Geoderma 193–194 (2013) 180–188

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Soil water prediction based on its scale-specific control using multivariate empiricalmode decomposition

Wei Hu, Bing Cheng Si ⁎University of Saskatchewan, Department of Soil Science, Saskatoon, SK, Canada S7N 5A8

⁎ Corresponding author.E-mail address: bing.si@usask.ca (B.C. Si).

0016-7061/$ – see front matter © 2012 Elsevier B.V. Allhttp://dx.doi.org/10.1016/j.geoderma.2012.10.021

a b s t r a c t

a r t i c l e i n f o

Article history:Received 13 March 2012Received in revised form 29 September 2012Accepted 13 October 2012Available online 17 November 2012

Keywords:Soil waterMultivariate empirical mode decompositionScaleHilbert transformIntrinsic mode functions

Soilwater (SW) is controlled by different factors operating in different intensities and scales. The objective of thisstudy was to apply multivariate empirical mode decomposition (MEMD) in revealing scale-specific control ofSW. Two data sets from different climates were used. One data set was soil water storage (SWS) of 0–140 cmmeasured at two different periods (recharge and discharge periods) from a transect at St. Denis NationalWildlifeArea in a Canadian prairie area (SDNWA). The other data set was soil water content (SWC) of 0–6 cm from twotransects (bunge needlegrass and korshinsk peashrub) in the Laoyemanqu watershed on the Chinese LoessPlateau (LYMQ). In both areas, five environmental factors including elevation, sand, silt, clay, and organic carbon(OC) contents were measured at each sampling location. SW and environmental factors were separated intodifferent intrinsic mode functions (IMFs) and residue representing different scales. The dominant componentsin terms of the percentages of total variations in SW were identified. At each scale, SW was controlled by oneor multiple factors. Each IMF of SW or residue can be predicted with the corresponding IMF or residue of someenvironmental factors. The summation of all predicted IMFs and residue predicted well the SW at the measure-ment scale, which outperformed SWprediction based on simple linear regression between SWand environmen-tal factors and regression between IMFs of SW and factors at the measurement scale. Organic carbon was themajor predictor for SWS in SDNWA for both periods and soil particle composition was the major predictor forSWC in LYMQ. MEMD has a great potential in revealing the scale-specific control of other soil properties.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

Soil water (SW) is influenced by different factors operating in dif-ferent intensities and scales (Biswas and Si, 2011c; Brocca et al., 2009;Gómez-Plaza et al., 2001; Hu et al., 2008; Seyfried, 1998). It is impor-tant to understand the scale dependent relationship between SW andvarious environmental factors for flood forecasting, ecosystems man-agement, and sustaining food production (Blöschl and Sivapalan,1995; Xu et al., 2004).

Many methods have been proposed to elucidate the scale-dependent relationships of SW and environmental factors. The Pearsonlinear correlation analysis explores the linear relationships only at themeasurement scale. However, the scale of the processes controllingSW distribution may not always be the same as the measurementsscale (Blöschl and Sivapalan, 1995). Spectral analysis and spatial coher-ency have been used to reveal the scale-dependent relationshipsbetween two or more variables, but are only applicable to stationarysystems (Kachanoski et al., 1985; Si, 2008). Because the majorityof soil spatial series behave non-stationary, discrete wavelet (LarkandWebster, 1999) and continuouswavelet (Si and Farrell, 2004) trans-forms have been introduced. However, discrete wavelet transform can

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only work with scales of power of two, which may not be in line withthe process scales inherent in the data series. The continuous wavelettransform and wavelet packet analysis allow interpolation between thediscrete scales obtained from the discrete wavelet transform, but theresults are dependent on the selection of the type of mother wavelets.Assuming the power–law relationship between the statistics of soilproperty (for example, probability and partition function) and scale,jointmultifractal analysis was used to identify the scale-specific relation-ships between two spatial variables (Kravchenko et al., 2000; Zeleke andSi, 2005, 2006). However, not all soil properties obey the power–lawrelationships. In addition, all the above mentioned methods assumesoil properties and related processes to be linear which follow the prin-ciple of superposition. In nature, however, the effects from differentprocesses as represented by different frequency components are notadditive and do not follow the principle of superposition (Biswas andSi, 2011d; Yan and Gao, 2007), indicating the system to be nonlinear.

With the aim to reveal the scale-specific control of non-stationaryand nonlinear system of soil water storage (SWS), Biswas and Si(2011d) employed empirical mode decomposition (EMD) combinedwith Hilbert spectral analysis (HSA). Instead of fitting a function(e.g., sine and cosine function for spectral analysis and mother wave-lets for wavelet analysis) to the data, EMD is a data-driven approachand looks for scales inherent in the data. EMD separates SWS into dif-ferent scales-intrinsic mode functions (IMFs). Based on the IMFs of

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SWS separated by EMD, Biswas and Si (2011d) correlated differentIMFs independently with the environmental factors (elevation, OC,sand, silt, and clay content) at the measurement scale, and obtainedimproved overall prediction of SWS. However, in Biswas and Si(2011d), EMD was applied only to SWS, but not to the environmentalfactors. They correlated different IMFs of SWS directly to the environ-mental factors at the measurement scale. Their hypothesis was thateach IMF of SWS was controlled by one or more environment vari-ables at the measurement scale. However, SW at one scale may becontrolled by the environmental factors from scales different fromthe measurement scale. Separation of variations in SW as well as inenvironmental factors at similar scales may provide enhanced infor-mation on the scale-specific control and overall SW prediction.

Recently, multivariate empirical mode decomposition (MEMD)has been extended from univariate EMD (Rehman and Mandic,2010b). It is a fully adaptive, data-driven approach and has beenused in signal processing (Fleureau et al., 2011) and neural signalanalysis (Hu and Liang, 2011). Similarly to bivariate (Rilling et al.,2007) and trivariate (Rehman and Mandic, 2010a) extensions ofEMD, MEMD has the ability to align “common scales” present withinmultivariate data. Each “common scale” is manifested in the commonoscillatory modes in all the variates within an n-variate intrinsicmode function (IMF). Such mode alignment property helps to makeuse of similar scales in different data sources. Thus, the obtained com-mon scales represent the true scales of underlying processes.

The objectives were to use MEMD method to examine the scale-specific control of SW and predict SW distribution at the measurementscale using the scale-specific SW and environmental factors. Two SWdata sets from different climates were used to show the applicabilityof the new method at different environmental backgrounds. Onedata set was SWS from a transect measured at two different periods(recharge and discharge periods) in a Canadian prairie area. The otherdata setwas soilwater content (SWC) from two transectswith differentland uses in a watershed on the Chinese Loess Plateau.

