Journal of Hydrology (2007) 341, 1– 11

ava i lab le a t www.sc iencedi rec t . com

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

Stochastic analysis of soil moisture to understandspatial and temporal variations of soil wetness at asteep hillside

Sanghyun Kim a,*, Hyeonjun Kim b

a Department of Environmental Engineering, Pusan National University, P.B. 609-735, Jangjun-dong san 30 Kumjungku,Pusan 609-735, Republic of Koreab Korea Institute of Construction Technology, 2311, Daewha-Dong, Ilsangu Goyangsi 411-712, Republic of Korea

Received 15 June 2006; received in revised form 25 February 2007; accepted 16 April 2007

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*

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KEYWORDSSoil moisture;Time series modeling;Terrain analysis;Hillslope hydrology

22-1694/$ - see front mattei:10.1016/j.jhydrol.2007.04

Corresponding author. Tel74.E-mail addresses: kimsang

.kr (H. Kim).

r ª 200.012

.: +82 05

h@pusan

Summary Systematic analysis procedures for point scaled data are required to explainspatial and temporal distributions of soil moisture during storm flows and subsequentlow flow periods. In this study, field and simulated data of soil wetness were used toimprove the understanding of hydrological processes at the hillslope scale. Using multipleTDR sensors on a steep hillside, soil moisture responses to sequential rainfall events wereobtained as multiple time series data. Mathematical descriptions of the soil moisturetransfer processes can provide the hydrological basis for time series analysis with mea-sured soil moisture at the hillslope scale. The spatial distribution of the models selectedfor monitoring points of the study area was obtained by applying time series modeling pro-cedures to the measured data. The variation of soil moisture can be characterized usingthe final model structures in conjunction with terrain attributes such as topographic wet-ness indices or the distances between monitoring points and the outlet.ª 2007 Elsevier B.V. All rights reserved.

Introduction

Reliable in situ measurements by means of TDR (e.g. Walkeret al., 2004) can be represented as a multiple time series ofsoil moisture. Time series modeling has been widely used in

7 Elsevier B.V. All rights reserved

1 510 2479; fax: +82 051 514

.ac.kr (S. Kim), hjkim@kict.

hydrologic data analysis to understand and forecast stream-flow variation (Salas et al., 1988). The simple structure ofautoregressive moving average modeling has been inter-preted in various hydrologic parameters such as river waternitrate (Worrall and Burt, 1999), water table depths (Knot-ters and de Gooijer, 1999), water consumption patterns(Fullerton and Nava, 2003), and even with bulk sap flow datafor canopy transpiration (Ford et al., 2005). The variationsin the characteristics patterns were configured based on soilmoisture data with an uneven time interval using spatial

.

2 S. Kim, H. Kim

inference and the cross-correlation relationship betweenthe soil water content and meteorological variables suchas rainfall, temperature, and wind speed (Thierfelderet al., 2003). Stochastic characteristics of soil moisturedynamics in a hypothetical hillslope were studied by usingnumerical simulation at the daily time scale (Ridolfi et al.,2003).

Soil moisture is one of the most difficult hydrologicalvariables to predict because its temporal and spatial distri-bution depends upon several interrelated factors such assoil properties, vegetation cover, and topography (Wilsonet al., 2005). Topography seems to have the most dominantimpact at intermediate levels of soil saturation (Wilsonet al., 2004; Anderson and Kneale, 1980). Several re-searches have shown that the topographic attribute sug-gested by Beven and Kirkby (1979) explains the spatialvariability of soil moisture (Chaplot and Walter, 2003; Gunt-ner et al., 2004; Wilson et al., 2004). The spatial and tem-poral distributions of soil moisture during storm flow andsubsequent low flow periods have not been investigatedfrom the viewpoint of the stochastic aspects of soil moistureat the hillslope scale.

To develop a better understanding of the hydrologic pro-cess in the context of a series of rainfall events, and soilwater redistribution, the soil moisture of a mountainoushillside needs to be intensively monitored. Soil moistureat the hillslope-scale can be understood by systematicallyevaluating time series of measurements. In order to providea deterministic basis to the time series model developmentprocess, the mass balance of soil moisture transfer processaround installed TDR sensors was derived for a hillslope. Asoil monitoring system was installed and operated for afew sequential rainfall events. A sensitive multiplex TDRsystem (Soil Moisture Equipment Corp., 2003) was used forintensive monitoring of the soil moisture during November2003. The monitoring results were interpreted by analyzingthe spatial distribution of the temporal variation features ofsoil moisture. The modeling characterized the variations inthe patterns of soil moisture in terms of the stochasticstructures of the selected models in conjunction with thetopographic surrogates for a hillside in a mountainouscatchment.

Method and data

Hydrological basis of time series modeling for soilmoisture at a hillslope

Most hydrologic variables are frequently represented as acombination of deterministic components and an additionaluncertainty term, namely a residual (Salas et al., 1988). Thestochastic process of streamflow provides a sound basis forARIMA modeling (Salas et al., 1988). In this paper, a justifi-cation of the soil moisture transfer process at the hillslope-scale for time series analysis of measured soil moisture willbe attempted, similar to the work of Salas et al. (1988).Fig. 1 shows a schematic diagram of the soil moisture pro-cess on a steep hillslope. Mass balance equations of soilmoisture are introduced to the stochastic process of soilmoisture (Ridolfi et al., 2003). Effective rainfall input maybe associated with the development of vegetation and its

resulting interception by leaves or stem flow generation.Uncertainty in the vertical or lateral pipeflow distributioncan be a primary random component in the modeling pro-cess (Uchida et al., 2001).

