Geos 567: Inverse Problems Fall 2007
Problem Set # 6: Mapping Data Noise to Solution Error
Due: Thursday, 11 October 2007
This homework builds on HW#5. Assume that the b
1
,b
2
data from HW#5 are actually
noisy y
1
,y
2
values to which you are finding the least squares straight line solution y = m
1
+ m
2
z. Assume that the y
1
and y
2
values were taken at z
1
=2, and z
2
=7, respectively.
This homework should help you see how noise propagates into the solution, and the
important role that data covariance and correlation coefficient matrices play.
1. Find (and print) the (unweighted) least squares solutions (m
1
,m
2
)
T
and predicted data
(y
1
,y
2
)
Tpredicted
for the mean values of y
1
,y
2
for Experiments 1-50, 51-100, and 1-100,
respectively, recalling that unweighted implies unit data covariance matrices for each
case. Calculate (and print) the a posteriori model covariance matrix (see Equation
2.63 (page 25) of the Notes) for each case. Calculate (and print) the model
correlation coefficient matrix (see Equation 2.60, page 24) for each case.
2. Find (and print) the weighted least squares solutions (m
1
,m
2
)
T
and predicted data
(y
1
,y
2
)
Tpredicted
for the mean values of the y
1
,y
2
values for Experiments 1-50, 51-100,
and 1-100, respectively, using the data covariance matrices from HW#5 for the
weighting. Calculate (and print) the a posteriori model covariance matrix and model
correlation coefficient matrices for each case (Equations 2.63 and 2.60). Are they
different from the ones you calculated in Problem 1? Are the ones from Problem 1 or
2 ‘better’, and why? Finally, are your solutions (m
1
,m
2
)
T
and predicted data
(y
1
,y
2
)
Tpredicted
different from those calculated in Problem 1 above? Does this surprise
you? Explain.
3. Now generate (m
1
,m
2
)
T
data sets for Experiments 1-50, 51-100, and 1-100,
respectively, using m = G
LS-1
d. That is, you will get a set of 50 (m
1
,m
2
)
T
pairs for
Experiments 1-50, etc. Use G
LS-1
from problem 1, and note that while d and m will
vary from experiment to experiment, G
LS-1
does not. Calculate (and print) the mean
value for (m
1
,m
2
)
T
for Experiments 1-50, 51-100, and 1-100, respectively, and
compare with your results in Problem 1.
4. Using Equation 2.59 of the Notes, calculate (and print) the a posteriori model
covariance matrix for each case (i.e., Experiments 1-50, 51-100, and 1-100).
Compare the results with those obtained in Problem 2 above. Do the results surprise
you? Explain.