2. Materials and methods

2.1. Multivariate empirical mode decomposition

MEMD is themultivariate extensions of standard EMD (Huang et al.,1998). An important step for MEMD is the computation of the localmean. Moreover, the notion of “oscillatory model” defining an IMF israther confusing for multivariate spatial data. To deal with these prob-lems, Rehman and Mandic (2010b) produced multiple n-dimensionalenvelopes by taking projections ofmultiple inputs along different direc-tions in an n-dimensional space. The IMFs were calculated using theenvelopes.

Assuming V(s)={v1(s), v2(s),…, vn(s)} being the n spatial data setsas a function of space (s), and Xθk ¼ xk1; x

k2;…; xkn

� �denoting the direc-

tion vector along the direction given by angles θk={θ1k,θ2k, …,θn−1k } in

a direction set, X (k=1, 2, …, K, K is the total number of direction).Then, IMFs of the n spatial data sets can be obtained by MEMD usingalgorithm 1 (Rehman and Mandic, 2010b):

Algorithm 1. MEMD algorithm

(1) Generate a suitable set of direction vectors, X.(2) Calculate a projection, pθk sð Þ, of the spatial data sets V(s) along

the direction vector Xθk , for all k.(3) Find the spatial instants siθk corresponding to the maxima of

projection for all k.(4) Interpolate siθk ;V siθk

� �� �to obtain multivariate envelope curves

eθk sð Þ for all k.(5) The mean M(s) of the envelope curves is calculated by M sð Þ ¼

1K∑K

k¼1eθk sð Þ

(6) Extract the “detail” D(s) using D(s)=V(s)−M(s). If the“detail” D(s) fulfills the stoppage criterion for a multivariateIMF, apply the above procedure to V(s)−D(s), otherwiseapply it to D(s).

The stoppage criterion for multivariate IMFs is similar to that pro-posed by Huang et al. (2003), the difference being that the conditionfor equality of the number of extrema and zero crossings is not imposed,as extrema cannot be properly defined for multivariate signals.

Hilbert transformation (Huang et al., 1998) was carried out for eachIMF to obtain instantaneous frequencies, i.e., the derivative of the phasefunction. The instantaneous frequencies of SW and environmentalfactors were converted to period (1/frequency) and the period wasfurther converted to the spatial scale after being multiplied by thesampling interval.

2.2. Data acquisition

One data set was obtained from a transect at St. Denis NationalWildlife Area (52°12′N latitude, 106°50′W longitude) in Saskatchewan,Canada (SDNWA) (Fig. 1b). This area has a humid continental climate(Dfb) based on the Köppen–Geiger climate classification (Peel et al.,2007). The dominant soils are Dark Brown Chernozem (Canada SoilSurvey Committee, 1978) or Mollisols in the U.S. Soil Taxonomy(USDA, 1975). The landscape presents a very complex sequence ofslopes extending fromdifferent size of rounded depressions to irregularto complex knolls and knobs (Pennock et al., 1987). A sampling transectwas established over several rounded knolls and depressionsrepresenting different landform cycles. This transect was 576 m longwith 128 sampling points located at 4.5 m intervals. In thisstudy, one typical measurement of SWS of 0–140 cm from the re-charge period (2nd May 2008) and discharge period (23rd August2008) was used. In addition, five environmental factors at each sam-pling location were obtained. These included relative elevation, sand(>0.05 mm), silt (0.002–0.05 mm), clay (b0.002 mm), and organiccarbon (OC) contents of surface soil (0 to 20 cm). Detailed descriptionof the experimental site, background information, and measurementscan be found in Biswas and Si (2011a, 2011b, 2011c, 2011d, 2011e,2012).

The other soil water data set was from a small watershed calledLaoYeManQu (110°23′ E and 38°46′ N), Yulin, Shaanxi Province,China (LYMQ) (Fig. 1c). Climate was cold semi-arid (Bsk) based onthe Köppen–Geiger climate classification (Peel et al., 2007). Thisarea was noted by deep gullies and undulating slopes. Two transectstermed as bunge needlegrass transect (BNT) and korshinsk peashrubtransect (KPT) were set up along the toposequence in the LYMQ wa-tershed. Each transect was 100 m long with 51 sampling points at2 m intervals. In each sampling point, SWC of 0–6 cm was measuredby an impedance soil moisture probe (Delta-T Devices Theta probe,ML2x) at 136 times from 28th June 2007 to 30th August 2008 (Huet al., 2011). In this study, the average SWC over time at each pointwas used to refer to the spatial pattern of SWC. In addition, elevation,sand (>0.05 mm), silt (0.002-0.05 mm), clay (b0.002 mm), andorganic carbon (OC) contents of surface soil (0 to 10 cm) were alsoobtained. Detailed information on the experimental site, background,and measurements can be found in Hu et al. (2010, 2011).

2.3. Data analysis

The SW data series (each of the recharge and discharge periodsin SDNWA or each of the two transects in LYMQ) along with thefive environmental factors formed a multi-variate data series. Eachmulti-variate data series was decomposed into different IMFs withMEMD using a MATLAB (MathSoft Inc.) program written byRehman and Mandic (2009). The number of directions should begreater than the dimensionality of the spatial data sets, so we

Fig 1. Geographic location of study site and the transect position at (b) St. Denis National Wildlife Area in Saskatchewan, Canada (SDNWA) and (c) LaoYeManQu watershed on theLoess Plateau, China (LYMQ) and their locations in the (a) world. From south to north, the sampling locations are marked as 1, 2, …, 128 in SDNWA. From up-slope to down-slope,the sampling locations are marked as 1, 2, …, 51 in LYMQ. BNT: bunge needlegrass transect; KPT: korshinsk peashrub transect.

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used the default value of 64. The stop criteria of sifting was selectedas the default value of [0.075, 0.75, and 0.075] for [θ1, θ2, and α] re-spectively. The percent variance contributed by each IMF or residuewas calculated as the percent variance of each IMF or residue overthe total variance of the spatial data series. Instantaneous frequen-cies were calculated by Hilbert transformation using a MATLAB(MathSoft Inc.) program written by Rilling (2007).

