Mixing models provide a useful null hypothesis against which to evaluate processes controlling stream water chemical data. Because conservative mixing of end-members with constant concentration is a linear process, a number of simple mathematical and multivariate statistical methods can be applied to this problem. Although mixing models have 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 need for explicitly identified end-members. Diagnostic tools are developed to determine the approximate rank of the data set and to assess lack of fit of the data. This permits identification of processes that violate the assumptions of the mixing model and can suggest the dominant processes controlling stream water chemical variation. These same diagnostic tools can be used to assess the fit of the chemistry of one site into the mixing subspace of a different site, thereby permitting an assessment of the consistency of controlling end-members across sites. This technique is applied to a number of sites at the Panola Mountain Research Watershed located near Atlanta, Georgia.
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 Mixing models of surface water chemistry have been extensively employed for hydrograph separation [Genereux and Hooper, 1998]. In geographic-source separations, measured soil water and groundwater solutions are evaluated 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 examining data sets and generating hypotheses because they link small-scale, internal catchment measurements (groundwater and soil water solution chemistry, in this case) with the phenomenon of interest (catchment runoff chemistry) that occurs at a much larger scale [Hooper, 2001].
Christophersen and Hooper  stated that only the number of end-members can be determined from the stream mixture, but not their chemistry. They determined the number of end-members based upon a subjective consideration of the variance explained by each principal component of the solute correlation matrix in a manner consistent with standard approaches in principal component analysis (PCA). In this paper, a different approach is developed that is based more upon the geometry underlying the correlation matrix rather than on a simple consideration of variance explained. Beyond providing further guidance for determining the appropriate number of end-members, this analysis also helps to identify violation of the assumptions of standard mixing models, including conservative mixing and constant end-member composition. Thus it can also help to determine which solutes can be used as tracers and which should not be used.
 The mathematics developed here uses only the stream chemistry and does not require an explicit identification of end-members. Therefore this analysis also can be used at sites where only stream chemistry data are available, thus expanding the utility of this technique beyond intensively studied research catchments to water quality monitoring and assessment programs. A direct extension of the theory developed here permits a quantitative comparison of chemistry among multiple sites that can be used, for example, to identify sites where the stream chemistry is controlled by the same processes or to examine changes in water quality across spatial scales.
 The foundation for this analysis is the mathematical distinction between mixing and equilibrium chemistry. Conservative mixing of end-members with fixed composition is a linear process, but equilibrium reactions among solutes of different charge are higher-order polynomials. Nordstrom and Ball  noted that the relation between aluminum activity and hydrogen ion, although cubic at low hydrogen ion activities (high pH) as expected from equilibrium considerations, became linear at high hydrogen ion activities (low pH). They attributed this linear behavior to mixing processes. (Note that the cubic relation, although taken as evidence of equilibrium control, is a mathematical necessity because aluminum activity is calculated as a function of hydrogen ion. Observing a cubic relation at high pH is not a strong test of equilibrium control [Neal et al., 1987].) In this paper, a series of diagnostic tools using linear mathematics are developed to test the hypothesis of mixing control on stream water chemical variation. The geometry of the “data cloud” formed by plotting solutes against one another is explored to determine whether the relation among the solutes is essentially linear in this highly dimensional space or exhibits curvature.
2. Data Set
 The following site description is a summary of that reported by Huntington et al. . The Panola Mountain Research Watershed (PMRW) is one of five Water, Energy, and Biogeochemical Budget (WEBB) watersheds operated by the U.S. Geological Survey (USGS). PMRW, a 41-ha catchment within the Panola Mountain State Conservation Park, is located in the Piedmont physiographic province approximately 25 km southeast of Atlanta, Georgia (Figure 1). The watershed is underlain primarily by granodiorite 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 the catchment, which accounts for much of the 51 m relief at PMRW. Soils are classified as entisols near the base of the outcrop and as ultisols on hillslopes and ridge tops. Kaolinite, hydroxy-interlayered vermiculite, mica, and gibbsite are the dominant secondary minerals. Soils have developed to a thickness of 1–2 m, and saprolite underlies the soils ranging from a depth of 0 to 5 m below land surface. The catchment, with the exception of the outcrop, is completely forested with a cover of approximately 70% mixed southern hardwoods (oak (Quercus spp.), hickory (Carya spp.) and tulip poplar (Liriodendron tulipfera)) and 30% loblolly pine (Pinus taeda). The stream draining the catchment is perennial. The climate in the area is classified as warm temperate 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% of precipitation.
