Diagnostic tools for mixing models of stream water...

Diagnostic tools for mixing models of stream water chemistry

Richard P. Hooper

U.S. Geological Survey, Northborough, Massachusetts, USA

Received 17 June 2002; revised 7 January 2003; accepted 7 January 2003; published 14 March 2003.

[1] Mixing models provide a useful null hypothesis against which to evaluate processescontrolling stream water chemical data. Because conservative mixing of end-memberswith constant concentration is a linear process, a number of simple mathematical andmultivariate statistical methods can be applied to this problem. Although mixing modelshave been most typically used in the context of mixing soil and groundwater end-members, an extension of the mathematics of mixing models is presented that assesses the‘‘fit’’ of a multivariate data set to a lower dimensional mixing subspace without the needfor explicitly identified end-members. Diagnostic tools are developed to determine theapproximate rank of the data set and to assess lack of fit of the data. This permitsidentification of processes that violate the assumptions of the mixing model and cansuggest the dominant processes controlling stream water chemical variation. These samediagnostic tools can be used to assess the fit of the chemistry of one site into the mixingsubspace of a different site, thereby permitting an assessment of the consistency ofcontrolling end-members across sites. This technique is applied to a number of sites at thePanola Mountain Research Watershed located near Atlanta, Georgia. INDEX TERMS: 1045

Geochemistry: Low-temperature geochemistry; 1806 Hydrology: Chemistry of fresh water; 1871 Hydrology:

Surface water quality; KEYWORDS: hillslope hydrology, streamflow generation, end-member mixing analysis,

stream chemistry

Citation: Hooper, R. P., Diagnostic tools for mixing models of stream water chemistry, Water Resour. Res., 39(3), 1055,

doi:10.1029/2002WR001528, 2003.

1. Introduction

[2] Mixing models of surface water chemistry have beenextensively employed for hydrograph separation [Genereuxand Hooper, 1998]. In geographic-source separations,measured soil water and groundwater solutions are eval-uated as potential ‘‘end-members’’ for the stream water,which is assumed to be a mixture of these end-members.These models have been particularly effective at examiningdata sets and generating hypotheses because they link small-scale, internal catchment measurements (groundwater andsoil water solution chemistry, in this case) with the phe-nomenon of interest (catchment runoff chemistry) thatoccurs at a much larger scale [Hooper, 2001].[3] Christophersen and Hooper [1992] stated that only

the number of end-members can be determined from thestream mixture, but not their chemistry. They determinedthe number of end-members based upon a subjective con-sideration of the variance explained by each principalcomponent of the solute correlation matrix in a mannerconsistent with standard approaches in principal componentanalysis (PCA). In this paper, a different approach isdeveloped that is based more upon the geometry underlyingthe correlation matrix rather than on a simple considerationof variance explained. Beyond providing further guidancefor determining the appropriate number of end-members,this analysis also helps to identify violation of the assump-tions of standard mixing models, including conservative

mixing and constant end-member composition. Thus it canalso help to determine which solutes can be used as tracersand which should not be used.[4] The mathematics developed here uses only the stream

chemistry and does not require an explicit identification ofend-members. Therefore this analysis also can be used atsites where only stream chemistry data are available, thusexpanding the utility of this technique beyond intensivelystudied research catchments to water quality monitoring andassessment programs. A direct extension of the theorydeveloped here permits a quantitative comparison of chem-istry among multiple sites that can be used, for example, toidentify sites where the stream chemistry is controlled bythe same processes or to examine changes in water qualityacross spatial scales.[5] The foundation for this analysis is the mathematical

distinction between mixing and equilibrium chemistry.Conservative mixing of end-members with fixed composi-tion is a linear process, but equilibrium reactions amongsolutes of different charge are higher-order polynomials.Nordstrom and Ball [1986] noted that the relation betweenaluminum activity and hydrogen ion, although cubic at lowhydrogen ion activities (high pH) as expected from equili-brium considerations, became linear at high hydrogen ionactivities (low pH). They attributed this linear behavior tomixing processes. (Note that the cubic relation, althoughtaken as evidence of equilibrium control, is a mathematicalnecessity because aluminum activity is calculated as afunction of hydrogen ion. Observing a cubic relation athigh pH is not a strong test of equilibrium control [Neal etal., 1987].) In this paper, a series of diagnostic tools using

This paper is not subject to U.S. copyright.Published in 2003 by the American Geophysical Union

HWC 2 - 1

WATER RESOURCES RESEARCH, VOL. 39, NO. 3, 1055, doi:10.1029/2002WR001528, 2003

linear mathematics are developed to test the hypothesis ofmixing control on stream water chemical variation. Thegeometry of the ‘‘data cloud’’ formed by plotting solutesagainst one another is explored to determine whether therelation among the solutes is essentially linear in this highlydimensional space or exhibits curvature.