In Fig. 1, Xt is the drainage to the soil moisture sensorfrom above; St is the soil moisture at time step t. Sn is thesoil moisture of n + 1th order of a lateral soil pixel; e isthe fraction of soil moisture that evaporates back to theatmosphere; p is the fraction of soil moisture leaking down-ward to the lower layer, which constitutes the verticallosses in soil water balance; and l is the fraction of soilmoisture content contributed to an adjacent downslope soilpixel. Therefore, pSt is downward leakage, eSt is evapo-transpiration, lSt is lateral subsurface flow through the soilmatrix or interconnected macroporosities (Beven and Ger-mann, 1982; Bronstert and Plate, 1997) in Eq. (1).

Soil moisture equilibrium for the first pixel (i = 1), S, inFig. 1 can be expressed as:

St ¼ St�1 � ðlþ eþ pÞSt þ Xt ð1Þ

Eq. (1) can be rearranged as:

St ¼ ½1=ð1þ lþ eþ pÞ� � St�1 þ nt ð2Þ

where nt = Xt/(1 + l + e + p) can be assumed to be a randomvariable due to the complicated interactions of evaporationparameters, radiation, temperature, wind speed, and tran-spiration functions. Functions can include the non-lineardependence of plant water potential and uptake (Laioet al., 2001), drainage from above, leakage below, and lat-eral downslope flux. These functions are associated withlayers of different hydraulic properties, and macroporedevelopment due to roots, worm holes, animal burrows(Beven and Germann, 1982; Uchida et al., 2001) and evensoil erosion (Anderson, 1988; Noguchi et al., 1999).

The soil moisture in the second pixel (i = 2) in Fig. 1, S 0, isdetermined not only by vertical recharge, feedback, andhorizontal drainage, but also by a contribution from the firstupslope pixel (i = 1). The soil moisture mass balance equa-tion of the second pixel can be expressed as:

S0t ¼ S0t�1 þ lSt � ðl0 þ e0 þ p0ÞS0t þ Xt ð3Þ

where l0S0t is the soil moisture outflow from the second tothird pixel; and l 0, e 0, and p 0 are fractions of downslopeflux, evapotranspiration, and percolation to bedrock,respectively.

One time step lag of Eqs. (3) and (1) provide a relation-ship among St; S

0t�1; and S0t�2. A model for the second pixel

can be defined by combining this relationship with Eq. (3)as:

S0t � ½1=A0 þ 1=A�S0t�1 þ ½1=ðAA

0Þ�S0t�2¼ ðl=Aþ 1Þnt � 1=A � nt�1 ð4Þ

where nt = Xt/(1 + l 0 + e 0 + p 0) is the random variable for thesecond pixel; and A and A 0 are 1 + l + e + p and1 + l 0 + e 0 + p 0, respectively.

The soil moisture equilibrium for the third pixel (i = 3)and one and two time lags of the corresponding equationcan be combined with Eq. (4) and defined as:

S00t þ a001S00t�1 þ a002S

00t�2 þ a003S

00t�3 ¼ b000nt þ b001nt�1 þ b002nt�2 ð5Þ

Figure 1 A schematic diagram of soil moisture process in a hillslope; en, pn, ln < 1, 0 < en + pn + ln < 1.

Stochastic analysis of soil moisture to understand spatial and temporal variations of soil wetness 3

where a001; a002; a

003; b

000; b

001; b

002 are constants which consist of

A, A 0, and A00, which is equal to 1 + l00 + e00 + p00.Similarly, the general model for the nth pixel can be ex-

pressed as

Xn

j¼0an�1j Sðn�1Þt�j ¼

Xn�1

k¼0bn�1k nt�k ð6Þ

where an�10 ¼ 1.The general univariate model used in this study assumes

that soil moisture is a stochastic process which can be ex-pressed as a weighted sum of current and past values of soilmoisture process and random noise process, which can beexpressed as:

UðBÞ � St ¼ HðBÞ � nt ð7Þ

where B is the backward operator such that Bknt = nt�k; U(B)is the general autoregressive polynomial; and H(B) is thegeneral moving average polynomial. An analogy can befound between the general formulation of the univariatemodel in Eq. (7) and in Eqs. (2) and (4)–(6).

Study area

The study area is a hillside in the Sulmachun catchment, lo-cated at the headwaters of the Imjin River in northern SouthKorea. Fig. 2 shows the location of the Sulmachun catch-ment and the study area on the Korean peninsula. The an-nual precipitation is about 1600 mm, and the temperaturevaries from �10 to 35 �C in a year. The study area consistsof gneiss composites underlain by granite bedrock. The pri-mary soil component in the study area is Leptosol (from FAOsoil classification) as is expected when soil formation is con-ditioned by the physiography of the terrain. An intermediatesoil depth between 40 cm to 120 cm, steep topography, andweak water holding capacity of bedrock are the primarycharacteristics of the study area. The primary community

of the study area forest is a mixture of Polemoniales andshrubby Quercus species. Pinus koraiensis and Princip-isruprechtii can be observed in the lower hillslope zone.The average height of vegetation ranged from 2 m to 10 mtoward the downslope direction of the hillslope. Visualinspection of the study area indicates that macropore devel-opment resulted mainly from biological activity such asworm hole creation and root channels (Bouma et al.,1982). Further details of the Sulmachun catchment can befound in Kim and Lee (2004) or Kim et al. (2007).

