1. 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

2. 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

3. 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

4. HYDROLOGIC FLOWPATHS

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

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

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

8. MIXING MODEL: Generalization Using Matrices
Simultaneous Equations
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
Where
Solutions
Note:
Cx-1 is the inverse matrix of Cx
This procedure can be generalized to N tracers for N+1 components

9. 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.
fx can be sought geometrically, but more difficult than algebraically.

10. 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.

11. 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.

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

13. 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]

14. 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

15. 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)

16. 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

17. 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

18. 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

19. This slide is from Hooper, 2001

20. 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

21. 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.

22. 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!

23. 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!

24. 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.

25. 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!

26. 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!

27. 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.

28. 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.

29. 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.

30. 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]:

31. 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 18O and major solutes

32. GL4: d18O IN SNOW AND STREAM FLOW

33. 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.

34. GL4: NEW WATER AND OLD WATER

35. STREAM CHEMISTRY AND DISCHARGE

36. MIXING DIAGRAM: PAIRED TRACERS

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

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

39. FLOWPATHS: EMMA

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

41. EMMA VALIDATION: TRACER PREDICTION

42. SUMMARY:EMMA IDENTIFY MULTIPLE SOURCE WATERS AND FLOWPATHS
QUANTITATIVELY SELECTS NUMBER AND TYPE OF END-MEMBERS
QUANTITATIVELY EVALUATE RESULTS
IDENTIFY MISSING END-MEMBERS