Short Course—Mixing Models
Problem Set.
Part A.
For the following data set, determine the proportion of old water using first δ18O and then
δD. Recall that the proportion of old water, pold, is given by the following formula:
pold =
Qold
C
− Cnew
= stream
Qstream
Cold − Cnew
Streamwater data are:
delta-O18
delta-D
-6.3
-5.4
-7
-6.5
-7
-6.1
-5.7
-7
-6.3
-5.8
-6.1
-5.7
-5.4
-36
-33
-42
-39
-42
-36
-35
-43
-40
-38
-38
-36
-37
For this river basin, the average rainwater and baseflow isotopic composition are as
follows:
d-O18
Rainwater
Baseflow
d-D
-3
-7
-20
-50
Assume that the “new” water is rain and the “old” water is baseflow.
(i)
Does the proportion of old water agree between the two tracers? Why?
(ii)
To explore this further, plot the streamwater and end-member data on the same
axes. Does this plot help you see what the problem is?
Part B.
Assume that the following three end members control stream chemistry by mixing in
different proportions.
δO-18
End Member
1
2
3
Si
-5
-8
-6
20
80
150
i.
Where in the silica-O-18 plane must stream samples lie?
ii.
How many dimensions do the data “fill”?
iii.
How many non-zero eigenvalues are there for stream samples in this case?
Assume that the following three end members control stream chemistry:
δO-18
End Member
1
2
3
Si
-6
-5
-8
40
20
80
i.
Where do stream samples lie in this case?
ii.
How many dimensions do the data “fill”?
iii.
How many non-zero eigenvalues are there for stream samples in this case?
Part C.
Perform a “graphical” PCA analysis of the following data set by indicating
•
The origin of the new coordinate system
•
The orientation of the axes
Simply draw the location of these items on the graphs themselves. Estimate the
proportion of the variance explained by each component.
Graph 1.
500
450
400
350
300
250
200
150
100
50
0
0
100
200
Sodium
300
400
500
Graph 2.
300
200
100
0
0
100
200
300
Sodium
•What is the relative magnitude of the variance explained by the first component in
Graph 1 and Graph 2?
Graph 3. Here the data in Graph 1 are “standardized” by subtracting their mean and
dividing by their variance. How does this change the figure? Is this equivalent to
factoring the correlation or the covariance matrix?
4
3
2
1
0
-4
-2
0
-1
-2
-3
Sodium, Standardized
2
4