2. Data Set

[6] The following site description is a summary of thatreported by Huntington et al. [1993]. The Panola Moun-tain Research Watershed (PMRW) is one of five Water,Energy, and Biogeochemical Budget (WEBB) watershedsoperated by the U.S. Geological Survey (USGS). PMRW,a 41-ha catchment within the Panola Mountain State Con-servation Park, is located in the Piedmont physiographicprovince approximately 25 km southeast of Atlanta, Georgia(Figure 1). The watershed is underlain primarily by gran-odiorite emplaced 320 Myr ago in a host rock of biotite-plagioclase-gneiss that contains amphibolite and K-feldspar.

There is a 3-ha granodiorite outcrop in the headland of thecatchment, which accounts for much of the 51 m relief atPMRW. Soils are classified as entisols near the base of theoutcrop and as ultisols on hillslopes and ridge tops. Kao-linite, hydroxy-interlayered vermiculite, mica, and gibbsiteare the dominant secondary minerals. Soils have developedto a thickness of 1–2 m, and saprolite underlies the soilsranging from a depth of 0 to 5 m below land surface. Thecatchment, with the exception of the outcrop, is completelyforested with a cover of approximately 70% mixed southernhardwoods (oak (Quercus spp.), hickory (Carya spp.) andtulip poplar (Liriodendron tulipfera)) and 30% loblolly pine(Pinus taeda). The stream draining the catchment is per-ennial. The climate in the area is classified as warm temper-ate subtropical, and the annual average temperature is 16�C.The long-term annual average precipitation is 1.24 m;annual runoff, although highly variable, averages 30% ofprecipitation.[7] Features of the chemical data considered in this paper

are summarized in Table 1. Both the lower gage and the

Figure 1. Site map of Panola Mountain Research Watershed.

Table 1. Description of Data Set

Site Sampling Approach Sample Size Duration Stream Characteristics

Lower gage weekly + event samples 2819 10/1985–12/1999 perennial, gagedSite 201 weekly + event samples 584 7/1990–10/1996 perennial, gagedSite 210 weekly 223 1/1990–9/1996 perennial, ungagedSite 404 weekly + event 424 9/1992 –10/1996 perennial, gagedSite 420 weekly 301 9/1992–3/1998 perennial, ungaged

Upper gage weekly + event 1223 10/1985–12/1999 seasonal, gaged

HWC 2 - 2 HOOPER: DIAGNOSTIC TOOLS FOR MIXING MODELS

upper gage have permanent control structures in place. Atthe numbered sites, either temporary structures wereinstalled for some period, or no gaging was done at thesite. Stream water samples were collected manually everyweek at all sites. The duration of the sampling is indicatedin Table 1. For some period of the record, many of the siteswere equipped with stage-activated automatic samplers tobetter characterize chemical variation with discharge. Sam-ples were analyzed for major anions, cations, alkalinity, andpH using standard methods by a project laboratory. Hun-tington et al. [1993] provide further details on analyticalmethods, precision, and the quality assurance program atPMRW. The coefficient of variation for concentrations ofstandard reference materials is generally less than 5%;sodium has the highest coefficient of variation at 7.7%.The standard deviation of repeated determinations for mostsolutes is less than 1 meq/L, except for magnesium, forwhich it is 2 meq/L.

3. Previous Mixing Models at PMRW

[8] Hooper et al. [1990] identified measured soil waterand ground solutions that could be used as end-members in

a three-component mixing model for the lower gage atPMRW. These end-members explained between 83 and97% of the variation in the concentration of six solutes orsolute combinations: alkalinity, calcium, magnesium,sodium, sulfate, and silica. These solutes were assumed tomix conservatively. Hooper et al. [1990] used all six solutessimultaneously to determine the proportion of the three end-members when only two tracers are required. A leastsquares error criterion was employed to solve the over-determined system of equations.[9] Christophersen and Hooper [1992] demonstrated that

this solution to the end-member mixing problem wasequivalent to making an orthogonal projection of the streamsamples into the subspace defined by the end-members. Thelogic of this mathematical solution is reversed because thestream is the observed mixture and the soil water andgroundwater solutions are being tested as potential end-members. Instead, Christophersen and Hooper [1992] rec-ommended defining the mixing subspace of the streamsamples using an eigenvalue analysis and projecting thepotential end-members into that subspace. The dimension-ality of the subspace is determined by the number ofeigenvectors retained to define the mixing subspace, andthis dimensionality is related to the number of end-membersrequired in a mixing model. The number of eigenvectors toretain is a subjective decision, although a number ofheuristic rules have been proposed that are based upon therelative magnitude of the eigenvalues associated with eacheigenvector [Preisendorfer et al., 1981]. Christophersenand Hooper [1992] noted that the maximum amount ofvariation in the data that can be explained is limited to thatexplained by the eigenvectors retained, but provided nofurther guidance in determining the number of componentsto retain. In this paper, a more formal analysis is proposed todetermine the dimensionality of the stream data, mathemati-cally known as the ‘‘rank’’ of the data set.

