Geosciences JournalVol. 15, No. 4, p. 423 − 431, December 2011DOI 10.1007/s12303-011-0034-8ⓒ The Association of Korean Geoscience Societies and Springer 2011

Time series modeling of the spatial distribution of soil moisture in a moun-tainous hillslope headwater

ABSTRACT: The spatial distribution of soil moisture depth pro-files on a hillslope at the headwater of a mountainous forest water-shed in Korea was obtained from time series measurementsperformed using a refined soil moisture monitoring system. Digitalterrain analysis based on intensive field surveying was applied toconfigure the spatial distribution of soil moisture along the steepregolith. The upslope contributing area and topographic wetnessindex were employed to determine the optimal locations of sensorsfor time domain reflectometry (TDR) monitoring along two transects.In the early autumn of 2007, 37 time series were recorded overthree weeks and used as data for modeling, accounting for theknown reliability of the data and the preliminary evaluation ofmeasured soil moisture histories in the context of time series mod-eling assumptions. A systematic modeling procedure involvingpretreatment, investigation of stochastic structure, modeling prac-tice with heuristic repetition, and diagnostic checking was appliedto selected soil moisture time series. The two transects showed dif-ferences in the distribution of the explanatory models. Transect A,covering a region of relatively fine soil texture, showed a systematicmodel distribution in the context of soil depth, whereas transect Bacross coarser soil occasionally produced inconsistent depth pro-file modeling results. The predictability of the models tends toreduce substantially at the outlet of the hillslope region, possiblydue to active subsurface runoff generation near the channel.

Key words: soil moisture, time series modeling, hydrological processes,digital terrain analysis, subsurface water transport

1. INTRODUCTION

Soil moisture in the hillslope areas contributes to runoffgeneration and acts as the ultimate source of water foropen channels (Lee et al., 2007; Reggiani et al., 1999). Thespatial and temporal distributions of soil wetness determinethe hydrological processes that may be important in thedevelopment of a physically-based hydrologic model(Beven, 2002). Temporal and spatial variations of soil watercontent govern the eco-hydrological system, and play animportant role in infiltration, evapotranspiration, and thetransport of contaminants (Georgakaos, 1996; Rodriguez-Iturbe, 2000).

Several drivers determine moisture in soil layers, includ-ing hydro-meteorological components (e.g., precipitationand evapotranspiration), geophysical components (e.g., soil

texture, terrain of soil surface, and bedrock topography),and biological components (e.g., impact of plant or otherbiological communities in the soil layer) (Canton et al., 2004;Teuling and Troch, 2005; Wilson et al., 2005). In humid cli-mates, topography has been reported to be the predominantdriver in the determination of spatial wetness at the hills-lope scale (Anderson and Kneale 1980). The redistributionof soil moisture on the hillside is driven by changing in theupslope contributing area, the topographic wetness index,and the soil depth distribution along the transect (Barling etal., 1994; Beven and Kirkby, 1979; Western et al., 1999;Tromp van Meerveld and McDonnell 2006).

Vertical and lateral water transport patterns can be ascer-tained through investigations of the stochastic features ofsoil moisture responses along transects of varying depths. Inthe present study, the time series of soil moisture variationon two transects on a mountainous hillside were intensivelymonitored. A Trase multiplex time domain reflectometry(TDR) system was used to obtain multiple time series mea-surements of soil moisture.

Various hydrologic parameters, such as stream flow gen-eration, have been interpreted in the past from the simplestructure of autoregressive moving average modeling of soilmoisture time series (Salas et al., 1998) and water tabledepths (Knotters and de Gooijer, 1999). The stochasticcharacteristics of soil moisture dynamics in a hypotheticalhillslope domain have also been studied by numerical sim-ulation on a daily time scale (Ridolfi et al., 2003).

The characteristics of soil moisture measurements canbe understood via systematic time series modeling. Inorder to represent the stochastic feature of soil moistureseries, this study used the univariate autoregressive inte-grated moving-average modeling procedure (Salas et al.,1988). The spatial distribution of modeling results pro-vides information on the intrinsic soil water transfer pro-cess along the hillslope transect. The present studyconsiders how the stochastic structure of the spatial dis-tribution of soil moisture responses can be expressed in aunivariate model, and how the model should be inter-preted in terms of hydrological processes such as infil-tration, lateral flow, vertical flow, and the distribution ofmatrix and macropore fluxes.