Soil moisture monitoring system

Configuring the spatial distribution of soil water movementis important for the design of an efficient soil moisture mon-itoring system using multiple TDR sensors. Since the area issmall, the spatial variability of soil properties and vegeta-tion type is negligible.

A topographic wetness index (Beven and Kirkby, 1979)has been used to predict the spatial patterns of the satu-rated area (Guntner et al., 2004) and water tables (Blazkovaet al., 2002). A high-resolution digital elevation model(DEM) seems to be a critical requirement for representingthe connectivity features of the surface processes such asspatial distribution of soil saturation (Western et al.,2002). An intensive topographical survey of the study areawas performed with a theodolite (DT-208P, TOPCON). Thedepth to bedrock was measured manually using an iron poleat each corresponding survey point. A refined DEM with a1 m resolution was delineated using the distances and thevertical or horizontal angles from a reference point locatedat the outlet of the hillslope. Fig. 3 shows the surface andsubsurface (bedrock surface) DEMs obtained from the inten-sive survey.

The spatial distribution of water accumulation can becalculated based on the assumption that a local lateral

Figure 2 The location of Sulmachun catchment with the study area.

Figure 3 Surface DEM and subsurface DEM of the study areaand monitoring positions.

4 S. Kim, H. Kim

hydraulic gradient can be approximated by the local slope(Beven and Kirkby, 1979; O’Callaghan and Mark, 1984; Quinnet al., 1991). The single flow direction algorithm, SFD, by

O’Callaghan and Mark (1984) and multiple flow directionalgorithm, MFD, by Quinn et al. (1991) were used to com-pute the spatial distribution of the topography based wet-ness index. The terrain of the subsurface topography wasalso analyzed. Differences between the surface and subsur-face terrain analysis are negligible because the slope of thehill is steep, between 30� and 45�, and the soil depth is rel-atively shallow. Based upon the computed upslope areas,ln(a/tanb), where a is the upslope area per unit width andtanb is the local topographic slope, and field conditions,11 points were selected, where monitoring locations weredictated by the variations found at site as shown in Fig. 3.The installation of the monitoring system and its represen-tation of variation patterns of the moisture flow was basedon flow divergence from the MFD topographic wetness indexand the primary converging flow path from the SFD topo-graphic wetness index. The Cartesian coordinates of themeasurement positions in Fig. 3 were reconverted to dis-tance and angle vectors. The ground truthing from a refer-ence position using the distance and angle vectors made theaccurate installation of TDR wave guides possible. Surfacesensors were installed vertically without disturbing the soilmatrix. Waveguides were located approximately 10 cm be-low the ground surface.

Temporal variations of soil moisture were recordedhourly from November 6 to 22, 2003. Rainfall was measuredby an Automatic Rain Gauge System (ARGS), located 50 maway from the study area as shown in Fig. 2. Three rainfallevents, of 28 mm, 19 mm and 8 mm, were recorded onNovember 8, 11, and 20, 2003, respectively.

Fig. 4 shows the soil moisture variations of several pointsin the study area. For clarity points B, D, and I are not shownin Fig. 4. Three primary rainfalls occurred, at 29, 112, and331 h, as shown in Fig. 4. The immediate recharge and fastrecession after a peak are the primary features of soil mois-

Time (hr)0 100 200 300 400

Soi

l Moi

stur

e (%

)

10

15

20

25

30

A

A

C

C

F F

GG

H

H

K

K

Figure 4 Time series of soil moisture dotted lines for points Aand C as; dashed lines for points F and G; solid lines for points Hand K.

Stochastic analysis of soil moisture to understand spatial and temporal variations of soil wetness 5

ture at points A, B, C and D. Stability and significant storageincrease are distinct characteristics of soil moisture atpoints E, F, I, and G and at points H, J, and K, respectively(Kim et al., 2007).

Model development and results

Pretreatment of data and identification ofcorrelation structure

Preliminary evaluation of soil moisture statistics indicatesthat a transformation seems necessary to improve normalityand stationarity. An extension of power transformation,y = a(x � c)b, is introduced and then centralized to trans-form the time series (Salas et al., 1988). This process pro-vides normality of the data, which is prerequisite for theapplication of a parameter estimation procedure, i.e. themaximum likelihood method.

Examination of the time series plots for each soil mois-ture series indicated that each had an increasing trend. Inorder to develop the stationary input for the data series,all soil moisture series were transformed by altering onelag, which removed an increasing trend. Autocorrelation(ACF) and partial autocorrelation (PACF) functions of thecorrected data yielded temporal correlation structures asshown in Fig. 5. The ACF of the corrected soil moisturefor points A, B, C, E and G showed that autocorrelationswere significant through lags 3–5, after which the functionstailed off (Fig. 5a). ACF is the correlation between a timeseries and its lags. PACF represents the amount of correla-tion between a time series and lag that is not explainedby the correlations of all of the lower order lags. The PACFsof points A, B, C, E, and G also revealed responses beyondthe lags 3–6 as shown in Fig. 5b. The difference betweenthe actual correlations at lags 3–6 and the expected corre-lations were due to the correlation propagation at lower or-ders being significant. This result indicated a possibleautoregressive integrated moving average model, ARIMA(p,1,q) model with p = 2, 3, 4, 5, or 6 and q = 1, 2, 3 or 4depending upon the correlation structure of each point.