4. Methods

4.1. Conceptual Basis

[10] To illustrate the central concepts of this paper,consider a site where two solutes have been measured on50 samples that are, to some extent, collinear (Figure 2a).This site is called the ‘‘reference site’’ and, in practicalapplications, is a well-characterized site against which othersites’ chemistry will be compared. Using an eigenvalueanalysis that will be described below, the ‘‘mixing sub-space’’ for these data is one-dimensional (a line), and thisline accounts for 91% of the variation in the data. Theorthogonal projections of the data onto the line (Figure 2b)are the ‘‘best’’ one-dimensional representation of the dataset (in the sense of being geometrically closest) and can beused instead of the original values to reduce the dimension-ality of the data set. Residuals between the original valuesand the projected values can be calculated, and examinedfor structure. Just as in the case of a regression model, a‘‘good’’ mixing subspace is indicated by a random patternof residuals plotted against the concentration of the originalsample (Figure 2c). Structure in this plot indicates a lack offit in the mixing subspace and hence a deficiency in themixing model that might arise for a number of reasons,including nonconservative behavior of tracers. Scalar meas-

Figure 2. Solute mixing diagram for reference site(artificial data). (a) Mixing diagram. (b) Mixing diagramwith orthogonal projections. (c) Residual between measuredand projected concentrations against measured concentra-tion for solutes 1 and 2.

HOOPER: DIAGNOSTIC TOOLS FOR MIXING MODELS HWC 2 - 3

ures of fit, such as bias and root-mean-square error (RMSE)can also be calculated from the residuals. Outliers can beidentified and may indicate either errors in the data ordifferent processes controlling the chemistry for thosesamples.[11] Data from other sites, referred to as sites ‘‘A’’ and

‘‘B,’’ can be projected into the ‘‘best fit’’ line from thereference site, and the residuals between the projected andoriginal values examined. Site A appears to fit the line well(Figure 3). The residuals (Figure 4) are small and have nosystematic pattern, and the bias and RMSE are small. Bycontrast, site B is not consistent with the reference site, asevidenced by the structure and size of the residuals (Figure 5).[12] Physically, the mixing of the same two end-members

may control site A and the reference site, with lowerproportions of the more concentrated end-member exhibitedat the reference site. When the sites are at different basinscales, such an analysis suggests the areal influence ofcontrolling end-members. Alternately, the more concen-trated end-member at site A may be water that has beenin contact with the same minerals as at the reference site fora longer time and hence has a higher concentration. A sitewill ‘‘fit’’ so long as the solutes are added to solutions withthe same initial chemistry in the same ratio, as would beexpected for congruent or incongruent dissolution of min-erals. For example, if water acquires the chemical signatureof the organic soil horizon and then gradually evolves withthe same mineral assemblage in the lower soil horizons, all

of these waters would ‘‘fit’’ in the same mixing subspace. Ifthe water either starts with a different chemistry, contacts adifferent mineral assemblage, exchanges solutes with thesolid phase in a manner that violates the weatheringstoichiometry, or has a different set of controlling end-members, these solutions would not fit the same mixingsubspace. Site B exhibits lack of fit. Note that solutions witha different concentration range from the reference site may‘‘fit’’ the mixing subspace (site A) and a site with the sameconcentration range may exhibit lack of fit (site B). The keypoint is that the ratio among all solutes must be preserved tofit the subspace; their absolute concentration is not relevant.[13] This approach is a refinement of a simple classifica-

tion of solutions by ion ratios that has long been done ingeochemistry using trilinear ‘‘piper’’ diagrams [Piper,1944]. The triangular portions at the bottom of the figuredisplay the relative proportion of three cations (left side) andanions (right side), or combinations of ions. The diamonddisplays the relative proportion of two groups of anionsalong one axis and cations along the other. The four sites inFigure 6 are compared by examining the areas of each ofthese subplots the sites occupy. The contribution of thispaper can be viewed as a generalization of these concepts toan arbitrary number of dimensions and providing quantita-tive measures of similarity.

4.2. Mathematical Development

[14] Let X represent an (n � p) matrix of stream chemicaldata consisting of n samples on which p solutes weremeasured. Most typically, these data will be standardizedby subtracting the mean concentration of each solute anddividing by the standard deviation of each solute so that eachsolute has equal weight in the analysis. See Christophersenand Hooper [1992] for further discussion on data stand-ardization in mixing models. Let X* be the matrix of stand-ardized values. Letting �xj be the mean of the jth solute and sjbe its standard deviation, each element of this matrix is

x*ij ¼ xij � �xj� �

=sj: ð1Þ

X*TX* is related to the correlation matrix of the p solutesby a constant. The ‘‘best’’ m-dimensional subspace (m < p)that fits X* (in terms of explaining the maximum variance,which is equivalent to Euclidean distance for standardizeddata) is simply the first m eigenvectors of this solutecorrelation matrix. V, the matrix formed by the first m

Figure 3. Solute mixing diagram showing data fromreference site and two additional sites, labeled A and B,along with orthogonal projection into mixing subspace ofreference site.