Sanghyun Kim* School of Civil and Environmental Engineering, Pusan National University, Busan 609-735, Republic of Korea

*Corresponding author: kimsangh@pusan.ac.kr

424 Sanghyun Kim

2. MATERIALS AND METHODS

2.1. Study Area

The study area is the headwater of the Sulmachun trib-utary of the Imjin River in north-western South Korea (Fig.1). Hydrologic parameters such as rainfall, runoff, andmeteorological variables in the area have been monitoredby the Korea Institute of Construction since 1996. Rainfallis approximately 1,500 mm/year, and the annual tempera-ture range is –10 to 35 °C. The primary hillslope vegetationis a mixture of Polemoniales and shrubby Quercus, with nosystematic variation over the small scale of the area. Usingknowledge gained from an earlier soil moisture study on anadjacent hillslope (Kim et al., 2007) and from an extensivefield survey in spring 2007, the monitoring program andnetwork were established in the present study to allow formultiple transects to be monitored simultaneously.

Geologically, the study area is mainly made up of gneisscomposites underlain by granite bedrock, with a soil layerdepth of 20–90 cm. Particle analysis of soil samples indi-cate loamy sand and sandy loam to be the main soil textureson the hillslope (Fig. 2). Soil samples from upslope showeda higher sand composition than those taken in mid- anddownslope points. Bulk density increases with soil depth atthe sampled sites. The mineral compositions of collectedsamples were determined by X-ray diffraction analysis atall sites at depths of 10 and 30 cm. Figure 3a, b, c, and dshow the classifications of minerals using a quantifyinganalysis program (Siroquant, Sietronics Australia) accord-ing to depth and location along two transects (A and B).The proportion of primary rock-forming minerals in sam-ples was 92–99.5%, and no spatial pattern could be observedin terms of mineral composition.

2.2. System Design and Data Acquisition

Refined digital elevation models (DEMs) with a 0.5 mresolution for surface and bedrock were constructed by con-ducting an intensive topographical survey of the study areausing a theodolite (DT-208P, TOPCON, USA). The corre-sponding soil depths were measured using multiple ironpoles. Figure 2 shows the delineated DEM for the studyarea and its channels. The hillslope area is about 4000 m2,and the slope varies between 20° and 35°.

An evaluation of the spatial distribution of water accu-mulation is necessary for sound design of the soil moisturemonitoring network. The present evaluation rests on theassumption that the local lateral hydraulic gradient can beapproximated by the local slope (Beven and Kirkby, 1979;O’Loughlin, 1986). In this study, the multiple flow directionalgorithm, MD8 (Quinn et al., 1991), was employed tocompute the saturation characteristics from the DEM. Inthis algorithm, flow from a central pixel is applied to anadjacent downslope pixel by calculating the ratio of theproduct of directional slope and contour length to the prod-uct of the countour lengths and the sum of the eight direc-tional slopes (Quinn et al., 1991). MD8 was used to calculateboth the upslope contributing area and the topographic wet-ness index, ln(a/tanβ), where a is the upslope area per unitwidth and tanβ is the local topographic slope. The terrain ofthe subsurface topography was also analyzed, although thedifferences between surface and subsurface terrain analysis

Fig. 1. Location of the Sulmachun watershed and study area show-ing hydrological monitoring systems.

Fig. 2. DEM of hillslope area with locations of monitoring pointsand the channel in the downslope region.

Spatial distribution of soil moisture 425

were found to be negligible. As shown in Figure 4a and b, the monitoring locations

were determined such that the flow accumulation patternwould apply to two different transects, A and B. Two orthree TDR sensors were inserted laterally, heading upslope,depending on the depth of soil. The numbers of soil mois-ture monitoring points along transects A and B were 17 and20, respectively.

2.3. Procedure for Soil Moisture History Analysis

The time series analysis provides a basic understandingof the system response and serves as a basis for modelingto forecast the minimum mean square deviations betweenmeasurements and predictions (Salas et al., 1988). Aunivariate model relates the behaviors of a particular timeseries to the present and past values of the same series. Thegeneral univariate model used in this study assumes thatsoil moisture is a stochastic process that can be expressed asa weighted sum of current and past values of soil moisturewith random noise:

(1)

where St is the time series of soil moisture, xt is the timeseries of random noise, B is the backshift operator Bkxt= xt–k, F(B) is the general autoregressive polynomial, andQ(B) is the general moving-average polynomial.