Parameter estimates were made for multiple candidatemodels in order to determine the best model. The autocor-relation structures of the corrected soil moisture of points Dand F were simpler than those of the other points. ACF andPACF of point D tailed off after lags 2 and 3, respectively.The corrected soil moisture series of point F did not showany significant stochastic structure. In order to evaluatethe possible model structures, the candidate models forpoints D and F had to be similar to other models. The ACFof the corrected soil moisture series for points H, J, I, andK eliminated lag 4 and minor autocorrelation functions ap-peared between lags 11 and 14 (Fig. 5e). PACF were overthe confidence limits until lag 4 and appeared occasionallybetween lags 8 and 12 as shown in Fig. 5f. This result sug-gested that the time series for the time series for pointsH, J, I, and K could have correlation structures up to lags12. Depending upon ACF and PACF of each point, generalmodels with parameter orders p from lags 4 to 12 and ordersq from lags 6 to 1 were made for initial identification.

Estimation

The maximum likelihood estimates (Box and Jenkins, 1976)and conditional likelihood estimates were used to find cor-responding parameters. Model estimates were made usingthe autoregressive integrated moving average (ARIMA) pro-cedure of the SCA package (Box and Jenkins, 1976; Linet al., 1994). The marquardt optimization algorithm of theSCA drives the parameters in directions that minimize thesum of squares function and updates through the iterationprocedure.

Diagnostic check and selection of model

The adequacy of the model must be determined through aseries of diagnostic checks to ensure that the assumptionsof the modeling process are correct. One diagnostic methodfor optimum model structure is the overfitting of models.Progressively more complicated models are fitted to the soilmoisture series to determine whether additional parame-ters may improve the model performance. Table 1 illus-trates one of the applications using this procedure.

Based on initial estimations of the parameters for severalcandidate models, both the statistical index and hillslopehydrological process were used to determine whether amodel with estimated parameters appropriately repre-sented the soil moisture variation of a certain point. TheStudent’s t statistics indicate the significance of the param-eter. If the t statistics are within an absolute value of 2,then the null hypothesis of the Student’s t-test is not re-jected at the 5% significance level. Continuity of the param-eter structure in the time series model is another importantcriterion for explaining the hillslope hydrological processesof soil moisture. Continuity of the parameter indicates thatthe contribution of consecutive parameters of a model toexplain soil moisture time series is well distributed and nomissing component can be found.

Table 1 shows parameter estimates of several candidatemodels for the soil moisture of point G. Models (1,1,1),(2,1,1), and (3,1,1) were invalid because the absolutes ofthe primary parameter’s t statistics are less than 2. Model

Lag (hr)0 5 10 15 20 25 30 35

AC

F

-0.4

-0.2

0.0

0.2

0.4A

Confidential Limit

B C

E G

Lag (hr)0 5 10 15 20 25 30 35

PA

CF

-0.4

-0.2

0.0

0.2

0.4A

Confidential Limit

B C

E G

Lag (hr)0 5 10 15 20 25 30 35

AC

F

-0.4

-0.2

0.0

0.2

0.4D

Confidential Limit

F

Lag (hr)0 5 10 15 20 25 30 35

PA

CF

-0.4

-0.2

0.0

0.2

0.4D

Cofidential Limit

F

Lag (hr)0 5 10 15 20 25 30 35

AC

F

-0.4

-0.2

0.0

0.2

0.4 H

Confidential Limit

J I

K

Lag (hr)0 5 10 15 20 25 30 35

PA

CF

-0.4

-0.2

0.0

0.2

0.4 H

Confidential Limit

J I

K

Figure 5 Autocorrelation functions (a), (c), (e); and partial autocorrelation functions (b), (d), (f) for transformed soil moisturetime series.

6 S. Kim, H. Kim

(4,1,3) resulted in a negligible U1 estimation, �0.034, withinsignificant t statistics, �0.09. This result indicates thatmodel (4,1,3) violated the continuity of the parameterstructure for an acceptable model structure consideringthe hillslope hydrological process. The intermediate autore-gressive parameters of model (6,1,4), such as, U2 or U3,and one of the middle moving average parameters, H3, ofthe identical model, were also very minor. Similarly, model

(6,1,4) was rejected after further consideration. Theparameter estimate of model (5,1,3) provided highlyskewed weightings of the contribution of U2, and the mov-ing average terms, H2, appeared to constitute unacceptablemodel. Model (4,1,2) produced well distributed t statisticsof the parameter estimates in most autoregressive and mov-ing average parameters. If multiple candidates were ob-tained through this procedure, a parsimony test was

Table 1 ARIMA model parameters and t statistics for soil moisture of point G

Models (p,1,q) U1 U2 U3 U4 U5 U6 H1 H2 H3 H4

(1,1,1) 0.580 0.2175.8 1.82

(2,1,1) 0.336 0.116 �0.170.8 0.7 �0.04

(3,1,1) 0.338 0.117 �0.001 �0.0150.4 0.4 �0.01 �0.02

(3,1,2) 0.738 �0.983 0.358 0.377 �0.94112.0 �17.4 6.82 9.53 �31.4

(4,1,2) 0.588 0.729 �0.273 �0.125 0.284 0.733.9 �3.76 �3.97 �2.31 2.09 5.05

(4,1,3) �0.034 �0.43 �0.39 0.30 �0.387 �0.663 �0.71�0.09 �1.66 �1.25 2.34 �0.96 �5.83 �2.17

(5,1,3) �0.22 1.24 0.271 �0.333 �0.107 �0.563 1.022 0.573�4.02 24.8 3.31 �6.63 �2.09 �19.8 74.1 29.1

(6,1,4) 0.411 0.056 0.029 0.802 �0.339 �0.124 0.101 �0.119 0.06 0.965.59 0.65 0.42 12.44 �6.38 �2.42 1.9 �2.19 1.07 17.4

Numbers in boldface indicates t statistics.