Figure 5. Residual between measured and projectedconcentrations at site B against measured concentrationsfor solutes 1 and 2.

Figure 4. Residual between measured and projectedconcentrations at site A against measured concentrationsfor solutes 1 and 2.


eigenvectors, is the basis of a new m-dimensional sub-space. In standard PCA, the ‘‘scores’’ of the observationsare expressed in the coordinates of this subspace. However,one can also express these points in terms of the originalsolutes. Geometrically, this is equivalent to the orthogonalprojection of X* given by

X̂* ¼ X*VT VVT� ��1

V ð2Þ

X̂* can be destandardized by multiplying by the standarddeviation of each solute and adding the mean to yield X̂; thatis, each element of this matrix is x̂ij = x̂*ij sj + �xj. Residualsbetween the projected and original data are simply

E ¼ X̂� X: ð3Þ

Diagnostic plots (such as Figures 4 or 5) can be made byplotting the residuals (ej) against the observed concentration(xj) for each solute j. A well-posed model is indicated by arandom pattern in the residuals; any structure in this plotsuggests a lack of fit in the model, which can arise from theviolation of any of the assumptions inherent in the mixingmodel. However, the magnitude of the residuals, both

absolute and relative to the observed concentration, shouldbe considered. The residuals have units of concentration andshould be evaluated in light of analytical precision todetermine if any apparent patterns are meaningful or withinthe noise of the laboratory analysis.[15] Two useful scalar measures of fit are the bias and

root-mean-square error. Because the solutes often differmarkedly in concentration, these measures are scaled bythe mean concentration of each solute to make them unit-less. The relative bias (RB) for each solute j is defined as

bj ¼

Xni¼1

x̂ij � xij� ��xj

: ð4Þ

Similarly, the relative root-mean-square error (RRMSE) forsolute j is

rj ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXni¼1

x̂ij � xij� �2s

n � �xj: ð5Þ

Figure 6. Piper diagrams for selected solutes at three sites in Panola Mountain Research Watershed.


[16] For a site projected into a subspace defined by itsown eigenvectors, as for the reference site, the bias willalways be zero. The relative root-mean-square error pro-vides an indication of the ‘‘thickness’’ of the data cloudoutside the lower dimensional subspace. The dimension ofthe mixing subspace (m) should be simply the rank of X(that is, the number of dimensions that X spans). However,because real data contain error, there is always an elementof subjectivity in this decision. How much of the variancein the data set should be explained and how much is‘‘noise’’? An eigenvalue analysis is the first step in thewidely used multivariate statistical technique known asprincipal components analysis (PCA). The same questionis faced in PCA: How many components should beretained? A number of rules have been proposed [Prei-sendorfer et al., 1981], all dependent on the eigenvalues.A common rule is to retain all eigenvectors associatedwith eigenvalues greater than or equal to 1 called the ‘‘ruleof 1.’’ Because the correlation matrix is being factored,each original variable has a variance equal to 1. Thereforethis rule is based on the principle that each new compo-nent should explain at least as much variance as theoriginal variables. This argument is consistent with theuse of PCA as a technique to reduce the dimensionality ofthe data set.[17] In this analysis, a different test is suggested. The

dimension of the mixing subspace should be the smallestpossible such that the residuals exhibit no structure whenplotted against observed concentration; that is, the residualsappear to be random noise. There are two qualifications tothis rule. First, the higher the dimension of the mixingspace, the fewer ‘‘degrees of freedom’’ exist for testingwhether or not mixing processes hold; a mixing space withthe same number of dimensions as solutes entered in theanalysis will, of course, perfectly fit the data. Second, thisrule assumes that all solutes mix conservatively. Curvaturein the residuals could indicate chemical reactions or variableend-member concentrations under some circumstances. Dis-criminating between these processes and the need for anadditional dimension in the mixing space requires consid-eration of the chemical and biological properties of thesolute to determine the likelihood of these processes.[18] The rank of the data set is directly related to the

number of end-members required in a mixing model. Fordata that have been centered on their mean (as a correlationmatrix), one more end-member than the rank is required[Miesch, 1976]. Thus, if the rank of the data set is two (i.e.,a plane), a minimum of three end-members is required for amixing model.[19] To project another site, Y, into the subspace defined

by the reference site, the data must first be standardized bythe means and standard deviation of the reference site, andnot by its own mean and standard deviation, because thereference site defines the basis for the mixing subspace; thatis,

y*ij¼ yij � �xj

� �=sj: ð6Þ

The data can be projected using equation (2) (substitutingY* for X*), and destandardized. Residuals, relative bias,and relative root-mean-square error can be calculated asabove with the appropriate substitutions.

