solute transport scales in an unsaturated stony soil

Advances in Water Resources 34 (2011) 747–759

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Advances in Water Resources

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Solute transport scales in an unsaturated stony soil

Antonio Coppola a,⇑, Alessandro Comegna a, Giovanna Dragonetti a,b, Miles Dyck b, Angelo Basile c,Nicola Lamaddalena d, Mohamed Kassab d, Vincenzo Comegna a

a Department for Agricultural and Forestry Systems Management (DITEC), Hydraulics Division, University of Basilicata, Potenza, Italyb Department of Renewable Resources, University of Alberta, Edmonton, Alberta, Canadac Institute for Mediterranean Agricultural and Forestry Systems (ISAFoM), National Research Council (CNR), Ercolano (Napoli), Italyd Mediterranean Agronomic Institute, Land and Water Division, IAMB, Bari, Italy

a r t i c l e i n f o

Article history:Received 14 November 2010Received in revised form 24 March 2011Accepted 24 March 2011Available online 29 March 2011

Keywords:Solute transportDispersivityTransport scaleSpectral analysisvan Wesenbeeck–Kachanoski method

0309-1708/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.advwatres.2011.03.006

⇑ Corresponding author.E-mail address: [email protected] (A. Cop

a b s t r a c t

Solute transport parameters are known to be scale-dependent due mainly to the increasing scale of het-erogeneities with transport distance and with the lateral extent of the transport field examined. Based ona transect solute transport experiment, in this paper we studied this scale dependence by distinguishingthree different scales with different homogeneity degrees of the porous medium: the observation scale,transport scale and transect scale. The main objective was to extend the approach proposed by van Wes-enbeeck and Kachanoski to evaluating the role of textural heterogeneities on the transition from theobservation scale to the transport scale. The approach is based on the scale dependence of transportmoments estimated from solute concentrations distributions. In our study, these moments were calcu-lated starting from time normalized resident concentrations measured by time domain reflectometry(TDR) probes at three depths in 37 soil sites 1 m apart along a transect during a steady state transportexperiment. The Generalized Transfer Function (GTF) was used to describe the evolution of apparent sol-ute spreading along the soil profile at each observation site by analyzing the propagation of the momentsof the concentration distributions. Spectral analysis was used to quantify the relationship between thesolid phase heterogeneities (namely, texture and stones) and the scale dependence of the solute transportparameters. Coupling the two approaches allowed us to identify two different transport scales (around 4–5 m and 20 m, respectively) mainly induced by the spatial pattern of soil textural properties. The analysisshowed that the larger transport scale is mainly determined by the skeleton pattern of variability. Ouranalysis showed that the organization in hierarchical levels of soil variability may have major effectson the differences between solute transport behavior at transport scale and transect scale, as the transectscale parameters will include information from different scales of heterogeneities.

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1. Introduction

As a tracer flows through a porous medium, it will spread outboth ahead of and behind its center of mass. This phenomenon isknown as hydrodynamic dispersion and it is the result of mechan-ical dispersion and molecular diffusion. The property of a mediumwhich quantifies mechanical dispersion is called the dispersivity, a,which is known to be scale-dependent as its value may varydepending on the scale of the transport. This dependence is re-ferred to as a scale effect and is generally ascribed to the increasingscale of heterogeneities with transport distance and with thelateral extent of the transport field examined. Vanderborght andVereecken [36] derived a database of dispersivities from severallaboratory [33,34,2,26,3,4] and field-scale leaching experiments[27,1,9,11,31,32,24,38,15,7,8]. Besides, this database contains

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pola).

information about experimental parameters: transport distance,experimental scale, flow rate, boundary conditions, soil texture,pore water velocity, transport velocity and measurement method.

Solute transport at laboratory scale through a homogenized soilcolumn and through natural soil at field-scale differs chiefly in thevastly increased spatial variability of the transport volume found inthe latter. Such large-scale variations are generally described byspatially variable parameter fields [27]. However, small-scale vari-ations frequently occur, which would be classified as noise. Exam-ples of effects of variations comprise preferential flow or enhanceddispersivity with travel distance. Such effects are usually incorpo-rated into the model itself, say, by adopting a stochastic-convec-tion (SC) instead of a convection–dispersion (CD) model [24].

An appropriate model to describe field-scale solute transportthrough soil is generally difficult to establish a priori. For a givenapplication, a suitable type of model depends mainly on the homo-geneity of the transport volume. Two spatial scales pertinent to atracer experiment may be distinguished: (i) local scale, defined

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748 A. Coppola et al. / Advances in Water Resources 34 (2011) 747–759

by the active volume of the measuring device and (ii) transportscale, defined by the spread of a solute pulse originating from apoint-source at the surface. Although we can usually determinethe degree of homogeneity at the local scale from the data, thesame cannot be achieved for the transport scale.

Looking at local scale concentrations, the porous medium gen-erally appears to be heterogeneous. Nevertheless, averaging con-centrations over the transport scale may produce a mediumbehaving as homogeneous [24]. At that scale, the effects on thetransport of the local heterogeneities have been completelysmoothed. Parameters to be used in 1D descriptions of solutetransport process should be transport scale-parameters. Concen-tration measurements should first be averaged over the transportscale, before looking for any relationships between local-scaleand field-scale transport parameters. Identifying the spatial extent(representative elementary length, the averaging scale for localscale concentrations) at which the transport scale may be consid-ered homogeneous is thus decisive for correct parameterizationof solute transport (and water flow) models to be used at largescale.

A very effective method for determining the transport scale wasdeveloped by van Wesenbeeck and Kachanoski [31]. The methodidentifies the transition from the local to the transport scale forthe solute travel-time/depth variance in a given soil by averaginglocal scale concentrations across variable distance. The methodwas successfully used by van Wesenbeeck and Kachanoski [32]and by Woods et al. [39].

Because the transport scale varies with the soil heterogeneitiesand the corresponding scale of variability, for prediction purposesit is essential to establish to what extent soil heterogeneitiesdetermine the length for local-to-transport scale transition. vanWesenbeeck and Kachanoski [32] used spectral analysis to deter-mine the extent to which lateral redistribution of solutes at thefield scale was controlled by subsurface conditions such as thevarying depth of the B/C horizon interface by measuring the resi-dent concentration plume in the soil resulting from a pulse inputat the soil surface, which was subsequently leached through thesoil profile at a steady water application rate. As expected, theyfound that soil layering influences the representative elementarylength as it induces three-dimensional transport processes, whichin turn determine the magnitude of the lateral redistribution andhence the dispersivity. Similar results were observed by Hamlinand Kachanoski [11]. The issue was amply discussed by Javauxand Vanclooster [15] and explained mainly in terms of texturechanges at the layer interface.