In this study, the analysis was carried out by the follow-ing procedure: pretreatment of data, model identification,parameter estimation, refinement of model structure, anddiagnostic checking. Pretreatment includes the sub-step ofdata improvement, during which the normality and station-ary nature of the time series were estimated. If necessary,the data were converted by differencing or by several trans-formations, such as the Box-Cox transformation or central-ization. The Box-Cox transformation can be expressed as(Box and Cox, 1964):

: (2)

where, λ is the transformation parameter. Investigation ofthe time series autocorrelation structure using an autocor-relation function (ACF) and partial autocorrelation func-tion (PACF) provides useful information for identifying therational model structure (Salas et al., 1988). Evaluation ofthe extended autocorrelation function (EACF) helps toΦ B( ) St⋅ Θ B( ) ξ t⋅=

y xλ 1–( )λ

----------------- λ 0≠,= y logx= λ 0=,

Fig. 3. Analyses of mineral composition by X-ray diffraction analysis for transect A at depths of 10 cm (a), and 30 cm (b), and transectB at depths of 10 cm (c), and 30 cm (d).

426 Sanghyun Kim

obtain multiple candidate models for final model identifi-cation (Tsay and Tiao, 1984). To estimate model parameters,maximum likelihood and conditional likelihood estimateswere used to find the sum of the square surface for a rangeof parameters and locate the minimum for the correspond-ing parameters. The structure of multiple candidate modelscan be refined by heuristic approaches in both modelparameter estimation and the evaluation procedure. Diag-nostic checks of the model were performed to reveal pos-sible model inadequacies. Student’s t-statistics were used

to check the significance of each parameter in cases wheretheir use did not influence the estimates of the other param-eters. If the t-statistics were within an absolute value oftwo, then the null hypothesis of Student’s t-test was notrejected at the 5% significance level. Overfitting of theadditional parameters over the selected models ensured theexistence of a more general model than the preliminaryselected models. The sensitivities of model convergencewere high and other general models frequently failed toconverge or resulted in negligible differences in data fittingcompared with the model being tested. The model resid-uals were tested to check the removal of the stochasticstructures through time series modeling, and both the indi-vidual of the residual series and group autocorrelation weretested. In all models, the ACF of the residuals was lessthan two standard deviations. Further details of the mod-eling procedure can be found in Salas et al. (1988). Whenthe identified model was not correct, the procedure needswas iterated until the appropriate model was obtained.

3. RESULTS

3.1. Soil Moisture Monitoring

Soil moisture time series were recorded bi-hourly in thesummer and autumn of 2007. Rainfall was measured usingan automatic rain gauge system installed, 50 m from thestudy area as shown in Figure 1. In the summer, soil satu-ration frequently resulted in wave transmission because ofhigh humidity in the TDR electrical circuit. The modelingperiod between 18 September and 8 October 2007 wasselected for stationary process. During the modeling period,rainfalls of 11.2, 10.1, 14.4 and 4 mm were recorded onSeptember 19, 21 and 27 and October 7, respectively. Fig-ure 5 shows the rainfalls and soil moisture time series atthree different depths on transects A and B. The soil depthsalong transects A and B are not constant. Although deeperdevelopment of soil depth was found in the downsloperegion of transect B, only three points on transect A (A2,A3, A4), showed soil depths greater than 60 cm.

Figure 5 shows that similar response patterns were obtainedfor all soil moisture records, characterized by a rapid responseto rainfall and a subsequent fast recession to the designatedsoil moisture. This feature fulfills one of the preliminaryconditions for successful time series modeling, i.e., theexistence of a stationary feature in the variation.

The soil moisture response to rainfall is fastest at a depthof 10 cm, and delayed responses can be found at greaterdepth. The scale of peaks tends to be amplified at pointsclose to the downslope region, as shown in Figure 5 (seeA6-20, A7-30, B7-60, and B8-60). The soil depths at pointA5, A6, and A7 are 25–35 cm. Peak responses also increasewith depth. Soil moisture responses were stronger at deeperand more downslope point, such as A6-20, A7-30, B7-60,

Fig. 4. Digital terrain analysis: upslope contributing area (a), andtopographic wetness index (b).