Pearson correlation analysis was employed to explore the linearrelationships among IMFs or residue of SW and environmental fac-tors. Stepwise multiple linear regressions were used to predict SWat each scale or residue from the corresponding IMFs or residue ofthe environmental variables. An IMF of a factor was added to the re-gression equation when the value of P≤0.05 and was taken outfrom the regression equation when the value of P≥0.10. Then thevalues of SW at the measurement scale were predicted by summingup the predicted IMFs and residue of SW. Coefficient of determination(r2) between the measured SW and predicted SW was used to judgethe quality of overall SW prediction. All the statistical analyses werecompleted using SPSS 16.0 software (SPSS Inc.).

3. Results and discussion

3.1. SW and environmental factors at the measurement scale

In SDNWA, soil water storage and its spatial variability during therecharge period (mean SWS=41.0 cm, variance=45.7 cm2) wasmuch higher than that of the discharge period (mean SWS=32.8 cm, variance=23.3 cm2) (Fig. 2a). This was due to the snow-melt runoff processes and much elevated SWS in depressions in therecharge period (Biswas and Si, 2011c).

In LYMQ, no significant differences of SWC were observedbetween the two land uses (10.7% in BNT versus 11.0% in KPT)(Fig 2b). However, spatial variability of SWC in KPT (variance=2.5%2) was larger than that in BNT (variance=1.3%2), which waspartly attributed to the larger variability of environmental factors inKPT than that in BNT (Hu et al., 2011).

Except for elevation in the discharge period in SDNWA and OC atboth transects in LYMQ, all environmental factors were significantlycorrelated to SW (Table 1), implying significant influences of these

Fig. 2. Spatial distribution of soil water (soil water storage in SDNWA and soil water content in LYMQ) and environmental factors (elevation, sand, silt, clay, and organic carboncontents) along the sampling transects. Horizontal axis is the distance between a sampling location and the origin of the transect. BNT: bunge needlegrass transect; KPT: korshinskpeashrub transect.

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factors on SW distribution at the measurement scale. According to thestepwise multiple linear regressions (Table 1), selected factors in themodels could explain 61.6% and 56.1% of variance of SWS in SDNWAand 43.6% and 62.5% of variance of SWC in LYMQ.

3.2. Multivariate empirical mode decomposition

Different numbers (seven in SDNWA and five in LYMQ) of IMFsfor SW and environmental factors were obtained by MEMD (Figs. 3and 4). Similar to univariate EMD, the IMFs with lower numericalnumbers corresponded to higher frequency (smaller scale) oscillations,whereas IMFs with higher numerical numbers corresponded to lowerfrequency oscillations at larger scales (Biswas and Si, 2011d; Si, 2003).For a given multi-variate data series, the numbers of oscillations forIMFs with the same numerical numbers were generally the same fordifferent variables and thus the widths of oscillations in an IMF were

Table 1Correlation coefficients between soil water (soil water storage in SDNWA and soilwater content in LYMQ) and environmental factors (elevation, sand, silt, clay, andorganic carbon (OC) contents) and stepwise multiple linear regression between soilwater and environmental factors at the measurement scale.

Elevation Sand Silt Clay OC

SDNWARecharge period −0.237⁎⁎ −0.593⁎⁎ 0.350⁎⁎ 0.472⁎⁎ 0.723⁎⁎

45.16+2.67∗OC−0.24∗Sand-1.29∗Elevation(F†=68.8, adjusted R2=0.616)

Discharge period −0.136 −0.696⁎⁎ 0.458⁎⁎ 0.491⁎⁎ 0.509⁎⁎45.41−0.29∗Sand−1.72∗Elevation+0.11∗Clay(F=55.1, adjusted R2=0.561)

LYMQBNT −0.307⁎ −0.586⁎⁎ 0.561⁎⁎ 0.412⁎⁎ 0.082

74.70−0.10∗Sand−0.06∗Elevation(F=20.3, adjusted R2=0.436)

KPT −0.410⁎⁎ −0.795⁎⁎ 0.741⁎⁎ 0.673⁎⁎ −0.01415.85−0.12∗Sand (F=84.4, adjusted R2=0.625)

BNT: bunge needlegrass transect; KPT: korshinsk peashrub transect.⁎⁎ Significant at Pb0.01.⁎ Significant at Pb0.05† The F statistics for the accepted models.

also similar. The accurate scales can be calculated from the instanta-neous frequencies of IMFs through Hilbert transform of the IMFs. Inter-estingly, the obtained scales for different variables varied for a givenIMF and the larger the scale, the larger the differences in scales amongdifferent environmental variables (Table 2). This implied that exactlycommon scales did not present among all variables at great scales.This was attributed partly to the Heisenberg uncertainty principle(Lewis andMayer, 1929), due to the limited number of cycles containedin IMFs at great scales (e.g., about one cycle for IMF7 in SDNWA). Similarproblem also exists in the spectral and the wavelet method (Singer,1999; Stegel, 2000).

The scales of all the variables corresponding to a particular IMFwere averaged to represent the scale of that IMF. The average scalesfor each IMF were very similar between the recharge and dischargeperiods in SDNWA. Furthermore, scales of the two SWS series werealso very similar, implying that processes controlling SWS occur atsimilar scales, irrespective of time. This may due to that both snow-melt runoff in the recharge period and evapotranspiration in the dis-charge period were related to the alternating knolls and depressions.The average values of the characteristic scales were 13, 22, 41, 71,122, 204, and 416 m for IMF1, IMF2, IMF3, IMF4, IMF5, IMF6, andIMF7, respectively. In LYMQ, except for IMF1 and IMF2, the averagescales on the two transects differed much (Table 2). Especially forIMF5, the scale of SWC in BNT was much larger than that in KPT(104 m versus 42 m). This implied that land use could change thescales at which soil moisture operated.

The distribution of variance over different IMFs was different(Figs. 3 and 4). In SDNWA, about 50% of the total variation of SWSwas separated in IMF5 (scale of 122 m) and IMF7 (scale of 416 m)for both periods. The scale of IMF5 was just about the distance ofone topographic cycle (i.e., one catchment size) (Fig. 2). The scale ofIMF7 may represent linear trend of elevation over the distance be-tween a sampling location and the origin of the sampling transect(r=−0.294, Pb0.01). The distributions of variance were similar be-tween the two periods for all the environmental factors in SDNWA.In LYMQ, the distribution of variance of SWC were slightly differentbetween the two transects. For BNT, 66% of the total variation wasseparated in IMF2 (scale of 11 m) and IMF4 (scale of 43 m). ForKPT, the major variations (about 50%) were separated in IMF1(scale of 7 m), IMF2 (scale of 11 m), and IMF4 (scale of 30 m).