These studies mostly focused on the effects of layer thicknessand of change in hydraulically effective porosity at the interface,implicitly assuming a within-layer homogeneity. Little attentionwas paid to the role of internal heterogeneity of single soil horizonsin terms of textural changes and, in stony soils, of the skeleton. Sig-nificant textural discontinuities may be found in soils at short hor-izontal distances and even distributed in disorderly zones ratherthan homogeneously organized in horizons along the vertical soilprofile. In this case, mixing mechanisms active in stratified soilsat the transition between two layers, hence along a fairly distinctsurface, apply diffusely at the interfaces between zones of differenttexture. For a given flow rate, diffuse textural heterogeneities andthe corresponding heterogeneity of hydraulic properties inducedifferent water contents in different zones of the soil porous sys-tem, thus resulting in a heterogeneous water-filled pore network.At steady state, water content is higher in the finer-textured zonesand lower in zones with a coarser texture. This affects the soluteparticle velocity field and lateral solute redistribution by inducingdivergence and/or convergence of streamlines to ensure flux conti-nuity. Consequently, particle velocities, and thus dispersivity, willcontinuously change along their trajectories. Similar effects may

be ascribed to stones distributed along the flow paths in the soilprofile. The latter could explain the results obtained by Schulinet al. [27] who observed strong variations in local velocity and dis-persion along a transect in a stony soil, as well as a significant lat-eral redistribution of the solutes applied.

That said, the role of texture on dispersivity is not so obvious. Itdepends on the flow rates and the related water contents establish-ing in the porous system. Some reported a dispersivity increasingwith saturation degree [35] due to larger pores involved at higherflow rates. Others [20,23] observed a reverse behavior, ascribed toless tortuous paths, and hence decreasing dispersivity at higherwater content. Comegna et al. [3] carried out 17 leaching experi-ments through three undisturbed soil columns of differenttextures. They found a rather constant dispersivity under bothsaturated and unsaturated conditions and different flow rates.Coppola et al. [5] carried out miscible experiments at differentwater contents on soils with bimodal pore-size distributions andobserved constant dispersivity values only when matrix waterflows were induced. However, it is clear that the lateral redistribu-tion of solute depends in a complex way on the interaction texture-water content. This interaction becomes far more complex in thepresence of significant stone content, which makes it extremelydifficult to identify a representative elementary length due todispersivity.

The main objective of this study was to extend the approachoriginally proposed by van Wesenbeeck and Kachanoski [31] toevaluating the role of stones and texture on the transition of localscale to the transport scale. The soil in question contains a signifi-cant internal heterogeneity in terms of texture and stones, chang-ing with depth and along the horizontal extent of the experimentalarea. Spectral analysis was used to quantify the relationship be-tween the solid phase heterogeneities and the scale dependenceof the solute transport parameters. van Wesenbeeck and Kachano-ski based their analysis on moments of BTCs. In our study, thesemoments were calculated starting from time-normalized residentconcentrations (as defined by Vanderborght et al. [34]), obtainedby time domain reflectometry (TDR) probes at different depths inseveral soil sites along a transect. The Generalized Transfer Func-tion (GTF) proposed by Zhang [40] was used to describe the evolu-tion of solute spreading along the soil profile at each observationsite by analyzing the propagation of the moments of the concentra-tion distributions.

2. Materials and methods

The experiment was carried out at the research station of theMediterranean Agronomic Institute (Bari – Italy) in the south-eastof Italy. The soil was pedologically classified as Colluvic Regosolconsisting of a silty loam layer of an average depth of 70 cm on afractured calcarenite rock.

An experimental plot 3.2 � 43 m2 (Fig. 1a and b) was equippedwith a drip irrigation system, consisting of:

– Two pumping stations, for fresh water and for solute applica-tions, respectively.

– Head control unit with valves, irrigation scheduler, pressureregulator, manometers, disk-filter, flow-meter.

– Polyethylene manifold, 40 mm diameter, 26 m length, equippedwith one valve, and one manometer.

– Dripper polyethylene pipes, 16 mm in diameter, 43 m long andspaced 0.15 m apart.

– Pressure self-compensating drippers, with a flow rate of4.4 l h�1, 0.15 m apart.

The drip irrigation system was evaluated from the pressure-dis-charge point of view in terms of the pressure-discharge character-

Fig. 1. (a) Layout of the irrigation system in the greenhouse used for the field tracer experiment and (b) vertical cross section showing the installation of the time domainreflectometry (TDR) probes along the transect.

A. Coppola et al. / Advances in Water Resources 34 (2011) 747–759 749

istics, emission uniformity, absolute emission uniformity and man-ufacturing coefficient of variation. A pressure of one bar was theproper operating pressure for the drip system. To determine theuniformity of irrigation, several buckets were placed under se-lected drippers. The emission and absolute emission uniformitycoefficients [21] were found to be 93% and 91%, respectively. Thesevalues are excellent based on different classification criteria.

Twenty undisturbed soil samples were collected at the soil sur-face along the transect using cylindrical steel samplers (15.0 cmdiameter and 15.0 cm high) for measuring bulk density and somehydraulic characteristics. In the laboratory, the samples wereslowly saturated from the bottom in four water level incrementsuntil the top of the sample was reached. We then determinedthe saturated water content hs gravimetrically and the saturatedhydraulic conductivity Ks by the falling-head permeameter [18].The main statistics are given in Table 1.

The experimental test area was covered by a plastic sheet toavoid soil evaporation and to ensure only downward movementof the applied water and salt solution. Drip irrigation was thenadapted to apply the constant flux density of 10 mm/day at 7a.m. An automatic irrigation scheduler was used to maintain thedesired rate.

The field plot was pre-irrigated with fresh water with an EC of1.05 dS/m until steady state conditions were attained. At steadystate, a depth of 3.87 mm of KCl solution (23.5 g/l with specificmass of 50 g/m2 of Cl�) was applied through the drip irrigation sys-

Table 1Average and coefficient of variation of the water content h andhydraulic conductivity at saturation Ks.

hs Ks (cm/h)

Average 0.492 34.6CV% 11.8 131.7

tem as a pulse application (d-Dirac type top boundary condition),while retaining the same supply rate of 10 mm/day. Fresh waterwas then applied once again to shift the KCl solution downwardinto the soil.

TDR probes were used for measuring the water content h, andimpedance, Z, along a transect in the middle of the experimentalplot. The Z values were used for estimating time-normalized resi-dent concentrations according to Vanderborght et al. [34]. TheTDR transmission line consisted of an antenna cable (RG58, 50 Xcharacteristic impedance, 210 cm long and with 0.2 X connectorimpedance) and of three-wire probes, 15 cm long, 2 cm internaldistance, 0.3 cm in diameter. After digging a trench, the TDR probeswere inserted at three different depths (7.5, 25, 40 cm) in 37 sites,at 1 m horizontal intervals, along the transect. Soil samples to beanalyzed for texture and stones were collected at the same depthsand horizontal intervals.

