Outline overview of mixing model overview of end-member mixing analysis (emma)



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OUTLINE

  • 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

  • -- Panola Mountain Research Watershed (Burns et al., 2001)

  • -- Green Lakes Valley (Liu et al., 2004)



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















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



EMMA NOTATION (2)

  • Measure p solutes;

  • p= # of solutes

  • Assume that there are k linearly independent end-members (k < p)

  • k = # of end-members

  • B, matrix of end-members, (kp);

  • each row bj (1  p)

  • X, matrix of streamflow samples, (np);

  • each row xi (1  p)



PROBLEM STATEMENT

  • Find a vector fi of mixing proportions such that

  • Stream chemistry can be defined as a function of end-member contribution

  • Note that this equation is the same as generalized one for mixing model; the re-symbolizing is for simplification and consistency with EMMA references



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]:



















STEP 4 - SCREEN END-MEMEBRS













SUMMARY:EMMA

  • IDENTIFY MULTIPLE SOURCE WATERS AND FLOWPATHS

  • QUANTITATIVELY SELECTS NUMBER AND TYPE OF END-MEMBERS

  • QUANTITATIVELY EVALUATE RESULTS

  • IDENTIFY MISSING END-MEMBERS



Burns et al. (2001)



Objectives

  • Use EMMA to derive three-component model during 2 rain storms to answer:

  • 1) What is the relative importance of each end-member to stream runoff?

  • 2)How do runoff processes vary with storm size, rain intensity, and antecedent wetness conditions?

  • 3)Are EMMA modeling results consistent with physical hydrologic measurements?



Site Description

  • Study Site: PMRW

  • 10ha

  • 3 assumed

  • End-members:

  • 1)Outcrop

  • 2) Hillslope

  • 3)Riparian Area



Field Methods

  • Chemical Analysis

  • Stream water

  • Runoff from outcrop

  • Subsurface stormflow from hillslope trench

  • Riparian ground water

  • Physical Meaurements

  • Stream runoff rate

  • Rainfall amount/intensity

  • Runoff rate from hillslope trench

  • Riparian water table levels



EMMA Modeling

  • Five solutes used as tracers

  • Data Standardized into correlation matrix

  • PCA

  • Concentrations of end-members projected into U-space

  • Examine extent to which end-members bound stream water observations in U-space.

  • Solute concentrations predicted by EMMA compared with measured concentrations during 2 storms



Storm Characteristics



Mixing Diagrams



EMMA Results



End-member contributions



End Member Contributions cont.



Test of Mixing Model

  • Using fraction of Flow from EMMA (fo, fh, and fr) with measured end-member concentrations calculate predicted stream flow concentrations.

  • Linear Regression of Predicted vs. Measured Concentrations…

  • r^2 = 0.95-0.99



Predicted Outcrop Runoff vs Measured Rainfall Intensity



Predicted Hillslope Runoff and Measured Trench Outflow



Predicted Riparian Groundwater Runoff and Observed Riparian Water Table Levels





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