Fig. 3. Intrinsic mode functions (IMFs) and residues of soil water storage and environmental factors (elevation, sand, silt, clay, and organic carbon contents) based on multivariateempirical mode decomposition in the recharge period (2nd May 2008, in black line) and discharge period (23rd August 2008, in red line) in SDNWA. Horizontal axis is the distancebetween a sampling location and the origin of the transect. Vertical solid bar in Y-axis shows the scale of IMFs. The numbers in each subplot are the percent of variance explained byeach intrinsic mode function (IMF) in the recharge period (in black) and discharge period (in red), respectively.

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These scales usually corresponded to those with major variations ofsoil particle composition.

Two points should be noted during the separation with MEMD.First, a certain amount of variations were also observed in residuefor some variables. For example, 17.0% of total variance of SWC, prob-ably come from greater scales which cannot be discovered by MEMDdue to the relatively short sampling distance, existed in residue inKPT. Second, the sum of percent of variances of all IMFs and residuewas not necessarily equal to 100%. This implied that related processesmay not operate independently at different scales, or lack of perfectorthogonality between different IMF of a variable.

3.3. Scale-specific controlling of SW

Relationships between SW and environmental factors varied withscales (Table 3). In SDNWA, SWS and elevation were significantly

Fig. 4. Intrinsic mode functions (IMFs) and residues of soil water content and environmentaempirical mode decomposition on the bunge needlegrass transect (BNT, in black line) andbetween a sampling location and the origin of the transect. Vertical solid bar in Y-axis showseach intrinsic mode function (IMF) on the BNT (in black) and KPT (in red), respectively.

correlated at all IMFs except IMF1 in the recharge period, and at allIMFs except IMF1 and 6 in the discharge period. SWS and sand weresignificantly correlated at all but IMF1 and 2 for the recharge period.However, the relations between SWS and silt were more complex:the correlation was not statistically significant at IMF2 in the rechargeperiod and at IMF1, 2, 3, and 6 in the discharge period. Correlationbetween SWS and clay was not statistically significant at IMF2 and 3in the recharge period and IMF1, 3, and 4 in the discharge period.There was nonsignificant correlation between SWS and OC at IMF1and 2 in the recharge period and at IMF3 in the discharge period. InLYMQ, SWC and soil particle compositions were significantly correlat-ed at all IMFs for both transects, indicating a dominant influence ofsoil texture on soil moisture (Hu et al., 2011). Significant relationshipbetween SWC and elevation was found at IMF5 in BNT and IMF3 and5 in KPT. Except for IMF3 and 4 in BNT and IMF1 in KPT, correlationsbetween SWC and OC were significant at all IMFs. Furthermore, the

l factors (elevation, sand, silt, clay, and organic carbon contents) based on multivariatekorshinsk peashrub transect (KPT, in red line) in LYMQ. Horizontal axis is the distancethe scale of IMFs. The numbers in each subplot are the percent of variance explained by

Table 2Scale of each intrinsic mode function (IMF) of soil water (soil water storage in SDNWAand soil water content in LYMQ) and environmental factors (elevation, sand, silt, clay,and organic carbon (OC) contents).

Soilwater

Elevation Sand Silt Clay OC Meana CV(%)b

SDNWARechargeperiod

IMF1 14 14 14 13 12 13 13 6%IMF2 25 24 22 23 21 22 23 6%IMF3 44 43 38 38 40 42 41 6%IMF4 74 83 67 66 65 72 71 9%IMF5 122 120 120 97 125 125 117 9%IMF6 228 204 196 106 218 204 180 23%IMF7 575 382 376 225 379 580 379 32%

Dischargeperiod

IMF1 13 14 14 13 13 14 13 4%IMF2 22 23 21 21 21 22 22 4%IMF3 38 48 41 36 37 42 40 11%IMF4 70 86 72 64 65 62 69 13%IMF5 120 120 120 148 125 123 125 9%IMF6 222 90 311 310 90 273 163 48%IMF7 576 380 380 379 381 381 403 19%

LYMQBNT IMF1 6 7 6 6 7 6 6 8%

IMF2 11 12 11 11 13 9 11 11%IMF3 26 27 22 22 26 22 24 9%IMF4 35 35 55 55 54 26 43 29%IMF5 104 63 62 65 105 56 76 30%

KPT IMF1 6 7 7 8 7 7 7 9%IMF2 10 10 10 10 12 10 11 10%IMF3 15 21 17 16 15 20 17 14%IMF4 29 35 33 33 24 29 30 14%IMF5 42 57 41 51 42 55 48 16%

BNT: bunge needlegrass transect; KPT: korshinsk peashrub transect.a Mean scale of SWS and environmental factors.b CV: coefficient of variation of scales of SWS and environmental factors.

Table 3Correlation coefficients between soil water (soil water storage in SDNWA and soilwater content in LYMQ) and environmental factors (elevation, sand, silt, clay, andorganic carbon (OC) contents) at each intrinsic mode function (IMF) and residuebased on multivariate empirical mode decomposition for all the variables.

Elevation Sand Silt Clay OC

SDNWARechargeperiod

IMF1 0.055 0.030 −0.203⁎⁎ 0.218⁎⁎ −0.044IMF2 −0.394⁎⁎ −0.158 −0.028 0.142 −0.151IMF3 −0.723⁎⁎ −0.588⁎⁎ 0.357⁎⁎ 0.120 0.806⁎⁎IMF4 −0.906⁎⁎ −0.693⁎⁎ 0.703⁎⁎ 0.201⁎⁎ 0.776⁎⁎IMF5 −0.775⁎⁎ −0.508⁎⁎ 0.259⁎⁎ 0.770⁎⁎ 0.945⁎⁎IMF6 −0.720⁎⁎ 0.351⁎⁎ −0.624⁎⁎ 0.852⁎⁎ 0.765⁎⁎IMF7 0.987⁎⁎ −0.975⁎⁎ 0.952⁎⁎ 0.990⁎⁎ 0.968⁎⁎Residue 0.913⁎⁎ −0.951⁎⁎ 0.929⁎⁎ 0.997⁎⁎ 0.987⁎⁎

Dischargeperiod

IMF1 −0.036 0.230⁎⁎ −0.149 0.078 −0.436⁎⁎IMF2 −0.239⁎⁎ −0.257⁎⁎ −0.118 0.286⁎⁎ −0.362⁎⁎IMF3 −0.452⁎⁎ −0.255⁎⁎ 0.121 0.098 0.001IMF4 −0.851⁎⁎ −0.358⁎⁎ 0.443⁎⁎ 0.002 0.502⁎⁎IMF5 −0.932⁎⁎ −0.851⁎⁎ 0.716⁎⁎ 0.759⁎⁎ 0.908⁎⁎IMF6 0.025 −0.308⁎⁎ 0.038 0.837⁎⁎ 0.726⁎⁎IMF7 0.994⁎⁎ −0.994⁎⁎ 0.993⁎⁎ 0.992⁎⁎ 0.999⁎⁎Residue 0.412⁎⁎ −0.500⁎⁎ 0.431⁎⁎ 0.711⁎⁎ 0.991⁎⁎