Stochastic analysis of soil moisture to understand spatial and temporal variations of soil wetness 7

applied to select a final model. Akaike’s information crite-rion (AIC) was used to find a balance between the varianceof the residuals and the number of autoregressive and mov-ing average parameters (Akaike, 1974):

AICðp; qÞ ¼ N � lnðr2e Þ þ 2 � ðpþ qÞ ð8Þ

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

estimate of the residuals variance, and p and q are ordersof autoregressive and moving average parameters,respectively.

Schwartz has proposed an alternative Bayesian criterion,SBC, for model selection as Wei (1990):

SBC ¼ N � lnðr2e Þ þ ðpþ qÞ � lnN ð9Þ

For those points in which the results of several models ful-filled the statistical and physical requirements, the modelswith the minimum AIC or SBC were chosen. AICs for model(5,1,3) and (4,1,2) are �594 and �589 and SBCs for the cor-responding models are �562.5 and �565.2, respectively.Considering the distribution of parameter estimates andparsimony tests, model (4,1,2) was selected as the finalmodel for point G. This procedure of model selection wasalso used with the other monitoring points.

Table 2 presents the parameters, residual standard devi-ation estimates and t statistics of the selected models forthe 11 points of the study area. All of the models met thecondition for model stability (Box and Jenkins, 1976). Diag-nostic checks for model residuals were performed to testwhiteness, for the nonexistence of stochastic structures,and normality. ACF of the residual series provided correla-tions among individual residuals. None of the residuals ofACF had significant stochastic structures over the confi-dence intervals. Group correlation of the residuals can betested using the portmanteau lack of fit test (Box and Jen-kins, 1976). None of the selected models were rejected at

the 0.05 level of significance by the portmanteau test.Residual histograms were plotted to check the modelassumptions. The residuals of all of the models were foundto be normally distributed except a few outliers associatedwith peak soil moisture at the beginning of the primary rain-fall events. However, normality assumption does not seemto be a crucial issue for identifying and estimating reason-able parameters (Hipel et al., 1977). The accuracy of themodels was checked by examining the residual standarddeviation estimates, r, and R2 in Table 2, in order to deter-mine the percentage of variation in the soil moisture thatwas determined by the models.

Discussion

Model structures of monitoring points

Point A was located at the highest elevation of the monitor-ing points with a negligible upslope contributing area. Table2 indicates that ARIMA(1,1,1) was the model selected to ex-plain the variations in soil moisture at point A. The higher tstatistic of the moving average parameter, 4.66, than thatof the autoregressive component, 2.35, might be associatedwith severe fluctuations in the soil moisture series of point Aas shown in Fig. 4. The soil moisture variation of point Amight be derived from the unsteady vertical flux of waterand vapor and might be due to the intensive macroporedevelopment associated with subsurface erosion as shownin Fig. 3. Table 2 also indicates that the model structuresof points B and C were identical to point A’s model(1,1,1). However, the models for points B and C had highert statistics for their autoregressive terms than for the mov-ing average parameters. Estimates of the autoregressiveparameters tended to increase as the monitoring points

Table 2 Selected ARIMA models (p,1,q), topographic surrogates, and parameters with t statistics and the coefficients ofdetermination

