12q21 q22

0:79 0:130:22 0:59

�:13 :14�:17 0:36

2�SE 0:17 0:230:12 0:17

0:20 0:260:14 0:19

SS þ :þ þ

: :� þ

(b)q11 q12q21 q22

1:11 �:090:56 0:35

�:19 0:20�:20 0:59

2�SE 0:13 0:280:07 0:14

0:23 0:300:11 0:14

SS þ :þ þ

: :� þ

⁄ The partial autocorrelation function when A1–30 leads A1–10 is q12 and vthe standard errors of the corresponding components, 2 SE for corresponinsignificant. In order to determine an effective summary of the pattern fofor the identification of candidate model.

tionship between points A4, A5 and A7 also appears to be onedirectional from the upper to lower layers. The vector model for

nd A5-30 (b); SE: standard error; SS: significance signal; +: significantlation.

3 4 5

0:11 0:21�:05 0:14

�:11 0:01�:07 0:12

0:16 �:34�:03 0:08

0:20 0:270:14 0:20

0:20 0:270:14 0:19

0:17 0:230:12 0:16

: :: :

: :: :

: �: :

0:18 �:050:03 �:15

�:17 �:15�:11 �:05

�:01 0:11�:03 0:02

0:23 0:320:11 0:16

0:23 0:290:11 0:14

0:17 0:200:09 0:10

: :: :

: :� :

: :: :

ice versa is q21. If all the coefficients of a matrix are lower than twiceding partial autocorrelation, then the matrix for lag k is considered

r the correlation structure, symbols such as +, �, and ., are introduced

Table 4Delineated vector models for all points along the vertical profile of the soil layer in transect A.

No. Pt Parameters x y Const.Backshift order Backshift order

x y Lag 1 Lag 2 Lag 3 Lag 4 Lag 1 Lag 2 Lag 3 Lag 4 C

1 A1 10 0.94 �0.0012 A1 30 0.24 �.13 �.13 0.6 0.37 �0.0163 A2 10 0.96 �0.0024 A2 30 0.44 0.57 �0.0055 A2 30 1.33 �.36 �0.0056 A2 60 0.70 �.47 0.72 0.1107 A3 10 0.95 �0.0028 A3 30 0.81 0.14 0.0149 A3 30 0.77 0.21 �0.1 0.013

10 A3 60 0.98 0.01011 A4 10 0.90 0.00612 A4 30 0.28 �.13 0.66 0.19 0.01213 A4 30 0.96 0.00614 A4 60 0.16 1.02 �0.2 0.00915 A5 10 0.95 �0.00916 A5 30 0.42 0.32 0.42 �.15 0.00517 A6 10 0.72 0.48 �.29 0.15 �0.1 �0.00218 A6 20 1.13 �.18 �0.00619 A7 10 0.93 �0.00520 A7 30 0.15 0.95 �.22 �0.002

Table 5Delineated vector models for all points along the vertical profile of the soil layer in transect B.

No. Pt Parameters x y ConstBackshift order Backshift order

x y Lag 1 Lag 2 Lag 3 Lag 4 Lag 1 Lag 2 Lag 3 Lag 4

21 B1 10 0.41 0.29 0.29 0.02222 B1 30 0.65 0.28 0.00523 B1 30 0.64 0.30 0.61 �.62 �.00124 B1 60 0.95 0.01525 B2 10 0.96 0.01526 B2 30 0.81 0.1 �.01127 B3 10 1.33 �.34 �.20 0.15 �.00428 B3 30 0.76 0.55 0.51 0.26 0.01729 B3 30 1.18 �.19 �.19 0.13 0.00230 B3 60 0.74 �.48 0.25 0.46 �.01531 B4 10 0.90 �.00332 B4 30 0.55 �.27 �.17 0.36 0.30 0.20 �.00933 B5 10 0.94 �.00634 B5 30 0.53 �.23 �.22 0.4 0.23 0.23 �.01535 B6 10 1.25 �.50 �.32 0.81 �.32 �.00636 B6 30 0.63 �.49 �0.1 0.35 0.59 �.01337 B6 30 1.05 �.11 �.01238 B6 60 0.52 �0.3 0.50 0.24 �.00239 B7 10 1.05 �.13 �.01140 B7 30 0.91 �.21 0.23 �.01141 B7 30 0.97 �.11 �.01242 B7 60 0.44 �.16 0.59 �.00643 B8 10 0.91 �.01144 B8 30 0.86 �.39 0.29 �.00645 B8 30 0.81 0.3 0.11 �.47 �.00846 B8 60 0.38 0.48 0.002

358 S. Kim et al. / Journal of Hydrology 399 (2011) 353–363

point A6 was somewhat different to the other points as the surfacesoil moisture was affected at a deeper part of the soil layer.