[20] Two possible benchmarks exist for the RB andRRMSE of the test site. The best possible fit of these datato an m-dimensional subspace would be defined by its ownm eigenvectors. The RRMSE for the samples projected intothe test site can be compared with this benchmark todetermine how much poorer the fit is. (The RB for a siteprojected into a subspace defined by its own eigenvectorsis always zero.) A second benchmark is to compare theRRMSE of the projected data to the RRMSE of the re-ference site. This statistic indicates how ‘‘noisy’’ thereference site itself is. Thus both of these benchmarksprovide a lower bound for the RRMSE. The RB indicateswhether the mixing subspace slices through the center ofthe data cloud or lies to one side of it. If the RB is ofthe same order of the RRMSE, the data all lie to oneside of the mixing subspace, and hence a poor fit isindicated.[21] This analysis can be easily implemented using a

combination of statistical packages to extract the eigenvec-tors and spreadsheets to perform the matrix manipulations.Custom programs are not necessary, although they canspeed the analysis. Advanced programming languages canperform these analyses and make the suggested diagnosticplots with a few dozen commands.[22] In this application, the lower gage at PMRW is used

as the ‘‘reference’’ site and data from five upstream sites(Figure 1) are considered the ‘‘test’’ sites.

5. Results and Discussion

5.1. Approximate Rank

[23] The percentage of variance explained by each eigen-vector for the six sites at PMRW is shown in Table 2. Thefirst two eigenvectors explain more than 85% of thevariance of the data for all sites except the upper gage.Some additional insight into the structure of the data setcan be gained using matrix plots. Samples from the lowergage (Figure 7) have a nearly linear structure among allsolutes, except for sulfate, while samples from the uppergage (Figure 8) show a more complex pattern, such as twodifferent populations of samples, one with positive corre-lation between alkalinity and calcium and one with norelation between alkalinity and calcium. If the data arelinear in all projections, the rank of the data is approx-imately 1. However, if there is scatter in projections formore than one solute, the rank is at least 2, but could behigher.[24] Because there are six solutes in this analysis of the

correlation matrix, an eigenvector must explain more than16.7% (one sixth) of the variation of the data set to beretained using the rule of 1. On the basis of that reasoning,only one eigenvector should be retained for the lowergage, site 210, and site 420; two should be retained forsites 201 and site 404; and three should be retained for theupper gage. To assess the utility of that rule, data from thelower gage were projected into a one-, two-, and three-dimensional subspace defined by the first one, two orthree eigenvectors using equation (2). The projected valueswere destandardized and residuals calculated using theapproach described above. The RRMSE is contained inTable 3, and the residual plots are in Figure 9. Sulfateclearly does not fit the one-dimensional (linear) subspace,as evidenced by both the high RRMSE and the pattern of


the residuals. However, even solutes with modestRRMSEs, such as calcium and magnesium, exhibit struc-ture in the residuals. When the dimensionality of themixing subspace is increased to 2, the fit for sulfate,calcium, and magnesium markedly improves. (This is thedimensionality of the past mixing analyses at PMRW.)However, some curvature is observed for sodium andsilica at concentrations less than 50 meq/L and 100mmol/L, respectively. This problem is corrected if a thirdeigenvector is retained.[25] The curvature apparent in two dimensions could

arise because of nonconservative processes. This seemedunlikely given that sodium is not biologically active andthere is no evidence for precipitation of silica at PMRW,unlike in more northerly catchments. Therefore retention ofa third eigenvector is indicated, implying that a minimum offour end-members, rather than three as had been previouslyused by Hooper et al. [1990], are required.

[26] Thus the ‘‘best-fit’’ plane (i.e., two-dimensional sub-space) to these data systematically over-predicts sodium andsilica at these low concentrations, which correspond to high-discharge samples. Because these are the two major weath-ering products at PMRW, these results suggest that there isanother source of the other weathering products ( principallycalcium and magnesium) at high discharge. Examination ofthe eigenvectors for this site indicates that the third eigen-vector has coefficients of the same sign for calcium andsulfate, whereas the first two eigenvectors had oppositesigns for these solutes. Mobile groundwater collected from atrench face on a hillslope 50 m from the stream is a solutionwhose dominant cation/anion pair was calcium and sulfate[Hooper, 2001], very different from the sodium-bicarbonatestream water. Hooper [2001] concluded that there was noexpression of the hillslope chemistry during the storm event.This analysis would suggest, however, that there is evidenceof a limited expression of this water at high discharge.