Point r MFDsf DFO U1 U2 U3 U4 H1 H2 H3 R2

MFDsub (m) U5 U6 U7 U8 H4 H5

A 0.318 2.89 97.88 0.35 0.59 0.902.86 2.35 4.66

B 0.199 3.62 85.59 0.6 0.34 0.963.67 4.82 2.36

C 0.124 4.51 80.18 0.69 0.19 0.994.58 10.74 2.25

D 0.323 4.50 70.80 0.1 0.904.82 2.0

E 0.120 5.70 53.81 0.40 0.28 0.995.78 8.06 5.67

F 0.221 5.56 61.94 �0.37 �0.44 0.955.57 �1.87 �2.26

G 0.206 6.02 48.54 0.59 0.73 �0.27 �0.13 0.28 0.73 0.966.29 3.94 �3.76 �3.97 �2.31 2.09 5.05

H 0.354 5.79 50.44 �0.58 �0.32 �0.24 0.21 �0.31 0.885.82 �3.86 �3.98 �3.19 2.93 �2.06

I 0.267 5.73 35.74 0.43 0.66 �0.81 �0.16 0.56 0.57 �0.98 0.925.83 7.5 11.7 �16.9 �2.9 26.2 28.7 �56.2

J 0.151 6.32 39.18 �0.16 �0.16 �0.36 0.53 �0.68 �0.7 �0.77 0.986.21 �2.82 �2.97 �8.17 11.9 �27.9 �28.7 �36.4

�0.15 �0.14 0.24�2.95 �2.73 9.12

K 0.202 6.48 28.08 1.59 �1.01 0.99 �1.28 1.42 �0.89 0.93 0.966.66 15.38 �7.02 6.00 �8.09 15.16 �8.38 7.01

0.85 �0.38 0.44 �0.29 �0.96 0.514.92 �2.87 3.94 �2.86 �9.28 5.92

Numbers in boldface indicates t statistics. r – residual standard deviations; wetness indices by MFD – multiple flow direction algorithmfrom Quinn et al. (1991); DFO – distance from outlet of study area; U1, . . .,U8 – autoregressive parameters; H1, . . .,H5 – moving averageparameters; R2 – the coefficients of determination; subscripts sf and sub stand for surface and subsurface topography, respectively.

8 S. Kim, H. Kim

moved downslope, but the converse was true for the movingaverage parameters. The fluctuations at point C were not asgreat as those at point A in the time series as can be seen inFig 4. An increase in the upslope area with a stable horizon-tal flux component might cause the soil moisture levels tobe relatively stable and might cause the weightings of theautoregressive parameters tend to increase. The standarddeviations of the residuals also tended to diminish frompoints A to C.

The impact of the moving average parameter was insig-nificant in the selection of the model for point D. Mod-el(1,1,0) indicated that recession is the primarycharacteristic explaining the variations in the soil moistureof point D. Abrupt increases and immediate decreases aftera rainfall event are the primary characteristics of soil mois-ture at point D. The relatively high standard deviation,0.323, compared with the higher points might be associatedwith the absence of a moving average component.

The dominance of the autoregressive components overthe moving average terms was pronounced at point E. Themodel for point E was (2,1,0) and the parameter estimates,orders, and t statistics of the autoregressive parameters in-creased with no moving average components. Furthermore,the standard deviation of the residual appears to be a min-imum among the other points. It was difficult to select can-didate models for point F because the correlation structuresin Fig. 5c and d were negligible. Model (1,1,1) was applica-ble even with relatively low t statistics for the parameterestimates. This may be associated with the instantaneousreductions of soil moisture after a rainfall event near timesteps 43 and 70 h in Fig. 4. A possible explanation for theseabrupt reductions in soil moisture could be the changes inthe water retaining structure at the soil layer such as an in-stant development of macropore flow such as erosion(Anderson, 1988; Noguchi et al., 1999) or pipeflow (Uchidaet al., 2001).

Stochastic analysis of soil moisture to understand spatial and temporal variations of soil wetness 9

The stochastic models for points G and H were (4,1,2)and (4,1,1), respectively, as illustrated in Table 2. The tstatistics for the estimates were evenly distributed overall the parameters. The higher orders of the model struc-tures than those of the other upper points indicate thatthe soil moisture mechanism can be more complicatedand depend more upon the history of the series. Continuousinflow from the upper soil layer or lateral flow through bed-rock and outflow to downslope areas even after a rainfallevent is one possible explanation of the well distributedparameters. Even more complicated models, (4,1,3) and(6,1,4) were found in points I and J, respectively. Unlikethe models for points G and H, the t statistics of the movingaverage parameters were higher than those for the autore-gressive components. Visual inspection revealed substantialdevelopment of macropores in this area and this might berelated to the significance of the random components inpoints I and J. Point K was located at the lowest elevationof the monitoring points as can be seen in Fig. 3. ARIMAmodel (8,1,5) was the final model selected for point K.The autoregressive and moving average components hadsimilar weightings of statistical significance. The substantialdevelopment of flow paths with the largest upslope areaprovides the most complicated stochastic structure of allof the selected models in Table 2.

Selected models and hydrological basis of soilmoisture process

The time series analysis of the measured soil moistures pro-vided a spatial distribution for the stochastic structures fora steep hillside in the study area. Use of Eqs. (2) and (4)–(6)in the time series modeling of soil moisture partially ex-plains why the selected models tend to increase the numberof parameters as for the down slope monitoring points. Theselected ARIMA(p,1,q) models can be transformed to iden-tical polynomials of ARIMA(p+1,q) processes (Salas et al.,1988). The difference between the final model selectionsand the general formulations of the physical process canbe explained by the simplified procedure that was intro-duced to derive the soil moisture transfer process. Theone dimensional array of soil pixels in Fig. 1 is different fromthe two dimensional soil layer distribution in the study areaas can be seen in Fig. 2. Multiple lateral components froman upslope pixel to adjacent lower pixels or multiple inflowsto a lower pixel can be generated and more complicatedflow diversion and converging patterns can be expected(Kim and Lee, 2004). This is also because the spatial distri-bution of the macropore development is substantially dif-ferent among soil pixels. In the hydrological process, thiswill affect not only the fractional parameter between adja-cent soil pixels, l, in Eqs. (2) and (4)–(6), but also those ofthe leakage and evapotranspiration parameters such as pand e in Fig. 1.

The moving average terms of the selected models inTable 2 are believed to express the random process insidethe soil layer. High frequency variation in soil moisture con-tent would be caused by the interaction of moisture fromprecipitation, lateral flow from the upslope area and en-trapped air in macropores generating fast water flow inthe porous media. Macropore flow may cause non-uniform

distribution of hydraulic conductivity temporally as well asspatially. A downslope pixel will be affected by a large num-ber of macropores which may or may not be completely con-nected to a corresponding pixel (Vervoort and Cattle, 2003).Substantial weightings of the moving average components inthe final selected models indicated the importance of therandom components associated with macropore develop-ment in the study area. The lower the monitoring point onthe hillslope, the more intensive the development of rootzone. This is because the majority of roots are located inthe upper 0.5–1.0 m of soil (Baird and Wilby, 1999).