The delineated vector models along transect B are illustrated inTable 5. The soil moistures at points B1 and B2 were not associatedwith those of the other pair depth, but strong relationships wereobserved between different depths at points B3, B4 and B5. The im-pact of past soil moisture and the contribution structures were mu-tual between the two depths at point B6. A greater influence fromthe upper layer than for its past history was shown from the modelstructure in points B6, B7 and B8.

The model distributions for region C are presented in Table 6.In region C, the soil moistures at a depth of 10 cm were notinfluenced or were negligible (see C6-10) from those at a depth

of 30 cm, but all the model structures for deeper layers weresignificantly correlated to their past histories from the upperlayer.

The cross correlation matrices for the residuals of all modelswere checked by comparing them with the confidence intervalswhich were estimated from the approximated standard error,1=

ffiffiffinp

, where n is the number of observations. For all models, dou-ble the approximated standard errors with both signs were com-pared up to lag 24 of the residual cross correlation matrices. Theplots of all the residual series showed no significant stochasticstructure over the confidence limits. Therefore, the diagnosticcheck indicated that the delineated models in Tables 4–6 appropri-ately represent the stochastic structures of the vector series.

Table 6Delineated vector models for all points along the vertical profile of the soil layer in region C.

No. Pt Parameters x y ConstBackshift order Backshift order

x y Lag 1 Lag 2 Lag 3 Lag 4 Lag 1 Lag 2 Lag 3 Lag 4

47 C1 10 0.93 �.01348 C1 30 0.75 �.25 �.29 0.37 0.22 0.30 �.12 �.00749 C1 30 0.88 0.23 �.17 �.01650 C1 60 0.52 �.28 0.51 0.17 0.01951 C2 10 0.94 �.01152 C2 30 0.52 �.26 0.46 0.25 �.01153 C2 30 0.95 �.01154 C2 60 0.23 0.97 �.22 �.00255 C5 10 0.94 �.01056 C5 30 0.64 �0.4 0.72 �.00457 C5 30 0.90 �.00958 C5 60 0.41 0.64 �0.1 �.00559 C6 10 1.23 �.18 �0.1 �.00160 C6 30 0.52 �.43 0.38 �.32 0.82 0.00961 C6 30 1.18 �.18 �.00162 C6 60 0.24 �.16 0.81 0.016

Fig. 4. Observed and predicted soil moistures with 10% (a), 30% (b) and 50% (c)missing rates for the modeling period. Root mean square errors are 1.051, 1.035,and 1.186 for (a), (b), and (c), respectively.

S. Kim et al. / Journal of Hydrology 399 (2011) 353–363 359

5. Discussion

5.1. Gap filling for data acquisition

Field operation of the soil moisture monitoring system was achallenging task, not only because of the uncertainty associatedwith the soil matrix but also due to the variable natural conditionssuch as humidity and temperature and their corresponding im-pacts on the critical electrical equipments (i.e. the data acquisitionboard and TDR machine). A soil moisture time series can sufferfrom unpredictable interruptions. Corruptions of the soil moisturesignal originated for various reasons: human error, contact prob-lems in the waveguide, electrical circuit problems in high humidityconditions, wildlife disturbance of the connection cables and, themost serious damage, from a thunderbolt. Occasionally, an appro-priate explanation for a Time Measurement Failure (TMF) error isdifficult to find, with the corresponding data correction being ex-tremely difficult. A systematic method for filling the missing datais required.

The performance of all the models developed in Tables 4–6were tested using soil moisture measurements as a dependent var-iable, with the success of the retrieval of all measurements con-firmed via inverse transformation and centralization procedures.In order to introduce abnormality into soil moisture measure-ments, the error was assumed to be distributed randomly with auniform distribution, and the error generation rates were assumedto be 10%, 30%, and 50% of all data. A conventional random numbergenerator (Press et al., 1994) was introduced to deliberately assignmissing data within all the possible data. The measured soil mois-ture data used for gap filling were replaced with those predicted,and then used to predict the future soil moisture.