Spatial distribution of soil moisture 427

and B8-60, showing that soil water flux progresses in thedepth and downslope directions.

3.2. Modeling and Checking

Skewness in the soil moisture statistics indicates that atransformation is required. The pretreatment of data byBox-Cox transformation and centralization normalizes thedata as a prerequisite for application of the maximum like-lihood method as required for parameter estimation.

The ACFs and PACFs of the data yielded the temporalcorrelation structures shown in Figure 6. The ACF of thecorrected soil moisture for points A1-10, A1-30, A2-10,and A2-30 showed that autocorrelations were significantthrough lags 7 to 14, after which the functions tailed off(see Fig. 6a). Whereas the ACF represents the correlationbetween a time series and its lags, PACF represents the cor-relation between a time series and the lag that is not

explained by the correlations of all lower order lags. ThePACFs of the corresponding points revealed significantresponses on lags 1 to 3, as shown in Figure 6b. Taken asa whole, these results suggest that multiple models appli-cable at these points.

The EACF (Tsay and Tiao, 1984) offers another way toidentify the possible structure of the model. Table 1 lists theEACF tables for points A1-10 and A1-30. The boldfaceEACF for A1-10 indicates that the model structure could beeither ARMA (1, 3) or (2, 3), and for A1-30 that it could beARMA (1, 1) or (2, 1).

Multiple candidate models were delineated for all pointsalong transects A and B by ACF, PACF and EACF usingthe corresponding data. The parameters for all candidatemodels were estimated using both maximum and condi-tional likelihood methods. Both the t statistics and Akaike’sinformation criterion (AIC) were applied to complete esti-mates of all candidate model parameters in order to eval-

Fig. 5. Rainfall and soil moisture timeseries at three depths along transects Aand B.

428 Sanghyun Kim

uate the significance of each parameter and to check modelparsimony. The AIC can be expressed as

(3)

where, N is the sample size, σε2 is the maximum likelihood

estimate of the variance of residuals, and p and q are theorders of the autoregressive and moving-average parame-ters, respectively.

The final models were delineated for all monitoring points,based on the comprehensive evaluation of model perfor-mance using t-statistics and the AIC. All selected models

were checked to establish whether the stochastic structureof residuals was sufficiently negligible. The ACF and PACFwere computed with confidence limits for all points. Figure7a and b show the ACFs and PACFs for A1-10, A1-30, A2-10, and A2-30. All the residual diagnostic checks indicatedthat the selected models reasonably represented the stochas-tic process of all corresponding points.

Tables 2 and 3 list the parameters of the final models fortransects A and B, respectively. In both tables, the coeffi-cients of determination, R2, and the residual standard error(RSE), demonstrate that the models successfully describethe variation of soil moisture at all points except point A4-

AIC p q,( ) N σε2( )ln 2 p q+( )+=

Fig. 6. Soil moisture ACFs and PACFs for pretreated data at A1-10, A1-30, A2-10, and A2-30.

Table 1. Extended autocorrelation function table for points A1-10 (a) and A1-30 (b)(a)

P | Q 0 1 2 3 4 50 –0.03 –0.11 0.12 0.02 –0.08 –0.021 –0.26 –0.07 0.13 0.03 –0.09 0.032 0.50 0.38 0.04 –0.05 –0.1 –0.033 –0.14 0.35 0.16 –0.02 –0.1 0.034 0.28 0.39 –0.14 0.15 –0.36 –0.025 –0.45 –0.29 0.22 0.15 –0.3 –0.02

Boldface is the possible models identified from EACF Table. P and Q are potential orders for autoregressive and moving average operators.

(b)P | Q 0 1 2 3 4 5

0 0.96 0.92 0.89 0.85 0.81 0.771 –0.14 0.10 0.11 –0.05 0.06 –0.132 0.44 0.11 0.14 0.04 –0.02 –0.133 –0.44 0.47 0.16 0.03 –0.02 –0.134 0.16 –0.13 0.23 –0.06 0.05 –0.145 0.49 0.02 0.29 0.34 –0.03 –0.11

Boldface is the possible models identified from EACF Table. P and Q are potential orders for autoregressive and moving average operators.