Table 2. Variance Explained by Eigenvectors

First Eigenvector,%

Second Eigenvector,%

Third Eigenvector,%

Fourth Eigenvector,%

Fifth Eigenvector,%

Sixth Eigenvector,%

Lower gage 83.50 11.67 2.38 1.30 0.85 0.30Site 201 61.42 24.80 5.21 4.11 2.93 1.53Site 210 74.43 12.79 8.55 2.12 1.25 0.84Site 404 54.87 35.79 4.23 3.23 1.15 0.74Site 420 74.46 16.12 5.47 2.76 0.71 0.47Upper gage 40.29 38.04 17.41 2.24 1.19 0.83

Figure 7. Matrix plot of solutes for lower gage.


[27] Viewed in this light, the pattern in the calcium-sulfate bivariate plot (Figure 7) becomes more interpretable.Sulfate increases with increasing discharge at PMRW, sothat high sulfate concentrations correspond to high dis-charge. As sulfate concentrations increase, calcium concen-trations initially decrease, but at the highest sulfateconcentrations (above 140 meq/L), calcium concentrationsincrease again. This is further evidence that the hillslopecalcium/sulfate water does contribute to the stream at thehighest discharge.[28] In this case, the third eigenvector, explaining only

about 2% of the stream water chemical variation, appears tobe physically meaningful based upon ancillary data. Theresults of the multivariate analysis must be interpreted withan understanding of site characteristics, hydrologic mecha-nisms, and biogeochemical processes. In particular, caremust be exercised in interpreting eigenvectors explainingsmall amounts of variance of the data that these patterns didnot just arise by chance. The larger the data set, the lesslikely this is to occur.[29] A rank analysis of the upper gage data, where the

rule of 1 indicates that three eigenvectors should beretained, was performed for a two-, three-, and four-dimen-sional subspace. Structure in the residuals (Figure 10) wasobserved for the two-dimensional case for many solutes,which disappeared at three dimensions. Adding a fourthdimension reduced the magnitude of the residuals forcalcium and magnesium but otherwise did not change theresidual pattern markedly. For this site, a rank of 3 isindicated by this analysis, consistent with the rule of 1.

[30] The number of eigenvectors to retain based upon theresidual plots did not correspond to that obtained from therule of 1 for any of the other sites, however. Site 201required one additional eigenvector (three instead of two),and the other sites required two more eigenvectors thanindicated by the rule of 1. (Residual plots are not shown.)[31] The residual plots therefore provide more informa-

tion for deciding the approximate rank of a data set than theeigenvalues or scalar diagnostics alone. The number of end-members required to explain the data with a mixing modelis one greater than the rank. Structure in the residuals canhighlight particular samples. When combined with otherinformation, such as the hydrologic conditions under whichthe samples were collected, further insight into controllingmechanisms can be gained.

5.2. Projection of Test Sites Into Reference Site

[32] The five sites upstream of the lower gage wereprojected into the three-dimensional mixing space defined

Figure 8. Matrix plot of solutes for upper gage.

Table 3. Relative Root-Mean-Square Error (RRMSE) for Project-

ing Lower Gage Samples Into Mixing Subspaces of Lower

Dimensionality

Rank ofMixingSpace

Alkalinity,%

Calcium,%

Magnesium,%

Sodium,%

Sulfate,%

Silica,%

1 9.79 15.8 9.61 9.52 41.3 10.72 7.36 7.46 7.19 9.50 9.33 10.73 7.19 5.31 6.34 5.87 3.38 6.81


by the first three eigenvectors of the lower-gage stream data.The scalar measures of fit are displayed as bar charts inFigure 11. The top panel depicts the RB; the lower gage,being the reference site, exhibits no bias. The middle paneldisplays the RRMSE for the projections into the mixingspace of the lower gage; the bottom panel is the RRMSE foreach site projected into a three-dimensional subspace

defined by its own eigenvectors. The values of the upstreamsites can be compared with reference site (the lower gage) bycomparing the height of the bars with those in the leftmostgroup of the second panel and with the at-site ‘‘best fit’’ bycomparing with the corresponding group in the lowest panel.The RRMSE is always higher for the upstream projectedsamples than either the lower gage projected into itself or for

Figure 9. Residual plot for lower gage.


the at-site projections, as expected. However, the patterns ofboth the RB and RRMSE indicate that the sites on thesoutheastern tributary (404 and 420) best fit into the lowergage mixing subspace, followed by the eastern tributary sites(202 and 210), with the upper gage fitting worst.