 Features of the chemical data considered in this paper are summarized in Table 1. Both the lower gage and the upper gage have permanent control structures in place. At the numbered sites, either temporary structures were installed for some period, or no gaging was done at the site. Stream water samples were collected manually every week at all sites. The duration of the sampling is indicated in Table 1. For some period of the record, many of the sites were equipped with stage-activated automatic samplers to better characterize chemical variation with discharge. Samples were analyzed for major anions, cations, alkalinity, and pH using standard methods by a project laboratory. Huntington et al.  provide further details on analytical methods, precision, and the quality assurance program at PMRW. The coefficient of variation for concentrations of standard reference materials is generally less than 5%; sodium has the highest coefficient of variation at 7.7%. The standard deviation of repeated determinations for most solutes is less than 1 μeq/L, except for magnesium, for which it is 2 μeq/L.
Table 1. Description of Data Set
weekly + event samples
weekly + event samples
weekly + event
weekly + event
3. Previous Mixing Models at PMRW
Hooper et al.  identified measured soil water and ground solutions that could be used as end-members in a three-component mixing model for the lower gage at PMRW. These end-members explained between 83 and 97% of the variation in the concentration of six solutes or solute combinations: alkalinity, calcium, magnesium, sodium, sulfate, and silica. These solutes were assumed to mix conservatively. Hooper et al.  used all six solutes simultaneously to determine the proportion of the three end-members when only two tracers are required. A least squares error criterion was employed to solve the over-determined system of equations.
Christophersen and Hooper  demonstrated that this solution to the end-member mixing problem was equivalent to making an orthogonal projection of the stream samples into the subspace defined by the end-members. The logic of this mathematical solution is reversed because the stream is the observed mixture and the soil water and groundwater solutions are being tested as potential end-members. Instead, Christophersen and Hooper  recommended defining the mixing subspace of the stream samples using an eigenvalue analysis and projecting the potential end-members into that subspace. The dimensionality of the subspace is determined by the number of eigenvectors retained to define the mixing subspace, and this dimensionality is related to the number of end-members required in a mixing model. The number of eigenvectors to retain is a subjective decision, although a number of heuristic rules have been proposed that are based upon the relative magnitude of the eigenvalues associated with each eigenvector [Preisendorfer et al., 1981]. Christophersen and Hooper  noted that the maximum amount of variation in the data that can be explained is limited to that explained by the eigenvectors retained, but provided no further guidance in determining the number of components to retain. In this paper, a more formal analysis is proposed to determine the dimensionality of the stream data, mathematically known as the “rank” of the data set.
4.1. Conceptual Basis
 To illustrate the central concepts of this paper, consider a site where two solutes have been measured on 50 samples that are, to some extent, collinear (Figure 2a). This site is called the “reference site” and, in practical applications, is a well-characterized site against which other sites' chemistry will be compared. Using an eigenvalue analysis that will be described below, the “mixing subspace” for these data is one-dimensional (a line), and this line accounts for 91% of the variation in the data. The orthogonal projections of the data onto the line (Figure 2b) are the “best” one-dimensional representation of the data set (in the sense of being geometrically closest) and can be used instead of the original values to reduce the dimensionality of the data set. Residuals between the original values and the projected values can be calculated, and examined for structure. Just as in the case of a regression model, a “good” mixing subspace is indicated by a random pattern of residuals plotted against the concentration of the original sample (Figure 2c). Structure in this plot indicates a lack of fit in the mixing subspace and hence a deficiency in the mixing model that might arise for a number of reasons, including nonconservative behavior of tracers. Scalar measures of fit, such as bias and root-mean-square error (RMSE) can also be calculated from the residuals. Outliers can be identified and may indicate either errors in the data or different processes controlling the chemistry for those samples.