The numbers used for gap filling in the data acquisition were1417, 4274, and 7128 for missing rates of 10%, 30% and 50%,respectively. Fig. 4a shows the predicted and observed soil mois-tures for gap filling for 10% of the data. Predictions for 30% and50% missing rates are shown in Fig. 4b and c, respectively. The rootmean square errors (RMSEs) between the model predictions andobservation were estimated to be 1.051, 1.035, and 1.186 inFig. 4a–c with the coefficients of determinations (R2) of 0.96, 0.96and 0.95, respectively.

The performance of the delineated models, as shown in Tables4–6 for the gap filling was evaluated using the soil moisture datafrom the extended monitoring period between October 09 and25, 2007. Gap filling of the data miss rates of 10%, 30% and 50%for the extended monitoring period are shown in Fig. 5a–c, respec-

tively. The RMSEs for the corresponding predictions were 0.805,0.832 and 0.963, an R2 of 0.98, 0.98 and 0.97, respectively. The

360 S. Kim et al. / Journal of Hydrology 399 (2011) 353–363

better prediction of results during the extended period comparedto those during the model delineation period may be associatedwith the shorter period with less variable soil moisture duringthe extended measurements.

The potential of the vector autoregressive model for gap fillingin the retrieval of the soil moisture was significant, even with ahigh missing rate of up to 50%. The differences in the RMSE andR2 between the specified ranges of errors were also negligible forboth the modeled and extended periods, which was because theobservation intermittently corrects the model behavior and mini-mizes the generation of potential error.

5.2. Sensitivity analysis for ungauged soil moisture

Variation in soil moisture is a key characteristic in understand-ing hydrological processes at a hillslope scale, as it is significantlyassociated with infiltration, evapotranspiration and vegetationdynamics (Brolsma and Bierkens, 2007; Rodriguez-Iturbe andPorporato, 2004; Teuling et al., 2006; Tromp-van Meerveld andMcDonnell, 2006). The generation of subsurface storm flow, animportant component in the generation of hillslope runoff, is also

Fig. 5. Observed and predicted soil moistures with 10% (a), 30% (b) and 50% (c)missing rates for the extended monitoring data between October 08 and 25 in 2007.Root mean square errors are 0.805, 0.832, and 0.963 for (a), (b), and (c), respectively.

significantly affected by both temporal and spatial distributionsof soil moisture (Freer et al., 2002; Montgomery and Dietrich,2002). However, the prediction of soil moisture in a deep soil layeris a more challenging issue than its measurement, mainly due tothe substantial heterogeneity in water conductance associatedwith the existence of macropores, soil structure, bedrock interface,and interface of different soil horizons (Beven and Germann, 1982;Gish et al., 2005; Greco, 2002; Haga et al., 2005; Lin et al., 2006;Rezanzhad et al., 2006; Weiler and McDonnell, 2007; Zhu andLin, 2009). Practically, the operation of TDR monitoring also de-mands a soil moisture prediction model for two reasons. One isdue to the damage of connection cables by rodents (3–4 cablesout of 50 cables per year), which systematically contaminates thesignal, but are usually found later the timing and degree of inter-ruption are not available. The other is the capital cost for extensivecables, sensors and multiplex boards. If the response for any desig-nated position could be forecast, the cost could be substantiallyreduced.

Considering the model structure, models in Tables 4–6 can beclassified into three different categories (see Table 7). The firstclass (class 1) is basically a univariate model, such as A1-10(model 1) and A2-10 (model 3) in Table 4, where no correlationcan be found to its conjugate component. The models of class 2show some impact from its soil moisture pair, but the autocorre-lation structure seems stronger than the other influences, such asA1-30 (model 2) and A2-30 (model 4), as shown in Table 4. Themodels represent an even greater contribution from its soil mois-ture pair, and can be classified as class 3 (i.e. B3-60 (model 28) inTable 5).