LYMQBNT IMF1 0.238 0.492⁎⁎ 0.543⁎⁎ −0.010 −0.277⁎

IMF2 0.246 −0.857⁎⁎ 0.807⁎⁎ 0.541⁎⁎ −0.357⁎IMF3 −0.041 −0.738⁎⁎ 0.801⁎⁎ 0.468⁎⁎ 0.034IMF4 −0.255 −0.538⁎⁎ 0.469⁎⁎ 0.724⁎⁎ 0.163IMF5 0.719⁎⁎ −0.419⁎⁎ 0.731⁎⁎ −0.978⁎⁎ −0.375⁎⁎Residue −0.959⁎⁎ 0.956⁎⁎ −0.941⁎⁎ −0.994⁎⁎ 0.964⁎⁎

KPT IMF1 −0.188 −0.362⁎⁎ 0.157 0.417⁎⁎ −0.102IMF2 −0.270 −0.697⁎⁎ 0.700⁎⁎ 0.572⁎⁎ −0.312⁎IMF3 −0.279* −0.729⁎⁎ 0.743⁎⁎ 0.624⁎⁎ −0.432⁎⁎IMF4 0.091 −0.855⁎⁎ 0.765⁎⁎ 0.692⁎⁎ −0.790⁎⁎IMF5 −0.712⁎⁎ −0.899⁎⁎ 0.826⁎⁎ 0.982⁎⁎ −0.952⁎⁎Residue −0.850⁎⁎ −0.986⁎⁎ 0.992⁎⁎ 0.955⁎⁎ 0.881⁎⁎

BNT: bunge needlegrass transect; KPT: korshinsk peashrub transect.⁎⁎ Significant at Pb0.01.⁎ Significant at Pb0.05

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relationships between SW and all the five environmental factors weresignificant for residue in both areas.

Moderate to strong correlation (|r|>0.5) between SW and theenvironmental factors were generally observed at IMF3, 4, 5, 6, and7 and residue in SDNWA and at IMF2, 3, 4, and 5, and residue inLYMQ (Table 3). This implied that SWS was distributed in a more ran-dom way at small scales (Biswas and Si, 2011d), while it was moredeterministic as influenced by the environmental factors at greaterscales. The correlation coefficients between SW and some environ-mental variables changed from strong positive correlation to strongnegative correlation. This is remarkable because negative correlationat one scale would cancel positive correlation at another scale,resulting low overall correlation (e.g., correlation between elevationand SWS in SDNWA and correlation between OC and SWC in LYMQ).

At the dominant scale of 122 m and 416 m in SDNWA, the corre-lation coefficients between SWS and the five environmental factorswere also significant and higher than that at any other scales inboth periods. SWS was influenced more strongly by controllingfactors at the scale of 416 m than at the scale of 122 m (Table 3).This implied that SWS was strongly correlated to the environmentalfactors at the dominant scale, especially at the scale of 416 m. Fur-thermore, the correlation coefficients were greater in the dischargeperiod than those in the recharge periods at the scale of 416 m. Thisimplied a stronger influence of controlling factors in the discharge pe-riod than those in the recharge period at this scale. For IMF5, correla-tion coefficients between SWS and silt were lower than others, andthe disparity of their scales (Table 2) may be the main reason. Atthe dominant scale of 11 m (IMF2) and 43 m (IMF4) in BNT, the cor-relation coefficients between SWC and sand (silt) or clay were usuallyhigher than other scales. The main reason was that majority ofvariance of soil particle compositions existed in these two scales

(Fig. 4). In KPT, however, the correlation coefficients between SWCand environmental factors usually increased with scale. This impliedthat SWC in KPT was more random at shorter scale as influenced bythe scattered grass among irregular shrub, which agreed with theshorter range of soil water content in KPT (9.1 m) in relation to thatin BNT (15.1 m) (Hu et al., 2011). Although the scales of SW differedfrom the scales of environmental factors (e.g., IMF7 in SDNWA andIMF5 in LYMQ), there were large correlation coefficients betweenthem. This suggested that SW was strongly associated with the envi-ronmental factors at similar scales, which makes it possible to predictSW at a particular scale from environment factors at similar scales.

Correlations at the dominant scale between SW and the environ-mental factors obtained using MEMD were stronger than thoseusing univariate EMD. For example, the correlation coefficient be-tween SWS and elevation at the dominant scale in SDNWA was −0.90 using MEMD (Table 3), and was only about −0.70 using univar-iate EMD (Biswas and Si, 2011d). The correlation coefficient betweenSWC and clay at the dominant scale of 43 m in BNT was 0.72 usingMEMD (Table 3), and was only 0.34 using univariate EMD. Therefore,MEMD revealed stronger influences of controlling factors on SWS atdifferent scales, mainly because the variations in SW at a particularscale was contributed from the controlling factors operating at thatscale, while there was still a scale disparity between SW data at ascale and controlling factors at the measurement scales (Biswas andSi, 2011d). Soil moisture and environmental factors are usually mea-sured at the same scale, but the measurement scale is not necessarilythe inherent scales they operate, which makes it difficult to identifythe controls with a single measurement scale (Blöschl and Sivapalan,1995; Tallon and Si, 2004).

186 W. Hu, B.C. Si / Geoderma 193–194 (2013) 180–188

3.4. SW prediction

Soil water at each scale was predicted from the controlling factorsat similar scales using stepwise multiple linear regression (Table 4).The predictors and their relative importance differed with scales.The relative importance of each variable can be judged by the abso-lute value of standardized regression coefficients. Therefore, themost important predictor was OC for IMF5 and clay for IMF7 in therecharge period, while it was OC for both IMF5 and IMF7 in the dis-charge period in SDNWA. In LYMQ, the most important predictorsat both transects were soil particle compositions at all scales. Interest-ingly, adjusted R2 values between SW and controlling factors usuallyincreased with scales. This further confirmed that soil hydrologicalprocesses could be more deterministic at larger scales.

