mixing models and end-member mixing analysis: principles and examples matt miller and nick sisolak...
Post on 19-Dec-2015
296 views
TRANSCRIPT
![Page 1: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/1.jpg)
MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES
Matt Miller and Nick Sisolak
Slides Contributed by: Mark Williams and Fengjing Liu
![Page 2: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/2.jpg)
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)
![Page 3: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/3.jpg)
PART 1: OVERVIEW OF MIXING MODEL
Definition of Hydrologic Flowpaths 2-Component Mixing Model 3-Component Mixing ModelGeneralization of Mixing ModelGeometrical Definition of Mixing ModelAssumptions of Mixing Model
![Page 4: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/4.jpg)
HYDROLOGIC FLOWPATHS
![Page 5: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/5.jpg)
MIXING MODEL: 2
COMPONENTS
• One Conservative Tracer
• Mass Balance Equations for Water and Tracer
![Page 6: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/6.jpg)
MIXING MODEL: 3
COMPONENTS(Using Specific
Discharge)
• Two Conservative Tracers
• Mass Balance Equations for Water and Tracers
tQQQQ =++ 321
tt QCQCQCQC 13
132
121
11 =++
tt QCQCQCQC 23
232
221
21 =++
ttt Q
CCCCCCCC
CCCCCCCCQ
))(())((
))(())((23
21
13
12
23
22
13
11
23
213
12
23
22
13
1
1 −−−−−−−−−−
=
113
12
13
11
13
12
13
1
2 QCC
CCQ
CC
CCQ t
t
−−
−−−
=
213 QQQQ t −−=
Simultaneous Equations
Solutions
Q - Discharge
C - Tracer Concentration
Subscripts - # Components
Superscripts - # Tracers
![Page 7: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/7.jpg)
MIXING MODEL: 3
COMPONENTS(Using Discharge
Fractions)
• Two Conservative Tracers
• Mass Balance Equations for Water and Tracers
1321 =++ fff
13
132
121
11 tCfCfCfC =++
23
232
221
21 tCfCfCfC =++
))(())((
))(())((23
21
13
12
23
22
13
11
23
213
12
23
22
13
1
1 CCCCCCCC
CCCCCCCCf tt
−−−−−−−−−−
=
113
12
13
11
13
12
13
1
2 fCC
CC
CC
CCf t
−−
−−−
=
213 1 fff −−=
Simultaneous Equations
Solutions
f - Discharge Fraction
C - Tracer Concentration
Subscripts - # Components
Superscripts - # Tracers
![Page 8: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/8.jpg)
MIXING MODEL: Generalization Using Matrices
• One tracer for 2 components and two tracers for 3 components
• N tracers for N+1 components? -- Yes
• However, solutions would be too difficult for more than 3 components
• So, matrix operation is necessary
1321 =++ fff1
3132
121
11 tCfCfCfC =++
23
232
221
21 tCfCfCfC =++
Simultaneous Equations
Where
txx CfC =
1−= xtx CCf
23
22
21
13
12
11
111
CCC
CCCCx =
3
2
1
f
f
f
f x =2
1
1
t
tt
C
CC =
Solutions
Note:
• Cx-1 is the inverse matrix of Cx
• This procedure can be generalized to N tracers for N+1 components
![Page 9: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/9.jpg)
MIXING MODEL:
Geometrical Perspective
• For a 2-tracer 3-component model, for instance, the mixing subspaces are defined by two tracers.
• If plotted, the 3 components should be vertices of a triangle and all streamflow samples should be bound by the triangle.
• If not well bound, either tracers are not conservative or components are not well characterized.
• fx can be sought geometrically, but more difficult than algebraically.
0
30
60
90
120
150
180
0 20 40 60 80 100
Tracer 1
Tracer 2
Streamflow
Component 1
Component 2
Component 3
![Page 10: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/10.jpg)
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.
![Page 11: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/11.jpg)
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.
![Page 12: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/12.jpg)
PART 2: EMMA AND PCA
EMMA NotationOver-Determined SituationOrthogonal ProjectionNotation of Mixing SpacesSteps to Perform EMMA
![Page 13: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/13.jpg)
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]
![Page 14: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/14.jpg)
EMMA NOTATION (1)
Hydrograph separations using multiple tracers simultaneously;
Use more tracers than necessary to test consistency of tracers;
Typically use solutes as tracers
Modified from Hooper, 2001
![Page 15: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/15.jpg)
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)
![Page 16: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/16.jpg)
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
Bfx ii
![Page 17: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/17.jpg)
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]:
1)( TTii BBBxf
Constraint: all proportions must sum to 1 Solutions may be > 1 or < 0; this issue will be
elaborated later
![Page 18: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/18.jpg)
ORTHOGONAL PROJECTIONS
Following the normal equation, the predicted streamflow chemistry is [Christophersen and Hooper, 1992]:
Geometrically, this is the orthogonal projection of xi into the subspace defined by B, the end-members
BBBBxBfx TTiii
1* )(
![Page 19: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/19.jpg)
This slide is from Hooper, 2001
![Page 20: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/20.jpg)
OUR GOALS ACHIEVED SO FAR?• We measure chemistry of streamflow and end-members.