 Data from other sites, referred to as sites “A” and “B,” can be projected into the “best fit” line from the reference site, and the residuals between the projected and original values examined. Site A appears to fit the line well (Figure 3). The residuals (Figure 4) are small and have no systematic pattern, and the bias and RMSE are small. By contrast, site B is not consistent with the reference site, as evidenced by the structure and size of the residuals (Figure 5).
 Physically, the mixing of the same two end-members may control site A and the reference site, with lower proportions of the more concentrated end-member exhibited at the reference site. When the sites are at different basin scales, such an analysis suggests the areal influence of controlling end-members. Alternately, the more concentrated end-member at site A may be water that has been in contact with the same minerals as at the reference site for a longer time and hence has a higher concentration. A site will “fit” so long as the solutes are added to solutions with the same initial chemistry in the same ratio, as would be expected for congruent or incongruent dissolution of minerals. For example, if water acquires the chemical signature of the organic soil horizon and then gradually evolves with the same mineral assemblage in the lower soil horizons, all of these waters would “fit” in the same mixing subspace. If the water either starts with a different chemistry, contacts a different mineral assemblage, exchanges solutes with the solid phase in a manner that violates the weathering stoichiometry, or has a different set of controlling end-members, these solutions would not fit the same mixing subspace. Site B exhibits lack of fit. Note that solutions with a different concentration range from the reference site may “fit” the mixing subspace (site A) and a site with the same concentration range may exhibit lack of fit (site B). The key point is that the ratio among all solutes must be preserved to fit the subspace; their absolute concentration is not relevant.
 This approach is a refinement of a simple classification of solutions by ion ratios that has long been done in geochemistry using trilinear “piper” diagrams [Piper, 1944]. The triangular portions at the bottom of the figure display the relative proportion of three cations (left side) and anions (right side), or combinations of ions. The diamond displays the relative proportion of two groups of anions along one axis and cations along the other. The four sites in Figure 6 are compared by examining the areas of each of these subplots the sites occupy. The contribution of this paper can be viewed as a generalization of these concepts to an arbitrary number of dimensions and providing quantitative measures of similarity.
4.2. Mathematical Development
 Let X represent an (n × p) matrix of stream chemical data consisting of n samples on which p solutes were measured. Most typically, these data will be standardized by subtracting the mean concentration of each solute and dividing by the standard deviation of each solute so that each solute has equal weight in the analysis. See Christophersen and Hooper  for further discussion on data standardization in mixing models. Let X* be the matrix of standardized values. Letting j be the mean of the jth solute and sj be its standard deviation, each element of this matrix is
X*TX* is related to the correlation matrix of the p solutes by 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 standardized data) is simply the first m eigenvectors of this solute correlation matrix. V, the matrix formed by the first m eigenvectors, is the basis of a new m-dimensional subspace. In standard PCA, the “scores” of the observations are expressed in the coordinates of this subspace. However, one can also express these points in terms of the original solutes. Geometrically, this is equivalent to the orthogonal projection of X* given by
* can be destandardized by multiplying by the standard deviation of each solute and adding the mean to yield ; that is, each element of this matrix is ◯ij = ◯*ijsj + j. Residuals between the projected and original data are simply
Diagnostic plots (such as Figures 4 or 5) can be made by plotting the residuals (ej) against the observed concentration (xj) for each solute j. A well-posed model is indicated by a random pattern in the residuals; any structure in this plot suggests a lack of fit in the model, which can arise from the violation of any of the assumptions inherent in the mixing model. However, the magnitude of the residuals, both absolute and relative to the observed concentration, should be considered. The residuals have units of concentration and should be evaluated in light of analytical precision to determine if any apparent patterns are meaningful or within the noise of the laboratory analysis.