The volumetric water content, h, and impedance, Z, monitoredbefore applying KCl solution, were considered as initial values forthe experiment. After KCl solution application, daily measurementof volumetric soil water content h and impedance Z were taken inthe morning and evening to follow the KCl solution propagationthrough the soil profile.

During the experiment, air temperatures recorded inside thegreenhouse at the time of TDR readings were used for correctingthe impedances at a reference temperature of 25 �C asZl;25ðtÞ ¼

Zl;T ðtÞfT

with fT ¼ 11þcðT�25Þ where Zl,T(t) is the TDR impedance

of a waveguide at depth l measured at time t and temperature T,and fT is the temperature correction factor.

Heimovaara et al. [12] found that the temperature coefficientc = 0.0191 �C�1 used for electrical conductivity of soil extractswas accurate for soil electrical conductivity measured with TDR.We used this value of c and assumed that the measured air temper-ature adequately approximated soil temperatures under irrigationconditions. The temperature-corrected impedances were con-verted to resident concentrations of solute.


Detailed texture and distribution of stones (average equivalentdiameter = 27 mm) along the transect were derived from the dis-turbed samples taken during TDR probe insertion. The distributionof stones, clay and sand contents are depicted in Fig. 2a–c.

The very high skeleton content stems from previous deepploughing breaking the calcarenite rock and spreading stonesalong the soil profile. Soil tillage produced a sort of columnar dis-tribution of skeleton, sand and clay with a fairly regular size of 4–6 m. Columns showing decreasing clay contents with depth alter-nate with columns with increasing clay contents. Columns withhigh stone content increasing with depth alternate with uniformlow stone content columns.

3. Theory

3.1. Generalized Transfer Function (GTF)

At local scale, physical heterogeneities force the water and thedissolved phase to move in all directions through the pore space.Thus, because of local scale heterogeneities, the solute velocityhas components in the three directions and solute transport shouldbe dealt with as a 3D process, even if in the presence of advectionthe particles move preferentially in the direction of the main flow.Since the direct observation of the 3D velocity distribution pattern

a

c

b

Fig. 2. Skeleton (a), clay (b) and sand (c)

is practically impossible, predictions have to rely on models. Solutetransport parameters have to be derived by inverse modeling,where simulated concentration curves are fitted to either residentor flux concentrations from transport experiments. These parame-ters are related only implicitly to the local physical propertiesthrough the specific structure of the model. Their physical signifi-cance depends on how realistically the model represents the actualtransport process. Different models with different complexity maybe adopted depending on the data set available. Sophisticated 3Ddeterministic models have the ability to simulate solute transportin complex flow geometry porous media. Many codes are evenbased on tensor formulations for the hydraulic conductivity anddispersion parameter. However, as discussed elsewhere ([30];among others), these are merely formal representations of the por-ous medium characteristics, as tensor properties cannot be mea-sured by any existing methodology. What is more, obtainingtransport parameter fields with a field scale 3D approach requiresthe inverse problem to be solved using a 3D field of measuredwater contents and concentrations at different times [25]. Such asolution is rather difficult or even intractable in the event of lim-ited information about the changing solute concentration field. Inrecognition of such difficulties, many research efforts have beendevoted to developing 1D models, mainly assuming eitheradvective–dispersive (AD) or stochastic convective (SC) transport

distributions along the soil transect.


mechanisms and vertical solute transport. These approaches as-sume different depth-propagation of the travel time statistical mo-ments. The AD and SC models assume a linear and quadraticincrease in travel time variance with depth, respectively. However,transport cannot always be conceptualized as an AD or SC process.

Zhang [40] proposed a Generalized Transfer Function (GTF) inwhich the travel time moments may propagate over depth withany power value. The GTF [40] assumes a lognormal distributionof the travel times:

f f ðt; zÞ ¼ 1rzt

ffiffiffiffiffiffiffi2pp exp �ðln t � lzÞ

2

2r2z

" #ð1Þ

where lz and rz are the moments of the ln(t) at a depth z.The mean, E, and the variance, VAR, of the travel times, t, are re-

lated to the mean (l) and variance (r2) of the ln(t) through:

Eðt; zÞ ¼ exp lz þ 0:5r2z

� �;

Vðt; zÞ ¼ exp 2lz þ r2z

� �exp r2

z

� �� 1

� �ð2Þ

The model scales the moments of travel time moments between areference depth l and depth z according to the followingrelationships:

Eðt; zÞEðt; lÞ ¼

zl

� �s1

;Vðt; zÞVðt; lÞ ¼

zl

� �2s2

;CVðt; zÞ2

CVðt; lÞ2¼ l

z

2ðs1�s2Þ

with

CVðt; zÞ2 ¼ Vðt; zÞEðt; zÞ2

ð3Þ

where s1 and s2 are additional parameters accounting for the prop-agation of the solute travel time moments.

Zhang [40] classified the transport processes as a function of thedifference s1 � s2. (i) if s1 � s2 = 0, that is, var[ln(t)] is constantwith the travel distance, the process is SC; (ii) if the difference isnegative, var[ln(t)] increases with travel distance, the process isscale-dependent; (iii) if this difference is positive, var[ln(t)] de-creases with travel distance, as in the CD model (s1 � s2 = 0.5).

The average solute velocity, vs, and dispersivity, k, may be calcu-lated from the median and the coefficient of variation of the traveltime distribution, respectively, as

vsðzÞ ¼z

expðlzÞfor a lognormal distribution ðEq:ð1ÞÞ

aðzÞ ¼ z2

CVðt; zÞ2ð4Þ

Many studies use the evolution of travel time moments with depthobserved at the local scale for establishing the solute propagationmechanism and the corresponding approach to be used for its inter-pretation. However, the transport moments and related parametersobtained from these models are ‘‘vertical’’ moments, which may bemisleading if used to deduce the actual transport process in the soilprofile. By using 1D models, it is implicitly assumed that, on aver-age, the transport process is 1D and occurring only in the verticaldirection and the actual 3D local velocity distribution is replacedby the 1D distribution of the vertical velocity components. By con-sidering only the vertical component of the velocity in a 3D trans-port process, the local velocity and the spreading will beunderestimated. The evolution with depth of the moments of theconcentration distribution obtained by using 1D models may thusbe used to deduce the transport mechanism along the soil profileonly if the transport is actually vertical. In the other cases, any 1Dmodel should be considered as a regression model and no physicalmeaning should be attached to the model parameters. In this sense,we emphasize that the 1D transfer function used in this paper hasto be viewed merely as an analytical tool to quantitatively charac-terize the variability of the observed tracer distribution. If the trans-

port process is 2–3D, the calculated values of the dispersivity (andpore water velocity) only reflect the variance (and the center ofmass) of the vertical tracer distributions.