• Then, we can derive fractions of end-members contributing to streamflow using equations above.
• So, our goals achieved?
• Not quite, because we also want to test end-members as well as mixing model.
• We need to define the geometry of the solute “cloud” (S-space) and project end-members into U-space!
• How? Use PCA to determine number and orientation of axes in U-space.
Modified from Hooper, 2001
![Page 21: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/21.jpg)
EMMA PROCEDURES• Identification of Conservative Tracers - Bivariate solute-solute plots to screen data;
• PCA Performance - Derive eigenvalues and eigenvectors;
• Orthogonal Projection - Use eigenvectors to project chemistry of streamflow and end-members;
• Screen End-Members - Calculate Euclidean distance of end-members between their original values and S-space projections;
• Hydrograph Separation - Use orthogonal projections and generalized equations for mixing model to get solutions!
• Validation of Mixing Model - Predict streamflow chemistry using results of hydrograph separation and original end-member concentrations.
![Page 22: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/22.jpg)
STEP 1 - MIXING
DIAGRAMS
• Look familiar?
• This is the same diagram used for geometrical definition of mixing model (components changed to end-members);
• Generate all plots for all pair-wise combinations of tracers;
• The simple rule to identify conservative tracers is to see if streamflow samples can be bound by a polygon formed by potential end-members or scatter around a line defined by two end-members;
• Be aware of outliers and curvature which may indicate chemical reactions!
0
30
60
90
120
150
180
0 20 40 60 80 100
Tracer 1
Tracer 2
Streamflow
End-member 1
End-member 2
End-member 3
![Page 23: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/23.jpg)
STEP 2 - PCA PERFORMANCE
• For most cases, if not all, we should use correlation matrix rather than covariance matrix of conservative solutes in streamflow to derive eigenvalues and eigenvectors;
• Why? This treats each variable equally important and unitless;
• How? Standardize the original data set using a routine software or minus mean and then divided by standard deviation;
• To make sure if you are doing right, the mean should be zero and variance should be 1 after standardized!
![Page 24: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/24.jpg)
APPLICATION OF EIGENVALUES• Eigenvalues can be used to infer the number of end-members that should be used in EMMA.
How?
• Sum up all eigenvalues;
• Calculate percentage of each eigenvalue in the total eigenvalue;
• The percentage should decrease from PCA component 1 to p (remember p is the number of solutes used in PCA);
• How many eigenvalues can be added up to 90% (somewhat subjective! No objective criteria for this!)? Let this number be m,
m = # of PCA components retained (explains 90% of variance)
(m +1) = # of end-members we use in EMMA.
![Page 25: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/25.jpg)
STEP 3 - ORTHOGONAL PROJECTION
• X - Standardized data set of streamflow, (n p);
• V - Eigenvectors from PCA, (m p); Remember only the first m eigenvectors to be used here!
TXVU =
• Use the same equation above;
• Now X represents a vector (1 p) for each end-member;
• Remember X here should be standardized by subtracting streamflow mean and dividing by streamflow standard deviation!
Project End-Members
![Page 26: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/26.jpg)
STEP 4 - SCREEN END-MEMEBRS
• Plot a scatter plot for streamflow samples and end-members using the first and second PCA projections;
• Eligible end-members should be vertices of a polygon (a line if m = 1, a triangle if m = 2, and a quadrilateral if m = 3) and should bind streamflow samples in a convex sense;
• Calculate the Euclidean distance between original chemistry and projections for each solute using the equations below:
Algebraically
Geometrically
*jjj bbd −= VVVVbb TT
jj1* )( −=
• j represent each solute and bj is the original solute value
Those steps should lead to identification of eligible end-members!
![Page 27: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/27.jpg)
STEP 4 - SCREEN END-MEMEBRS
![Page 28: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/28.jpg)
STEP 5 - HYDROGRAPH SEPARATION
• Use the retained PCA projections from streamflow and end-members to derive flowpath solutions!
• So, mathematically, this is the same as a general mixing model rather than the over-determined situation.
tQQQQ =++ 321
tt QUQUQUQU 13
132
121
11 =++
tt QUQUQUQU 23
232
221
21 =++
Solve for Q’s
![Page 29: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/29.jpg)
STEP 6 - PREDICTION OF STREAMFLOW CHEMISTRY
• Multiply results of hydrograph separation (usually fractions) by original solute concentrations of end-members to reproduce streamflow chemistry for conservative solutes;
• Comparison of the prediction with the observation can lead to a test of mixing model.
