MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Mark Williams and Fengjing Liu Department of Geography and Institute of Arctic and Alpine Research, University of Colorado, Boulder, CO80309

OUTLINES OF LECTURE  OVERVIEW OF MIXING MODEL  OVERVIEW OF END-MEMBER MIXING ANALYSIS (EMMA) -- PRINCIPAL COMPONENT ANALYSIS (PCA) -- STEPS TO PERFORM EMMA  APPLICATIONS OF MIXING MODEL AND EMMA -- GREEN LAKES VALLEY -- LEADVILLE MINE SITUATION

PART 1: OVERVIEW OF MIXING MODEL  Definition of Hydrologic Flowpaths  2-Component Mixing Model  3-Component Mixing Model  Generalization of Mixing Model  Geometrical Definition of Mixing Model  Assumptions of Mixing Model

HYDROLOGIC FLOWPATHS

MIXING MODEL: 2 COMPONENTS One Conservative Tracer Mass Balance Equations for Water and Tracer

MIXING MODEL: 3 COMPONENTS (Using Specific Discharge) Two Conservative Tracers Mass Balance Equations for Water and Tracers Simultaneous Equations Solutions Q - Discharge C - Tracer Concentration Subscripts - # Components Superscripts - # Tracers

MIXING MODEL: 3 COMPONENTS (Using Discharge Fractions) Two Conservative Tracers Mass Balance Equations for Water and Tracers Simultaneous Equations Solutions f - Discharge Fraction C - Tracer Concentration Subscripts - # Components Superscripts - # Tracers

MIXING MODEL: Generalization Using Matrices One tracer for 2 components and two tracers for 3 components N tracers for N+1 components? -- Yes However, solutions would be too difficult for more than 3 components So, matrix operation is necessary Simultaneous Equations Where Solutions Note: C x -1 is the inverse matrix of C x This procedure can be generalized to N tracers for N+1 components

MIXING MODEL: Geometrical Perspective For a 2-tracer 3-component model, for instance, the mixing subspaces are defined by two tracers. If plotted, the 3 components should be vertices of a triangle and all streamflow samples should be bound by the triangle. If not well bound, either tracers are not conservative or components are not well characterized. f x can be sought geometrically, but more difficult than algebraically.

ASSUMPTIONS FOR MIXING MODEL  Tracers are conservative (no chemical reactions);  All components have significantly different concentrations for at least one tracer;  Tracer concentrations in all components are temporally constant or their variations are known;  Tracer concentrations in all components are spatially constant or treated as different components;  Unmeasured components have same tracer concentrations or don’t contribute significantly.

A QUESTION TO THINK ABOUT  What if we have the number of conservative tracers much more than the number of components we seek for, say, 6 tracers for 3 components?  For this case, it is called over-determined situation  The solution to this case is EMMA, which follows the same principle as mixing models.

PART 2: EMMA AND PCA  EMMA Notation  Over-Determined Situation  Orthogonal Projection  Notation of Mixing Spaces  Steps to Perform EMMA

DEFINITION OF END-MEMBER  For EMMA, we use end-members instead of components to describe water contributing to stream from various compartments and geographic areas  End-members are components that have more extreme solute concentrations than streamflow [Christophersen and Hooper, 1992]

EMMA NOTATION (1)  Hydrograph separations using multiple tracers simultaneously;  Use more tracers than necessary to test consistency of tracers;  Typically use solutes as tracers Modified from Hooper, 2001

EMMA NOTATION (2)  Measure p solutes; define mixing space (S- Space) to be p-dimensional  Assume that there are k linearly independent end-members (k < p)  B, matrix of end-members, (k  p); each row b j (1  p)  X, matrix of streamflow samples, (n observations  p solutes); each row x i (1  p)

PROBLEM STATEMENT  Find a vector f i of mixing proportions such that  Note that this equation is the same as generalized one for mixing model; the re- symbolizing is for simplification and consistency with EMMA references  Also note that this equation is over- determined because k < p, e.g., 6 solutes for 3 end-members

SOLUTION FOR OVER- DETERMINED EQUATIONS  Must choose objective function: minimize sum of squared error  Solution is normal equation [Christophersen et al., 1990; Hooper et al., 1990]:  Constraint: all proportions must sum to 1  Solutions may be > 1 or < 0; this issue will be elaborated later

ORTHOGONAL PROJECTIONS  Following the normal equation, the predicted streamflow chemistry is [Christophersen and Hooper, 1992]:  Geometrically, this is the orthogonal projection of x i into the subspace defined by B, the end- members

This slide is from Hooper, 2001

OUR GOALS ACHIEVED SO FAR? We measure chemistry of streamflow and end-members. Then, we can derive fractions of end-members contributing to streamflow using equations above. So, our goals achieved? Not quite, because we also want to test end-members as well as mixing model. We need to define the geometry of the solute “cloud” (S-space) and project end-members into S-space! How? Use PCA to determine number and orientation of axes in S-space. Modified from Hooper, 2001

EMMA PROCEDURES Identification of Conservative Tracers - Bivariate solute-solute plots to screen data; PCA Performance - Derive eigenvalues and eigenvectors; Orthogonal Projection - Use eigenvectors to project chemistry of streamflow and end-members; Screen End-Members - Calculate Euclidean distance of end- members between their original values and S-space projections; Hydrograph Separation - Use orthogonal projections and generalized equations for mixing model to get solutions! Validation of Mixing Model - Predict streamflow chemistry using results of hydrograph separation and original end-member concentrations.