Spatial distribution of soil moisture 429

10, for which no valid model was found. The order ofautoregressive operators along on transect A rangesbetween 1 and 2, and no particular distribution feature canbe seen in either depth or slope. However, the distributionof the moving average operator for transect A has a morecomplicated moving average contribution at depth (seeTable 2). The autoregressive operators for downslopepoints on transect B, such as points in B7 and B8, showeda longer memory effect. The apparent increasing trend ofsoil moisture and deep development of soil depth isresponsible for the distribution of autoregressive operatorsfor points B7 and B8 (see Table 3). A middle point alongtransect B (B4-30) and all points on B7 and B8, modeled

the difference of soil moisture due to a significant increas-ing trend.

4. DISCUSSIONS

4.1. Model Distribution for Vertical Soil Profile

Infiltration is a fundamental process in hillslope hydrol-ogy, and the variation in soil moisture in the vertical soilprofile reflects movement of soil water in response to rain-fall events. In this study, the water content variation in thesoil layer could be successfully represented using a timeseries model, as illustrated in Tables 2 and 3.

Fig. 7. The (a) ACF and (b) PACF of residuals for final models at A1-10, A1-30, A2-10, and A2-30.

Table 2. Delineated time series models for transect APt.| Par. θ1 θ2 θ3 θ4 θ5 φ1 φ2 φ3 R2 RSEA1-10 –0.01 0.04 –0.15 0.93 0.88 0.345A1-30 0.11 0.98 0.94 0.247A2-10 –0.1 –0.08 0.93 0.92 0.285A2-30 –0.34 –0.11 0.96 0.96 0.208A2-60 –0.25 0.98 0.96 0.201A3-10 0.04 –0.11 –0.20 0.92 0.90 0.313A3-30 –0.83 0.04 0.90 0.98 0.144A3-60 –0.20 –0.15 0.97 0.99 0.116A4-10 * * * * * * * * * *A4-30 –0.07 –0.08 –0.21 0.91 0.94 0.246A4-60 –0.38 –0.14 0.94 0.97 0.170A5-10 –0.14 0.40 0.01 0.31 0.91 0.302A5-30 0.46 –0.39 1.40 –0.43 0.95 0.218A6-10 –0.21 0.92 0.90 0.323A6-20 –0.24 0.92 0.91 0.299A7-10 –0.08 –0.08 0.07 –0.02 –0.15 0.90 0.87 0.356A7-30 –0.35 0.89 0.88 0.351

θi: moving average parameter; φi: autoregressive parameter; R2: the coefficient of determination between modeled and measured data;RSE: residual standard error, *: no model to fit.

430 Sanghyun Kim

The model distribution for depth on transect A (see Table2) revealed two notable features. Firstly, the structure of themodel at depth is simpler than that at shallow levels for allmonitoring points from A1 to A7. As expected, the impactof rainfall on soil moisture is therefore greater closer to thesurface due to filling of shallow pores before deeper pen-etration. The complexity of the model, with factors like thememory impact of past moisture and random noise, couldbe greater when disturbance (from rainfall) is greater. Themacropore distribution with depth (Beven and Germann,1982) also supports the substantial decrease with depth inthe moving-average operator, as shown in Table 2.

The coefficient of determination between the model andobservations tends to increase with depth. The vertical dis-tribution of RSE is also consistent results with R2. The bet-ter predictability at deeper depths may be explained by thesmaller uncertainty in the soil moisture transport pattern,attributable to the heterogeneity of soil conductivity, or bythe macropore or preferential flux, both of which are rela-tively weak at depth.

Table 3 shows a similar trend in vertical distribution,although two exceptions were found, at B3-10 and B3-30,and at B6-10 and B6-30. The order of the moving-averageoperator for these two couples is the reverse of that for allother relationships (10 couples on transect B and 10 ontransect A), even though the impact of the autoregressiveprocess is consistent. This may be associated with a par-ticular soil texture feature at these two points. Large-sized

gravel and weathered rock were found at several points,including points B3 and B6, on transect B. The infiltrationflow path may be significantly distorted by such barriers,which could affect the vertical distribution of the model.

However, the vertical distribution of the model in Tables2 and 3 indicates that the model can generally reproduce thesoil hydraulic characteristic distribution, and that the twocomponents of the model structure may correspond to theeffect of memory and disturbance history.