Soil water at the measurement scale was predicted by adding allthe predicted IMFs and residue. The coefficients of determination(r2) between the measured SWS and predicted SWS at the measure-ment scale were 0.89 and 0.86 for the recharge period and the dis-charge period, respectively in SDNWA. They were much higher thanthe corresponding values of r2 (0.63 and 0.57) when the overall pre-diction of SWSwere made directly from the SWS data at the measure-ment scale. They were also higher than the values of r2 (0.76 and0.65) when overall prediction of SWS were made based on the uni-variate EMD of SWS even interactions among controlling factorsbeing considered (Biswas and Si, 2011d). In LYMQ, the values of r2 be-tween the measured SWC and predicted SWC at the measurementscale were 0.68 for both transects. They were also higher than thoseobtained by univariate EMD (0.44 and 0.63) and by overall predictiondirectly at the measurement scale (0.46 and 0.63). This indicated thatSW prediction using MEMD outperformed the SW predictions basedon the original data or decomposed data using univariate EMD.

Table 4Predictive equations and regression statistics (F value and adjusted R2) of soil water (soil wa(IMF) using stepwise multiple linear regression based on multivariate empirical mode deco

Model

SDNWARecharge period IMF1 0.11+0.06 (0.22b)∗Clay

IMF2 −0.01–15.05 (−0.39)∗ElevationIMF3 −0.03+2.47 (0.57)∗OC-9.41 (−0.45)∗ElevaIMF4 −0.02–6.51 (−0.79)∗Elevation+0.29 (0.28)IMF5 −0.20+3.74 (0.57)∗OC+1.08 (0.50)∗Clay+IMF6 0.02+0.68 (0.72)∗Clay+0.17 (0.47)∗Sand+IMF7 −0.11+0.51 (0.52)∗Clay+3.53 (0.37)∗OC+Residue 13.24+4.58 (0.74)∗OC+0.56 (0.47)∗Clay −

Discharge period IMF1 0.02–1.29 (−0.48)∗OC+0.19 (0.30)∗SandIMF2 −0.01–1.54 (−0.40)∗OC+0.12 (0.33)∗ClayIMF3 −0.02–7.50 (−0.55)∗Elevation-1.17 (−0.43IMF4 −0.02–6.58 (0.33)∗Elevation-1.82 (0.20)∗OCIMF5 −0.12+1.76 (0.40)∗OC-0.20 (−0.34)∗SandIMF6 −0.08+0.54 (0.78)∗Clay-2.70 (−0.42)∗ElevIMF7 −0.02+15.48 (1.81)∗OC-0.52 (−0.69)∗Silt-Residue 6.46+9.87 (1.16)∗OC-0.12 (−0.22)∗Clay

LYMQBNT IMF1 0.02+0.08 (−0.53)∗Silt-0.42 (−0.26)∗OC

IMF2 −0.01–0.15 (−0.86)∗SandIMF3 −0.03+0.18 (0.93)∗Silt+0.52 (0.32)∗ElevaIMF4 0.05+0.67 (1.17)∗Clay-0.08 (−0.53)∗SiltIMF5 −0.81 (−0.84)∗Clay+0.23 (0.63)∗Silt-0.17Residue 12.98-0.88 (−1.47)∗Clay+0.07 (0.49)∗Silt

KPT IMF1 −0.01+0.11 (0.42)∗ClayIMF2 0.19 (0.70)∗SiltIMF3 0.01+0.37 (1.84)∗Silt+0.12 (0.97)∗Sand+IMF4 −0.01–0.43 (−1.84)∗Sand+21.70 (1.01)∗OIMF5 1.44 (3.00)∗Clay+0.19 (1.16)∗Sand+7.54 (Residue 30.04+0.20 (1.35)∗Silt-5.23 (−0.65)∗OC-0.

BNT: bunge needlegrass transect; KPT: korshinsk peashrub transect.a the F statistics for the accepted models.b Number in bracket is standardized regression coefficient.

Coefficients of determination between overall SW at the measure-ment scale and each predicted IMF (residue) were calculated to ex-plore the relative important role of each scale to the predictivemodel (Table 5). As can be seen, IMF5 and IMF7 contributed mainlyto the overall prediction of SWS in both periods in SDNWA. However,IMF5 was more important in the recharge period while IMF7 wasmore important in the discharge period. In addition, residue whichmay represent greater scales also explained a fair amount of variationof overall SWS. This again indicated that the dominant processes hadgreater scales in the discharge period in SDNWA. In LYMQ, predictedIMF4 and 2 in BNT and IMF4 and 3 in KPT explainedmajor variation ofoverall SWC. Predicted residue in KPT could explain 25% of the totalvariation of SWC. This may imply that much variation occurred atlarger scales (larger than measurement scale) in KPT.

Coefficients of determination between overall SW at the measure-ment scale and total predictions from different scales by each variablewere calculated to explore the relative important role of each variableto the predictive model (Table 5). Organic carbon was the major pre-dictor for SWS in SDNWA for both periods. This agreed with the rela-tively strong correlation coefficient between SWS and OC at themeasurement scale (0.723 in recharge period and 0.509 in dischargeperiod) as also observed by others (e.g., Hu et al., 2008; Wang et al.,2002). IMFs 5 and 7 of OC were generally the most important IMFsin predicting SWS (Table 4), therefore the greatest effect of OC inpredicting the SWS at the measurement scale mainly came from thegreater scales, such as 122 m and 416 m. Furthermore, clay in the re-charge period and silt in the discharge period also contributed to theoverall SWS prediction, mainly by their contributions at the greaterscales (Table 4). In the study of Biswas and Si (2011d), elevationwas recognized as the major predictor using univariate EMD. In ourstudy, the correlation coefficients between SWS and elevation were

ter storage in SDNWA and soil water content in LYMQ) for each intrinsic mode functionmposition.

