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

tQQQQ 321

tt QCQCQCQC 13

132

121

11

tt QCQCQCQC 23

232

221

21

ttt Q

CCCCCCCC

CCCCCCCCQ

))(())((

))(())((23

21

13

12

23

22

13

11

23

213

12

23

22

13

1

1

113

12

13

11

13

12

13

1

2 QCC

CCQ

CC

CCQ t

t

213 QQQQ t

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

1321 fff

13

132

121

11 tCfCfCfC

23

232

221

21 tCfCfCfC

))(())((

))(())((23

21

13

12

23

22

13

11

23

213

12

23

22

13

1

1 CCCCCCCC

CCCCCCCCf tt

113

12

13

11

13

12

13

1

2 fCC

CC

CC

CCf t

213 1 fff

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

1321 fff1

3132

121

11 tCfCfCfC

23

232

221

21 tCfCfCfC

Simultaneous Equations

Where

txx CfC

1 xtx CCf

23

22

21

13

12

11

111

CCC

CCCCx

3

2

1

f

f

f

f x 2

1

1

t

tt

C

CC

Solutions

Note:

• Cx-1 is the inverse matrix of Cx

• 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.

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

0

30

60

90

120

150

180

0 20 40 60 80 100

Tracer 1

Tra

cer

2

Streamflow

Component 1

Component 2

Component 3

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 bj (1 p)

X, matrix of streamflow samples, (n observations p solutes); each row xi (1 p)

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

Bfx ii

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]:

1)( TTii BBBxf

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 xi into the subspace defined by B, the end-members

BBBBxBfx TTiii

1* )(

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!

0

30

60

90

120

150

180

0 20 40 60 80 100

Tracer 1

Tra

cer

2

Streamflow

End-member 1

End-member 2

End-member 3

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!

TVXX '

• 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

*jjj bbd VVVVbb TT

jj1* )(

• j represent each solute and bj 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.

-2

-1

0

1

2

3

-10 -5 0 5 10

U 1

U2

A

B

C

D

E

ab

x

yd

• The basic rule is to force fc = 0, fA and fB are calculated below [Liu et al., 2003]:

222

211 )()( UUUU DADAa

222

211 )()( UUUU DBDBb

222

211 )()( UUUU ABABd

2

222

2d

bdaxfB

xyf A 1

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

Green Lake 4

GL4: 18O IN SNOW AND STREAM FLOW

-22

-18

-14

-10

-6

18 O

(‰)

Stream FlowSoil WaterSnowmeltBaseflow

0

10

20

30

40

100 125 150 175 200 225 250 275 300

Calendar Day (1996)

Q (1

03 m

3 day

-1)

VROF18O IN SNOWMELT

-22

-20

-18

-16

18 O

(‰

)

Original

Date-Stretched by Monte Carlo

0

50

100

150

100 125 150 175 200 225 250 275 300

Calendar Day (1996)

Snow

mel

t (m

m)

• 18O gets enriched by 4%o in snowmelt from beginning to the end of snowmelt at a lysimeter;

• Snowmelt regime controls temporal variation of 18O in snowmelt due to isotopic fractionation b/w snow and ice;

• Given f is total fraction of snow that have melted in a snowpack, 18O 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 18O in snowmelt measured at a point to a catchment scale using the streamflow 18O values.

GL4: NEW WATER AND OLD WATEROld Water = 64%

0

10

20

30

40

135 165 195 225 255 285

Calendar Day (1996)

Q (

103 m

3 day

-1)

New Water

Old Water

ST

RE

AM

CH

EM

IST

RY

A

ND

DIS

CH

AR

GE

Calendar Day (1996)

0

30

60

90

120

Sol

ute

s (

eq L

-1)

ANCCalciumNitrateSulphate

0

10

20

30

Sol

ute

s (

eq L

-1)

ChlorideMagnesiumSodiumPotassium

0

10

20

30

40

130 190 250 310 370

Q (

103 m

3 day

-1)

MIXING DIAGRAM: PAIRED TRACERS

0

10

20

30

40

50

60

-24 -20 -16 -12 -8

18O(‰)

Si (m m

ol L

-1)

Stream FlowIndex SnowpitSnowmeltTalus EN1-LTalus EN1-MTalus EN1-UTalus EN2-LTalus EN2-UTalus EN4-VTalus EN4-LTalus EN4-USoil WaterBase Flow

FLOWPATHS: 2-TRACER 3-COMPONENT MIXING MODEL

0

10

20

30

40

50

60

135 165 195 225 255 285

Calendar Day (1996)

Q (

103 m

3 day

-1)

0

40

80

120

160

200

240

280

320

Per

cen

tage

(%

)

Surface FlowTalus WaterBaseflow

MIXING DIAGRAM: PCA PROJECTIONS

-3

-1

1

3

5

-8 -3 2 7 12

U1

U2

Stream Flow

Snowpit

Snowmelt

Talus EN1-L

Talus EN1-M

Talus EN1-U

Talus EN2-L

Talus EN2-U

Talus EN4-V

Talus EN4-L

Talus EN4-U

Base Flow

Soil Water

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

FLOWPATHS: EMMA

0

10

20

30

40

50

60

135 165 195 225 255 285

Calendar Day (1996)

Q (

103 m

3 day

-1)

0

40

80

120

160

200

240

280

320

Per

cen

tage

(%

)

Surface Flow

Talus Flow

Baseflow

End-Members Cond ANC Ca2+

Mg2+

Na+

SO42-

S i* 18

O

Index Snowpit -17 -118 139 203 -260 -131 - -3

Snowmelt in Lysimeter 21 -66 4 -6 32 78 -168 -5

Talus EN1-L 39 -38 6 -1 -36 130 -48 -8

Talus EN1-M 22 -38 8 -11 -17 193 -53 -8

Talus EN1-U -13 35 6 -13 -20 11 85 3

Talus EN2-L -10 38 2 -26 -16 86 19 5

Talus EN2-M -22 65 -2 -26 18 34 67 7

Talus EN4-V -2 0 -16 -10 59 20 -16 -1

Talus EN4-L 0 -32 -10 -6 38 45 22 2

Talus EN1-U -17 3 2 -22 77 19 184 7

Soil Water -48 146 24 -10 66 65 114 43

Base Flow 0 -3 6 -3 14 -9 3 1

DISTANCE OF END-MEMBERS BETWEEN U-SPACE AND THEIR

ORIGINAL SPACE (%)

ANC

R2 = 0.64

20

40

60

80

100

20 40 60 80 100

Ca2+

R2 = 0.97

20

40

60

80

100

120

20 40 60 80 100 120

Na+

R2 = 0.88

5

10

15

20

25

30

5 10 15 20 25 30

SO42-

R2 = 0.88

10

30

50

70

90

10 30 50 70 90

Si

R2 = 0.85

0

10

20

30

40

50

0 10 20 30 40 50

18O

R2 = 0.81

-19

-18

-17

-16

-15

-14

-19 -18 -17 -16 -15 -14

Pre

dic

tion

(m

ol L

-1fo

r S

i an

d

eq L

-1 f

or o

ther

s)

Observation (units same as in y axis)

EMMA VALIDATION: TRACER PREDICTION

SUMMARY:EMMA

IDENTIFY MULTIPLE SOURCE WATERS AND FLOWPATHS

QUANTITATIVELY SELECTS NUMBER AND TYPE OF END-MEMBERS

QUANTITATIVELY EVALUATE RESULTS IDENTIFY MISSING END-MEMBERS