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SUR 330 Introduction to Least Squares Adjustment
Final Exam
(30 pts)
1. Consider the problem of fitting three points to a line. In this particular
application–the x coordinate is treated as a constant and the y coordinate is the
measurement (in feet). The method of indirect observations ( v + BΔ = f) is used to
solve the problem. Let the two parameters be m (slope) and b ( the y intercept) and
used in that order in the adjustment.
Data:
x1 = 1 y1 = 3.2 σ1 = ±0.05 feet
x2 = 2 y2 = 4.0 σ2 = ±0.01 feet
x3 = 3 y3 = 5.0 σ3 = ±0.03 feet
Calculations:
0011.0ˆ
0011.0
1263.
2
9421.0
6836.4
7947.9
1511.13733.2
3733.204.5
0001.0
2
2
=
=
=∆
==
==
=
o
t
t
t
o
Wvv
WfBt
WBBN
σ
σ
a) Is this a least squares problem? Support your answer.
b) Calculate the standard deviations of the parameters ( m and b).
c) Given that the a priori reference variance is (0.0001). Write the W
matrix used in the adjustment.
d) Write the hypotheses for a one-sided (Upper Bound) Chi-Square Test for
Goodness of Fit.
e) Compute the Chi-Square Test Statistic?
f) Conduct a one-sided Chi-Square Test (using the hypotheses and Test
Statistic computed in (e) and (f) for Goodness of Fit at a 95% Level of
Significance and interpret the results?
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(5 pts.)
2. The Traverse PC Least Squares Network Adjustment will report three different
types of stations: fixed, weighted, and free. Explain how you would use these
different stations in an adjustment, and the different treatment these points will
have in the adjustment?
( 5 pts.)
3. You have postprocessed the data from a static GPS survey resulting in a set of
baseline vectors. Explain why it is good practice to conduct an inner constraints
adjustment on the vectors before you use them in a network to determine the
coordinates of the unknown points in your network?
(20 pts.)
4. You are conducting an inner constraints adjustment on your GPS vectors to see
if they fit together at a 95% confidence level ( α = 0.05). You determine/compute the
following data:
8.15
32.1ˆ
1
2
2
2
=
=
=
r
o
o
χ
σ
σ
a) Write the hypothesis for a one-sided Chi Square (Upper Bound)
Goodness of Fit test.
b) Write the rejection criteria and state the conclusions of your test.
( 30 pts )
5. Two different quantities X1 and X2 were determined with a standard deviation of
±0.05 feet and ±0.04 feet respectively. The measurements are not independent.
They have a covariance of -0.20 square feet. A quantity Y is computed using X1 and
X2. The relationship between the variables is:
42 21 +−= XXY
Compute the standard deviation of Y.
( 10 pts.)
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6. Given the following variance-covariance matrix on the coordinates of Point P.
Assume the statistics are ordered in the variance-covariance matrix as the
coordinate triplet (x, y, z).
−−
−−
−−
11.167.044.0
67.000.133.0
44.033.078.0
a) Extract the standard deviation of Py.
b) Extract the covariance between Py and Pz.
(10 pts)
7. A surveyor computed the length of line AD using the following field data (all
measurements independent and uncorrelated):
AD = AB+BC+CD. AB = 506.754 083.0±=ABσ meters
BC = 289.361 104.0±=BCσ meters
CD = 911.645 236.0±=CDσ meters
a) Compute the length of the line AD and its standard deviation.
b) Using the standard deviation, write the length of the line to the correct
number of significant digits.
( 50 pts.)
8. A total station is taken out to an NGS Calibration Baseline and
twelve distance measurements made between the four monuments.
The particular baseline upon which the measurements were taken has
marks at the 0, 150, 405, and 1186 meters. The instrument is first set
on the 0 mark and three measurements made to the 150, 405, and 1186
marks; the instrument is moved up to the 150 mark and one
measurement made back to the 0 mark and two measurements made
ahead to the 405 and 1186 mark; the instrument is moved up to the 405
mark, two measurements are made back to the 0 and 150 marks and
one measurement made ahead to the 1186 mark; then the instrument is
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set on the 1186 mark and three measurements made back to the 0, 150,
and 405 meter marks.
The variable D
A
denotes the record distance between the marks.
The variable D
H
denotes the actual measurement between the marks.
A table containing the record distances and measurements follows:
The equation expressing the functional relationship between the record
distances, measurements, and parameters S (Scale) and C (Constant) is:
V = D
A
-D
H
-SD
A
-C
Calculate the precision of the measurements assuming a Leica T600
Total Station with a manufacturer’s specification of )33( ppmmm +± . Use
these standard deviations to construct a weight matrix using an a priori
reference variance of one (1).