This is true at local scale. At the transport scale, the effects of lo-cal scale heterogeneities have been smoothed and the porous med-ium may be dealt with as a homogeneous medium. At this scale,using a 1D model makes sense and the effective parameters fromthe moments of the concentration distributions averaged at thatscale assume a more robust physical meaning. It is at this scale thatthe evolution of travel time moments with depth should be cor-rectly used for deducing the transport mechanism active alongthe mean flow direction.

3.2. GTF and TDR time normalized resident concentrations

Travel time moments are straightforwardly obtained when fluxconcentrations are available. However, when resident concentra-tions are measured, as is the case for TDR-based concentrations,the flux have to be drawn from the resident ones. Some approachesexist [13] which derive travel time moments through the probabil-ity conservation relation [16] or by defining time integral normal-ized concentrations [34]. Time integral resident concentrations,Crt�, resulting from a pulse input of solute at the surface may be di-rectly related to a TDR probe signal without needing anycalibration:

Crt�ðt; zÞ ¼1

Z�ðt;zÞR10

1Z�ðt;zÞdt

ð5Þ

where Z⁄ is the difference between the actual and initial impedanceload of the TDR trace at long times, and Crt⁄ represents the instanta-neous fraction of the total solute mass actually passing through thesampling volume of the TDR probe [34].

Crt⁄ may be expressed in terms of GTF parameters [14]:

Crt�ðt; zÞ ¼ a lnðtÞ � ar2z � alz þ k1

tk1rz

ffiffiffiffiffiffiffi2pp exp �

lnðtÞ � r2z � lz

� �2

2r2z

" #with

a ¼ k2� k1r2

z1� 1

exp r2z

� �" #ð6Þ

The four parameters of the model may be estimated by fitting thenon-linear function in Eq. (6) to the measured Crt⁄. As parameterss1 and s2 describe the propagation of the moments between twodepths, they may be robustly estimated only if Eq. (6) is fittedsimultaneously to at least two experimental Crt⁄ measured at twodifferent depths. For a detailed discussion, see Javaux and Vancloo-ster [14].

3.3. The van Wesenbeeck–Kachanoski (vWK) method

Given a transect where M breakthrough curves (BTCs) havebeen measured at regular intervals Ds, van Wesenbeeck and Kac-hanoski [31] proposed estimating the transport parameters atany arbitrary spatial scale D (D = 0 to M � 1) from an average sol-ute concentration distribution, bCD, obtained by averaging all theadjacent BTCs measured in individual sites (local scale BTCs) inthe interval DbCDðt; zÞj ¼ hCðt; zÞdi ðh i spatial expectation operatorÞ ð7Þ

with d = 1–D + 1 the index for adjacent BTCs in the interval D,J = M � D–1 the maximum number of bCDðt; zÞ that can be calculatedat lag D from a transect of length M and j = 1–J the index for thebCDðt; zÞ. For example, for D = 3, the number of adjacent BTCs to be

Fig. 3. Water content variability along the transect at 7.5 (a), 25 (b) and 40 (c) cmdepth, at the initial (0 h), final (528 h) and middle (272 h) times.


averaged is 4, d = 1–4, the number of average bCDðt; zÞs along thetransect is J = M � 3, and j = 1–M � 3. For D = 0, J = M and thebCDðt; zÞs concide with the local scale BTCs. For D = M � 1, J = 1 andthe single bCDðt; zÞ coincide with the transect scale BTC [1].

The average value of the moments from all the JbCDðt; zÞ werecalculated for each D asbEDðt; zÞ ¼ hEDðt; zÞjibV Dðt; zÞ ¼ hVDðt; zÞji

ð8Þ

and give an estimate of the transport parameters at the spatial scaleD. According to van Wesenbeeck and Kachanoski [31], a graph of bV D

vs. DDs at each z should indicate how the dispersion processchanges as a function of the spatial scale. The presence of a stableflat value should indicate the occurrence of a transport scale vari-ance estimate that is no longer a function of the spatial scale.

In this paper, the moments in Eq. (8) were calculated by apply-ing Eq. (7) to the measured time normalized resident concentra-tions, so that bCDðt; zÞ ¼ bCrt�ðt; zÞ for each interval D. The bCDðt; zÞswere calculated for each of the three observation depths (7.5;25.0, 40.0 cm). Thus, for each D, we obtained J terns of averagecurves. Eq. (6) was then applied simultaneously to each jth ternof average curves. Then the estimated GTF transport parameterswere used for calculating the jth transport moments, ED(t,z)j andVD(t,z)j at the three depths according to Eqs. (2) and (3).

Once the transport scale has been identified (D = DTs), the evolu-tion of travel time moments with depth can be correctly used fordeducing the transport mechanism active along the mean flowdirection. The difference (s1 � s2) calculated for each of theJ = M � DTs terns of bCDðt; zÞ may be especially useful for identifyingthe transport process prevailing at that scale. vWK analysis wasperformed using specifically developed code in MatLab, which isavailable on request ([email protected]).

3.4. Spectral analysis

To quantify the scale dependence of the spatial variance of avariable and the scale-dependent spatial covariance between vari-ables, spectral analysis is used. The spectral analysis is probably anappropriate tool for quantifying spatial scale-dependent features ofsoil physical processes and hydraulic and hydrodispersive proper-ties. The bases of spectral analysis are: Fourier’s transform, period-ogram analysis, cross-spectral analysis and spectral coherencyanalysis.

The theory of spectral analysis is still the subject of research. Itwill suffice, in the context of our study, to give in Appendix A theformulae from which the autocorrelation and spectra werederived.

4. Results and discussion

The experiment was performed under steady-state water con-tents in the soil profile compatible with the characteristics of thedifferent sites and with the boundary conditions adopted. For eachposition where TDR probes were located, a temporal series ofwater content was obtained. Fig. 3a–c shows the evolution of watercontent along the transect for three different times.

Notwithstanding the variability along the transect, which isstrictly related to the variability of soil physical characteristicsand to some differences in the supply rate along the drip lines,what should be noted is the relatively constant water content foreach location. This is confirmed for each depth. This result suggeststhat the steady-state conditions required for the experiment wereeffectively fulfilled, but it also suggests the important role that thewater content variability might have on the transport along thetransect. Also, it can be seen that the experiment was carried out

under unsaturated conditions (see the saturated water contentsin Table 1).