 Two useful scalar measures of fit are the bias and root-mean-square error. Because the solutes often differ markedly in concentration, these measures are scaled by the mean concentration of each solute to make them unitless. The relative bias (RB) for each solute j is defined as
Similarly, the relative root-mean-square error (RRMSE) for solute j is
 For a site projected into a subspace defined by its own eigenvectors, as for the reference site, the bias will always be zero. The relative root-mean-square error provides an indication of the “thickness” of the data cloud outside the lower dimensional subspace. The dimension of the 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 element of subjectivity in this decision. How much of the variance in the data set should be explained and how much is “noise”? An eigenvalue analysis is the first step in the widely used multivariate statistical technique known as principal components analysis (PCA). The same question is faced in PCA: How many components should be retained? A number of rules have been proposed [Preisendorfer et al., 1981], all dependent on the eigenvalues. A common rule is to retain all eigenvectors associated with eigenvalues greater than or equal to 1 called the “rule of 1.” Because the correlation matrix is being factored, each original variable has a variance equal to 1. Therefore this rule is based on the principle that each new component should explain at least as much variance as the original variables. This argument is consistent with the use of PCA as a technique to reduce the dimensionality of the data set.
 In this analysis, a different test is suggested. The dimension of the mixing subspace should be the smallest possible such that the residuals exhibit no structure when plotted against observed concentration; that is, the residuals appear to be random noise. There are two qualifications to this rule. First, the higher the dimension of the mixing space, the fewer “degrees of freedom” exist for testing whether or not mixing processes hold; a mixing space with the same number of dimensions as solutes entered in the analysis will, of course, perfectly fit the data. Second, this rule assumes that all solutes mix conservatively. Curvature in the residuals could indicate chemical reactions or variable end-member concentrations under some circumstances. Discriminating between these processes and the need for an additional dimension in the mixing space requires consideration of the chemical and biological properties of the solute to determine the likelihood of these processes.
 The rank of the data set is directly related to the number of end-members required in a mixing model. For data that have been centered on their mean (as a correlation matrix), 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 a mixing model.
 To project another site, Y, into the subspace defined by the reference site, the data must first be standardized by the means and standard deviation of the reference site, and not by its own mean and standard deviation, because the reference site defines the basis for the mixing subspace; that is,
The data can be projected using equation (2) (substituting Y* for X*), and destandardized. Residuals, relative bias, and relative root-mean-square error can be calculated as above with the appropriate substitutions.
 Two possible benchmarks exist for the RB and RRMSE of the test site. The best possible fit of these data to an m-dimensional subspace would be defined by its own m eigenvectors. The RRMSE for the samples projected into the test site can be compared with this benchmark to determine how much poorer the fit is. (The RB for a site projected into a subspace defined by its own eigenvectors is always zero.) A second benchmark is to compare the RRMSE of the projected data to the RRMSE of the reference site. This statistic indicates how “noisy” the reference site itself is. Thus both of these benchmarks provide a lower bound for the RRMSE. The RB indicates whether the mixing subspace slices through the center of the data cloud or lies to one side of it. If the RB is of the same order of the RRMSE, the data all lie to one side of the mixing subspace, and hence a poor fit is indicated.
 This analysis can be easily implemented using a combination of statistical packages to extract the eigenvectors and spreadsheets to perform the matrix manipulations. Custom programs are not necessary, although they can speed the analysis. Advanced programming languages can perform these analyses and make the suggested diagnostic plots with a few dozen commands.
 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
 The percentage of variance explained by each eigenvector for the six sites at PMRW is shown in Table 2. The first two eigenvectors explain more than 85% of the variance of the data for all sites except the upper gage. Some additional insight into the structure of the data set can be gained using matrix plots. Samples from the lower gage (Figure 7) have a nearly linear structure among all solutes, except for sulfate, while samples from the upper gage (Figure 8) show a more complex pattern, such as two different populations of samples, one with positive correlation between alkalinity and calcium and one with no relation between alkalinity and calcium. If the data are linear in all projections, the rank of the data is approximately 1. However, if there is scatter in projections for more than one solute, the rank is at least 2, but could be higher.