[33] Alkalinity at the upper gage is the most poorlypredicted of the solutes. The absolute bias, as shown in theresiduals plots (Figure 12), is approximately the same as forcalcium and magnesium; the low concentration of alkalinityresults in a higher relative bias. Nonetheless, alkalinity is

Figure 10. Residual plot for upper gage.


under-predicted: Given the concentration of the other ions,alkalinity would be expected to be much lower than it wasobserved to be if the ion ratios observed at the lower gageheld. The atmosphere is the only source of sulfate atPMRW, where there are no sulfur-bearing minerals. Thesulfate is in the form of sulfuric acid. The primary chemicalvariation at PMRW during events is a decrease in alkalinityand an increase in sulfate. The under-prediction of alkalin-ity at the upper gage suggests that there are additional acid-neutralizing reactions in this subcatchment relative to theother parts of the catchment. Because calcium is also under-predicted at the upper gage site, perhaps the calcium-sulfatehillslope water accounts for this discrepancy.[34] However, the pattern of the calcium and magnesium

predictions is reversed for the eastern and southeasterntributary sites: Calcium is over-predicted and magnesium isunder-predicted. One hypothesis is that the hillslope waterobserved at the trench is not representative of hillslope watersthroughout the catchment. Further fieldwork is required totest this hypothesis.[35] The patterns of solute misfit also cannot readily be

explained by changes in the mineral assemblages encoun-

tered by the water. The mineral contacts that have beenmapped suggest that the upper gage, site 210, and site 420 alllie above the contact with the gneiss and should drain an areaof only granodiorite. The amphibole-rich gneiss influencesall sites below this, including the lower gage. However, thepattern of fit is better explained by the tributaries (south-eastern fitting best, southwestern worst) rather than by thelocation of the site relative to the contact.[36] The pattern between the sites may be explained by an

alternative hypothesis. Bullen et al. [1996] found thatweathering reactions in a laboratory column proceeded onlywhen waters were mobile. Stagnant zones interrupted theweathering reactions, perhaps by precipitation of clays ontoreactive surfaces. The upper gage is distinguished from theother sites as being the only ephemeral stream included inthis analysis. The wetting and drying in this subcatchmentcould interrupt the weathering sequences, much as observedin the laboratory column. The solute ratios at this sitetherefore may not be controlled by mineral dissolutionbut, rather, by surface exchange reactions.[37] Sulfate at the upper gage also appears to exhibit

some structure in its residuals (Figure 12), although theirsmall magnitude weakens this observation. More interest-ingly, distinct clusters of sulfate residuals are evident. Thecluster at 190 meq/L sulfate with positive residuals wascollected during a storm on 23 July 1986, whereas thecluster at 250 meq/L with essentially no bias was collectedduring a storm on 31 July 1987. Both of these storms aresmall summer thunderstorms with very low runoff ratios;the 31 July storm was somewhat larger and more intenseand occurred under wetter conditions (Table 4). Both eventsexhibited high sulfate concentrations at the upper gage, butonly the 31 July event had chemistry that was consistentwith the chemistry of the lower gage. The reason for thisdifference is not known, but different streamflow generationmechanisms between these storms seems likely. This multi-variate analysis enables the identification of this differencebetween these two storms that otherwise would not havebeen evident.[38] This discussion indicates the kinds of questions that

the multivariate analysis can raise, but answers to thesequestions require process knowledge and, no doubt, addi-tional data. The advantage of the analysis is that it providesa more sensitive tool than those currently available forquerying data.

5.3. Comparison of Time Periods

[39] Between 1989 and 1991 the calcium and sulfateconcentration of the vadose end-member at PMRW declinedto approximately half of the value observed between 1986and 1988 [Hooper, 2001]. The stream concentration alsochanged in a manner consistent with the changing concen-tration of this end-member.[40] The tools developed here can be used to evaluate

whether the ratios among the solutes have also shifted, or ifthe later data lie in the same mixing space, but just in adifferent portion of that mixing space. The data set for thelower gage is divided into those samples collected betweenWater Year (WY) 1986 and WY1988 and those collectedafter WY1992. (The U.S. Geological Survey defines a wateryear as running from 1 October to 30 September. WY1986began on 1 October 1985 and ended on 30 September1986.) A rank analysis of the WY1986–1988 data indicated

Figure 11. Diagnostic statistics for upstream sites pro-jected into three-dimensional mixing space of lower gage.


that three dimensions were necessary to fit the data, con-sistent with the findings reported above for the entire dataset of the lower gage. The RRMSEs of the WY1986–1988projected into this subspace are reported in Table 5. Datafrom the later period were projected into the three-dimen-sional mixing subspace defined by the WY1986–1988 data.The RB and RRMSE of the WY1992–1999 data are alsocontained in Table 5. Many of the solutes do exhibit a smallbias, with magnesium having the largest bias (in magnitude)at �8.1%. The residuals for alkalinity and magnesium(Figure 13) are the only two solutes that may exhibit somestructure, but it is very weak. The most striking aspect is thedecreased variability in all solutes, as can be seen in Figure13. For example, the magnesium RRMSE declines from7.9% for the earlier period to 4.7% for the later period wheneach set of data are fit to their own mixing space.[41] Thus these results indicate that the lower gage

samples have not markedly shifted in their ion ratios, but

are more nearly contained within three dimensions in thelater period than in the earlier period.