The bi-dimensional distribution of the time-average water con-tents is shown in Fig. 4. The plot shows a water content variabilitymainly along the vertical axis, which reflects only partially thecolumnar distribution of the textural components pictured inFig. 2. The plot of the bi-dimensional distribution of both the localdispersivity and the ratio of the local average pore water velocity,vw, to the local average solute velocity vs (see Eq. (4)) is representedin Fig. 5a and b.

vw was obtained as the ratio of the flux density q to the averagewater content measured at each of the three observation depths foreach site. The ratio vw/vs = hts/h, where hts is the transport volume,is defined as the volume fraction of the wetted pore space which isactive in the movement of the solute.

The two distributions contain some important information todecipher the role of textural and skeleton heterogeneities on thetransport process. First, they show a sort of columnar (horizontal)pattern of variability along the transect. Even with a different

Fig. 4. Distribution of the time-average water contents along the transect.

a

b

Fig. 5. Distribution of the local scale dispersivity (a) and vw/vs (b) along the transect.


behavior among columns, the observation verticals within a givencolumn seem to describe a similar transport process (see, forexample, the behavior in the interval 1–5 m, 9–13 m, 14–19 m).This pattern of variability reflects more that observed in the tex-tural components of the soil than the spatial pattern of the averagewater content (see Fig. 4). This highlights the role of clay and theskeleton as major factors inducing the spatial pattern of the trans-port process (and the transport scale) in the soil.

The large horizontal variability in the convective is evidencedby the alternating columns of increasing/decreasing vw/vs values,which are not compatible with the always increasing values ofthe water contents with depth. Also, the plot in Fig. 5 shows thatin our experiment vs was found systematically larger than vw. Insoils, vs may overcome vw because of micro/macro-scale processesoperating in the soil investigated. All these mechanisms may besimultaneously active during a solute transport process in a heter-ogeneous soil.

It seems more plausible that the individual sensors are viewinga three-dimensional transport process with a wide distribution oflocal velocities [15]. A three-dimensional heterogeneous flowcould have been generated by local heterogeneity of hydraulicproperties related to textural variability. As discussed in the

introduction section, at a given flow rate, this variability causes dif-ferent water contents and thus different water-filled pore net-works in different regions of the soil porous system. Thisproduces divergence and convergence of streamlines, such thatparticle velocities will continuously change along their trajectories.

In the soil examined, alternate zones with different textural andskeleton contents could also determine a pulse splitting, where theinitially uniform solute pulse may split into a series of fast andslow pulses moving in distinct transport zones [24,32,10]. The fastpulses may also aggregate at different depths along the soil profile.

Divergence and convergence of streamlines, as well as splitting,may also be induced by the skeleton scattered along the flow pathsin the soil profile. This is confirmed by the results of Schulin et al.[27] who also observed strong variations in local velocity and dis-persion along a transect in a stony soil. Obviously, skeleton andtextural heterogeneities interfere with the streamlines in a differ-ent way. The skeleton surface is an impermeable interface inducingstreamlines to bypass stones. On the other hand, textural changesforce streamlines to converge or diverge to ensure flux continuityaccording to the differences in their hydraulic properties [19].

The TDR observation window may not be large enough to sam-ple the representative transport volume for this type of transport


[15]. A TDR probe may be sampling mainly in the convergence ordivergence zones (fast or slow pulse zones, for the case of pulsesplitting). In the convergence (fast pulse) zones, for example, lowerestimation values of exp(l) (see Eq. (4)) are expected, with corre-sponding high solute velocities and low transport volume esti-mates. This also means that a TDR probe at a given depth may beseeing a transport behavior completely different from that ob-served by the TDR probe just below or above, with estimation ofthe solute velocity (and dispersivity) at different depths whichare not consistent with one-dimensional solute transport understeady state. In this case, as discussed in the introduction, the localsolute velocities and dispersivity obtained by the 1D GTF used inthis paper have no physical meaning and only reflect the medianand the variance of the vertical tracer distributions. Accordingly,in the following they will be defined as apparent parameters.

Of course, the similarity of the variability pattern of the appar-ent transport parameters with that of the textural componentsmay be only fortuitous and deserves quantification. A simple initialquantification of the correlation among the local (apparent) trans-port parameters and the texture is given in Table 2. The table alsogives the sample means and variances.

The only significant value (at the 0.05% and 0.1 probability lev-els) was found between dispersivity and skeleton at the surface(0.372). Simple correlations between apparent transport parame-ters and textural properties do not give any information aboutthe perceptible correspondence of their variability patterns. It isquite evident that a hidden co-structure exists that is difficult toobserve with classical statistical estimators in the spatial domain.Nielsen et al. [22] and Kachanoski et al. [17] have given examplesshowing that a non-significant overall correlation may mask signif-icant positive and negative correlations at different frequencies. Inthis sense, the analysis in the frequency domain may be a powerfultool for detecting and quantifying this co-structure.

Fig. 6 shows the power spectra of the clay, the skeleton and thelocal apparent dispersivity. The windows in the graphs indicate thesignificant peaks (at the 0.1 significance level), which indicate thatthe spatial variability of all the variables analyzed was dominatedby patterns that cycle at a scale around 2–5 m, with some differ-ences between clay and the skeleton (2–3 m for clay and 3–5 mfor the skeleton). Additionally, the skeleton and apparent disper-sivity spectra at depths of 7.5 and 25 cm have significant valuesat the scale of about 20 m, suggesting an organization of soil vari-ability into hierarchical levels. Later in this section, this relation-ship will become clearer when discussing the results of the vWKmethod (see Section 3.3) for identifying the transport scale.

The presence of a significant linear relationship between thevariables for given frequencies can be examined using the

Table 2Statistics of skeleton, clay and local apparent dispersivity. The value in bold is the only si

Skeleton (%) Clay (%)

z = 7.5 cmMean 14.086 10.882Variance 61.688 25.881Correlation with a 0.372 0.029Correlation with vw/vs �0.068 �0.189

z = 25 cmMean 15.022 12.499Variance 120.767 24.111Correlation with a �0.179 �0.217Correlation with vw/vs �0.017 0.150

z = 40 cmMean 24.235 11.692Variance 271.279 29.059Correlation with a �0.124 �0.120Correlation with vw/vs 0.169 0.209

coherency spectrum (Eq. (A13)). Indeed, coherency partitions thecorrelation between two variables with frequency (or equivalentlythe scale). The coherency spectra between textural componentsand apparent dispersivity are given in Fig. 7. As expected, signifi-cant coherency values may be found at the scale of 2–5 m, thesame already observed for the power spectra.

