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

EMMA ADVANTAGES

• Use more tracers than components• Quantitatively evaluate potential

end-members• Quantitatively evaluate results of the

mixing model

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

LEADVILLE CASE STUDY

Rich mining legacy Superfund site: over $100M so far Complicated hydrology:

Mine shaftsFaultsDrainage tunnelsWe know nothing about mountain groundwater!

What are water sources to drainage tunnel? Complicated, rigorous test

COMPLICATED GEOLOGY, HYDROLOGY

APPLICATION AT LEADVILLE

18O IN VARIOUS SAMPLES

RAINSNOWEMETINF-1

BMW3CT

ELKHORNMAB

NW5-CNW5-D

OG1TMW-1WCC PZ1

WO3YT

YT-BHCG-03CG-04EG-04

MARIONPWCWEFS-1SDDS

SDDS-2SPR-20SPR-23

SPR-23 (200)

0-5-10-15-20-25

• 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

RAINSNOWEMETINF-1

BMW3CT

ELKHORNMAB

NW5-CNW5-D

OG1TMW-1WCCPZ1

WO3YT

YT-BHCG-03CG-04EG-04

MARIONPWCWEFS-1SDDS

SDDS-2SPR-20SPR-23

SPR-23 (200)

2520151050

VARIATION OF TRITIUM

AND 18O

-19

-18

-17

-16

-15

-14

-13

11/02 02/03 04/03 06/03 07/03

18 O

(‰

) EMET

INF-1

9

10

11

12

13

14

11/02 02/03 04/03 06/03 07/03

Tri

tiu

m (

TU

) EMET

INF-1

• Seasonal variation of tritium and 18O 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.

November 2002

OG1TMW-1BMW-3

EMETYT

WCCPZ1WO3

INF-1

NW5-C

PWOF

CG-04

CG-03

EG-04

SPR-23 (GS)

SPR-20 (VS)

SPR-23 (200)

SDDS

0

2

4

6

8

10

12

14

16

18

-19 -18 -17 -16 -1518O (%o)

Tri

tiu

m (

TU

)

Groundwater

Surface Water

Spring Water

Febuary 2003

BMW-3

CT

WCCPZ1

YT

NW5-D

PWOF

PWCW

EMET

NW5-C

INF-1

WO3

ELKHORN

SDDS

CG-03

EFS1

CG-04

PW RES

OG1TMW-1

SPR-23 (200)

SPR-20 (VS)

0

2

4

6

8

10

12

14

16

18

-20 -19 -18 -17 -16 -1518O (%o)

Tri

tiu

m (

TU

)

Groundwater

Surface Water

Spring Water

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 18O at EMET in June 2003 may be due to analytical problem, or mixing with rainwater, or with water from Marion which generally has higher 18O.

April 2003CT

WCCPZ-1

WO3

EMETINF-1

NW5-D

NW5-C

Elkhorn

MAB

BMW-3 OG1TMW-1

WRIGHT

CG-04

MARION

CG-03

PWBEINF

EG-04

PWCW

SPR-20

EFS-1

SPR23

SDDS-2

SPR23(200)

0

2

4

6

8

10

12

14

16

-21 -20 -19 -18 -17 -16 -15 -14

18O (‰)

Tri

tium

(T

U)

Groundwater

Surface Water

Spring Water

June 2003CT

WCCPZ-1WO3

EMET

INF-1

NW5-D

NW5-C

ELKHORN

MAB

BMW-4

OG1TMW-1

LMDT-1

BMW-3

LEGH-01

YT-BH

SHG07A

PWCW

CG-04

SPR-23

CG-03

SPR-20

EG-04

PWRES

EFS-1

SDDS-2

0

2

4

6

8

10

12

14

16

-20 -19 -18 -17 -16 -15 -14 -13

18O (‰)

Tri

tium

(T

U)

Groundwater

Surface Water

Spring Water

Analytical problem?Effect of rainwater?Influence of Marion?(not measured in June)

MIXING DIAGRAMS

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

July 2003

SHGEMSP

SDDSSHG07A

BMW-4

LMDT-1

BMW-3

MAB

ELKHORN

NW5-C NW5-D

INF-1EMET

WO3

WCCPZ-1

CT

PWBEREG-04

SPR-20

SDDS-2SPR-23

CG-04

LEGH-01

MARIONYTPD

EFS-1

PWCW

0

2

4

6

8

10

12

14

16

-20 -19 -18 -17 -16 -15 -14 -13 -12

18O (‰)

Tri

tium

(T

U)

Groundwater

Surface Water

Spring Water

SUMMARY FOR MIXING DIAGRAMS OF TRITIUM AND 18O

• 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.

0

10

20

30

40

50

60

70

PCA1 PCA2 PCA3 PCA4

Per

cen

tage

of

Var

ian

ce E

xpla

ined

INF-1

EMET

PCA MIXING DIAGRAMS FOR INF-1• PCA conducted by 10 tracers: 18O, 3H, Alkalinity, Temperature, Conductance, Ca2+, Mg2+, Na+, SO4

2-, and Si;

• Note that conservativity of tracers used here are not justified by pair-wise mixing diagrams.

EG-04

ELKHORN

PWCW

PWOF

LEGH-01WRIGHT

SPR-23

BMW-4

SHG07ACT

SPR-23(200)EFS-1

LMDT-1

MAB

OG1TMW-1

CG-03

NW5-CNW5-DSPR-20 CG-04BMW-3WCCPZ-1

WO3

SDDS-2

SDDSYTPD

YT-BH

SHGEMSP

MARION

-70

-60

-50

-40

-30

-20

-10

0

10

20

-40 -20 0 20 40 60 80 100 120 140

U1

U2

INF-1

End-Member

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.

BMW-3

BMW-4

CG-03

CG-04CT

EFS-1

EG-04

ELKHORN

LEGH-01LMDT-1

MAB

NW5-CNW5-D

OG1TMW-1

PWCW

PWOF

SHG07-A

SPR-20

SPR-23

SPR-23 (200)

WCCPZ-1

WO3

WRIGHT

-15

-10

-5

0

5

10

15

-20 -10 0 10 20 30 40

U1

U2

Missing?

PCA MIXING DIAGRAMS FOR EMET

• Use 9 tracers without Alkalinity;

• Unique EMs cannot be identified this time.

EG-04

ELKHORN PWCWPWOF

LEGH-01

WRIGHT

SPR-23BMW-4

SHG07A

CTSPR-23(200)

EFS-1

LMDT-1

MABOG1TMW-1

CG-03

NW5-CNW5-D

SPR-20CG-04

BMW-3

WCCPZ-1

WO3

SDDS-2SDDS

YTPD

YT-BH

SHGEMSP

MARION

-6

-4

-2

0

2

4

6

-9 -7 -5 -3 -1 1 3 5 7

U1

U2

EMET

End-Member

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 18O 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: //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.