Use the Method of Indirect Observations to solve for the parameters S
and C. Answer the following questions:
a. Calculate n, n
o
, r, u, and c.
b. Write the condition equations and place in the ( v+BΔ=f) format.
c. Fill matrices B, W, and f.
Use Matlab to complete the solution:
d. Solve for the parameters S and C.
e. Compute the a posteriori reference variance.
f. Compute the variance-covariance matrix on the parameters S and C.
g. Conduct a two-sided Chi Square Goodness of Fit Test at a 95% level of
significance.
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Short Answer Questions ( Fill in the blanks ) 3 pts. each.
9. The tangent of 59˚27’40”is 1.69503124539. What are the
correct units on this answer_______________?
10. The output from a least squares adjustment that represents the
changes to the measurements necessary to fit the model are
_______________.
11. The condition that must exist before a least squares adjustment
is possible is called _____________________.
12. The Cofactor Matrix containing output statistics on parameters,
or residuals, or adjusted observations must be scaled by the
_________________________ before the statistics can be used.
13. The integer associated with the number of independent rows or
columns of a matrix is called the _______ of the matrix.
14. Always remember the rule: ____________ propagate, standard
deviations do not.
15. The interdependence between two variables is captured
numerically through the ____________________________.
Multiple Choice (4 pts each)
16. The most frequent reason for a least squares adjustment failing
the Goodness of Fit Test is:
a) the precision placed into the weight matrix is greater than
the actual precision of the measurements.
b) the data does not fit the model.
c) there is an error in the least squares adjustment such as an
incorrect sign on a number in the f matrix.
c) the least squares adjustment is non-linear and needed to be
Iterated.
17. The best definition of the a posteriori reference variance is:
a) It is the initial scalar used to scale the weight matrix.
b) It is the weighted sum of the residuals divided by the
adjustment redundancy.
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d) It is the weighted sum of the residuals divided by the a
priori reference variance.
e) It a number calculated from the residuals that is used to
determine the redundancy of the least squares adjustment.
18. The Variance matrix is inversed to form a Weight matrix
because:
a) measurement precision is placed into the Variance matrix
as numerators of fractions with the a priori reference
variance in the denominator.
c) the order of the measurement precision as initially placed
in the Variance matrix must be reversed for the
adjustment.
d) it contains the precisions of measurements whose
interdependence is expressed by covariances.
e) small variances must be converted into large weights and
large variances must be converted into small weights.
19. The primary rule that must be followed in using the Method of
Indirect Observations is that:
a) only one observation and its residual can be used in any
one condition equation.
b) the number of condition equations must equal the
redundancy.
c) only observations, associated residuals, and constants can
be used in any one condition equation.
d) only one parameter can be used in each condition equation.
20. The primary rule that must be followed in using the Method of
Observations Only is that:
a) only one observation and its residual can be used in any
one condition equation.
b) only observations, associated residuals, and constants can
be used in any one condition equation.
c) the number of parameters in the adjustment must equal no
— the minimum number of measurements necessary for a
unique determination of the model.
f) no parameters or constants can appear in any condition
equation.
21. Which of the following actions will change the values of your
residuals in a least Squares Adjustment?
a) Change the magnitude of the a priori reference variance.
b) Change the values of the weights in the Weight matrix, but
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leave their relative relationships unchanged.
g) Scaling the Cofactor matrices by the a posteriori reference
variance.
h) Fit the data to different model.
22. The most accurate statement regarding the meaning of the
look-up value for an upper bound Chi Square Test is:
a) the look-up value captures the difference between the
precision of the measurements in the weight matrix and
their actual precision.
b) the look-up value quantifies the acceptable error in the fit
of the data to the model based on the redundancy of the
adjustment.
c) the look-up value quantifies the allowable random error in
an adjustment with a specified redundancy and at a
specified level of statistical significance.
d) the look-up value captures the variability of the adjustment
due to its being based on the weighted sum of the squares
of the residuals divided by the redundancy.
23. The least squares adjustment has been run and you look at the
output. What is the first step that you should take upon finding one
large residual in the adjustment?
a) one should throw out the measurement associated with
the large residual as an outlier and redo the adjustment.
b) one should carefully look at the measurement associated
with the large residual to determine if it is a blunder.
c) one should check to see if the model used in the
adjustment is appropriate for the measurement data.
i) one should check the absolute value of the weight in the
weight matrix associated with the outlier to determine if it
is representative of the precision of the measurement?
- r = 12