Significant coherency may also be observed between dispersiv-ity and: (i) clay at 25 and 40 cm at a scale of 16 m; (ii) skeleton at7.5 and 25 cm. Nevertheless, coherency values at 16 m for the clayshould be considered with some prudence. Indeed, the squaredcoherency at a given frequency cannot be interpreted without apreliminary analysis of the univariate spectra at the same fre-quency. A large squared coherency is important only if the variancein the two series corresponding to the frequency is significant.Therefore, we should first check which frequency bands containenough of the variance in one or both spatial series to be interest-ing and then look at squared coherency only for those selected fre-quencies [29]. In this sense, comparing power and coherencyspectra in Figs. 6 and 7, it is apparent that there is a clear correla-tion at the spatial scale of 20 m only between dispersivity and skel-eton, while the role of clay in determining the transport is limitedto the 2–4 m scale.

These very important results cannot be detected by standardcorrelation analysis. As discussed by Kachanoski et al. [17], highlysignificant coherency does not necessarily mean significant overallcorrelation. The ordinary correlation coefficient is affected by thepresence of phase differences between the variables, while powerspectra and coherency are phase-independent. This can be madeclearer by looking at the cospectrum (the in-phase covariance)and the quadrature spectrum (the out-of-phase covariance) forthe pairs a-clay and a-skeleton in Fig. 8.

We recall that the cospectrum partitions the covariance be-tween two variables as a function of frequency. Thus, the integralof the cospectrum over all the frequencies is the standard covari-ance between the two variables.

For the pair a-clay, the significant negative and positive covari-ances cancel each other and the overall standard correlation be-tween the two variables is not significant (0.029 in Table 2). Thisis not the case for the pair a-skeleton, where the positive covari-ances clearly overcome the negative ones, thus resulting in a posi-tive and significant correlation value (0.372 in Table 2).

The quadrature spectrum for both the pairs was slightly smallerthan cospectrum only around the scale of significant coherency,thus indicating a substantial in-phase variability between the vari-ables only at that scale. For all the other scales, the out-of-phasecovariance is comparable to that of the cospectrum, thus indicatinga delay between the variables. If the delay is positive or negative

gnificant value at 0.05 and 0.1 significance level.

Silt (%) Sand (%) a (cm) vw/vs (–)

58.438 30.680 12.550 0.39919.718 72.747 111.967 0.133

0.053 �0.045�0.132 0.181

63.326 24.174 10.541 0.34316.059 55.498 82.406 0.039�0.034 0.162�0.034 �0.054

61.918 26.391 11.362 0.45130.953 78.106 102.715 0.033�0.032 0.093�0.032 �0.074

Fig. 6. Power spectra of the clay, the skeleton and the local apparent dispersivity. The rectangles identify the significant power spectrum values.

Fig. 7. Coherency spectra between clay, skeleton and local apparent dispersivity. The dashed line indicates the threshold value Ca for coherency (see (A14)).


(one variable is advanced or delayed to the other) it could be de-tected by plotting the phase spectrum (Eq. (A15)) of each pair(not given here for the sake of brevity).

The plot of the transport variances bV D calculated according tothe vWK method (see Section 3.3) as a function of the spatial scaleis depicted in Fig. 9a–c. The three plots refer to the three observa-tion depths (7.5, 25 and 40 cm, respectively). The first point on thegraph refers to the local scale variance (calculated for D = 0). All thegraphs indicate the existence of a flat region starting at 5 m. A

second flat region starts at about 18 m for the first two depthsand at about 30 m for the depth of 40 cm. It is quite interestingto see that the vWK method, though based on a different approachwith respect to the frequency domain method, confirms the resultsof spectral analysis and provides complementary information. Aspointed out by van Wesenbeeck and Kachanoski [31], the two flatregions indicate two different transport scales. Based on the spec-tral analysis above, it can be concluded that they are naturallyoccurring scales induced by the spatial pattern of soil textural

Fig. 10. Plot of average dispersivities a vs. soil depth at the local scale (D = 0),transport scale (D = 5) and transect scale (D = 37).

b

c

a

Fig. 9. Solute travel-time variance bV D (see Eq. (8)) vs. spatial scale at 7.5 cm (a),25 cm (b) and 40 cm (c) depth.

Fig. 8. Cospectrum and quadrature for the pair a-clay and a-skeleton.


properties. The same analysis showed that the larger transportscale is mainly determined by the skeleton pattern of variability.

The presence of more than one transport scale indicates differ-ent scales of heterogeneities and an organization in hierarchicallevels of soil variability. The last issue is amply discussed byCushman et al. [6] and Vogel and Roth [37]. Soil variability maybe organized in discrete levels (hierarchical levels), each with itsspecific parameters (and specific interpretative models). At eachof the two transport scales, the effects of the lower-level scaleheterogeneities have been smoothed and the porous mediummay be dealt with as a homogeneous medium. In other words,averaging concentrations at these scales produces a homogeneousmedium. At these scales, using a 1D model makes sense even ifdifferent transport models could be required at each scale.

In the following, with the transport scale, we will refer to the5 m scale. The plots of the local scale, transport scale and the tran-sect scale dispersivity are shown in Fig. 10. The local scale disper-sivities were obtained from the local scale GTF solute transportparameters calculated for each of the 37 measurement sites(D = 0 – see Section 3.3) and then by averaging the parameters oversites.

As already explained in detail in Section 3.3, in order to obtainthe transport scale average dispersivity, we averaged the time-normalized resident concentration measured in five (D = DTS = 4)adjacent observation sites. The GTF parameters estimated for eachof the M � DTS curves were then averaged to give the averagedispersivity at the transport scale. The transect scale average dis-persivity was obtained by the same procedure but averaging all the


time-normalized resident concentrations (D = M � 1) to get a sin-gle global curve for the whole transect and then evaluating thetransect scale GTF solute transport parameters.

In general, the comparison between local average and transectaverage dispersivity resulted in a global process which differed sig-nificantly, thus also resulting in very different predictions of solutetravel time distributions. Averaging concentration measurementsover the transport scale produced substantial effects only at thesurface, where the dispersivity shifted towards that at the transectscale. Compared to the local scale, the transport scale behavior be-came more similar to that at the transect scale. Our analysisshowed that the residual differences between transport scale andtransect scale may be only partly due to the non-linearity of theobserved process. The organization in hierarchical levels of soilvariability may have major effects, as the transect scale parameterswill include information from different scales of heterogeneities.

That said, looking at Fig. 9, the transport scale tends to remainconstant with depth, thus suggesting a constant spreading of thesolute plume with the travel distance. This appears contradictedby the decreasing dispersivities with depth observed for both thetransport and transect scales (Fig. 10). This is mainly because thevWK for identifying transport scales is variance-based. In thissense, decreasing dispersivities with depth may be explained bythe fact that, besides the variance, increasing average length D pro-duces decreasing ratios of VARD(z, t) to the square of mean traveltimes, ED(z, t), with depth.

