Part 2: emma and pca emma notation

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

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

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]:

ORTHOGONAL PROJECTIONS

• Following the normal equation, the predicted streamflow chemistry is [Christophersen and Hooper, 1992]:

GL4: 18O IN SNOW AND STREAM FLOW

VROF18O IN SNOWMELT

GL4: NEW WATER AND OLD WATER

STREAM CHEMISTRY AND DISCHARGE

MIXING DIAGRAM: PAIRED TRACERS

FLOWPATHS: 2-TRACER 3-COMPONENT MIXING MODEL

MIXING DIAGRAM: PCA PROJECTIONS

FLOWPATHS: EMMA

APPLICATION AT LEADVILLE

18O IN VARIOUS SAMPLES

TRITIUM IN VARIOUS SAMPLES

VARIATION OF TRITIUM AND 18O

MIXING DIAGRAMS

MIXING DIAGRAMS

MIXING DIAGRAMS

SUMMARY FOR MIXING DIAGRAMS OF TRITIUM AND 18O

PCA RESULTS: EIGENVALUES

PCA MIXING DIAGRAMS FOR INF-1

PCA MIXING DIAGRAMS FOR INF-1

PCA MIXING DIAGRAMS FOR EMET

SUMMARY FOR PCA AND EMMA

IMPLICATION FOR FUTURE SAMPLING SCHEME

SUMMARY: MIXING MODEL VS EMMA

• Easy to understand and manipulate!

• Doable with limited measurements of solutes!

• But different tracers may yield different results!

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.