4.2. Model Distribution for Hillslope Transect

The distribution of model structure along the hillslopetransect may reveal information on the hydrological processat the hillslope scale. No particular trend in model distri-bution between the upslope and downslope locations couldbe determined from Table 2, possibly indicating that thehydrological process in transect A is dominated by verticalinfiltration. A similar conclusion has arrived in a previousstudy on an adjacent hillside (Kim, 2009). However, themost downslope points, A6 and A7, showed weaker pre-dictabilities in R2 than did the upslope points. The predict-ability in fact tends to decrease downslope (see A3 to A7 inTable 2), suggesting that the soil water distribution processin the downslope region may not be completely explainedby vertical infiltration. The existence of a lateral flow atdownslope sites may explain the variation of R2. Theincreasing trend in upslope area in the downslope region

Table 3. Delineated time series models for transect BPt.| Par. θ1 θ2 θ3 θ4 θ5 φ1 φ2 φ3 R2 RSEB1-10 0.10 –0.14 0.01 0.02 –0.25 0.53 0.11 0.32 0.98 0.134B1-30 0.25 –0.10 –0.07 –0.06 0.16 0.92 0.90 0.313B1-60 0.95 0.97 0.181B2-10 0.88 1.89 –0.90 0.97 0.154B2-30 0.93 0.88 0.353B3-10 –0.33 0.92 0.91 0.294B3-30 –0.13 –0.04 –0.24 0.85 0.89 0.331B3-60 0.18 0.95 0.85 0.393B4-10 0.67 1.57 –0.6 0.80 0.443B4-30 0.66 0.60 0.85 0.382B5-10 –0.11 –0.06 –0.15 0.90 0.88 0.338B5-30 0.03 –0.04 –0.26 0.89 0.87 0.362B6-10 –0.19 0.89 0.87 0.367B6-30 –0.01 –0.03 –0.23 0.93 0.93 0.273B7-10 –0.14 –0.23 –0.67 –0.12 –0.31 –0.53 0.88 0.346B7-30 0.53 –0.37 0.82 0.42 –0.47 0.81 0.83 0.413B7-60 0.90 0.58 –0.51 0.75 0.43 –0.37 0.66 0.581B8-10 0.70 0.32 0.52 0.37 0.85 0.390B8-30 1.46 –1.1 0.49 0.95 –0.72 0.33 0.49 0.710B8-60 0.29 0.70 0.17 0.41 0.63 0.607

θi: moving average parameter; φi: autoregressive parameter; R2: the coefficient of determination between modeled and measured data;RSE: residual standard error, *boldface models soil moisture differencing.

Spatial distribution of soil moisture 431

(Quinn et al., 1991; Western et al., 1999) may generate alateral flux that is responsible for runoff generation.

The model distribution along the hillslope in Table 3shows compelling evidence for lateral flow. The boldfaceentries in Table 3 denote models with soil moisture differ-encing. The increasing trend of soil moisture is significantfor point B4-30 and for all depths at points B7 and B8; onlythe introduction of the differencing operator allows modelingto succeed. The contribution of upslope soil water throughlateral flow could be responsible for the models at thesepoints. The distribution in R2 toward the downslope regionindicates that the simple autoregressive and moving-aver-age process may not completely explain active hydrologicalprocesses in this area. Lateral flow and subsurface stormflow at locations close to the runoff generation region mayalso be present (see Table 3 and Fig. 2).

5. CONCLUSIONS

In the present study, a soil moisture monitoring systemwas installed on a mountainous hillslope to record timeseries of soil moisture obtained at 37 points along twotransects. The monitoring network was designed to considerthe impact of topography and soil depth as driving pro-cesses in the spatial and temporal distributions of soil mois-ture. Taking the preliminary evaluation of monitoring resultsas a basis, the data were modeled using a procedure that ful-fills the basic requirements for time series analysis.

The distribution of models indicates that both infiltrationand pore development in the vertical soil profile can beproperly resolved using the present modeling platform.Trend of decreasing predictability and increasing residualerror provide compelling evidence that the hydrologicalprocesses in the downslope region are more complicatedthan those in the upslope area. Lateral flow and subsurfacestorm flow may explain the modeling results, as may theexistence of channel initiation in the study area.

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Manuscript received December 23, 2010Manuscript accepted September 24, 2011