From a physical point of view, decreasing dispersivities withtravel distance have been mainly ascribed to changing water con-tents with depth. Maraqua et al. [20] and Nützmann et al. [23] ex-plained this behavior with less tortuous paths and thus decreasingdispersivity at higher water content. In this sense, in the soil exam-ined in this paper, despite the only minor and contradictory effectson local apparent dispersivity, the role of increasing water contentswith depth on transport can no longer be neglected at larger scales.

5. Conclusions

In this paper, we analyzed the role of distributed textural heter-ogeneities as the main sources of the scale dependence of solutetransport parameters. We distinguished three different scales, withdifferent homogeneity of the porous medium: the observationscale, transport scale and transect scale. At the observation scale,the porous medium generally appeared heterogeneous. The ap-proach used here allowed identification of the transport scale overwhich the effects on the transport of the local heterogeneities arecompletely smoothed and the corresponding transport scale-parameters may be used in 1D description of solute transportprocesses.

Our main results may be synthesised as follows:

– The vWK method proved to be a powerful tool for quantifyingthe scale dependence of transport variance estimated from sol-ute concentration distributions. The method allowed two trans-port scales to be identified at 5 m and at about 18 m.

– Spectral analysis enabled textural heterogeneities to be identi-fied as the main factors dominating the scale dependence of sol-ute transport parameters. The spatial variability of all thevariables analyzed was found to be dominated by clay and skel-eton patterns of variability that cycle at a scale around 2–5.Additionally, significant peaks in the power and coherencyspectra around 20 m suggested organization into hierarchicallevels of soil variability. Our analysis showed that the largertransport scale is mainly determined by the skeleton patternof variability. The correspondence between the outcomes ofspectral analysis and the vWK approaches is not coincidental

and indicates that the two transport scales are two naturallyoccurring scales induced by the spatial pattern of soil texturalproperties. It should be noted that the two methods are notalternative and should always be coupled as they provide com-plementary information.

– Similar scale dependence of the transport variance was found atthe three observation depths. This behavior appeared to contra-dict the decreasing dispersivities with depth observed for boththe transport and transect scales and was explained by consid-ering an increasing control of the water content on the trans-port parameters with the lateral extent of the transport fieldexamined.

– Analysis also showed that the organization of soil variabilityinto hierarchical levels may have major effects on the differ-ences between solute transport behavior at transport and tran-sect scales, as the transect scale parameters will includeinformation from different scales of heterogeneities.

Appendix A. Spectral analysis – frequency domain tools

A.1. The Fourier transform

The Fourier transform (FT) of discrete stationary series equi-spaced at intervals Ds {xs, s = 0,1, . . . ,M � 1} (with s being spatialor temporal location on the series) is defined as [28]:

XðkÞ ¼ M�1XM�1

s¼0

ðxs � xMÞ expð�2pimksÞ ðA1Þ

for k = 0,1, . . . ,M � 1. In Eq. (A1), X(k) are the Fourier coefficients,i ¼

ffiffiffiffiffiffiffi�1p

; mk ¼ k=M is the frequency (or wave number) in cyclesper unit distance (or time) and xM is the sample mean. For a discreteseries, the number of possible harmonics is finite. It should be notedthat, because at least three points are needed to draw a curve, theshortest wavelength a harmonic could have is 2Ds. Thus, the max-imum frequency mk that can be resolved by sampling at the rate ofone point per unit distance is mk = 1/2 cycles per point and is calledthe Nyquist frequency. It corresponds to a period of twice the sam-pling interval. The minimum frequency is mk = 0 and the corre-sponding period is the length of the transect. The period isfrequently called scale.

If the series is detrended, xs in Eq. (A1) is the detrended series.The FT in (A1) may be written in terms of sine and cosine trans-

form by noting that

expð�2pimksÞ ¼ cosð�2pmksÞ � i sinð�2pmksÞ ðA2Þ

This way, Eq. (A1) becomes

XðkÞ ¼ XCðkÞ � iXSðkÞwith

XCðkÞ ¼ M�1XM�1

s¼0

ðxs � xMÞ cosð�2pmksÞ

XSðkÞ ¼ M�1XM�1

s¼0

ðxs � xMÞ sinð�2pmksÞ

ðA3Þ

The Fourier coefficients X(k) are complex numbers. The advantageof representation in (A1) is that most software packages (MatLab,SAS, Microsoft Excel, . . .) have a built-in fast Fourier transform(FFT) algorithm that considerably speeds up the computation of(A1), with the sine and cosine transforms available immediatelyfrom the real and imaginary parts of the computed X(k).

Given a series of length M the MatLab FFT algorithm returns Mcomplex coefficients for positive and negative frequencies. Theordering of the frequencies is as follows: (0,1, . . . ,M/2,�M/


2 � 1,M/2 � 2, . . . ,�1). The first and second half of the list of num-bers correspond to the positive and negative frequencies, respec-tively. The real part of the FFT corresponds to the cosine seriesand the imaginary part corresponds to the sine. When taking anFFT of a real number data set (i.e. no complex numbers in the ori-ginal data) the positive and negative frequencies turn out to becomplex conjugates (the coefficient for frequency = 1 is the com-plex conjugate of frequency = �1). The MatLab FFT returns datathat can be used to get the following coefficients

ak ¼ �2M

imagðXðkÞÞ; 0 < k <M2

bk ¼ �2M

realðXðkÞÞ; 0 < k <M2

ðA4Þ

that can be used for recovering the original data signal by

xðsÞ ¼ a0 þXðM�1Þ=2

k¼0

ðak sinð2pmksÞ þ bk cosð2pmksÞÞ ðA5Þ

The notation imag(X(k)) means the imaginary part of X(k). The val-ues of the FFT in the negative frequencies provide no new informa-tion since the coefficients are simply complex conjugates.

The FFT works best on observed series where the number ofpoints M is highly composite. A power of 2 is the best situation[28]. For this reason it is common practice to extend a series oflength M to length M0 = 2n for some integer n by adding zeros tothe original series. In our case, M = 37 and M0 = 64. This onlychanges the frequencies at which the FT is calculated to mk = k/M0

for k = 0,1, . . . ,M0 � 1, while the FT values in (A1) are still scaledby M�1.

The power spectrum of the discrete stationary series xs, whichgives a representation of the variance of the series as a functionof frequency, is theoretically defined as

fxðmÞ ¼Xþ1

m¼�1CxðmÞ expð�2pimmÞ ðA6Þ

where Cx(m) = E[(xs+m � ls)(xs � ls)] is the autocovariance function,ls = E(xs) and m = 0,±1,±2, . . . is the lag on which the autocovarianceis calculated.