Topographic surrogates and time series model ofsoil moisture

Table 2 also presents the topographic wetness indices of themonitoring points, ln(a/tanb), depending upon the surfaceor subsurface DEM. The distance from the outlet, DFO, inTable 2 identifies the length of the surface flow path fromeach monitoring point to an outlet in the subcatchment asshown in Fig. 3. Even though the soil moisture time seriesmodels do not show a linear or proportional relationshipwith the topographic attributes; the complexity of the mod-el tends to increase substantially at points that have an MFDwetness index greater than 5.7. A similar boundary can befound for DFO between 53.81 and 50.44 m as shown in Table2. However, the topographic wetness index computed bySFD (not presented in Table 2) does not show any significantrelationship to the model structure. Fig. 6a and b show rela-tionships between the contributing areas and the number ofparameters for autoregressive and moving average opera-tors. Increasing parameter numbers near the contributingarea, 200 m2, indicates the potential for the existence ofa threshold in soil moisture transfer process at the hillslopescale. In this paper, threshold indicates an upslope area ascomputed from the MFD of surface and subsurface topogra-phy, which results in more complicated time series modelsthan simple model structure forms such as ARIMA (1,1,1),(2,1,0) or (1,1,0). A similar turning point for the stochasticprocess can be seen in Fig. 6c and d between the topo-graphic wetness index by MFD and the parameter complex-ity of autoregressive and moving average operators.

Assuming that the complexity of the soil moisture timeseries model depends heavily upon the complexity of thesubsurface flow path to the monitoring point, the contribut-ing area topographic wetness index by MFD and DFO couldprovide potential thresholds for explaining the stochasticstructures of soil moisture. The distribution of the final se-lected models along the hillside suggested that the spatialdistribution of macropore development can be one explana-tion for the variations in soil moisture.

Conclusions

Soil moisture is an important variable in understanding thehydrological processes at the hillslope scale. By characteriz-ing the spatial and temporal distribution of soil moistureduring a few rainfall runoff events, a better understandingof hydrologic processes as well as a sound basis for hydro-logic models at the hillslope scale can be achieved. Theintensive monitoring of soil moisture from multiple posi-

Figure 6 Contributing areas and number of parameters forautoregressive operators (a), and moving average operators (b);topographic wetness index and number of parameters forautoregressive operators (c), and moving average operators (d);MFD, multiple flow direction algorithm; Surface, surface DEM;Subsurface, subsurface DEM.

10 S. Kim, H. Kim

tions during several rainfall events was performed in thestudy area.

The simplified mechanisms of soil moisture transfer wereimplemented in the modeling procedure and a hydrologicalbasis for time series modeling was derived. The time series

analysis technique provided a systematic tool for interpret-ing soil moisture variations. The stochastic structures of thesoil moisture time series were explored in conjunction withtopographic attributes such as the topographic wetness in-dex, contributing upslope area, and pathway distance fromthe outlet. Even though the variations in soil moisture didnot have a significant linear or proportional relationshipwith the topographic attributes, the existence of a potentialthreshold in terms of contributing area or topographic wet-ness index for formulating time series models was found.The monitoring locations can be characterized by the finalmodel structures of soil moisture in conjunction with topo-graphic parameters. Distribution of the models on the hill-side suggested that the distribution and existence ofmacropores can be an important component for explainingthe soil moisture variations of a steep hillslope.

This research was performed using soil moisture duringthe late autumn season. Analysis of this restricted soil mois-ture data may need to be further generalized. Further anal-ysis of the extended monitoring of soil moisture and moreelaborate time series techniques are future research issuesfor configuring a complete soil moisture transfer process.

Acknowledgment

This study was partially supported by BK21 from Ministry ofEducation and Human Resources Management of Korea.

References

Akaike, H., 1974. A new look at the statistical model identification.IEEE Trans. Automat. Contr. 19 (AC6), 716–723.

Anderson, M.G., 1988. Modelling Geomorphological Systems. Wiley,New York.

Anderson, M.G., Kneale, P.E., 1980. Topography and hillslope soilwater relationships in a catchment at low relief. J. Hydrol. 47,115–128.

Baird, A.J., Wilby, R.L., 1999. Eco-hydrology Plants and Water inTerrestrial and Aquatic Environments. Routledge, New York.

Beven, K., Germann, P., 1982. Macropores and water flow in soils.Water Resour. Res. 18, 1311–1325.

Beven, K., Kirkby, M.J., 1979. A physically based variable contrib-uting area model of basin hydrology. Hydrol. Sci. Bull. 24, 43–69.

Blazkova, S., Beven, K., Tacheci, P., Kulasova, A., 2002. Testing thedistributed water table prediction of TOPMODEL (allowing foruncertainty in model calibration): the death of TOPMODEL?Water Resour. Res. 38 (11). Art. No. 1257.

Bouma, J., Belmans, C.F.M., Dekker, L.W., 1982. Water infiltrationand redistribution in a silt loam subsoil with vertical wormchannels. Soil Sci. Soc. Am. J. 46, 917–921.