STEP 1 - MIXING DIAGRAMS Look familiar? This is the same diagram used for geometrical definition of mixing model (components changed to end-members); Generate all plots for all pair-wise combinations of tracers; The simple rule to identify conservative tracers is to see if streamflow samples can be bound by a polygon formed by potential end-members or scatter around a line defined by two end-members; Be aware of outliers and curvature which may indicate chemical reactions!

STEP 2 - PCA PERFORMANCE For most cases, if not all, we should use correlation matrix rather than covariance matrix of conservative solutes in streamflow to derive eigenvalues and eigenvectors; Why? This treats each variable equally important and unitless; How? Standardize the original data set using a routine software or minus mean and then divided by standard deviation; To make sure if you are doing right, the mean should be zero and variance should be 1 after standardized!

APPLICATION OF EIGENVALUES Eigenvalues can be used to infer the number of end-members that should be used in EMMA. How? Sum up all eigenvalues; Calculate percentage of each eigenvalue in the total eigenvalue; The percentage should decrease from PCA component 1 to p (remember p is the number of solutes used in PCA); How many eigenvalues can be added up to 90% (somewhat subjective! No objective criteria for this!)? Let this number be m, which means the number of PCA components should be retained (sometimes called # of mixing spaces); (m +1) is equal to # of end-members we use in EMMA.

STEP 3 - ORTHOGONAL PROJECTION X - Standardized data set of streamflow, (n  p); V - Eigenvectors from PCA, (m  p); Remember only the first m eigenvectors to be used here! Use the same equation above; Now X represents a vector (1  p) for each end-member; Remember X here should be standardized by subtracting streamflow mean and dividing by streamflow standard deviation! Project End-Members

STEP 4 - SCREEN END-MEMEBRS Plot a scatter plot for streamflow samples and end-members using the first and second PCA projections; Eligible end-members should be vertices of a polygon (a line if m = 1, a triangle if m = 2, and a quadrilateral if m = 3) and should bind streamflow samples in a convex sense; Calculate the Euclidean distance between original chemistry and projections for each solute using the equations below: Algebraically Geometrically j represent each solute and b j is the original solute value Those steps should lead to identification of eligible end-members!

STEP 5 - HYDROGRAPH SEPARATION Use the retained PCA projections from streamflow and end- members to derive flowpath solutions! So, mathematically, this is the same as a general mixing model rather than the over-determined situation.

STEP 6 - PREDICTION OF STREAMFLOW CHEMISTRY Multiply results of hydrograph separation (usually fractions) by original solute concentrations of end-members to reproduce streamflow chemistry for conservative solutes; Comparison of the prediction with the observation can lead to a test of mixing model.

PROBLEM ON OUTLIERS PCA is very sensitive to outliers; If any outliers are found in the mixing diagrams of PCA projections, check if there are physical reasons; Outliers have negative or > 1 fractions; See next slide how to resolve outliers using a geometrical approach for an end-member model.

RESOLVING OUTLIERS A, B, and C are 3 end- members; D is an outlier of streamflow sample; E is the projected point of D to line AB; a, b, d, x, and y represent distance of two points; We will use Pythagorean theorem to resolve it. The basic rule is to force f c = 0, f A and f B are calculated below [Liu et al., 2003]:

APPLICATION IN GREEN LAKES VALLEY: RESEARCH SITE Sample Collection Stream water - weekly grab samples Snowmelt - snow lysimeter Soil water - zero tension lysimeter Talus water – biweekly to monthly Sample Analysis Delta 18 O and major solutes Green Lake 4

GL4:  18 O IN SNOW AND STREAM FLOW

V  R  OF  18 O IN SNOWMELT  18 O gets enriched by 4% o in snowmelt from beginning to the end of snowmelt at a lysimeter; Snowmelt regime controls temporal variation of  18 O in snowmelt due to isotopic fractionation b/w snow and ice; Given f is total fraction of snow that have melted in a snowpack,  18 O values are highly correlated with f (R 2 = 0.9, n = 15, p < 0.001); Snowmelt regime is different at a point from a real catchment; So, we developed a Monte Carlo procedure to stretch the dates of  18 O in snowmelt measured at a point to a catchment scale using the streamflow  18 O values.

GL4: NEW WATER AND OLD WATER Old Water = 64%

STREAM CHEMISTRY AND DISCHARGE

MIXING DIAGRAM: PAIRED TRACERS

FLOWPATHS: 2-TRACER 3- COMPONENT MIXING MODEL

MIXING DIAGRAM: PCA PROJECTIONS PCA Results: First 2 eigenvalues are 92% and so 3 EMs appear to be correct!