The periodogram, which may be written as the squared modu-lus of the FT

PxðmkÞ ¼ jXðkÞj2 ¼ X2CðkÞ þ X2

S ðkÞh i

¼ XðkÞXðkÞ ðA7Þ

where the overbar denotes complex conjugate, is approximately anunbiased estimator for the spectrum [28]. Each value of Px(mk) hastwo degrees of freedom and its interpretation is generally difficultwith excessive scatter of neighboring values and occurrence ofunexpected peaks. Also, the variance does not decrease to zerowhen the sample size tends to infinity. For this reason, it is commonpractice to average adjacent values of the periodogram to obtainestimates with higher degrees of freedom, and thus create asmoothed power spectrum.

The average spectral estimator in a frequency interval centeredon mk is defined as

f P;Bx ðmkÞ ¼ L�1

XðL�1Þ=2

l¼ðL�1Þ=2

P mk þl

M

¼ L�1

XðL�1Þ=2

l¼�ðL�1Þ=2

jXðkþ lÞj2 ðA8Þ

where L is some odd integer considerably less than M defining theaveraging window. In frequency terms, the averaging windowmay be expressed as a bandwidth B = L/M (cycles per point) cen-tered on mk. f P;B

x ðmkÞ is the periodogram-based power spectrum aver-aged on B. It is distributed approximately as a chi-squared v2

variable in which the degrees of freedom depend on the windowwidth L used.

The 100(1 � a) confidence interval for the smoothed spectrumcan be calculated as

2Lf P;Bx ðmkÞ

v22Lða=2Þ 6 f n

x ðmkÞ 62Lf P;B

x ðmkÞv2

2Lð1� a=2Þ ðA9Þ

where a is the significance level and f nx ðmkÞ is the background noise

power spectrum. The null hypothesis is f P;Bx ðmkÞ ¼ f n

x ðmkÞ vs.f P;Bx ðmkÞ – f n

x ðmkÞ. If f nx ðmkÞ falls within the interval in (A9), we fail to

reject the hypothesis. If not, the estimated power spectrum at a gi-ven frequency mk has to be considered significantly different fromthat of the assumed background noise. For the case of a white noise,implying a uniform distribution of the power spectrum across fre-quencies, f n

x ðmkÞ can be considered as the mean of all power spec-trum estimates.

A.2. Cross-spectral analysis

Cross-spectral analysis is useful for determining the structuredependence between two series {xs,s = 0,1, . . . ,M � 1} and{ys, s = 0,1, . . . ,M � 1} in the frequency domain. It allows us todetermine the relationship between the low- and high-frequencyfluctuations in the two series. Information on the relationships be-tween the frequencies in the spatial series obtained from cross-spectral analysis includes their phase differences (the phase isnot taken into account in the autocorrelation and spectral func-tions) and coherency, which is comparable with cross correlationin its classical sense. This analysis establishes inter-relationshipsbetween two spatial series and uses the relationships to forecastthe trend of a target series (y; output signal) from observationsmade on a primary series x; input signal).

The smoothed cross-spectrum estimator may be defined interms of FTs X(k) and Y(k) of two series xs and ys as

f P;Bxy ðmkÞ ¼ L�1

XðL�1Þ=2

l¼�ðL�1Þ=2

Pxy mk þl

M

¼ L�1XðL�1Þ=2

l¼�ðL�1Þ=2

jXðkþ lÞjjYðkþ lÞj ðA10Þ

where

PxyðmkÞ ¼ XðkÞYðkÞ ðA11Þ

The cross-spectrum is a complex quantity and is subdivided foroperational purposes into a real part called sample cospectrum[cxy(mk)] and an imaginary part called sample quadrature spectrum[qxy(mk)], which are the in-phase and out-of-phase covariances ofxs and ys, respectively

f P;Bxy ðmkÞ ¼ cxyðmkÞ � iqxyðmkÞ ðA12Þ

The smoothed sample squared coherency can be defined in terms ofcross-spectrum and single series spectra as

c2P;B

xy ðmkÞ ¼f P;Bxy ðmkÞ�� 2

f P;Bx ðmkÞf P;B

y ðmkÞðA13Þ

It is analogous to the coefficient of determination r2xy (ranging

from 0 to 1) and estimates the proportion of the spatial varianceof ys that can be explained by xs as a function of frequency (or ofthe spatial scale). In other words, it is a measure of the lineardependence of the two series at a given frequency and providesa measure of the degree of linear predictability of one series fromanother one.

The significance of sample squared coherency can be obtainedfrom the F distribution with 2 and 2(L � 1) degrees of freedom[28] as


Ca ¼F2;2L�2ðaÞ

L� 1þ F2;2L�2ðaÞðA14Þ

where Ca is the approximate value which must be exceeded for thesquared coherency to be able to reject the null hypothesisc2

xyðmkÞ ¼ 0 at an a priori specified frequency.

A.3. Spectral-coherency analysis

Coherency is a stationary process analog of the traditional cor-relation coefficient, taking on values between 0 and 1 at any givenfrequency. It is a measure of the time-invariant linear dependenceof the two series at a given frequency and provides a measure ofthe degree of linear predictability of one process from anotherprocess.

As the cospectrum and the quadrature represent the in-phaseand out-of-phase covariances of xs and ys, in the case of a signifi-cant high coherency, a low quadrature spectrum compared to thecospectrum will indicate no, or low, spatial delay between thetwo variables. In other words, the changes in the y spatial series oc-cur at the same distance (or time) as changes in the x spatial series.The eventual lag between the spatial series within each frequencyband can be calculated as phase spectrum, /P;B

xy

/P;Bxy ¼ tan�1 �qxyðmkÞ

cxyðmkÞ

� ¼ tan�1

imag f P;Bxy ðmkÞ

� �real f P;B

xy ðmkÞ� �

24 35 ðA15Þ

All estimates, including confidence intervals, require that we selectthe bandwidth B over which the spectrum will be essentially con-stant. With too broad a band there is the risk of smoothing out validpeaks, while too small a band will lead to confidence intervals thatare too wide for isolating and comparing significant peaks. Shum-way [28] recommends that a value of m equal to 1/20 of M be usedin the analysis. In the present paper, M = 37 measurement siteswere used in spectral estimation and we selected L = 3, with a band-width of 0.08 cycle/m. Spectral analysis was performed using a spe-cifically developed code in MatLab, which is available on request([email protected]).

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http://dx.doi.org/10.1016/j.jconhyd.2008.10.004

http://dx.doi.org/10.1016/j.jconhyd.2008.10.004

http://dx.doi.org/10.2136/vzj2007.0040

http://dx.doi.org/10.2136/vzj2007.0040

solute transport scales in an unsaturated stony soil

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