![Page 30: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/30.jpg)
ANC
R2 = 0.64
20
40
60
80
100
20 40 60 80 100
Ca2+
R2 = 0.97
20
40
60
80
100
120
20 40 60 80 100 120
Na+
R2 = 0.88
5
10
15
20
25
30
5 10 15 20 25 30
SO 42-
R2 = 0.88
10
30
50
70
90
10 30 50 70 90
Si
R2 = 0.85
0
10
20
30
40
50
0 10 20 30 40 50
δ 18O
R2 = 0.81
-19
-18
-17
-16
-15
-14
-19 -18 -17 -16 -15 -14
(Prediction
mol L
-1 for Si and
eq L
-1 )for others
Observation (units same as in y axis)
![Page 31: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/31.jpg)
PROBLEM ON OUTLIERS
• PCA is very sensitive to outliers;
• If any outliers are found in the mixing diagrams of PCA projections, check if there are physical reasons;
• Outliers have negative or > 1 fractions;
• See next slide how to resolve outliers using a geometrical approach for an end-member model.
![Page 32: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/32.jpg)
RESOLVING OUTLIERS• A, B, and C are 3 end-members;
• D is an outlier of streamflow sample;
• E is the projected point of D to line AB;
• a, b, d, x, and y represent distance of two points;
• We will use Pythagorean theorem to resolve it.
-2
-1
0
1
2
3
-10 -5 0 5 10
U 1
U2
A
B
C
D
E
ab
x
yd
• The basic rule is to force fc = 0, fA and fB are calculated below [Liu et al., 2003]:
222
211 )()( UUUU DADAa −+−=
222
211 )()( UUUU DBDBb −+−=
222
211 )()( UUUU ABABd −+−=
2
222
2d
bdaxfB
−+==
xyf A −== 1
![Page 33: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/33.jpg)
SUMMARY:EMMA
IDENTIFY MULTIPLE SOURCE WATERS AND FLOWPATHS
QUANTITATIVELY SELECTS NUMBER AND TYPE OF END-MEMBERS
QUANTITATIVELY EVALUATE RESULTS IDENTIFY MISSING END-MEMBERS
![Page 34: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/34.jpg)
Burns et al. (2001)
![Page 35: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/35.jpg)
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?
![Page 36: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/36.jpg)
Site Description
Study Site: PMRW 10ha 3 assumed
End-members:
1)Outcrop
2) Hillslope
3)Riparian Area
![Page 37: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/37.jpg)
Field Methods
Chemical Analysis1) Stream water
2) Runoff from outcrop
3) Subsurface stormflow from hillslope trench
4) Riparian ground water
Physical Meaurements1) Stream runoff rate
2) Rainfall amount/intensity
3) Runoff rate from hillslope trench
4) Riparian water table levels
![Page 38: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/38.jpg)
EMMA Modeling
1) Five solutes used as tracers2) Data Standardized into correlation matrix3) PCA 4) Concentrations of end-members projected into
U-space5) Examine extent to which end-members bound
stream water observations in U-space.6) Solute concentrations predicted by EMMA
compared with measured concentrations during 2 storms
![Page 39: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/39.jpg)
Storm Characteristics
![Page 40: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/40.jpg)
Mixing Diagrams
![Page 41: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/41.jpg)
EMMA Results
First 2 principal components explained 93% of variability (m = 2) 3 end-members
![Page 42: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/42.jpg)
End-member contributions
![Page 43: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/43.jpg)
End Member Contributions cont.Storm 1 Storm 2
![Page 44: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/44.jpg)
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
![Page 45: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/45.jpg)
Predicted Outcrop Runoff vs Measured Rainfall Intensity
*Good Fit
*Initial lag due to travel time of rain water
*Outcrop runoff continues due to
draining of soil
![Page 46: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/46.jpg)
Predicted Hillslope Runoff and Measured Trench Outflow
*Hillslope trench located halfway up hillslope
*Lower soil moisture before March Storm
*Subsurface storm flow likely provided recharge to riparian ground water and dry hillslope soils.
![Page 47: MIXING MODELS AND END-MEMBER MIXING ANALYSIS: PRINCIPLES AND EXAMPLES Matt Miller and Nick Sisolak Slides Contributed by: Mark Williams and Fengjing Liu](https://reader036.vdocuments.site/reader036/viewer/2022081420/56649d3e5503460f94a16ff1/html5/thumbnails/47.jpg)
Predicted Riparian Groundwater Runoff and Observed Riparian Water
Table Levels
*Outcrop runoff and hillslope subsurface flow mix in riparian aquifer during transport to the stream.