## OUTLINE ## 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 ## 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
## 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*) **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 ## 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]:
## 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 ## 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 ## 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
**Dostları ilə paylaş:** |