Fa Adjusted R2

6.3 0.0423.2 0.15

tion+0.21 (0.27)∗Silt 308.7 0.88∗Silt-0.12 (−0.09)∗Clay 356.7 0.890.40 (0.45)∗Sand-2.66 (−0.35)∗Elevation 785.4 0.960.85 (0.43)∗OC+2.66 (0.26)∗Elevation 401.1 0.920.08 (0.13)∗Silt 2.0E+4 1.000.35 (−0.20)∗Elevation 1.2E+8 1.00

24.0 0.27-4.46 (−0.19)∗Elevation 16.4 0.27)∗OC-0.20 (−0.34)∗Sand 20.8 0.32-0.21 (0.05)∗Sand-0.12 (0.04)∗Silt 195.2 0.86-1.61 (−0.31)∗Elevation+0.07 (0.04)*Clay 859.5 0.98ation+1.01 (0.40)∗OC+0.06 (0.21)∗Silt 520.8 0.941.07 (−0.12)∗Elevation 1.1E+5 1.00

2.1E+6 1.00

13.6 0.34135.9 0.73

tion 64.4 0.7236.1 0.58

(−0.42)∗Elevation-3.47 (−0.07)∗OC 4.1E+4 1.001.4E+7 1.0010.3 0.1647.1 0.48

3.38 (0.61)∗Elevation 74.9 0.82C+2.47 (0.72)∗Elevation+0.17 (0.16)∗Clay 193.7 0.940.85)∗OC+0.05 (0.13)∗Elevation 2.8E+3 1.0002 (−0.28)∗Elevation 2.9E+6 1.00

Table 5Coefficient of determination between overall soil water (soil water storage in SDNWAand soil water content in LYMQ) at the measurement scale and predicted IMFs (resi-due) or total soil water predicted by each variable.

IMF1 IMF2 IMF3 IMF4 IMF5 IMF6 IMF7 Residue

SDNWA Rechargeperiod

0.01 0.02 0.13 0.13 0.43 0.06 0.25 0.14

Dischargeperiod

0.02 0.02 0.04 0.05 0.33 0.24 0.46 0.47

LYMQ BNT 0.02 0.25 0.13 0.30 0.03 0.13KPT 0.01 0.15 0.21 0.24 0.13 0.25

Elevation Sand Silt Clay OC

SDNWA Recharge period 0.36 0.08 0.20 0.46 0.74Discharge period 0.09 0.18 0.47 0.23 0.58

LYMQ BNT 0.00 0.25 0.01 0.32 0.00KPT 0.00 0.07 0.42 0.16 0.38

BNT: bunge needlegrass transect; KPT: korshinsk peashrub transect.

187W. Hu, B.C. Si / Geoderma 193–194 (2013) 180–188

also high, especially at the scale of 122 m and 416 m. However, OCoutperformed elevation in terms of the contribution to the overallSWS prediction. Possible reason was that the influences of OC onSWS reflected the combined influences of soil, vegetation, and topog-raphy in this area.

In LYMQ, soil particle composition was obviously the main con-tributor to the overall SWC prediction. In addition, OC also contribut-ed much to the variation of overall SWC in KPT. However, thecorrelation coefficient between SWC and OC at the measurementscale was negligible. This implied that the contribution of OC wasignored by the traditional regressionmethod. Actually, the contributionof OC on SWCmainly occurred at larger scale (larger thanmeasurementscale). For elevation, however, although it showed significant relation-shipwith SWC at both transects at the measurement scale, it contrib-uted little to the overall SWC prediction. The main reason was thatalmost all the variations of elevation occurred at scales larger thanmeasurement scale (Fig. 4).

This study clearly showed that MEMD can reveal the common scalefor an IMF with smaller numerical numbers for different variables.However, it is difficult to find exactly the common scales for IMFswith higher numerical numbers (e.g., IMF6 and 7 in SDNWA and IMF5for BNT in LYMQ) due in part to the low resolution of MEMD in identi-fication of large scales. Nevertheless, the obtained mean scales for IMFswith higher numerical numbers are still meaningful to explore thecontrolling factors and their correlations at these scales.

For the application of MEMD, data sets along a sampling transectwith a regular interval are required. Longer sampling transect issuggested to obtain more accurate scale and deeper understanding ofpedogenetic processes at large scales. For data sets from regular grid,two-dimensional empiricalmode decompositionmethodmay be possi-ble (Xu et al., 2006). In case of measurements from irregular sampling,interpolation is needed to obtain regular data beforeMEMD application.

4. Conclusions

Different number (seven in SDNWA and five in LYMQ) of IMFs forSW and environmental factors were obtained using MEMD. The dom-inant scales in terms of the percentages of total variances in SW wereidentified. There were scale-specific relationships between SW andenvironmental factors. SW at a specific scale (or an IMF) or residuewas predictable from controlling factors at that scale or residue, andSW at the measurement scale was predictable by summing up all pre-dicted IMFs and residue. Organic carbon content was found to be themain contributor in explaining the SWS variability in SDNWA and soilparticle composition was the major predictor for SWC in LYMQ. Theoverall SW prediction based on MEMD outperformed that based onsimple linear regression between SW and controlling factors and re-gression between IMFs of SW and factors at the measurement scale.

Hummocky terrains with alternating knolls and depressions occupy80% of the Canadian Prairie landscape, while sloping land separated byerosion gully is the main geomorphology on the Chinese Loess Plateau.Furthermore, the two areas have drastically different soil, vegetation,and climate (humid continental climate versus cold semi-arid). There-fore, results obtained from this study can be useful to understandsoil water process and its controls in these two regions, as well asother regions with sloping topography. Because non-stationarity andnonlinearity are common to various pedologic and hydrological pro-cesses. MEMD has a great potential in revealing the scale-specific con-trol of soil hydrological processes and making overall prediction.

Acknowledgments

The project was funded by Natural Science and Engineering Council(NSERC) of Canada, the University of Saskatchewan, and the NationalNatural Science Foundation of China (41001131). We acknowledgeDr. Asim Biswas, who helped with preparing the data. We thank theanonymous reviewers for their constructive comments.

References

Biswas, A., Si, B.C., 2011a. Depth persistence of the spatial pattern of soil water storagein a Hummocky Landscape. Soil Science Society of America Journal 75, 1099–1109.

Biswas, A., Si, B.C., 2011b. Identifying effects of local and nonlocal factors of soil waterstorage using cyclical correlation analysis. Hydrological Processes. http://dx.doi.org/10.1002/hyp. 8459.

Biswas, A., Si, B.C., 2011c. Identifying scale-specific controls of soil water storage in ahummocky landscape using wavelet coherency. Geoderma 165, 50–59.

Biswas, A., Si, B.C., 2011d. Revealing the controls of soil water storage at different scalesin a Hummocky Landscape. Soil Science Society of America Journal 75, 1295–1306.

Biswas, A., Si, B.C., 2011e. Scale and location specific temporal stability of soil waterstorage in a hummocky landscape. Journal of Hydrology 408, 100–112.

Blöschl, G., Sivapalan,M., 1995. Scale issues in hydrologicalmodeling: a review. HydrologicalProcesses 9, 251–290.

Brocca, L., Melone, F., Moramarco, T., Morbidelli, R., 2009. Soil moisture temporal stabilityover experimental areas in Central Italy. Geoderma 148, 364–374.

