
Outline overview of mixing model overview of endmember mixing analysis (emma)

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OUTLINE OVERVIEW OF ENDMEMBER 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 2Component Mixing Model 3Component Mixing Model Generalization 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 overdetermined situation The solution to this case is EMMA, which follows the same principle as mixing models.
PART 2: EMMA AND PCA EMMA Notation OverDetermined Situation Orthogonal Projection Notation of Mixing Spaces
DEFINITION OF ENDMEMBER For EMMA, we use endmembers instead of components to describe water contributing to stream from various compartments and geographic areas Endmembers 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 endmembers (k < p) B, matrix of endmembers, (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 Note that this equation is the same as generalized one for mixing model; the resymbolizing is for simplification and consistency with EMMA references
SOLUTION FOR OVERDETERMINED 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]:
SUMMARY:EMMA IDENTIFY MULTIPLE SOURCE WATERS AND FLOWPATHS QUANTITATIVELY SELECTS NUMBER AND TYPE OF ENDMEMBERS QUANTITATIVELY EVALUATE RESULTS IDENTIFY MISSING ENDMEMBERS
Burns et al. (2001)
Objectives Use EMMA to derive threecomponent model during 2 rain storms to answer: 1) What is the relative importance of each endmember 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 Endmembers: 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 Riparian water table levels
EMMA Modeling Five solutes used as tracers Data Standardized into correlation matrix PCA Concentrations of endmembers projected into Uspace Examine extent to which endmembers bound stream water observations in Uspace. Solute concentrations predicted by EMMA compared with measured concentrations during 2 storms
Storm Characteristics
Mixing Diagrams
EMMA Results
Endmember contributions
End Member Contributions cont.
Test of Mixing Model Using fraction of Flow from EMMA (fo, fh, and fr) with measured endmember concentrations calculate predicted stream flow concentrations. Linear Regression of Predicted vs. Measured Concentrations… r^2 = 0.950.99
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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