FLOWPATHS: EMMA

DISTANCE OF END-MEMBERS BETWEEN U-SPACE AND THEIR ORIGINAL SPACE (%)

EMMA VALIDATION: TRACER PREDICTION

LEADVILLE CASE STUDY  Rich mining legacy  Superfund site: over $100M so far  Complicated hydrology: Mine shafts Faults Drainage tunnels We know nothing about mountain groundwater!  What are water sources to drainage tunnel?  Complicated, rigorous test

COMPLICATED GEOLOGY, HYDROLOGY

APPLICATION AT LEADVILLE

 18 O IN VARIOUS SAMPLES GW: from BMW-3 to YT-BH; SFW: from CG-03 to PWCW; SPR: from EFS-1 to SPR-23 Note: * means outlier

TRITIUM IN VARIOUS SAMPLES GW: from BMW-3 to YT-BH; SFW: from CG-03 to PWCW; SPR: from EFS-1 to SPR-23

VARIATION OF TRITIUM AND  18 O Seasonal variation of tritium and  18 O is less marked at INF-1 than EMET; Hydrological regime (flowpath) appears to be different at INF-1 and EMET.

MIXING DIAGRAMS Potential end-members are clustered and circled; Unique end-members generally cannot be identified; The bigger the circle, the higher the uncertainty in identifying a unique end-member; Recall from the last slide that tritium has increased 4 TU from Nov’02 to Feb’03 at EMET; This leads to recognition of Elkhorn to be an unambiguous EM.

MIXING DIAGRAMS EM used in the triangle is a representative from the circle only and not our current recommendation; # of EM and EM themselves may change from time to time due to sampling problem; The value of  18 O at EMET in June 2003 may be due to analytical problem, or mixing with rainwater, or with water from Marion which generally has higher  18 O.

MIXING DIAGRAMS Mixing diagram of  18 O and tritium for July 2003 is somewhat troubled; the circles are inter-crossed.

SUMMARY FOR MIXING DIAGRAMS OF TRITIUM AND  18 O EMs may change from time to time within a water year; Except for Elkhorn, unique EMs cannot be identified at this time; However, EM clusters are usually consistent from time to time; One cluster includes: WO3, CT, YT, and WCCPZ-1; The other cluster generally includes: SPR-23, PWBEINF, SDDS, SDDS-2, SHG07A, EFS-1, BMW-4, CG-03, CG-04; Particularly, some EMs could be excluded from a potential EM list: OG1TMW-1, BMW-3, MAB, and SPR-20.

PCA RESULTS: EIGENVALUES The first 2 PCA components explain 80% and 85% of total variance at INF-1 and EMET, respectively; The first 3 PCA components explain 95% of total variance at both sites; Either 3 or 4 EMs appear to be appropriate in EMMA.

PCA MIXING DIAGRAMS FOR INF-1 PCA conducted by 10 tracers:  18 O, 3 H, Alkalinity, Temperature, Conductance, Ca 2+, Mg 2+, Na +, SO 4 2-, and Si; Note that conservativity of tracers used here are not justified by pair-wise mixing diagrams.

PCA MIXING DIAGRAMS FOR INF-1 Same as the last one, but enlarged by eliminating some EMs; Unique EMs still cannot be identified; One EM appears to be missing.

PCA MIXING DIAGRAMS FOR EMET Use 9 tracers without Alkalinity; Unique EMs cannot be identified this time.

SUMMARY FOR PCA AND EMMA Unique EMs cannot be identified at this time; However, some potential end-members are consistent with the mixing diagrams of tritium and  18 O such as Elkhorn, CT, and CG-03; Future work is needed to plot mixing diagrams for all tracers so that non-conservative tracers can be eliminated;

IMPLICATION FOR FUTURE SAMPLING SCHEME Monthly or bi-monthly sampling scheme does capture seasonal signal within a water year; But this scheme may miss temporal variation within all seasons; Hydrological regime may change from season to season and within seasons; So, temporally intensive sampling scheme may be needed to capture within-season variation in order to unanimously identify EMs using EMMA.

SUMMARY: MIXING MODEL VS EMMA  Easy to understand and manipulate!  Doable with limited measurements of solutes!  But different tracers may yield different results! General Mixing Model EMMA  Use more tracers than necessary to lead to consistent results;  Provide a framework for analyzing watershed chemical data sets;  Generate testable hypotheses that focus future field efforts!

REDERENCES  Hooper, R., 2001, http: //  Christophersen, N., C. Neal, R. P. Hooper, R. D. Vogt, and S. Andersen, Modeling stream water chemistry as a mixture of soil water end-members – a step towards second-generation acidification models, Journal of Hydrology, 116, ,  Christophersen, N. and R. P. Hooper, Multivariate analysis of stream water chemical data: the use of principal components analysis for the end- member mixing problem, Water Resources Research, 28(1), ,  Hooper, R. P., N. Christophersen, and N. E. Peters, Modeling stream water chemistry as a mixture of soil water end-members – an application to the Panola mountain catchment, Georgia, U.S.A., Journal of Hydrology, 116, ,  Liu, F., M. Williams, and N. Caine, in review, Source waters and flowpaths in a seasonally snow-covered catchment, Colorado Front Range, USA, Water Resources Research, 2003.