Canada Soil Survey Committee, 1978. The Canadian System of Soil Classification,Research Branch. Canada Department of Agriculture . Publication 1646.

Fleureau, J., Kachenoura, A., Albera, L., Nunes, J.C., Senhadji, L., 2011. Multivariateempirical mode decomposition and application to multichannel filtering. SignalProcessing 91, 2783–2792.

Gómez-Plaza, A., Martinez-Mena, M., Albaladejo, J., Castillo, V.M., 2001. Factors regulatingspatial distribution of soil water content in small semiarid catchments. Journal ofHydrology 253, 1261–1277.

Hu, M., Liang, H.L., 2011. Intrinsic mode entropy based on multivariate empirical modedecomposition and its application to neural data analysis. Cognitive Neurodynamics5, 277–284.

Hu, W., Shao, M.A., Wang, Q.J., Reichardt, K., 2008. Soil water content temporal-spatialvariability of the surface layer of a Loess Plateau hillside in China. Scientia Agricola65, 277–289.

Hu, W., Shao, M.A., Han, F.P., Reichardt, K., Tan, J., 2010. Watershed scale temporalstability of soil water content. Geoderma 158, 181–198.

Hu, W., Shao, M.A., Han, F.P., Reichardt, K., 2011. Spatio-temporal variability behaviorof land surface soil water content in shrub- and grass-land. Geoderma 162,260–272.

Huang, N.E., Shen, Z., Long, S.R., Wu, M.C.L., Shih, H.H., Zheng, Q., Yen, N.C., Tung, C.C.,Liu, H.H., 1998. The empirical mode decomposition and the Hilbert spectrum fornonlinear and non-stationary time series analysis. Proceedings of the Royal SocietyA 454, 903–995.

Huang, N.E., Wu, M.C.L., Long, S.R., Shen, S.S.P., Qu, W., Gloersen, P., Fan, K.L., 2003. Aconfidence limit for the empirical mode decomposition and Hilbert spectral analy-sis. Proceedings of the Royal Society A 459, 2317–2345.

Kachanoski, R.G., Rolston, D.E., de Jong, E., 1985. Spatial and spectral relationships ofsoil properties and microtopography: I. Density and thickness of a horizon. SoilScience Society of America Journal 49, 804–812.

Kravchenko, A.N., Bullock, D.G., Boast, C.W., 2000. Joint multifractal analysis of cropyield and terrain slope. Agronomy Journal 92, 1279–1290.

Lark, R.M., Webster, R., 1999. Analysis and elucidation of soil variation using wavelets.European Journal of Soil Science 50, 185–206.

Lewis, G.N., Mayer, J.E., 1929. The quantum laws and the uncertainty principle ofHersenberg. Proceedings of the National Academy of Sciences of the United Statesof America 15, 127–139.

Peel, M.C., Finlayson, B.L., McMahon, T.A., 2007. Updated world map of the Köppen–Geiger climate classification. Hydrology and Earth System Sciences 11, 1633–1644.

Pennock, D.J., Zebarth, B.J., de Jong, E., 1987. Landform classification and soil distribu-tion in hummocky terrain, Saskatchewan, Canada. Geoderma 40, 297–315.

Rehman, N., Mandic, D.P., 2009. http://www.commsp.ee.ic.ac.uk/~mandic/research/emd.htm.

188 W. Hu, B.C. Si / Geoderma 193–194 (2013) 180–188

Rehman, N., Mandic, D.P., 2010a. Empirical mode decomposition for trivariate signals.IEEET Transactions on Signal Procesing 58, 1059–1068.

Rehman, N., Mandic, D.P., 2010b. Multivariate empirical mode decomposition.Proceedings of the Royal Society A 466, 1291–1302.

Rilling, G., 2007. http://perso.ens-lyon.fr/patrick.flandrin/emd.html.Rilling, G., Flandrin, P., Goncalves, P., Lilly, J.M., 2007. Bivariate empirical mode decom-

position. IEEE Signal Processing Letters 14, 936–939.Seyfried, M., 1998. Spatial variability constraints to modeling soil water at different

scales. Geoderma 85, 231–254.Si, B.C., 2003. Spatial and scale dependent soil hydraulic properties: a wavelet approach.

In: Pachepsky, Y., et al. (Ed.), Scaling Method in Soil Physics. CRC Press, New York,NY, pp. 163–178.

Si, B.C., 2008. Spatial scaling analyses of soil physical properties: a review of spectraland wavelet methods. Vadose Zone Journal 7, 547–562.

Si, B.C., Farrell, R.E., 2004. Scale-dependent relationship between wheat yield and topo-graphic indices: a wavelet approach. Soil Science Society of America Journal 68,577–587.

Singer, P., 1999. Uncertainty inequalities for the continuous wavelet transform. IEEETransactions on Information Theory 45, 1039–1042.

Stegel, G., 2000. An uncertainty principle for convolution operators on discrete groups.Procedings of the American Mathematical Society 128, 1807–1812.

Tallon, L.K., Si, B.C., 2004. Representative soil water benchmarking for environmentalmonitoring. Journal of Environmental Informatics 4, 31–39.

USDA, 1975. Soil Taxonomy: A Basic System of Soil Classification for Making andInterpreting Soil Surveys. Agricultural Handbook 436, Soil Conservation Service.US Department of Agriculture, Washington, DC.

Wang, H.Q., Hall, C.A.S., Cornell, J.D., Hall, M.H.P., 2002. Spatial dependence and therelationship of soil organic carbon and soil moisture in the Luquillo ExperimentalForest, Puerto Rico. Landscape Ecology 17, 671–684.

Xu, M., Qi, Y., Chen, J.Q., Song, B., 2004. Scale-dependent relationships between land-scape structure and microclimate. Plant Ecology 173, 39–57.

Xu, Y., Liu, B., Liu, J., Riemenschneider, S., 2006. Two-dimensional empirical mode decom-position by finite elements. Proceedings of the Royal Society A 462, 3081–3096.

Yan, R., Gao, R.X., 2007. A tour of the Hilbert–Huang transform: an empirical tool forsignal analysis. IEEE Instrumentation and Measurement Magazine 1094–6969(07), 40–45.

Zeleke, T.B., Si, B.C., 2005. Scaling relationships between saturated hydraulic conductivityand soil physical properties. Soil Science Society of America Journal 69, 1691–1702.

Zeleke, T.B., Si, B.C., 2006. Characterizing scale-dependent spatial relationships betweensoil properties using multifractal techniques. Geoderma 14, 440–452.