Fig. 6a shows the prediction using models of class 1 (seeTable 7) during the modeling period. Almost identical predictionsfor a wide range of observations indicate that predictions tend toconverge due to the univariate structure of the models in class 1.Prediction with class 2 models (see Table 7), as in Fig. 6b, showsa greater adaptive behavior than those for class 1 models. This isbecause the observations of conjugate soil moisture sequentiallyimprove the prediction. Similar behavior can be seen for class 3models (see Table 7) as shown in Fig. 6c. The RMSEs for Fig. 6a–cwere 1.864, 2.068 and 2.611.

Fig. 7a illustrates the modeling for the extended period usingclass 1 models. The prediction starting point was identical to thatfor the observation, but the predicted soil moisture evolves in a dif-ferent direction due to the limitation of the class 1 model structure.The misleading or diverged model in Table 7 produced more un-matched modeling results to observations as the time step is in-creased. The behavior of class 2 models during the extendedperiod was more dispersed. Models 20, 27, 29, 37, 39, 45 and 59were increased or decreased up to physically meaningless rangesand Fig. 7b shows the prediction of class 2 models without di-verged models. Two models (models 2 and 20) in class 2 couldnot properly describe the variation in the field observed soil mois-ture because the contribution of the conjugate soil moisture maynot be sufficient and the information from the autoregressive com-ponent appears to be required for a reliable prediction. Fig. 7c illus-trates the prediction of class 3 models, with the exception of model28, which was a misleading model (see Table 7). The RMSEs forFig. 7a–c were 4.011, 4.461 and 1.787, respectively. The modelstructure with a stronger impact from its conjugate observationmay be highly associated with the best performance of class 3models of the three model classifications. The prediction of unga-uged soil moisture with the vector autoregressive model requirespreliminary observation, modeling practice and validation suchas class 3 showing convergence to observations. The correspondingsoil moisture sensor can be eliminated or relocated, and the mon-itoring system can reduce the redundancy in the sensor networkand improve efficiency in monitoring.

Table 7Classification of the delineated vector models from Tables 4–6.

Class Model No. Diverged or miss leading forecasting models for the extended period

1 1, 3, 5, 7, 8, 9, 10, 11, 13, 15, 18, 19, 21, 22, 24, 25, 26, 31,33, 43, 47, 51, 53, 55, 57, 61

1, 3, 5, 7, 8, 9, 10, 11, 13, 15, 18, 19, 21, 22, 24, 25, 26, 31, 33, 43,47, 51, 53, 55, 57, 61

2 2, 4, 6, 12, 14, 17, 20, 23, 27, 29, 35, 37, 39, 41, 42, 45, 46, 49,54, 56, 58, 59, 60, 62

27, 29, 37, 39, 45, 59, 2, 20

3 16, 28, 30, 32, 34, 36, 38, 40, 44, 48, 50, 52 28

⁄ Italicized numbers are misleading models, which provide more unmatched results to observations as the time step is increased.

Fig. 6. Forecasting for the modeling period using class 1 (a), class 2 (b) and class 3(c) models. Root mean square errors are 1.864, 2.068, and 2.611 for (a), (b), and (c),respectively.

Fig. 7. Forecasting for the extended period using class 1 models (a), class 2 withoutmodels 27, 29, 37, 39, 45 and 59 (b), and class 3 without model 28 (c). Root meansquare errors are 4.011, 4.461, and 1.787 for (a), (b), and (c), respectively.

S. Kim et al. / Journal of Hydrology 399 (2011) 353–363 361

5.3. Hydrological process and time series modeling

As illustrated in Tables 4–6, the variation in soil moisture can becategorized into three distinct vector autoregressive models. Gen-erally, class 1 models are mainly distributed for relatively upper orlower locations within the soil profile, where its conjugate modelwill also be a univariate model. The only exception was point A6of models 17 and 18, where the soil depth was unusually shallow(25 cm) and the concave topography caused relatively high satura-tion (Quinn et al., 1991). The majority of the models at the depth of10 cm were class 1 (14/18), indicating that the downward flux wasthe dominant hydrological process (McGlynn and McDonnell,

