END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES

1. 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

2. EMMA ADVANTAGES • Use more tracers than components • Quantitatively evaluate potential • end-members • Quantitatively evaluate results of the • mixing model

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

4. 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]

5. 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

6. 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 bj (1  p) • X, matrix of streamflow samples, (n observations  p solutes); each row xi (1  p)

7. PROBLEM STATEMENT • Find a vector fi 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

8. 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

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

10. This slide is from Hooper, 2001

11. 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

12. 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.

13. 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!

14. 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!

15. 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.

16. 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! Project End-Members • 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!

17. STEP 4 - SCREEN END-MEMEBRS Geometrically • 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; Algebraically • Calculate the Euclidean distance between original chemistry and projections for each solute using the equations below: • j represent each solute and bj is the original solute value Those steps should lead to identification of eligible end-members!

18. 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.

19. 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.

20. 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.

21. 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 fc = 0, fA and fB are calculated below [Liu et al., 2003]:

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

23. GL4: d18O IN SNOW AND STREAM FLOW

24. VARIATION OF d18O IN SNOWMELT • d18O gets enriched by 4%o in snowmelt from beginning to the end of snowmelt at a lysimeter; • Snowmelt regime controls temporal variation of d18O in snowmelt due to isotopic fractionation b/w snow and ice; • Given f is total fraction of snow that have melted in a snowpack, d18O values are highly correlated with f (R2 = 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 d18O in snowmelt measured at a point to a catchment scale using the streamflow d18O values.

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

26. STREAM CHEMISTRY AND DISCHARGE

27. MIXING DIAGRAM: PAIRED TRACERS

28. FLOWPATHS: 2-TRACER 3-COMPONENT MIXING MODEL

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

30. FLOWPATHS: EMMA

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

32. EMMA VALIDATION: TRACER PREDICTION

33. 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

34. COMPLICATED GEOLOGY, HYDROLOGY

35. APPLICATION AT LEADVILLE

36. d18O 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

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

38. VARIATION OF TRITIUM AND d18O • Seasonal variation of tritium and d18O is less marked at INF-1 than EMET; • Hydrological regime (flowpath) appears to be different at INF-1 and EMET.

39. 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.

40. 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 d18O at EMET in June 2003 may be due to analytical problem, or mixing with rainwater, or with water from Marion which generally has higher d18O.

41. MIXING DIAGRAMS • Mixing diagram of d18O and tritium for July 2003 is somewhat troubled; the circles are inter-crossed.

42. SUMMARY FOR MIXING DIAGRAMS OF TRITIUM AND d18O • 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.

43. 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.

44. PCA MIXING DIAGRAMS FOR INF-1 • PCA conducted by 10 tracers: d18O, 3H, Alkalinity, Temperature, Conductance, Ca2+, Mg2+, Na+, SO42-, and Si; • Note that conservativity of tracers used here are not justified by pair-wise mixing diagrams.

45. 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.

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

47. 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 d18O 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;

48. 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.

49. SUMMARY: MIXING MODEL VS EMMA General Mixing Model • Easy to understand and manipulate! • Doable with limited measurements of solutes! • But different tracers may yield different results! 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!

50. REDERENCES • Hooper, R., 2001, http: //www.cof.orst.edu/cof/fe/watershed/shortcourse/schedule.htm • 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, 307-320, 1990. • 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), 99-107, 1992. • 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, 321-343, 1990. • 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.