6. Conclusions

[42] The tools developed in this paper can be used toexamine the structure of multivariate water quality data andto evaluate the consistency of the ion ratios among sites, andbetween different time periods. The residual plots devel-oped here can indicate both general patterns in soluteconcentrations and outliers that may have been collectedduring unusual conditions or may be erroneous. The nullhypothesis of conservative mixing of stable end-membersprovides a benchmark against which to compare data, thus

Figure 12. Residual plots for upstream sites projected into three-dimensional mixing space of lowergage.

Table 5. Statistics for Projection of Samples From Two Different

Periods Into Mixing Space Defined by Samples Collected Between

WY1986 and WY1988

SoluteWY1986–1988RRMSE, %

WY1992–1999RB, %

WY1992–1999RRMSE, %

Alkalinity 9.3 2.2 7.2Calcium 4.8 5.0 6.0Magnesium 7.9 �8.1 9.4Sodium 7.4 1.4 5.1Sulfate 3.3 4.7 5.8Silica 8.5 5.1 8.9

Table 4. Characteristics of Two Storms at PMRW

Characteristic23 July

1986 Storm31 July

1987 Storm

Precipitation volume, mm 18 28Maximum 15-min volume, mm 11 21

Runoff ratio 0.5% 1.5%Antecedent 30-day precipitation, mm 38 84


providing greater insight and allowing more incisivehypotheses to be posed. The methods can handle anarbitrary number of solutes and large data sets.[43] Because this analysis does not require soil water or

groundwater solutions, as traditional end-member mixinganalysis does, it may be applied to studies where onlystream water samples are available. Particularly wherestable end-members are suspected (but not measured), thisanalysis can assess whether the same end-members arecontrolling different sampling sites.

[44] Acknowledgments. This work was completed while the authorwas a visiting scholar at the Department of Forest Engineering, OregonState University. Travel support was provided by Hydrological Science

Program of the National Science Foundation and salary support by theOffice of Water Quality, U.S. Geological Survey. Research at PMRW issupported by the Water, Energy and Biogeochemical Budgets program ofthe U.S. Geological Survey.

ReferencesBullen, T. D., D. H. Krabbenhoft, and C. Kendall, Kinetic and mineralogiccontrols on the evolution of groundwater chemistry and 87Sr/86Sr in asandy silicate aquifer, northern Wisconsin, USA, Geochim. Cosmochim.Acta, 60(10), 1807–1821, 1996.

Christophersen, N., and R. P. Hooper, Multivariate analysis of streamwaterchemical data: The use of principal components analysis for the end-member mixing problem, Water Resour. Res., 28(1), 99–107, 1992.

Genereux, D. P., and R. P. Hooper, Oxygen and hydrogen isotopes inrainfall-runoff studies, in Isotope Tracers in Catchment Hydrology, edi-ted by C. Kendall and J. J. McDonnell, pp. 319–346, Elsevier Sci., NewYork, 1998.

Hooper, R. P., Applying the scientific method to small catchment studies: Areview of the Panola Mountain experience, Hydrol. Proc., 15, 2039–2050, 2001.

Hooper, R. P., N. Christophersen, and N. E. Peters, Modelling streamwaterchemistry as a mixture of soil-water end members—An application to thePanola Mountain watershed, GA, USA, J. Hydrol., 116, 321–343, 1990.

Huntington, T. G., R. P. Hooper, N. E. Peters, T. D. Bullen, and C. Kendall,Water, energy, and biogeochemical budgets investigation at PanolaMountain Research Watershed, Stockbridge, GA—A research plan,U.S. Geol. Surv. Open File Rep., 93-55, 39 pp., 1993.

Miesch, A. T., Q-mode factor analysis of compositional data, Comput.Geosci., 1, 147–159, 1976.

Neal, C., R. A. Skeffington, R. Williams, and D. J. Roberts, Aluminumsolubility controls in acid waters; the need for a radical reappraisal, EarthPlanet. Sci. Lett., 86(1), 105–112, 1987.

Nordstrom, D. K., and J. W. Ball, The geochemical behavior of aluminumin acidified waters, Science, 232(4746), 54–56, 1986.

Piper, A. M., A graphical procedure in the geochemical interpretation ofwater analyses, Eos Trans. AGU, 25, 914–923, 1944.

Preisendorfer, R. W., F. W. Zweirs, and T. P. Barnett, Principal componentselection rules, SIO Ref. Ser. 81-4, 200 pp., Scripps Inst. of Oceanogr., LaJolla, Calif., 1981.

��R. P. Hooper, U.S. Geological Survey, 10 Bearfoot Road, Northborough,

MA 01532, USA. ([email protected])

Figure 13. Residual plots of alkalinity and magnesium forsamples from WY1986–1988 and WY1992–1999 pro-jected into the three-dimensional mixing subspace definedby samples collected from WY1986–1988.


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