Table 2. Variance Explained by Eigenvectors
First Eigenvector, %
Second Eigenvector, %
Third Eigenvector, %
Fourth Eigenvector, %
Fifth Eigenvector, %
Sixth Eigenvector, %
 Because there are six solutes in this analysis of the correlation matrix, an eigenvector must explain more than 16.7% (one sixth) of the variation of the data set to be retained using the rule of 1. On the basis of that reasoning, only one eigenvector should be retained for the lower gage, site 210, and site 420; two should be retained for sites 201 and site 404; and three should be retained for the upper gage. To assess the utility of that rule, data from the lower gage were projected into a one-, two-, and three-dimensional subspace defined by the first one, two or three eigenvectors using equation (2). The projected values were destandardized and residuals calculated using the approach described above. The RRMSE is contained in Table 3, and the residual plots are in Figure 9. Sulfate clearly 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 modest RRMSEs, such as calcium and magnesium, exhibit structure in the residuals. When the dimensionality of the mixing subspace is increased to 2, the fit for sulfate, calcium, and magnesium markedly improves. (This is the dimensionality of the past mixing analyses at PMRW.) However, some curvature is observed for sodium and silica at concentrations less than 50 μeq/L and 100 μmol/L, respectively. This problem is corrected if a third eigenvector is retained.
Table 3. Relative Root-Mean-Square Error (RRMSE) for Projecting Lower Gage Samples Into Mixing Subspaces of Lower Dimensionality
Rank of Mixing Space
 The curvature apparent in two dimensions could arise because of nonconservative processes. This seemed unlikely given that sodium is not biologically active and there is no evidence for precipitation of silica at PMRW, unlike in more northerly catchments. Therefore retention of a third eigenvector is indicated, implying that a minimum of four end-members, rather than three as had been previously used by Hooper et al. , are required.
 Thus the “best-fit” plane (i.e., two-dimensional subspace) to these data systematically over-predicts sodium and silica at these low concentrations, which correspond to high-discharge samples. Because these are the two major weathering products at PMRW, these results suggest that there is another source of the other weathering products (principally calcium and magnesium) at high discharge. Examination of the eigenvectors for this site indicates that the third eigenvector has coefficients of the same sign for calcium and sulfate, whereas the first two eigenvectors had opposite signs for these solutes. Mobile groundwater collected from a trench face on a hillslope 50 m from the stream is a solution whose dominant cation/anion pair was calcium and sulfate [Hooper, 2001], very different from the sodium-bicarbonate stream water. Hooper  concluded that there was no expression of the hillslope chemistry during the storm event. This analysis would suggest, however, that there is evidence of a limited expression of this water at high discharge.
 Viewed in this light, the pattern in the calcium-sulfate bivariate plot (Figure 7) becomes more interpretable. Sulfate increases with increasing discharge at PMRW, so that high sulfate concentrations correspond to high discharge. As sulfate concentrations increase, calcium concentrations initially decrease, but at the highest sulfate concentrations (above 140 μeq/L), calcium concentrations increase again. This is further evidence that the hillslope calcium/sulfate water does contribute to the stream at the highest discharge.
 In this case, the third eigenvector, explaining only about 2% of the stream water chemical variation, appears to be physically meaningful based upon ancillary data. The results of the multivariate analysis must be interpreted with an understanding of site characteristics, hydrologic mechanisms, and biogeochemical processes. In particular, care must be exercised in interpreting eigenvectors explaining small amounts of variance of the data that these patterns did not just arise by chance. The larger the data set, the less likely this is to occur.
 A rank analysis of the upper gage data, where the rule of 1 indicates that three eigenvectors should be retained, was performed for a two-, three-, and four-dimensional subspace. Structure in the residuals (Figure 10) was observed for the two-dimensional case for many solutes, which disappeared at three dimensions. Adding a fourth dimension reduced the magnitude of the residuals for calcium and magnesium but otherwise did not change the residual pattern markedly. For this site, a rank of 3 is indicated by this analysis, consistent with the rule of 1.
