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

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, (k  p);

• each row bj (1  p)

• X, matrix of streamflow samples, (n  p);

• 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

• First 2 principal components explained 93% of variability (m = 2)

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