Box, G., Jenkins, G., 1976. Time Series Analysis: Forecasting andControl, revised ed. Prentice-Hall, Englewood Cliffs, NJ.

Bronstert, A., Plate, E.J., 1997. Modelling of runoff generation andsoil moisture dynamics for hillslopes and micro-catchments. J.Hydrol. 198, 177–195.

Chaplot, V., Walter, C., 2003. Subsurface topography to enhancethe prediction of the spatial distribution of soil wetness. Hydrol.Process. 17, 2567–2580.

Ford, C.R., Goranson, C.E., Mitchell, R.J., Will, R.E., Teskey, R.O.,2005. Modeling canopy transpiration using time series annlysis: acase study illustrating the effect of soil moisture deficit in Pinustaeda. Agric. Forest. Meteorol. 130 (3–4), 163–175.

Stochastic analysis of soil moisture to understand spatial and temporal variations of soil wetness 11

Fullerton, T.M., Nava, A.C., 2003. Short-term water dynamics inChihuahua City, Mexico. Water Resour. Res. 39 (9). Art. No.1258.

Guntner, A., Seibert, J., Uhlenbrook, S., 2004. Modeling spatialpatterns of saturated area: An evaluation of different terrainindices. Water Resour. Res. 40 (5). Art. No. W05114.

Hipel, K.W., Mcleod, A.I., Lennox, W.C., 1977. Advances in Box–Jenkins modeling, I, model construction. Water Resour. Res. 13(5), 567–575.

Kim, S., Lee, H., 2004. A digital elevation analysis: a spatiallydistributed flow apportioning algorithm. Hydrol. Process. 18,1777–1794.

Kim, S., Lee, H., Woo, N., Kim, J., 2007. Soil moisture monitoringon a steep hillside. Hydrol. Process, doi:10.1002/hyp.6508.2007.02.020.

Knotters, M., de Gooijer, J.G., 1999. TARSO modeling of watertable depths. Water Resour. Res. 35 (3), 695–705.

Laio, F., Porporato, A., Ridolfi, L., Rodriguez-Iturbe, I., 2001.Plants in water-controlled ecosystems: active role in hydrolog-ical processes and response to water stress. II. Probabilistic soilmoisture dynamics. Adv. Water Res. 24, 707–723.

Lin, L.M., Hudak, G.B., Box, E., Muller, M., Tiao, G., 1994.Forecasting and Time Series Analysis Using the SCA StatisticalSystem. Scientific Computing Associates Corp., Oak Brook, IL.

Noguchi, S., Tsuboyama, Y., Sidle, R., Hosoda, I., 1999. Morpho-logical characteristics of macropores and the distribution ofpreferential flow pathways in a forest slope segment. Soil Sci.Soc. Am. J. 63, 1413–1423.

O’Callaghan, J.F., Mark, D.M., 1984. The extraction of drainagenetworks from digital elevation data. Comput. Vision Graph.Image Process. 28, 323–344.

Quinn, P., Beven, K., Chevallier, P., Planchon, O., 1991. Theprediction of hillslope flow paths for distributed hydrologicalmodeling using digital terrain models. Hydrol. Process. 5, 59–79.

Ridolfi, L., D’Odorico, P., Porporato, A., Rodriguez-Iturbe, I., 2003.Stochastic soil moisture dynamics along a hillslope. J. Hydrol.272 (1–4), 264–275.

Salas, J.D., Delleur, J.W., Yevjeuch, V., Lane, W.L., 1988. AppliedModeling of Hydrologic Time Series. Water Resource Publication,Chelsea, MI.

Soil Moisture Equipment Corp., 2003. TRASE Operating Instruction,pp. 2–5.

Thierfelder, T.K., Grayson, R.B., von Rosen, D., Western, A.W.,2003. Inferring the location of catchment characteristic soilmoisture monitoring sites. Covariance structures in the temporaldomain. J. Hydrol. 280, 13–32.

Uchida, T., Kosugi, K., Mizuyana, T., 2001. Effect of pipeflow onhydrological process and its relation to landslide: a review ofpipeflow studies in forested headwater catchments. Hydrol.Process. 15, 2151–2174.

Vervoort, R.W., Cattle, S.R., 2003. Linking hydraulic conductivityand tortuosity parameters to pore space geometry and pore sizedistribution. J. Hydrol. 272 (1–4), 36–49.

Walker, J.P., Willgoose, G.R., Kalma, J.D., 2004. In situ measure-ment of soil moisture: a comparison of techniques. J. Hydrol.293, 85–99.

Wei, W.S., 1990. Time Series Analysis. Addison-Wesley, Redwoodcity, CA.

Western, A.W., Bloschl, G., Grayson, R.B., 2002. Toward capturinghydrologically significant connectivity in spatial patterns. WaterResour. Res. 37, 83–97.

Wilson, D.J., Western, A.W., Grayson, R.B., 2004. Identifying andquantifying sources of variability in temporal and spatial soilmoisture observations. Water Resour. Res. 40 (2), W02507.

Wilson, D.J., Western, A.W., Grayson, R.B., 2005. A terrain anddata-based method for generating the spatial distribution of soilmoisture. Adv. Water Res. 28, 43–54.

Worrall, F., Burt, T.P., 1999. A univariate model of river waternitrate time series. J. Hydrol. 214 (1–4), 74–90.