 The number of eigenvectors to retain based upon the residual plots did not correspond to that obtained from the rule of 1 for any of the other sites, however. Site 201 required one additional eigenvector (three instead of two), and the other sites required two more eigenvectors than indicated by the rule of 1. (Residual plots are not shown.)
 The residual plots therefore provide more information for deciding the approximate rank of a data set than the eigenvalues or scalar diagnostics alone. The number of end-members required to explain the data with a mixing model is one greater than the rank. Structure in the residuals can highlight particular samples. When combined with other information, such as the hydrologic conditions under which the samples were collected, further insight into controlling mechanisms can be gained.
5.2. Projection of Test Sites Into Reference Site
 The five sites upstream of the lower gage were projected into the three-dimensional mixing space defined by the first three eigenvectors of the lower-gage stream data. The scalar measures of fit are displayed as bar charts in Figure 11. The top panel depicts the RB; the lower gage, being the reference site, exhibits no bias. The middle panel displays the RRMSE for the projections into the mixing space of the lower gage; the bottom panel is the RRMSE for each site projected into a three-dimensional subspace defined by its own eigenvectors. The values of the upstream sites can be compared with reference site (the lower gage) by comparing the height of the bars with those in the leftmost group of the second panel and with the at-site “best fit” by comparing with the corresponding group in the lowest panel. The RRMSE is always higher for the upstream projected samples than either the lower gage projected into itself or for the at-site projections, as expected. However, the patterns of both the RB and RRMSE indicate that the sites on the southeastern tributary (404 and 420) best fit into the lower gage mixing subspace, followed by the eastern tributary sites (202 and 210), with the upper gage fitting worst.
 Alkalinity at the upper gage is the most poorly predicted of the solutes. The absolute bias, as shown in the residuals plots (Figure 12), is approximately the same as for calcium and magnesium; the low concentration of alkalinity results in a higher relative bias. Nonetheless, alkalinity is under-predicted: Given the concentration of the other ions, alkalinity would be expected to be much lower than it was observed to be if the ion ratios observed at the lower gage held. The atmosphere is the only source of sulfate at PMRW, where there are no sulfur-bearing minerals. The sulfate is in the form of sulfuric acid. The primary chemical variation at PMRW during events is a decrease in alkalinity and an increase in sulfate. The under-prediction of alkalinity at the upper gage suggests that there are additional acid-neutralizing reactions in this subcatchment relative to the other parts of the catchment. Because calcium is also under-predicted at the upper gage site, perhaps the calcium-sulfate hillslope water accounts for this discrepancy.
 However, the pattern of the calcium and magnesium predictions is reversed for the eastern and southeastern tributary sites: Calcium is over-predicted and magnesium is under-predicted. One hypothesis is that the hillslope water observed at the trench is not representative of hillslope waters throughout the catchment. Further fieldwork is required to test this hypothesis.
 The patterns of solute misfit also cannot readily be explained by changes in the mineral assemblages encountered by the water. The mineral contacts that have been mapped suggest that the upper gage, site 210, and site 420 all lie above the contact with the gneiss and should drain an area of only granodiorite. The amphibole-rich gneiss influences all sites below this, including the lower gage. However, the pattern of fit is better explained by the tributaries (southeastern fitting best, southwestern worst) rather than by the location of the site relative to the contact.
 The pattern between the sites may be explained by an alternative hypothesis. Bullen et al.  found that weathering reactions in a laboratory column proceeded only when waters were mobile. Stagnant zones interrupted the weathering reactions, perhaps by precipitation of clays onto reactive surfaces. The upper gage is distinguished from the other sites as being the only ephemeral stream included in this analysis. The wetting and drying in this subcatchment could interrupt the weathering sequences, much as observed in the laboratory column. The solute ratios at this site therefore may not be controlled by mineral dissolution but, rather, by surface exchange reactions.