2003; Haga et al., 2005; Kim, 2009b). In terms of the soil moistureprocess, the points of class 1 models may not be required to con-sider the lower soil moisture and which may be useful and feasiblefor a simplified approach in surface-atmosphere modeling (Bro-lsma and Bierkens, 2007; Chen and Hu, 2004; Hotta et al., 2010;Montaldo et al., 2001; Teuling et al., 2006). The points categorizedas class 2 models were usually located in deeper soil layers, such asat depths of 30 or 60 cm. Redistribution towards the downslope ora lateral contribution from the upslope may result in higher auto-regressive weightings than its counterpart soil moisture. Espe-cially, the soil moistures of diverged models in forecasting(models 20, 27, 29, 37, 39, 45 and 59) may be determined by

362 S. Kim et al. / Journal of Hydrology 399 (2011) 353–363

significant contributions from other mechanisms, such as lateralflow or return flow. Soil moistures from class 3 models were alllower layer points from the vector models, which illustrates thatthe vertical contribution mainly determines the saturation of thecorresponding points. The vertical contribution for all soil profiledepths can be observed for points B3, B6 and C1. Modeling results,such as those for classes 1 and 3, that the vertical infiltration is theprimary process for water movement, but the existence of modelsof class 2 indicates that a lateral upslope contribution or down-slope drainage are not negligible. The model of class 2 could notbe found in even deep soil layer when the soil depths were deeperthan the deepest sensor such as points A3 and A5 with soil depths80 and 90 cm, respectively. In this study, the impact of soil depth tosoil moisture (Tromp-van Meerveld and McDonnell, 2006) is notappeared, since the study period showed relatively stationarybehavior in soil moisture.

The vector modeling results using the 31 soil moisture pairs ofthe vertical profile, indicated that the majority of the soil moistureresponse (22 out of 31) were a one-directional function, mostlyfrom the upper to lower direction, or that there was no causalityrelationship at all (i.e. models 7, 8, 9, 10, 21 and 22). However,the sensitive response of the upper soil moisture, with the varia-tion in the lower soil moisture, for points B3, B6 and B7 indicatedthe significant stochastic structure of the mutual interaction alongthe entire soil profile, as illustrated in Table 5. The existence ofcrack or macropore flow could be a possible explanation for thefast response model structure between two different depths(Beven and Germann, 1982; Greco, 2002; Rezanzhad et al., 2006;Weiler and McDonnell, 2007; Zehe and Flüher, 2001). Points B7,B8 and C1, in deeper soil layers (between 30 cm and 60 cm) alsoshowed substantial interaction between two different depths.The generation of subsurface lateral flow over the bedrock maybe concentrated on the downslope of the study area such as atpoints B7, B8 and C1, which is probably due to the model drivenstructure for corresponding points.

Based on the modeling and distribution along the hillslope, thehydrological process would primarily be driven by the verticalinfiltration of the surface soil and the lateral redistribution in thelower soil layer, with the subsequent generation of lateral flow(Gish et al., 2005; Zhu and Lin, 2009) and its concentration onthe downslope towards the stream initiation position (Hagaet al., 2005), as shown in Fig. 2.

6. Conclusion

Based on several weeks monitoring of 50 soil moisture time ser-ies for a steep hillslope for several weeks, a vector time series anal-ysis was performed to find the stochastic relationships betweentwo different depths along a vertical soil profile. The vector model-ing approach provides a tool to diagnose the existence of a possibleindependent univariate structure, a one-directional transfer func-tion, and a mutual feedback relationship between two soil mois-ture histories. The spatial distribution of stochastic models for 31soil moisture pairs was delineated through through a systematicmodeling procedure. The distribution of the models indicates thatthe hydrological process within study area was mainly governedby vertical infiltration for the upper soil layer, and that lateralredistribution seems a possible explanation for the soil moisturebehavior in the deeper soil layers. The area of subsurface stormflow generation and the deep soil layer at the downslope locations,can be also identified from the modeling results.

In terms of its application to field measurement management,the vector autoregressive modeling of soil moisture shows a poten-tial as a gap filling method for improving the data acquisition. Pre-diction of ungauged soil moisture only seems successful when the

vector model is determined by the strong causality due to its con-jugate soil moisture. However, the success at both gap filling andprediction using driven models, is based on the observation, mod-eling practice and validation. The applicability of delineated mod-els may be restricted to the autumn season as there trends to be noheavy rainfall or an extreme drought. Further research to relax therestrictions of modeling could be a possible future research topic.

Acknowledgment

This paper was supported from the research grant from Hydro-logic Survey Center. We thank anonymous reviewers for their con-structive review of the paper.

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