 Sulfate at the upper gage also appears to exhibit some structure in its residuals (Figure 12), although their small magnitude weakens this observation. More interestingly, distinct clusters of sulfate residuals are evident. The cluster at 190 μeq/L sulfate with positive residuals was collected during a storm on 23 July 1986, whereas the cluster at 250 μeq/L with essentially no bias was collected during a storm on 31 July 1987. Both of these storms are small summer thunderstorms with very low runoff ratios; the 31 July storm was somewhat larger and more intense and occurred under wetter conditions (Table 4). Both events exhibited high sulfate concentrations at the upper gage, but only the 31 July event had chemistry that was consistent with the chemistry of the lower gage. The reason for this difference is not known, but different streamflow generation mechanisms between these storms seems likely. This multivariate analysis enables the identification of this difference between these two storms that otherwise would not have been evident.
Table 4. Characteristics of Two Storms at PMRW
23 July 1986 Storm
31 July 1987 Storm
Precipitation volume, mm
Maximum 15-min volume, mm
Antecedent 30-day precipitation, mm
 This discussion indicates the kinds of questions that the multivariate analysis can raise, but answers to these questions require process knowledge and, no doubt, additional data. The advantage of the analysis is that it provides a more sensitive tool than those currently available for querying data.
5.3. Comparison of Time Periods
 Between 1989 and 1991 the calcium and sulfate concentration of the vadose end-member at PMRW declined to approximately half of the value observed between 1986 and 1988 [Hooper, 2001]. The stream concentration also changed in a manner consistent with the changing concentration of this end-member.
 The tools developed here can be used to evaluate whether the ratios among the solutes have also shifted, or if the later data lie in the same mixing space, but just in a different portion of that mixing space. The data set for the lower gage is divided into those samples collected between Water Year (WY) 1986 and WY1988 and those collected after WY1992. (The U.S. Geological Survey defines a water year as running from 1 October to 30 September. WY1986 began on 1 October 1985 and ended on 30 September 1986.) A rank analysis of the WY1986–1988 data indicated that three dimensions were necessary to fit the data, consistent with the findings reported above for the entire data set of the lower gage. The RRMSEs of the WY1986–1988 projected into this subspace are reported in Table 5. Data from the later period were projected into the three-dimensional mixing subspace defined by the WY1986–1988 data. The RB and RRMSE of the WY1992–1999 data are also contained in Table 5. Many of the solutes do exhibit a small bias, 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 some structure, but it is very weak. The most striking aspect is the decreased variability in all solutes, as can be seen in Figure 13. For example, the magnesium RRMSE declines from 7.9% for the earlier period to 4.7% for the later period when each set of data are fit to their own mixing space.
Table 5. Statistics for Projection of Samples From Two Different Periods Into Mixing Space Defined by Samples Collected Between WY1986 and WY1988
WY1986–1988 RRMSE, %
WY1992–1999 RB, %
WY1992–1999 RRMSE, %
 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 the later period than in the earlier period.
 The tools developed in this paper can be used to examine the structure of multivariate water quality data and to evaluate the consistency of the ion ratios among sites, and between different time periods. The residual plots developed here can indicate both general patterns in solute concentrations and outliers that may have been collected during unusual conditions or may be erroneous. The null hypothesis of conservative mixing of stable end-members provides a benchmark against which to compare data, thus providing greater insight and allowing more incisive hypotheses to be posed. The methods can handle an arbitrary number of solutes and large data sets.
 Because this analysis does not require soil water or groundwater solutions, as traditional end-member mixing analysis does, it may be applied to studies where only stream water samples are available. Particularly where stable end-members are suspected (but not measured), this analysis can assess whether the same end-members are controlling different sampling sites.
 This work was completed while the author was a visiting scholar at the Department of Forest Engineering, Oregon State University. Travel support was provided by Hydrological Science Program of the National Science Foundation and salary support by the Office of Water Quality, U.S. Geological Survey. Research at PMRW is supported by the Water, Energy and Biogeochemical Budgets program of the U.S. Geological Survey.